11 April 2021
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Entangling power and quantum circuit complexity

Quantum circuit complexity is one of the most crucial factors to take into consideration in quantum computation. It provides an account of computation with a minimal number of steps and time needed in order to implement a given unitary. One can associate quantum circuit complexity with the complexity involved in the preparation of a given quantum fiducial state from its initial state. For instance, a quantum state generated by a quantum chaotic Hamiltonian evolution will be highly complex, if the quantum circuit preparing it requires a lot of time on a quantum computer. When determining the overall complexity of a quantum circuit, an essential factor that needs to be accounted for, is the cost of such circuit while keeping into consideration the circuit design.

While the above factors can contribute to analyzing the complexity of quantum circuits, it is not trivial to calculate these factors quantitatively. Although there are (classical) algorithmic procedures that can find a decomposition of a unitary into a quantum circuit including Clifford and T-gates, which can be done in exponential run-time in circuit size, computing the complexity still requires optimization over these decompositions. During such decompositions, cancellations of gates occur that make the impact of a unitary gate to be partially compensated by the application of a following similar gate. One potential correlation to explore is whether the entanglement (created by quantum gates) has any impact on circuit complexity or at the cost of a unitary.

The author in this work analyzes such a relationship when both the entanglement of a state and the cost of a unitary take small values, based on how values of entangling power of quantum gates add up. It provides a simple lower bound for the cost of a quantum circuit that is tight for small values of the cost. The idea of these bounds comes from the entanglement capabilities of the quantum gates. Quantum gates that are close to the identity in operator norm have little capability to create entanglement from product states. It is however theorized that their contribution of entanglement to a given entangled state is also very little. The bound presented in this work implies that, assuming linear growth of entanglement entropies with time, cost also increases linearly.

Furthermore, for Gaussian continuous-variable systems, there is a small incremental entanglement bound as well as a harmonic equivalent of the above relationship between entanglement and quantum circuit cost. This bound can also be applied both to spin systems and Gaussian bosonic continuous-variable settings, which are important in approximating non-interacting bosonic theories. A noteworthy observation is that when a quantum many-body system undergoes non-equilibrium dynamics leading to a linear increase in the entanglement entropy, the quantum state complexity also increases.

An important result from the presented bounds, is that one can derive the required depth of a quantum circuit that can produce a given entanglement pattern in a desired final state, for either pure or mixed states. The presented bounds can also help in assessing the computational performance of analog quantum simulators and their direct comparison to their digital counterparts. One can argue that for a precisely defined computational effort, both analog and digital quantum simulators can achieve similar results. These simple bounds can provide a useful and versatile tool in various studies of such comparisons and relations.

While the above factors can contribute to analyzing the complexity of quantum circuits, it is not trivial to calculate these factors quantitatively. Although there are (classical) algorithmic procedures that can find a decomposition of a unitary into a quantum circuit including Clifford and T-gates, which can be done in exponential run-time in circuit size, computing the complexity still requires optimization over these decompositions. During such decompositions, cancellations of gates occur that make the impact of a unitary gate to be partially compensated by the application of a following similar gate. One potential correlation to explore is whether the entanglement (created by quantum gates) has any impact on circuit complexity or at the cost of a unitary.

The author in this work analyzes such a relationship when both the entanglement of a state and the cost of a unitary take small values, based on how values of entangling power of quantum gates add up. It provides a simple lower bound for the cost of a quantum circuit that is tight for small values of the cost. The idea of these bounds comes from the entanglement capabilities of the quantum gates. Quantum gates that are close to the identity in operator norm have little capability to create entanglement from product states. It is however theorized that their contribution of entanglement to a given entangled state is also very little. The bound presented in this work implies that, assuming linear growth of entanglement entropies with time, cost also increases linearly.

Furthermore, for Gaussian continuous-variable systems, there is a small incremental entanglement bound as well as a harmonic equivalent of the above relationship between entanglement and quantum circuit cost. This bound can also be applied both to spin systems and Gaussian bosonic continuous-variable settings, which are important in approximating non-interacting bosonic theories. A noteworthy observation is that when a quantum many-body system undergoes non-equilibrium dynamics leading to a linear increase in the entanglement entropy, the quantum state complexity also increases.

An important result from the presented bounds, is that one can derive the required depth of a quantum circuit that can produce a given entanglement pattern in a desired final state, for either pure or mixed states. The presented bounds can also help in assessing the computational performance of analog quantum simulators and their direct comparison to their digital counterparts. One can argue that for a precisely defined computational effort, both analog and digital quantum simulators can achieve similar results. These simple bounds can provide a useful and versatile tool in various studies of such comparisons and relations.

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31 March 2021
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Modular photonic quantum computing

Rapid progress has been made in recent years in the development of fault-tolerant quantum computing (FTQC) - across theoretical foundations, architectural design and experimental implementations. Most of the proposed architectures are based on an array of static qubits, where relevant large-scale computation with for example superconducting qubits is expected to require vast numbers of physical qubits taking up a lot of space and control machinery. Directly translating that paradigm to photonic FTQC architectures, implies that photons serve as the ‘static qubits’ implementing gates and measurements. However, the implementation of long sequences required by FTQC protocols, becomes difficult to process as photons are short-lived, easily lost, and destroyed after measurement. This makes conventional FTQC description not suitable to photonic quantum computing.

Fusion-based quantum computing (FBQC) is an alternative to standard photon-based FTQC architectures that can overcome such limitations. In FBQC, quantum information is not stored in a static array of qubits, but periodically teleported from previously generated resource states to currently generated photons. Hence, even when the measured photons are destroyed, their quantum information is preserved and teleported accordingly. In this work, the authors present a modular and scalable architecture for FBQC, which can provide the computational power of thousands of physical qubits. The unit module of the architecture consists of a single resource-state generator (RSG), a few fusion devices, and macroscopic fiber delays with low transmission loss rates connected via waveguides and switches. The networks of such modules execute the operations by adding thousands of physical qubits to the computational Hilbert space for executing computation. The authors argue that, pragmatically, “a static qubit-based device and a dynamic RSG-based device (can be considered) equally powerful, if they can execute the same quantum computation in the same amount of time”. A single RSG is shown to be much more ‘powerful’ than a single physical qubit.

The qubits produced by RSGs are encoded as photonic qubits and are combined using a stabilizer code such as a Shor code. The photonic qubits are then transported by waveguides to n-delays, which delay (store) photons for n RSG cycles therefore acting as a fixed-time quantum memory by temporarily storing photonic qubits. This photonic memory is used to increase the number of simultaneously existing resource states available in the total computation space. Fusion devices further perform entangling fusion measurements of photon pairs that enter the device. Finally, switches reroute the incoming photonic qubits to one of the multiple outgoing waveguides. Switch settings can be adjusted in every RSG cycle, thereby deciding the operations to be performed.

In contrast to circuits in circuit-based quantum computation (CBQC), photonic FBQC uses fusion graphs to describe the execution of a specific computation. The authors review the structure of simple cubic fusion graphs using 6-ring graph states which is a six-qubit ring-shaped cluster state as resource states. Each resource state is fused with six other resource states allowing one fusion per constituent qubit of the resource state. Another direction being explored is interleaving, which involves allocating the same RSG to successively produce different fusion-graph resource states. Exploiting different arrangements of RSGs and using longer delay lines can lead to larger fusion graphs. Furthermore, it is demonstrated that interleaving modules with n-delays can increase the number of available qubits by n, but inevitably decrease the speed of logical operations by the same factor. To avoid that, it is recommended increasing the number of interleaving modules and investigating different arrangements.

These photonics-based FBQC architectures are not only modular and highly scalable, but are cost-efficient as well, since they reduce the cost of logical operations. Combining this with the interleaving approach further improves feasibility, where instead of million-qubit arrays of physical qubits, arrays of disconnected few-qubit devices can be turned into a large-scale quantum computer, provided that their qubits are photonic qubits. Such hybrid architecture repeatedly generates identical few qubit resource states from matter-based qubits, connecting them to a large-scale fault-tolerant quantum computer. Moreover, it also handles the classical processing associated with error correction along with providing high-capacity memory. As quantum technology evolves, larger number of high-quality qubits are going to be available, allowing a transition from small-scale FTQC devices to fully scalable devices. These early FTQC devices are expected to be similar in design to the current NISQ devices albeit much more powerful. Utilizing such approaches in photonic FBQC along with the developments in highly efficient photonic hardware can make the transition to large-scale fault-tolerant quantum computers a reality in the near future.

Fusion-based quantum computing (FBQC) is an alternative to standard photon-based FTQC architectures that can overcome such limitations. In FBQC, quantum information is not stored in a static array of qubits, but periodically teleported from previously generated resource states to currently generated photons. Hence, even when the measured photons are destroyed, their quantum information is preserved and teleported accordingly. In this work, the authors present a modular and scalable architecture for FBQC, which can provide the computational power of thousands of physical qubits. The unit module of the architecture consists of a single resource-state generator (RSG), a few fusion devices, and macroscopic fiber delays with low transmission loss rates connected via waveguides and switches. The networks of such modules execute the operations by adding thousands of physical qubits to the computational Hilbert space for executing computation. The authors argue that, pragmatically, “a static qubit-based device and a dynamic RSG-based device (can be considered) equally powerful, if they can execute the same quantum computation in the same amount of time”. A single RSG is shown to be much more ‘powerful’ than a single physical qubit.

The qubits produced by RSGs are encoded as photonic qubits and are combined using a stabilizer code such as a Shor code. The photonic qubits are then transported by waveguides to n-delays, which delay (store) photons for n RSG cycles therefore acting as a fixed-time quantum memory by temporarily storing photonic qubits. This photonic memory is used to increase the number of simultaneously existing resource states available in the total computation space. Fusion devices further perform entangling fusion measurements of photon pairs that enter the device. Finally, switches reroute the incoming photonic qubits to one of the multiple outgoing waveguides. Switch settings can be adjusted in every RSG cycle, thereby deciding the operations to be performed.

In contrast to circuits in circuit-based quantum computation (CBQC), photonic FBQC uses fusion graphs to describe the execution of a specific computation. The authors review the structure of simple cubic fusion graphs using 6-ring graph states which is a six-qubit ring-shaped cluster state as resource states. Each resource state is fused with six other resource states allowing one fusion per constituent qubit of the resource state. Another direction being explored is interleaving, which involves allocating the same RSG to successively produce different fusion-graph resource states. Exploiting different arrangements of RSGs and using longer delay lines can lead to larger fusion graphs. Furthermore, it is demonstrated that interleaving modules with n-delays can increase the number of available qubits by n, but inevitably decrease the speed of logical operations by the same factor. To avoid that, it is recommended increasing the number of interleaving modules and investigating different arrangements.

These photonics-based FBQC architectures are not only modular and highly scalable, but are cost-efficient as well, since they reduce the cost of logical operations. Combining this with the interleaving approach further improves feasibility, where instead of million-qubit arrays of physical qubits, arrays of disconnected few-qubit devices can be turned into a large-scale quantum computer, provided that their qubits are photonic qubits. Such hybrid architecture repeatedly generates identical few qubit resource states from matter-based qubits, connecting them to a large-scale fault-tolerant quantum computer. Moreover, it also handles the classical processing associated with error correction along with providing high-capacity memory. As quantum technology evolves, larger number of high-quality qubits are going to be available, allowing a transition from small-scale FTQC devices to fully scalable devices. These early FTQC devices are expected to be similar in design to the current NISQ devices albeit much more powerful. Utilizing such approaches in photonic FBQC along with the developments in highly efficient photonic hardware can make the transition to large-scale fault-tolerant quantum computers a reality in the near future.

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28 March 2021
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Experimental Quantum Reinforcement Learning

With the advent of more powerful classical computational power, machine learning and artificial intelligence research has made a recent resurgence in popularity and massive progress has been made in recent years in developing useful algorithms for practical applications. Meanwhile, quantum computing research has advanced to a stage where quantum supremacy has been shown experimentally, and theoretical algorithmic advantages in, for instance, machine learning have been theoretically proven. One particularly interesting machine learning paradigm is Reinforcement Learning (RL), where agents directly interact with an environment and learn by feedback exchanges. In recent years, RL has been utilized to assist in several problems in quantum information processing, such as decoding of errors, quantum feedback and adaptive code-design with significant success. Turned around, implementing ‘quantum’ RL using quantum computers has been shown to make the decision making process for RL agents quadratically faster than on classical hardware.

In most protocols so far, the interaction between the agent and the environment has been designed to occur entirely via classical communication in most RL applications. However, there is a theoretically suggested possibility of gaining increased quantum speedup, if this interaction can be transferred via quantum route. In this work, the authors propose a hybrid RL protocol that enables both quantum as well as classical information transfer between the agent and the environment. The main objective is to evaluate the comparative impact of this hybrid model on agent’s learning time with respect to RL schemes based on solely classical communication. The work uses a fully programmable nanophotonic processor interfaced with photons for the experimental implementation of the protocol. The setup implements an active feedback mechanism combining quantum amplitude amplification with a classical control mechanism that updates its learning policy.

The setup consists of a single-photon source pumped by laser light leading to the generation of a pair of single photons. One of these photons is sent to a quantum processor to perform a particular computation, while the other one is sent to a single-photon detector for heralding. Highly efficient detectors with short dead time response serve as fast feedback. Both detection events at the processor output and photon detector are recorded and registered with a time tagging module (TTM) as coincidence events. The agent and the environment are assigned different areas of the processor, performing the prior steps of the Grover-like amplitude amplification. The agent is further equipped with a classical control mechanism that updates its learning policy.

Any typical Grover-like algorithm faces a drop in the amplitude amplification after reaching the optimal point. Each agent reaches this optimal point at different epochs, therefore one can identify the probability up to which it is beneficial for all agents to use a quantum strategy over the classical strategy. The average number of interactions until the agent accomplishes a specific task is the learning time. The setup allows the agents to choose the most favorable strategy by switching from quantum to classical as soon as the second becomes more advantageous. Such combined strategy is shown to outperform the pure classical scenario.

Such a hybrid model represents a potentially interesting advantage over previously implemented protocols which are purely quantum or classical. Photonic architectures in particular are put forward by the authors to be one of the most suitable candidates for implementing these types of learning algorithms, by providing advantages of compactness, full tunability and low-loss communication which easily implements active feedback mechanisms for RL algorithms even over long distances. However, the theoretical implementation of such protocols is general and shown to be applicable to any quantum computational platform. Their results also demonstrate the feasibility of integrating quantum mechanical RL speed-ups in future complex quantum networks.

Finally, through the advancement of integrated optics towards the fabrication of increasingly large devices, such demonstration could be extended to more complex quantum circuits allowing for processing of high-dimensional states. This raises hopes for achieving superior performance in increasingly complex learning devices. Undoubtedly in the near future, AI and RL will play an important role in future large-scale quantum communication networks, including a potential quantum internet.

In most protocols so far, the interaction between the agent and the environment has been designed to occur entirely via classical communication in most RL applications. However, there is a theoretically suggested possibility of gaining increased quantum speedup, if this interaction can be transferred via quantum route. In this work, the authors propose a hybrid RL protocol that enables both quantum as well as classical information transfer between the agent and the environment. The main objective is to evaluate the comparative impact of this hybrid model on agent’s learning time with respect to RL schemes based on solely classical communication. The work uses a fully programmable nanophotonic processor interfaced with photons for the experimental implementation of the protocol. The setup implements an active feedback mechanism combining quantum amplitude amplification with a classical control mechanism that updates its learning policy.

The setup consists of a single-photon source pumped by laser light leading to the generation of a pair of single photons. One of these photons is sent to a quantum processor to perform a particular computation, while the other one is sent to a single-photon detector for heralding. Highly efficient detectors with short dead time response serve as fast feedback. Both detection events at the processor output and photon detector are recorded and registered with a time tagging module (TTM) as coincidence events. The agent and the environment are assigned different areas of the processor, performing the prior steps of the Grover-like amplitude amplification. The agent is further equipped with a classical control mechanism that updates its learning policy.

Any typical Grover-like algorithm faces a drop in the amplitude amplification after reaching the optimal point. Each agent reaches this optimal point at different epochs, therefore one can identify the probability up to which it is beneficial for all agents to use a quantum strategy over the classical strategy. The average number of interactions until the agent accomplishes a specific task is the learning time. The setup allows the agents to choose the most favorable strategy by switching from quantum to classical as soon as the second becomes more advantageous. Such combined strategy is shown to outperform the pure classical scenario.

Such a hybrid model represents a potentially interesting advantage over previously implemented protocols which are purely quantum or classical. Photonic architectures in particular are put forward by the authors to be one of the most suitable candidates for implementing these types of learning algorithms, by providing advantages of compactness, full tunability and low-loss communication which easily implements active feedback mechanisms for RL algorithms even over long distances. However, the theoretical implementation of such protocols is general and shown to be applicable to any quantum computational platform. Their results also demonstrate the feasibility of integrating quantum mechanical RL speed-ups in future complex quantum networks.

Finally, through the advancement of integrated optics towards the fabrication of increasingly large devices, such demonstration could be extended to more complex quantum circuits allowing for processing of high-dimensional states. This raises hopes for achieving superior performance in increasingly complex learning devices. Undoubtedly in the near future, AI and RL will play an important role in future large-scale quantum communication networks, including a potential quantum internet.

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Quantum computers possess unique potential to outperform their classical counterparts with algorithms based on real-time evolution. Given the intrinsically quantum-mechanical relation between the time and energy domains, more focus is on quantum algorithms which focus on using a time-dependent perspective to solve time-independent problems. Hence, the simulation of the time-dependent Schrödinger equation is a more ideal framework to implement.

Presently, there are plenty of quantum algorithms which are based on solving the time-independent Schrödinger equation to determine the Hamiltonian eigenvalues and eigenstates. The majority of these algorithms are limited classically by the exponential scaling of Hilbert space with the system size and require considerable quantum resources to run. So far, methods such as Approximate Imaginary Time Evolution and Krylov diagonalization are more widely used in classical simulation of static phenomena than real-time evolution as the latter has computational limitations. There also exists practical limitations in getting closer to the ground state during the evolution. However, the states generated through real time evolution can provide a basis from which one can extract ground and excited states. In some cases, this method may be faster than other quantum methods that use time evolution as a subroutine for goals other than dynamical behaviour, such as using QPE for spectral analysis.

The authors in this work propose analyze variational quantum phase estimation (VQPE), a method based on real-time evolution for computing ground and excited states, using states generated by real-time evolution. The work consists of theoretical derivations using this method to solve strongly correlated Hamiltonians. The proposed VQPE method has a set of equations that specifies conditions for time evolution with a simple geometrical interpretation. These conditions decouple eigenstates out of the set of time evolved expansion states and connect the method to the classical filter diagonalization algorithm. Furthermore, the authors introduce a unitary formulation of VQPE that allows for a direct comparison to iterative phase estimation. In this formulation, the number of matrix elements that need to be measured scales linearly instead of quadratic with the number of expansion states thereby reducing the number of quantum measurements.

The authors also provide an analysis of the effects of noise on the convergence properties showing that simple regularization techniques suffice to mitigate the effects of noise. Also, the VQPE approach was demonstrated on a wide range of molecules of different complexities, simulating the algorithm classically, as well as the transverse field Ising model on IBM’s quantum simulator and hardware. For several weakly and moderately correlated molecules as well as strongly correlated transition metal dimer Cr2, the chemical accuracy for ground state energies is attained in less than ~50 real-time timesteps. This is comparatively faster than ~106 timesteps required by the state-of-the-art classical methods with orders of magnitude fewer variational parameters.

The results show VQPE as a natural and efficient quantum algorithm for ground and excited state calculations of general many-body systems. On one hand, QPE utilizes its deeper circuits to achieve Heisenberg-limited energy resolution which is comparatively more efficient for achieving high accuracy in overall run time. On the other hand, for the same number of time steps per circuit, VQPE is able to achieve higher accuracy than idealized QPE. It can be concluded that VQPE has near-term advantages as compared to long-term benefits of QPE. This makes it a better candidate for utilizing near-term hardware with shorter circuit depths and fewer available qubits being used as ancillae. By choosing an optimal time step size to generate linearly independent expansion states with each new time evolution, the variational Ansatz can be made compact which sets a lower bound to the time step size. This also minimizes the total simulation time as required by NISQ hardware. This compactness, together with their NISQ compatibility makes VQPE approaches as some of the most promising platforms to perform quantum simulations for many body systems beyond the reach of classical computations. Since real-time evolution is natural to implement on quantum hardware, this approach holds immense promise for NISQ implementation.

Presently, there are plenty of quantum algorithms which are based on solving the time-independent Schrödinger equation to determine the Hamiltonian eigenvalues and eigenstates. The majority of these algorithms are limited classically by the exponential scaling of Hilbert space with the system size and require considerable quantum resources to run. So far, methods such as Approximate Imaginary Time Evolution and Krylov diagonalization are more widely used in classical simulation of static phenomena than real-time evolution as the latter has computational limitations. There also exists practical limitations in getting closer to the ground state during the evolution. However, the states generated through real time evolution can provide a basis from which one can extract ground and excited states. In some cases, this method may be faster than other quantum methods that use time evolution as a subroutine for goals other than dynamical behaviour, such as using QPE for spectral analysis.

The authors in this work propose analyze variational quantum phase estimation (VQPE), a method based on real-time evolution for computing ground and excited states, using states generated by real-time evolution. The work consists of theoretical derivations using this method to solve strongly correlated Hamiltonians. The proposed VQPE method has a set of equations that specifies conditions for time evolution with a simple geometrical interpretation. These conditions decouple eigenstates out of the set of time evolved expansion states and connect the method to the classical filter diagonalization algorithm. Furthermore, the authors introduce a unitary formulation of VQPE that allows for a direct comparison to iterative phase estimation. In this formulation, the number of matrix elements that need to be measured scales linearly instead of quadratic with the number of expansion states thereby reducing the number of quantum measurements.

The authors also provide an analysis of the effects of noise on the convergence properties showing that simple regularization techniques suffice to mitigate the effects of noise. Also, the VQPE approach was demonstrated on a wide range of molecules of different complexities, simulating the algorithm classically, as well as the transverse field Ising model on IBM’s quantum simulator and hardware. For several weakly and moderately correlated molecules as well as strongly correlated transition metal dimer Cr2, the chemical accuracy for ground state energies is attained in less than ~50 real-time timesteps. This is comparatively faster than ~106 timesteps required by the state-of-the-art classical methods with orders of magnitude fewer variational parameters.

The results show VQPE as a natural and efficient quantum algorithm for ground and excited state calculations of general many-body systems. On one hand, QPE utilizes its deeper circuits to achieve Heisenberg-limited energy resolution which is comparatively more efficient for achieving high accuracy in overall run time. On the other hand, for the same number of time steps per circuit, VQPE is able to achieve higher accuracy than idealized QPE. It can be concluded that VQPE has near-term advantages as compared to long-term benefits of QPE. This makes it a better candidate for utilizing near-term hardware with shorter circuit depths and fewer available qubits being used as ancillae. By choosing an optimal time step size to generate linearly independent expansion states with each new time evolution, the variational Ansatz can be made compact which sets a lower bound to the time step size. This also minimizes the total simulation time as required by NISQ hardware. This compactness, together with their NISQ compatibility makes VQPE approaches as some of the most promising platforms to perform quantum simulations for many body systems beyond the reach of classical computations. Since real-time evolution is natural to implement on quantum hardware, this approach holds immense promise for NISQ implementation.

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14 March 2021
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Strongly Universal Hamiltonian Simulators

The “Analog quantum simulation” paradigm of quantum computing aims to develop simpler models of a complex quantum system while reproducing all the physical attributes of the system in the operational domain of interest, such as its spectrum or phase diagram. The main idea is to simulate a rather complex target Hamiltonian H using a simpler Hamiltonian H’ that can be more easily implemented on practical analog quantum-computational hardware. One of the advantages of analog quantum simulation is the expected lesser requirement of quantum error correction or precise controls. Hence, it is considered to be an important practical direction in the era of NISQ technology.

The concept of universality when seeking analog simulators is based on the existence of a Hamiltonian H’ in the family that can be used to simulate any local Hamiltonian H. General universal models such as spin-lattice model Hamiltonians can potentially be inefficient to simulate directly as they for example require interaction energy that scales exponentially with system size. This exponential scaling holds true if the original Hamiltonian has higher-dimensional, long-range, or even all-to-all interactions. In this work, the authors provide an efficient construction of these strongly universal families in which the required interaction energy and all other resources in the 2D simulator scale polynomially instead of exponentially. The scaling occurs in the size of the target Hamiltonian and precision parameters and is independent of the target’s connectivity. The work involves the conversion of the target Hamiltonian to a quantum phase estimation circuit embedded in 1D. This circuit is then mapped back to a low-degree simulating Hamiltonian, using the Feynman-Kitaev circuit to-Hamiltonian construction.

The authors extend this method to simulate any target Hamiltonian with a 1D or 2D Hamiltonian using some of the existing techniques in the literature. Combinations of techniques such as the quantum phase estimation algorithm and circuit-to-Hamiltonian transformations were used in a non-perturbative way, which allows to overcome the exponential overhead common to previous constructions. The results show that only polynomial overheads in both particle number and interaction energy are sufficient to simulate any local Hamiltonian with arbitrary connectivity by some universal Hamiltonians embedded in 1D or 2D.

This work establishes the possibility of efficient universal analog quantum simulation using simple 1D or 2D systems, which we know can be built in practice with good control. The required constructions known so-far have been far from optimal. For example, existing hardware has limited types of interactions available, so in order to consider also general interactions, these need to be simulated using those single type of interactions together with ancilla qubits placed in more than one dimension. Polynomial-sized encoding and decoding circuits can be used to simulate 1D analog Hamiltonians which can be explored further towards achieving strong universality. In this work, it is shown that strongly universal analog quantum simulation is possible and can efficiently simulate any target Hamiltonian using 1D and 2D universal systems using polynomial qubits and interaction energies, which they show is tight since it is impossible to lower interaction energy to constant. However, the encoding circuits inducing non-local correlations can affect the desirable properties of analog Hamiltonian simulations such as preservation of locality of observables, as well as considerations of noise. As an alternative approach, the translation-invariance can be relaxed by letting Hamiltonian interactions have more free parameters to encode the target Hamiltonian.

One interesting takeaway from this research is that analog quantum simulation is actually relevant for many more systems than previously thought, and digital gate-based quantum simulation may not always be the best way to go in the described cases. Further experimental realizations of analog quantum simulators are required to develop methods to simulate all physical systems and tackle classically intractable problems in a practical and efficient way.

The concept of universality when seeking analog simulators is based on the existence of a Hamiltonian H’ in the family that can be used to simulate any local Hamiltonian H. General universal models such as spin-lattice model Hamiltonians can potentially be inefficient to simulate directly as they for example require interaction energy that scales exponentially with system size. This exponential scaling holds true if the original Hamiltonian has higher-dimensional, long-range, or even all-to-all interactions. In this work, the authors provide an efficient construction of these strongly universal families in which the required interaction energy and all other resources in the 2D simulator scale polynomially instead of exponentially. The scaling occurs in the size of the target Hamiltonian and precision parameters and is independent of the target’s connectivity. The work involves the conversion of the target Hamiltonian to a quantum phase estimation circuit embedded in 1D. This circuit is then mapped back to a low-degree simulating Hamiltonian, using the Feynman-Kitaev circuit to-Hamiltonian construction.

The authors extend this method to simulate any target Hamiltonian with a 1D or 2D Hamiltonian using some of the existing techniques in the literature. Combinations of techniques such as the quantum phase estimation algorithm and circuit-to-Hamiltonian transformations were used in a non-perturbative way, which allows to overcome the exponential overhead common to previous constructions. The results show that only polynomial overheads in both particle number and interaction energy are sufficient to simulate any local Hamiltonian with arbitrary connectivity by some universal Hamiltonians embedded in 1D or 2D.

This work establishes the possibility of efficient universal analog quantum simulation using simple 1D or 2D systems, which we know can be built in practice with good control. The required constructions known so-far have been far from optimal. For example, existing hardware has limited types of interactions available, so in order to consider also general interactions, these need to be simulated using those single type of interactions together with ancilla qubits placed in more than one dimension. Polynomial-sized encoding and decoding circuits can be used to simulate 1D analog Hamiltonians which can be explored further towards achieving strong universality. In this work, it is shown that strongly universal analog quantum simulation is possible and can efficiently simulate any target Hamiltonian using 1D and 2D universal systems using polynomial qubits and interaction energies, which they show is tight since it is impossible to lower interaction energy to constant. However, the encoding circuits inducing non-local correlations can affect the desirable properties of analog Hamiltonian simulations such as preservation of locality of observables, as well as considerations of noise. As an alternative approach, the translation-invariance can be relaxed by letting Hamiltonian interactions have more free parameters to encode the target Hamiltonian.

One interesting takeaway from this research is that analog quantum simulation is actually relevant for many more systems than previously thought, and digital gate-based quantum simulation may not always be the best way to go in the described cases. Further experimental realizations of analog quantum simulators are required to develop methods to simulate all physical systems and tackle classically intractable problems in a practical and efficient way.

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05 March 2021
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Nanophotonic Quantum Computing

In recent years, there have been a number of promising breakthroughs in engineering quantum processors, for example in ion-trap systems and superconducting systems. Physical platforms include programmable machines that can deliver automation, stability, and repeatability to implement quantum algorithms. These machines can now be remotely accessed and loaded with algorithms written in high-level programming languages by users having no in-depth knowledge of the low-level quantum hardware details of the apparatus. These capabilities have rapidly accelerated research that targets application development for near-term quantum computers.

One major limitation of present-day systems is the high level of noise affecting the qubits. Noise severely restricts the efficient application of quantum algorithms even if the algorithms are in-principle compatible with large scale implementations. Some of these algorithms are efficiently encoded in binary modes called qubits, while others are more efficiently expressed in a model in which each independent quantum system is described by a state in an infinite-dimensional Hilbert space. Typical applications that include those are the ones implementing bosonic error correction codes and gaussian boson sampling applications. Physical platforms offered by photonic hardware possess great potential to explore the large-scale physical implementation of such quantum algorithms. An ideal system should be dynamically programmable and readily scalable to hundreds of modes and photons. Also, it should be able to access a class of quantum circuits that are exceedingly harder to be efficiently simulated by classical hardware when the system size is increased. Presently, no such system is known that can achieve all these simultaneously. On one hand, a large photonic cluster state has been demonstrated, with the caveat of being limited to all-Gaussian states, gates, and measurements. On the other, single-photon-based experiments on integrated platforms suffer from non-deterministic state preparation and gate implementation which hinders their scalability.

The authors of this paper offer a full-stack solution consisting of hardware-software co-design based on a programmable nanophotonic chip that includes the capabilities of an ideal system in a single scalable and unified machine. The work involves both experimental research and theoretical modeling of the proposed hardware in order to enhance the required capabilities; programmability, high sampling rate, and photon number resolving. The programmable chip operates at room temperature and is interfaced with an automated control system for executing many-photon quantum circuits. The procedure involves the initial state preparation, gate sequence implementation, and readout followed by verifying the non-classicality of the device’s output. The photonic chip was used to generate squeezed states coupled with a custom modulated pump laser source with active locking system for the on-chip squeezer resonators. Further digital-to-analog converters were used for tuning the asymmetric Mach-Zehnder interferometer filters and programming the four-mode interferometer. This is coupled with a real-time data acquisition system for detector readout. Finally, a master controller (conventional server computer) is used to run a custom-developed control software coordinating the operation of all the hardware. By means of strong squeezing and high sampling rates, multi-photon detection events with photon numbers and rates are observed to be improved, exceeding previous quantum optical demonstrations.

The authors used the platform to carry out proof-of-principle implementations such as Gaussian boson sampling (GBS), resolving molecular vibronic spectra, and solving graph similarity problems which use samples from the device to infer a property of the object central to the application. For GBS, the samples provide information about the nonclassical probability distribution produced by the device. The vibronic spectra algorithm uses outputs from the device to obtain molecular properties, while for graph similarity, the samples reveal information on graph properties. In all demonstrations, the device is programmed remotely using the Strawberry Fields Python library. The authors also theoretically predicted a more detailed model of the device involving two Schmidt modes per squeezer, non-uniform loss before the unitary transformation, and excess noise. Such noise-modelling is still relatively rare in the nanophotonic quantum computing community and their additions to the field are potentially valuable for the understanding of algorithmic performance of such systems as compared to very different hardware implementation.

The proposed device marks a significant advance in scaling such nanophotonic chips to a larger number of modes. One of the greatest challenges in scaling to a system of this size is maintaining acceptably low losses in the interferometer. With precise chip fabrication tools, new designs for integrated beam splitters and phase shifters, one can achieve an order-of-magnitude improvement in the loss per layer in the interferometer. Furthermore, the inclusion of tunable single-mode (degenerate) squeezing and displacement can add a significant upgrade, permitting the generation of arbitrary Gaussian states. Such scaling and upgrades constitute the next steps for near-term photonic quantum information processing demonstrations. Combined with rapid advancements in photonic chip fabrication, such demonstrations coincide with new optimism towards photonics as a platform for advancing the frontier of quantum computation.

One major limitation of present-day systems is the high level of noise affecting the qubits. Noise severely restricts the efficient application of quantum algorithms even if the algorithms are in-principle compatible with large scale implementations. Some of these algorithms are efficiently encoded in binary modes called qubits, while others are more efficiently expressed in a model in which each independent quantum system is described by a state in an infinite-dimensional Hilbert space. Typical applications that include those are the ones implementing bosonic error correction codes and gaussian boson sampling applications. Physical platforms offered by photonic hardware possess great potential to explore the large-scale physical implementation of such quantum algorithms. An ideal system should be dynamically programmable and readily scalable to hundreds of modes and photons. Also, it should be able to access a class of quantum circuits that are exceedingly harder to be efficiently simulated by classical hardware when the system size is increased. Presently, no such system is known that can achieve all these simultaneously. On one hand, a large photonic cluster state has been demonstrated, with the caveat of being limited to all-Gaussian states, gates, and measurements. On the other, single-photon-based experiments on integrated platforms suffer from non-deterministic state preparation and gate implementation which hinders their scalability.

The authors of this paper offer a full-stack solution consisting of hardware-software co-design based on a programmable nanophotonic chip that includes the capabilities of an ideal system in a single scalable and unified machine. The work involves both experimental research and theoretical modeling of the proposed hardware in order to enhance the required capabilities; programmability, high sampling rate, and photon number resolving. The programmable chip operates at room temperature and is interfaced with an automated control system for executing many-photon quantum circuits. The procedure involves the initial state preparation, gate sequence implementation, and readout followed by verifying the non-classicality of the device’s output. The photonic chip was used to generate squeezed states coupled with a custom modulated pump laser source with active locking system for the on-chip squeezer resonators. Further digital-to-analog converters were used for tuning the asymmetric Mach-Zehnder interferometer filters and programming the four-mode interferometer. This is coupled with a real-time data acquisition system for detector readout. Finally, a master controller (conventional server computer) is used to run a custom-developed control software coordinating the operation of all the hardware. By means of strong squeezing and high sampling rates, multi-photon detection events with photon numbers and rates are observed to be improved, exceeding previous quantum optical demonstrations.

The authors used the platform to carry out proof-of-principle implementations such as Gaussian boson sampling (GBS), resolving molecular vibronic spectra, and solving graph similarity problems which use samples from the device to infer a property of the object central to the application. For GBS, the samples provide information about the nonclassical probability distribution produced by the device. The vibronic spectra algorithm uses outputs from the device to obtain molecular properties, while for graph similarity, the samples reveal information on graph properties. In all demonstrations, the device is programmed remotely using the Strawberry Fields Python library. The authors also theoretically predicted a more detailed model of the device involving two Schmidt modes per squeezer, non-uniform loss before the unitary transformation, and excess noise. Such noise-modelling is still relatively rare in the nanophotonic quantum computing community and their additions to the field are potentially valuable for the understanding of algorithmic performance of such systems as compared to very different hardware implementation.

The proposed device marks a significant advance in scaling such nanophotonic chips to a larger number of modes. One of the greatest challenges in scaling to a system of this size is maintaining acceptably low losses in the interferometer. With precise chip fabrication tools, new designs for integrated beam splitters and phase shifters, one can achieve an order-of-magnitude improvement in the loss per layer in the interferometer. Furthermore, the inclusion of tunable single-mode (degenerate) squeezing and displacement can add a significant upgrade, permitting the generation of arbitrary Gaussian states. Such scaling and upgrades constitute the next steps for near-term photonic quantum information processing demonstrations. Combined with rapid advancements in photonic chip fabrication, such demonstrations coincide with new optimism towards photonics as a platform for advancing the frontier of quantum computation.

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28 February 2021
###
Active Quantum Research Areas: Error Mitigation & Correction

In recent years, Noisy Intermediate-Scale Quantum (NISQ) systems are broadly studied, with a particular focus on investigating how near-term devices could outperform classical computers for practical applications. A major roadblock to obtaining a relevant quantum advantage is the inevitable presence of noise in these systems. Therefore, a major focus point of NISQ research is the exploration of noise in currently-available and realistic devices and how the effects of such noise can be mitigated. A growing body of work in this direction proposes various error correcting and error mitigating protocols with an objective to limit this unwanted noise and possibly achieve error suppression. As NISQ devices cannot support full error correction, analysis of the noise and finding ways to suppress it, will increase the chances of obtaining tangible benefits by NISQ computation. In this edition of Active Quantum Research Areas, we cover several recent and promising papers in this direction.

While techniques like Dynamical Decoupling have potential to partially suppress quantum errors, their effectiveness is still limited by errors that occur at unstructured times during a circuit. Furthermore, other commonly encountered noise mechanisms such as cross-talk and imperfectly calibrated control pulses can also decrease circuit execution fidelity. Recent work by [1] discusses an error mitigation strategy named `quantum measurement emulation' (QME), which is a feed-forward control technique for mitigating coherent errors. This technique employs stochastically applied single-qubit gates to ‘emulate’ quantum measurement along the appropriate axis, while simultaneously making this process less sensitive to coherent errors. Moreover, it uses the stabilizer code formalism in order to enable error suppression leading to improved circuit execution fidelity observed in this work. Since QME does not require the computation of correction gates as needed in randomized compiling, it can only protect against errors that rotate the qubit out of the logical codespace. This technique also seems to be effective against coherent errors occurring during twirling gates. For arbitrarily generated circuits, QME can outperform simple dynamical decoupling schemes by addressing discrete coherent errors. Moreover, it does not require costly measurements and feedback is cost-effective as well.

Apart from passively mitigating errors, another approach to rectify these errors will be effective active error-suppressing techniques. Out of the recently introduced methods, virtual distillation is capable of exponentially suppressing errors by preparing multiple noisy copies of a state and virtually distilling a more purified version. Although this technique requires additional (ancilla) qubits, qubit efficiency can be achieved by resetting and reusing qubits. One such method is proposed by [2] named Resource-Efficient Quantum Error Suppression Technique (REQUEST) which is an alternative to virtual distillation methods. For N qubit states, the total qubit requirement of REQUEST is 2N + 1 for any number of copies instead of MN + 1 qubits to use M copies as required by past approaches. The optimal number of these copies is then estimated by using near-Clifford circuits by comparing results mitigated with different values of M to exact quantities. It has been observed that with increasing the optimal number of copies, error suppression will also increase; perhaps exponentially. This suggests that the method can be relevant for larger devices where sufficient qubits and connectivity are available. However, one of the drawbacks of the method is the increase in the overall depth of the quantum circuit in order to achieve a reduction in qubit resources, so further research on this trade-off would be interesting.

Another recent work concerning error suppression is presented in [3]. This work proposes a technique to exponentially suppress bit or phase-flip errors with repetitive error correction. In this work, the authors implement 1D repetition codes embedded in a 2D grid of superconducting qubits. This technique requires the errors to be local and the performance needs to be maintained over many rounds of error correction - two major outstanding experimental challenges. The results demonstrate reduced logical error per round in the repetition code by more than 100× when increasing the number of qubits from 5 to 21. This exponential suppression of bit or phase-flip errors is shown to be stable over 50 rounds of error correction. Also, it was observed that a stable percentage of detection events was observed throughout the 50 rounds of error correction for the system with 21 superconducting qubits, which is important for showing the value of error correction. The authors also perform error detection using a small 2D surface code. Both experimentally implemented 1D and 2D codes agree with numerical simulations considering a simple depolarizing error model, which supports that superconducting qubits may be on a viable path towards fault-tolerant quantum computing. It would be interesting to compare the performance on other types of hardware also.

One of the potential benefits and long-term goals of error correction is attaining scalable quantum computing. However, logical error rates will only decrease with system size while using error correction when physical errors are sufficiently uncorrelated. One limiting factor in terms of scalability is the creation of leakage states, which are non-computational states created due to the excitation of unused high energy levels of the qubits during computation. Particularly for superconducting transmon qubits, this leakage mechanism opens a path to errors that are correlated in space and time. To overcome this, the authors of [4] propose a reset protocol that returns a qubit to the ground state from all relevant higher-level states. It employs a multi-level reset gate using an adiabatic swap operation between the qubit and the readout resonator combined with a fast return. The authors claim a fidelity of over 99% for qubits starting in any of the first three excited states while the gate error is predicted by an intuitive semi-classical model. During the experimentation, only currently existing hardware was used for normal operation and readout. Since there was no involvement of strong microwave drivers which might induce crosstalk, this reset protocol can be implementable on large scale systems. The performance of the protocol is tested with the bit-flip stabilizer code, investigating the accumulation and dynamics of leakage during error correction. The study reveals that applying reset reduces the magnitude of correlations leading to lower rates of logical errors and improved scaling and stability of error suppression as the number of qubits is increased. Therefore, optimizing gates and readout to have minimal leakage is a necessary strategy and the correlated nature of the leakage error makes reset protocols critical for quantum error correction.

Error correction and error mitigation strategies are both valid paradigms which will be required on the road to useful quantum computing. Current NISQ devices, however, cannot support full error correction for deep and wide enough circuits to be useful, therefore more attention has been given to error mitigation strategies that attempt to suppress any type of noise as much as possible. At the moment research is focused in the reduction of noise in gate, initialization and measurement operations, in order to have more reliable information about the state of the qubits during computation. Noise processes like leakage to non-computational states and crosstalk between neighbouring qubits are deemed as extremely important, which led to the proposal of active reset and other qubit control techniques. Experiments with small devices consisting of up to 20 qubits have been performed, in order to: a) show the advantages of error correction in combating leakage and crosstalk in the setting of repeated stabilizer measurements and b) show the advantages of error mitigation through techniques like reset and re-use of qubits and conversion of coherent errors into incoherent. Nevertheless, it is clear that achieving exponential noise suppression in large systems of relevant size is far from straightforward and will require advanced error correction and error mitigation techniques, even though there are indications through experiments with small systems that the aforementioned techniques will provide high-level suppression. As mentioned in [3], there are experimental results on both 1D and 2D codes that show evidence of being within a striking distance of noise suppression (as defined by the surface code threshold).

It will be particularly interesting to see in future research whether a fixed number of physical qubits should best be used for error correction or combined with error mitigation techniques.

References:

[1] Greene et. al., “Error mitigation via stabilizer measurement emulation” arXiv:2102.05767 (Feb. 2021)

[2] P. Czarnik et. al. “Qubit-efficient exponential suppression of errors” arXiv:2102.06056v1 (Feb. 2021)

[3] Z. Chen et. al. “Exponential suppression of bit or phase flip errors with repetitive error correction” arXiv:2102.06132v1 (Feb 2021)

[4] M. McEwen et. al. “Removing leakage-induced correlated errors in superconducting quantum error correction” arXiv:2102.06131v1 (Feb 2021)

While techniques like Dynamical Decoupling have potential to partially suppress quantum errors, their effectiveness is still limited by errors that occur at unstructured times during a circuit. Furthermore, other commonly encountered noise mechanisms such as cross-talk and imperfectly calibrated control pulses can also decrease circuit execution fidelity. Recent work by [1] discusses an error mitigation strategy named `quantum measurement emulation' (QME), which is a feed-forward control technique for mitigating coherent errors. This technique employs stochastically applied single-qubit gates to ‘emulate’ quantum measurement along the appropriate axis, while simultaneously making this process less sensitive to coherent errors. Moreover, it uses the stabilizer code formalism in order to enable error suppression leading to improved circuit execution fidelity observed in this work. Since QME does not require the computation of correction gates as needed in randomized compiling, it can only protect against errors that rotate the qubit out of the logical codespace. This technique also seems to be effective against coherent errors occurring during twirling gates. For arbitrarily generated circuits, QME can outperform simple dynamical decoupling schemes by addressing discrete coherent errors. Moreover, it does not require costly measurements and feedback is cost-effective as well.

Apart from passively mitigating errors, another approach to rectify these errors will be effective active error-suppressing techniques. Out of the recently introduced methods, virtual distillation is capable of exponentially suppressing errors by preparing multiple noisy copies of a state and virtually distilling a more purified version. Although this technique requires additional (ancilla) qubits, qubit efficiency can be achieved by resetting and reusing qubits. One such method is proposed by [2] named Resource-Efficient Quantum Error Suppression Technique (REQUEST) which is an alternative to virtual distillation methods. For N qubit states, the total qubit requirement of REQUEST is 2N + 1 for any number of copies instead of MN + 1 qubits to use M copies as required by past approaches. The optimal number of these copies is then estimated by using near-Clifford circuits by comparing results mitigated with different values of M to exact quantities. It has been observed that with increasing the optimal number of copies, error suppression will also increase; perhaps exponentially. This suggests that the method can be relevant for larger devices where sufficient qubits and connectivity are available. However, one of the drawbacks of the method is the increase in the overall depth of the quantum circuit in order to achieve a reduction in qubit resources, so further research on this trade-off would be interesting.

Another recent work concerning error suppression is presented in [3]. This work proposes a technique to exponentially suppress bit or phase-flip errors with repetitive error correction. In this work, the authors implement 1D repetition codes embedded in a 2D grid of superconducting qubits. This technique requires the errors to be local and the performance needs to be maintained over many rounds of error correction - two major outstanding experimental challenges. The results demonstrate reduced logical error per round in the repetition code by more than 100× when increasing the number of qubits from 5 to 21. This exponential suppression of bit or phase-flip errors is shown to be stable over 50 rounds of error correction. Also, it was observed that a stable percentage of detection events was observed throughout the 50 rounds of error correction for the system with 21 superconducting qubits, which is important for showing the value of error correction. The authors also perform error detection using a small 2D surface code. Both experimentally implemented 1D and 2D codes agree with numerical simulations considering a simple depolarizing error model, which supports that superconducting qubits may be on a viable path towards fault-tolerant quantum computing. It would be interesting to compare the performance on other types of hardware also.

One of the potential benefits and long-term goals of error correction is attaining scalable quantum computing. However, logical error rates will only decrease with system size while using error correction when physical errors are sufficiently uncorrelated. One limiting factor in terms of scalability is the creation of leakage states, which are non-computational states created due to the excitation of unused high energy levels of the qubits during computation. Particularly for superconducting transmon qubits, this leakage mechanism opens a path to errors that are correlated in space and time. To overcome this, the authors of [4] propose a reset protocol that returns a qubit to the ground state from all relevant higher-level states. It employs a multi-level reset gate using an adiabatic swap operation between the qubit and the readout resonator combined with a fast return. The authors claim a fidelity of over 99% for qubits starting in any of the first three excited states while the gate error is predicted by an intuitive semi-classical model. During the experimentation, only currently existing hardware was used for normal operation and readout. Since there was no involvement of strong microwave drivers which might induce crosstalk, this reset protocol can be implementable on large scale systems. The performance of the protocol is tested with the bit-flip stabilizer code, investigating the accumulation and dynamics of leakage during error correction. The study reveals that applying reset reduces the magnitude of correlations leading to lower rates of logical errors and improved scaling and stability of error suppression as the number of qubits is increased. Therefore, optimizing gates and readout to have minimal leakage is a necessary strategy and the correlated nature of the leakage error makes reset protocols critical for quantum error correction.

Error correction and error mitigation strategies are both valid paradigms which will be required on the road to useful quantum computing. Current NISQ devices, however, cannot support full error correction for deep and wide enough circuits to be useful, therefore more attention has been given to error mitigation strategies that attempt to suppress any type of noise as much as possible. At the moment research is focused in the reduction of noise in gate, initialization and measurement operations, in order to have more reliable information about the state of the qubits during computation. Noise processes like leakage to non-computational states and crosstalk between neighbouring qubits are deemed as extremely important, which led to the proposal of active reset and other qubit control techniques. Experiments with small devices consisting of up to 20 qubits have been performed, in order to: a) show the advantages of error correction in combating leakage and crosstalk in the setting of repeated stabilizer measurements and b) show the advantages of error mitigation through techniques like reset and re-use of qubits and conversion of coherent errors into incoherent. Nevertheless, it is clear that achieving exponential noise suppression in large systems of relevant size is far from straightforward and will require advanced error correction and error mitigation techniques, even though there are indications through experiments with small systems that the aforementioned techniques will provide high-level suppression. As mentioned in [3], there are experimental results on both 1D and 2D codes that show evidence of being within a striking distance of noise suppression (as defined by the surface code threshold).

It will be particularly interesting to see in future research whether a fixed number of physical qubits should best be used for error correction or combined with error mitigation techniques.

References:

[1] Greene et. al., “Error mitigation via stabilizer measurement emulation” arXiv:2102.05767 (Feb. 2021)

[2] P. Czarnik et. al. “Qubit-efficient exponential suppression of errors” arXiv:2102.06056v1 (Feb. 2021)

[3] Z. Chen et. al. “Exponential suppression of bit or phase flip errors with repetitive error correction” arXiv:2102.06132v1 (Feb 2021)

[4] M. McEwen et. al. “Removing leakage-induced correlated errors in superconducting quantum error correction” arXiv:2102.06131v1 (Feb 2021)

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21 February 2021
###
Layered circuit ansatze for combinatorial optimization

Optimizing quantum algorithms on near-term noisy-intermediate scale quantum (NISQ) devices is an essential requirement to demonstrate the quantum advantage over the existing classical computing. The capabilities of these devices are constrained by high noise levels and limited error mitigation. Combinatorial optimization on quantum processors is one such promising route to solve the problem created by noise in these systems. Out of various existing approaches for optimization, the most notable ones are Quantum Approximate Optimization Algorithm (QAOA) and variational quantum algorithms, especially for eigenvalue problems with high complexity.

The authors in this work propose an iterative “Layer VQE (L-VQE)” approach, inspired by the well-known Variational Quantum Eigensolver (VQE). The work conducts numerical studies, simulating circuits with up to 40 qubits and 352 parameters (which is a hard problem to simulate) using matrix product state (MPS) representation to perform large-scale simulations of the quantum circuits. The performance of L-VQE in this work is simulated using a noisy simulator of a trapped-ion quantum computer.

It has been proven in literature that for a graph with n vertices, solving the k-communities modularity maximization problem requires kn qubits which encode the problem using the well-known Ising model Hamiltonian. The authors of this paper propose a novel formulation that requires only n log(k) qubits. They further compare the performance of L-VQE with Quantum Approximate Optimization Algorithm (QAOA), which is widely considered to be a strong candidate for quantum advantage in applications with NISQ computers. However, the many-body terms in the Hamiltonian make it harder to implement in the QAOA setting. The numerical results suggest that QAOA comparatively achieves lower approximation ratios and requires significantly deeper circuits. This gap can be compensated using the L-VQE approach for NISQ devices.

The proposed L-VQE algorithm in this work starts from a relatively simple and shallow hardware efficient ansatz with a small number of parameterized gates and then adds layers to the ansatz systematically. This differs from most VQE approaches which have an ansatz fixed upfront. The work claims that this approach can make the ansatz more expressive along with reducing the optimization overhead. Furthermore, the numerical results suggest that adding layers of the ansatz increases the probability of finding the ground state or finding the state that is sufficiently close to the ground state. Such an approach is deemed simple enough to be generalized to different quantum architectures. It is numerically shown that the standard VQE is more likely to fail in the presence of sampling noise as compared to L-VQE which is shown to be more robust under sampling noise.

Lastly, further numerical studies in this work claim that the ansatz with entanglement performs better than the ansatz without entanglement, which is intuitive because entanglement is an important resource to quantum computing; even when the target optimal state is a product state, optimization to it may be guided better with an entangling ansatz. These results provide insight into the introduction of additional entangling parameters in VQE for classical problems. It is proposed to break down the barriers in the optimization landscape, making it more convex and therefore more amenable to simple local outer-loop optimizers to find a minimum. This contrasts with the previous results where no beneficial effects of entanglement are observed. This difference in results suggests the importance of the parameterization choice and the overall VQE procedure design contributing to the success of such methods.

The authors in this work propose an iterative “Layer VQE (L-VQE)” approach, inspired by the well-known Variational Quantum Eigensolver (VQE). The work conducts numerical studies, simulating circuits with up to 40 qubits and 352 parameters (which is a hard problem to simulate) using matrix product state (MPS) representation to perform large-scale simulations of the quantum circuits. The performance of L-VQE in this work is simulated using a noisy simulator of a trapped-ion quantum computer.

It has been proven in literature that for a graph with n vertices, solving the k-communities modularity maximization problem requires kn qubits which encode the problem using the well-known Ising model Hamiltonian. The authors of this paper propose a novel formulation that requires only n log(k) qubits. They further compare the performance of L-VQE with Quantum Approximate Optimization Algorithm (QAOA), which is widely considered to be a strong candidate for quantum advantage in applications with NISQ computers. However, the many-body terms in the Hamiltonian make it harder to implement in the QAOA setting. The numerical results suggest that QAOA comparatively achieves lower approximation ratios and requires significantly deeper circuits. This gap can be compensated using the L-VQE approach for NISQ devices.

The proposed L-VQE algorithm in this work starts from a relatively simple and shallow hardware efficient ansatz with a small number of parameterized gates and then adds layers to the ansatz systematically. This differs from most VQE approaches which have an ansatz fixed upfront. The work claims that this approach can make the ansatz more expressive along with reducing the optimization overhead. Furthermore, the numerical results suggest that adding layers of the ansatz increases the probability of finding the ground state or finding the state that is sufficiently close to the ground state. Such an approach is deemed simple enough to be generalized to different quantum architectures. It is numerically shown that the standard VQE is more likely to fail in the presence of sampling noise as compared to L-VQE which is shown to be more robust under sampling noise.

Lastly, further numerical studies in this work claim that the ansatz with entanglement performs better than the ansatz without entanglement, which is intuitive because entanglement is an important resource to quantum computing; even when the target optimal state is a product state, optimization to it may be guided better with an entangling ansatz. These results provide insight into the introduction of additional entangling parameters in VQE for classical problems. It is proposed to break down the barriers in the optimization landscape, making it more convex and therefore more amenable to simple local outer-loop optimizers to find a minimum. This contrasts with the previous results where no beneficial effects of entanglement are observed. This difference in results suggests the importance of the parameterization choice and the overall VQE procedure design contributing to the success of such methods.

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For the Noisy Intermediate Scale Quantum (NISQ) era, in the absence of large-scale quantum error correction, the number of gates that can be applied while maintaining computational coherence is at present strongly limited by hardware noise and decoherence. In an attempt to alleviate some of the detrimental effects, current generations of quantum algorithms often rely on a hybrid classical-quantum approach. Such approaches consider a trial quantum state (ansatz state) with a tractable number of parameters and relatively short circuit depth. These parameters are then optimized in order to approximate a target state as accurately as possible. In most of such applications shown hitherto, the target state was variationally optimized to represent a lowest-energy eigenstate (groundstate) of some quantum Hamiltonian.

However, one can also envision simulating unitary time evolution (or ‘dynamics’) with such variational algorithms. The authors of today’s paper first reference the Time-Dependent Variational Algorithm (TDVA) which encodes the state into a variational circuit and iteratively updates the parameters by solving the corresponding equation of motion. However, a significant drawback of that existing algorithm is that it suffers from an expensive quadratic cost in the total number of variational parameters.

To tackle this problem, the authors in this work introduce a novel hybrid algorithm to simulate the real-time evolution of quantum systems using parameterized quantum circuits. They propose a new method called "projected - Variational Quantum Dynamics" (p-VQD) which realizes an iterative, global projection of the exact time evolution onto the parameterized manifold. In the small time-step limit, this approach is equivalent to the McLachlan’s variational principle and uses it to optimize all parameters at once. This algorithm is shown to overcome the drawbacks of existing approaches as it is both global – it optimizes all parameters at once – and efficient – it scales linearly with the number of parameters. Moreover, it does not require auxiliary (ancilla) qubits and the depth of the circuit is constant throughout all the simulation. They use circuit differentiation to compute gradients analytically and use them for gradient descent optimization.

This global approach potentially extends the scope of existing efficient variational methods, that instead typically rely on the iterative optimization of a restricted subset of variational parameters. The authors claim that this approach can be particularly advantageous over existing global optimization algorithms based on the time-dependent variational principle.

They have shown that the algorithm is asymptotically more hardware efficient than the standard variational algorithm while retaining a higher accuracy. Currently a drawback of this method is that the circuit constructed on the quantum device is approximately twice as deep as the ansatz used to represent the system. However, by suitably controlling the number of two-qubit gates in the particular ansatz of choice, the authors comment that p-VQD is already implementable to simulate small quantum systems on available devices.

One possible application of the approach used in this work is to study the dynamical properties of two-dimensional interacting systems which is a notoriously difficult problem for classical computation. Similar to all other variational algorithms, the choice of the right parametrization is fundamental for the algorithm to succeed. In this sense, having an efficient quantum algorithm to perform variational time evolution is essential to compare to classical results obtained with variational states based on tensor or neural networks.

However, one can also envision simulating unitary time evolution (or ‘dynamics’) with such variational algorithms. The authors of today’s paper first reference the Time-Dependent Variational Algorithm (TDVA) which encodes the state into a variational circuit and iteratively updates the parameters by solving the corresponding equation of motion. However, a significant drawback of that existing algorithm is that it suffers from an expensive quadratic cost in the total number of variational parameters.

To tackle this problem, the authors in this work introduce a novel hybrid algorithm to simulate the real-time evolution of quantum systems using parameterized quantum circuits. They propose a new method called "projected - Variational Quantum Dynamics" (p-VQD) which realizes an iterative, global projection of the exact time evolution onto the parameterized manifold. In the small time-step limit, this approach is equivalent to the McLachlan’s variational principle and uses it to optimize all parameters at once. This algorithm is shown to overcome the drawbacks of existing approaches as it is both global – it optimizes all parameters at once – and efficient – it scales linearly with the number of parameters. Moreover, it does not require auxiliary (ancilla) qubits and the depth of the circuit is constant throughout all the simulation. They use circuit differentiation to compute gradients analytically and use them for gradient descent optimization.

This global approach potentially extends the scope of existing efficient variational methods, that instead typically rely on the iterative optimization of a restricted subset of variational parameters. The authors claim that this approach can be particularly advantageous over existing global optimization algorithms based on the time-dependent variational principle.

They have shown that the algorithm is asymptotically more hardware efficient than the standard variational algorithm while retaining a higher accuracy. Currently a drawback of this method is that the circuit constructed on the quantum device is approximately twice as deep as the ansatz used to represent the system. However, by suitably controlling the number of two-qubit gates in the particular ansatz of choice, the authors comment that p-VQD is already implementable to simulate small quantum systems on available devices.

One possible application of the approach used in this work is to study the dynamical properties of two-dimensional interacting systems which is a notoriously difficult problem for classical computation. Similar to all other variational algorithms, the choice of the right parametrization is fundamental for the algorithm to succeed. In this sense, having an efficient quantum algorithm to perform variational time evolution is essential to compare to classical results obtained with variational states based on tensor or neural networks.

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07 February 2021
###
Picosecond Quantum computing

Recently there has been increased efforts to build large-scale quantum computers for solving certain types of hard computational problems. These efforts are mainly motivated by the prospect of enabling quantum algorithms with a quadratic, polynomial or potentially exponential speedup. When the size of the problem is sufficiently large, this scaling advantage implies that a quantum computer will outperform its classical counterpart, independently of the time it takes to execute a single gate. However, for any real-world application, not only the scaling but also the total computation time will be of importance, hence the realization of faster gate operations becomes a necessity to further improve the fidelity of the computation.

In the work we highlight today, the authors discuss the realization of a universal set of ultrafast single- and two-qubit operations with superconducting quantum circuits along with investigating the physical and technical limitations for achieving faster gates. The work establishes a fundamental bound on the minimal gate time, which depends on the qubit nonlinearity and bandwidth of the control pulse over a large parameter range, being independent of the qubit design. The numerical results suggest that for highly anharmonic flux qubits and commercially available control electronics, elementary single- and two-qubit operations can be implemented in about 100 picoseconds with residual gate errors below 10^{-4}. Under the same conditions, authors estimate that the complete execution of a compressed version of Shor’s algorithm for factoring the number 15 would take about one nanosecond on such a hypothetical device.

The numerical results claim that there exists a lower bound for both single-qubit gates and two-qubit gates, which holds without the often assumed three-level approximation for the whole range of qubit parameters explored in this work. For very fast gates in the range of hundred picoseconds, additional limitations arise from the finite qubit oscillation time.

The authors also addressed the implementation of larger quantum circuits composed out of many ultrafast gates. A full multi-level simulation of a basic three-qubit circuit consisting of eleven elementary single- and two-qubit gates is performed taking the finite qubit rotation time into account which introduces a natural cycle time according to which gates must be clocked. For realistic qubit nonlinearities and control bandwidths, the simulated execution times for the whole circuit are observed to be about 1-2 ns, which is about two orders of magnitude faster than what is achievable in most superconducting quantum computing experiments today. The results demonstrate that significant improvements in this direction are still possible.

In their analysis, the authors have restricted the gate times down to about 50 picoseconds, which requires absolute nonlinearities and larger control bandwidth than contemporary state-of-the-art experiments. Although such parameters are highly non-standard for current superconducting qubit experiments, they are still within physical and technological bounds. For the implementation of even faster gates, additional physical constraints will come into play and also the applicability of the usual effective circuit model must be re-evaluated. For example, at a certain value, the energy of the third circuit level is comparable to twice the superconducting gap of aluminum. Any components of the control field above this frequency will excite quasiparticles and strongly degrade the qubit coherence. However, in other materials the superconducting gap can be substantially higher, which suggests that at least in principle, gate times in the range of 1-10 picoseconds could become accessible in the future.

These results demonstrate that, compared to state-of-the-art implementations with transmon qubits, a hundredfold increase in the speed of gate operations with superconducting circuits is still feasible. Despite its long-term relevance, the implementation of quantum gates in the picosecond regime still remains highly unexplored and the ultimate limit for the speed of superconducting quantum processors is still a research question to be addressed with a hope for pushing towards faster gates. In such a scenario, decoherence will be a less limiting factor, since gates will take less time to be applied. Furthermore, algorithms that require a large circuit depth will be able to be implemented, allowing researchers to solve more complex problems. Finally, processes like quantum error correction and decoding of errors will be much easier to implement, therefore taking us beyond the NISQ era.

In the work we highlight today, the authors discuss the realization of a universal set of ultrafast single- and two-qubit operations with superconducting quantum circuits along with investigating the physical and technical limitations for achieving faster gates. The work establishes a fundamental bound on the minimal gate time, which depends on the qubit nonlinearity and bandwidth of the control pulse over a large parameter range, being independent of the qubit design. The numerical results suggest that for highly anharmonic flux qubits and commercially available control electronics, elementary single- and two-qubit operations can be implemented in about 100 picoseconds with residual gate errors below 10

The numerical results claim that there exists a lower bound for both single-qubit gates and two-qubit gates, which holds without the often assumed three-level approximation for the whole range of qubit parameters explored in this work. For very fast gates in the range of hundred picoseconds, additional limitations arise from the finite qubit oscillation time.

The authors also addressed the implementation of larger quantum circuits composed out of many ultrafast gates. A full multi-level simulation of a basic three-qubit circuit consisting of eleven elementary single- and two-qubit gates is performed taking the finite qubit rotation time into account which introduces a natural cycle time according to which gates must be clocked. For realistic qubit nonlinearities and control bandwidths, the simulated execution times for the whole circuit are observed to be about 1-2 ns, which is about two orders of magnitude faster than what is achievable in most superconducting quantum computing experiments today. The results demonstrate that significant improvements in this direction are still possible.

In their analysis, the authors have restricted the gate times down to about 50 picoseconds, which requires absolute nonlinearities and larger control bandwidth than contemporary state-of-the-art experiments. Although such parameters are highly non-standard for current superconducting qubit experiments, they are still within physical and technological bounds. For the implementation of even faster gates, additional physical constraints will come into play and also the applicability of the usual effective circuit model must be re-evaluated. For example, at a certain value, the energy of the third circuit level is comparable to twice the superconducting gap of aluminum. Any components of the control field above this frequency will excite quasiparticles and strongly degrade the qubit coherence. However, in other materials the superconducting gap can be substantially higher, which suggests that at least in principle, gate times in the range of 1-10 picoseconds could become accessible in the future.

These results demonstrate that, compared to state-of-the-art implementations with transmon qubits, a hundredfold increase in the speed of gate operations with superconducting circuits is still feasible. Despite its long-term relevance, the implementation of quantum gates in the picosecond regime still remains highly unexplored and the ultimate limit for the speed of superconducting quantum processors is still a research question to be addressed with a hope for pushing towards faster gates. In such a scenario, decoherence will be a less limiting factor, since gates will take less time to be applied. Furthermore, algorithms that require a large circuit depth will be able to be implemented, allowing researchers to solve more complex problems. Finally, processes like quantum error correction and decoding of errors will be much easier to implement, therefore taking us beyond the NISQ era.

Tagged under

31 January 2021
###
Active Quantum Research Areas: Quantum Machine Learning

In this edition of Active Quantum Research Areas (AQRAs), we highlight recent research in the quantum community on quantum machine learning; in particular, we focus here on training, expressibility and potential for quantum advantage.

Quantum Machine Learning (QML) techniques have been rigorously researched in the past few years, showing great promise in a number of applications and using various quantum algorithmic techniques and potential speedups. However, the effectiveness of quantum- over classical machine learning is not always evident and still heavily being investigated. In this blog post, we briefly summarize and contextualize several recent arXiv contributions in the area of quantum machine learning, quantum neural networks, quantum generative adversarial networks, and training their variational quantum circuits.

Research, such as the one presented in [1], suggests that potentially exponential quantum advantage is possible for certain tasks when the goal is to achieve accurate prediction on all inputs. While training techniques differ from the classical machine learning realm, more focus is on developing new models and algorithms to exploit the quantum mechanical phenomena. For instance, hybrid quantum-classical QML protocols can offer a significantly enhanced potential. Similarly, features like the expressivity of quantum neural networks can be exploited to gain quantum advantage. More accurately, the relationship between expressibillity and trainability of highly expressive ansatze states can provide useful insight into QML.

It was shown that the potential advantage of quantum ML over classical ML is limited, in the specific case of predicting outcomes of quantum experiments with a specified average-case prediction error. The recent findings by Huang et al. [1] establish the fact that quantum ML can have an exponential advantage over classical ML for certain problems where the objective is achieving a specified worst-case prediction error. This work claims that for any input distribution, a classical ML model can provide accurate predictions on average by accessing a quantum process a number of times comparable to the optimal quantum ML model. These results and analysis complement recent studies of the computational complexity of classical ML models trained with data from quantum experiments. Together, these rigorous results indicate that classical ML trained models using quantum measurement data can be surprisingly powerful. Such a hybrid protocol can potentially address challenging quantum problems in physics, chemistry, and materials for near-term experimental platforms.

Another direction being investigated is exploration towards establishing a quantum neural network (QNN) analogy of the universal approximation theorem, which attempts to provide the foundation for expressivity of classical neural networks. Recent work by Wu Y. et al. [2] discusses the generalization of expressivity of QNN for learning targets (for example observables of input wave functions), while focusing on fully connected architectures. The findings suggest that an accurate approximation is possible only when the input wave functions (target function) in the dataset do not exhaust the entire Hilbert space that the quantum circuit acts on, and more precisely, the Hilbert space dimension of the former has to be less than half of the Hilbert space dimension of the latter. If this requirement cannot be satisfied by the dataset, the expressivity capabilities can be restored by adding one ancillary qubit where the wave function is always fixed at input. Hybrid quantum-classical QML protocols with mixed target states can be exploited in already available experimental platforms with potential promising applications in quantum computing and noise sensing.

Another aspect of quantum ML that is being explored is quantum generative adversarial learning (QGAN) which is a promising strategy to use quantum devices for (quantum) estimation or generative machine learning tasks. The recent work by Braccia et al. [3], discusses the convergence behaviors of a QGAN training process and proposes new algorithms for QGAN training for mixed states which are expected to work better than previously used techniques. It is observed that the states obtained from currently available NISQ devices may display a “chaotic” behaviour during training, without achieving convergence. Results from this work show that algorithms based on the adaptation of a gradient-based technique allow provable convergence with bilinear score functions while the algorithms based on convex optimization techniques are shown to be especially suited for non-parametric states that are iteratively updated via a quantum circuit. Also, endowing the scheme with the ability to process entangled copies of the target state results in enhanced performance.

In Ref. [4] it is shown how generative models can be enhanced via quantum correlations, possibly paving the way to a quantum advantage in a generative ML setting. They show how quantum correlations offer a powerful resource, using a numerical example on a practical problem, and even prove a separation in expressive power between a particular classical generative model and its quantum counterpart. The separation can be explained in terms of the quantum nonlocality and quantum contextuality. Furthermore, there may be classical ML improvements drawing from QI foundations (also known as ‘quantum inspired’ research, for example using MPS or TN). As an outlook, the authors show how their specific proof and example can be extended to more general model paradigms. It would be great to see practical implementations of generative modeling advantage on hardware in the coming years.

In variational quantum algorithms, it is known that highly expressive ansatze can access a close approximation of a desired solution to a number of variational problems and provide a flexible paradigm for programming near-term quantum computers. However, the circuit ansatz must have sufficiently large gradients to allow for proper training; much has been researched about the 'barren plateau' phenomenon in this context and it is seen as a big challenge observed in variational QML and QGAN. Holmes et al. [5] attempts to derive a fundamental relationship between two essential ansatze properties: expressibility and trainability. The resulting bounds in the work by Holmes et al. [5] indicates that highly expressive anatze exhibit flatter cost landscapes and therefore will be harder to train. Therefore, on one hand making the quantum neural network more expressive leads to better representation of a function. However, on the other hand, larger expressivity leads to smaller gradients, which lead to the barren plateau problem and convergence issues. Strategies such as using structured ansatze, physics informed circuits, correlating parameters and restricting rotation angles have been proposed to avoid or mitigate barren plateaus. Further exploring these and other strategies will be an important direction for future research.

Also recently, the connection between quantum machine learning models and kernel methods was explored, more rigorously than previously. Oftentimes, a variational quantum circuit in QML is compared to a neural network (and aptly named a ‘QNN’). Such circuits typically consist of encoding circuit, a variational ansatz, and a measurement. But conceptually, these quantum models really only consist of two parts: the data encoding/embedding and the measurement. Training a quantum model is then simply the problem of finding the measurement that minimises a data-dependent cost function. The author in [6] argues such quantum models are typically much more related to kernel methods than NN’s. Besides being conceptually very interesting, this view also yields several interesting benefits: for example, it refocuses the attention of a potential quantum advantage over classical methods to the way data is encoded into quantum states. Also, kernel-based training guarantees to find the global optimum, which is typically hard for variational circuit training. It is also shown that the main drawback of kernel-based methods is the same reason why NN methods superseded kernel methods in Big Data applications: while NN training is theoretically very hard, empirically O(N) training time can yield good results, while kernel training requires a runtime of at least O(N^2) in a classical feedback loop, even though the training is convex (‘efficient’). However, the model could also be trained using quantum algorithms, and is known to have a quadratic speedup to O(N), although such protocols require large fault-tolerant quantum hardware.

Quantum machine learning is a promising candidate for quantum advantage and recent results indicate the possibility of an advantage for certain tasks. However, this is still ongoing research with various aspects that need to be investigated. A number of algorithms are proposed claiming enhanced performance in numerics but in particular the training and learning problems are essential to explore under future scopes. Similarly, further generalizations on learning targets, regression, and classification tasks are required for implementation on experimental platforms.

References:

[1] Huang H. Y., Kueng R., J. Preskill, “Information-theoretic bounds on quantum advantage in machine learning” arXiv:2101.02464 (Jan. 2021)

[2] Y. Wu, J. Yao, P. Zhang, H. Zhai, “Expressivity of Quantum Neural Networks” arXiv:2101.04273 (Jan. 2021)

[3] P. Braccia, F. Caruso, L. Banchi, “How to enhance quantum generative adversarial learning of noisy information” arXiv:2012.05996 (Dec. 2020)

[4] Xun Gao, Eric R. Anschuetz, Sheng-Tao Wang, J. Ignacio Cirac, Mikhail D. Lukin, “Enhancing Generative Models via Quantum Correlations” arXiv:2101.08354 (Jan. 2021)

[5] Z.Holmes, K. Sharma, M. Cerezo, P.J. Coles, “Connecting ansatz expressibility to gradient magnitudes and barren plateaus” arXiv:2101.02138 (Jan. 2021)

[6] Maria Schuld, “Quantum Machine Learning Models are kernel methods” arXiv:2101.11020 (Jan. 2021)

Quantum Machine Learning (QML) techniques have been rigorously researched in the past few years, showing great promise in a number of applications and using various quantum algorithmic techniques and potential speedups. However, the effectiveness of quantum- over classical machine learning is not always evident and still heavily being investigated. In this blog post, we briefly summarize and contextualize several recent arXiv contributions in the area of quantum machine learning, quantum neural networks, quantum generative adversarial networks, and training their variational quantum circuits.

Research, such as the one presented in [1], suggests that potentially exponential quantum advantage is possible for certain tasks when the goal is to achieve accurate prediction on all inputs. While training techniques differ from the classical machine learning realm, more focus is on developing new models and algorithms to exploit the quantum mechanical phenomena. For instance, hybrid quantum-classical QML protocols can offer a significantly enhanced potential. Similarly, features like the expressivity of quantum neural networks can be exploited to gain quantum advantage. More accurately, the relationship between expressibillity and trainability of highly expressive ansatze states can provide useful insight into QML.

It was shown that the potential advantage of quantum ML over classical ML is limited, in the specific case of predicting outcomes of quantum experiments with a specified average-case prediction error. The recent findings by Huang et al. [1] establish the fact that quantum ML can have an exponential advantage over classical ML for certain problems where the objective is achieving a specified worst-case prediction error. This work claims that for any input distribution, a classical ML model can provide accurate predictions on average by accessing a quantum process a number of times comparable to the optimal quantum ML model. These results and analysis complement recent studies of the computational complexity of classical ML models trained with data from quantum experiments. Together, these rigorous results indicate that classical ML trained models using quantum measurement data can be surprisingly powerful. Such a hybrid protocol can potentially address challenging quantum problems in physics, chemistry, and materials for near-term experimental platforms.

Another direction being investigated is exploration towards establishing a quantum neural network (QNN) analogy of the universal approximation theorem, which attempts to provide the foundation for expressivity of classical neural networks. Recent work by Wu Y. et al. [2] discusses the generalization of expressivity of QNN for learning targets (for example observables of input wave functions), while focusing on fully connected architectures. The findings suggest that an accurate approximation is possible only when the input wave functions (target function) in the dataset do not exhaust the entire Hilbert space that the quantum circuit acts on, and more precisely, the Hilbert space dimension of the former has to be less than half of the Hilbert space dimension of the latter. If this requirement cannot be satisfied by the dataset, the expressivity capabilities can be restored by adding one ancillary qubit where the wave function is always fixed at input. Hybrid quantum-classical QML protocols with mixed target states can be exploited in already available experimental platforms with potential promising applications in quantum computing and noise sensing.

Another aspect of quantum ML that is being explored is quantum generative adversarial learning (QGAN) which is a promising strategy to use quantum devices for (quantum) estimation or generative machine learning tasks. The recent work by Braccia et al. [3], discusses the convergence behaviors of a QGAN training process and proposes new algorithms for QGAN training for mixed states which are expected to work better than previously used techniques. It is observed that the states obtained from currently available NISQ devices may display a “chaotic” behaviour during training, without achieving convergence. Results from this work show that algorithms based on the adaptation of a gradient-based technique allow provable convergence with bilinear score functions while the algorithms based on convex optimization techniques are shown to be especially suited for non-parametric states that are iteratively updated via a quantum circuit. Also, endowing the scheme with the ability to process entangled copies of the target state results in enhanced performance.

In Ref. [4] it is shown how generative models can be enhanced via quantum correlations, possibly paving the way to a quantum advantage in a generative ML setting. They show how quantum correlations offer a powerful resource, using a numerical example on a practical problem, and even prove a separation in expressive power between a particular classical generative model and its quantum counterpart. The separation can be explained in terms of the quantum nonlocality and quantum contextuality. Furthermore, there may be classical ML improvements drawing from QI foundations (also known as ‘quantum inspired’ research, for example using MPS or TN). As an outlook, the authors show how their specific proof and example can be extended to more general model paradigms. It would be great to see practical implementations of generative modeling advantage on hardware in the coming years.

In variational quantum algorithms, it is known that highly expressive ansatze can access a close approximation of a desired solution to a number of variational problems and provide a flexible paradigm for programming near-term quantum computers. However, the circuit ansatz must have sufficiently large gradients to allow for proper training; much has been researched about the 'barren plateau' phenomenon in this context and it is seen as a big challenge observed in variational QML and QGAN. Holmes et al. [5] attempts to derive a fundamental relationship between two essential ansatze properties: expressibility and trainability. The resulting bounds in the work by Holmes et al. [5] indicates that highly expressive anatze exhibit flatter cost landscapes and therefore will be harder to train. Therefore, on one hand making the quantum neural network more expressive leads to better representation of a function. However, on the other hand, larger expressivity leads to smaller gradients, which lead to the barren plateau problem and convergence issues. Strategies such as using structured ansatze, physics informed circuits, correlating parameters and restricting rotation angles have been proposed to avoid or mitigate barren plateaus. Further exploring these and other strategies will be an important direction for future research.

Also recently, the connection between quantum machine learning models and kernel methods was explored, more rigorously than previously. Oftentimes, a variational quantum circuit in QML is compared to a neural network (and aptly named a ‘QNN’). Such circuits typically consist of encoding circuit, a variational ansatz, and a measurement. But conceptually, these quantum models really only consist of two parts: the data encoding/embedding and the measurement. Training a quantum model is then simply the problem of finding the measurement that minimises a data-dependent cost function. The author in [6] argues such quantum models are typically much more related to kernel methods than NN’s. Besides being conceptually very interesting, this view also yields several interesting benefits: for example, it refocuses the attention of a potential quantum advantage over classical methods to the way data is encoded into quantum states. Also, kernel-based training guarantees to find the global optimum, which is typically hard for variational circuit training. It is also shown that the main drawback of kernel-based methods is the same reason why NN methods superseded kernel methods in Big Data applications: while NN training is theoretically very hard, empirically O(N) training time can yield good results, while kernel training requires a runtime of at least O(N^2) in a classical feedback loop, even though the training is convex (‘efficient’). However, the model could also be trained using quantum algorithms, and is known to have a quadratic speedup to O(N), although such protocols require large fault-tolerant quantum hardware.

Quantum machine learning is a promising candidate for quantum advantage and recent results indicate the possibility of an advantage for certain tasks. However, this is still ongoing research with various aspects that need to be investigated. A number of algorithms are proposed claiming enhanced performance in numerics but in particular the training and learning problems are essential to explore under future scopes. Similarly, further generalizations on learning targets, regression, and classification tasks are required for implementation on experimental platforms.

References:

[1] Huang H. Y., Kueng R., J. Preskill, “Information-theoretic bounds on quantum advantage in machine learning” arXiv:2101.02464 (Jan. 2021)

[2] Y. Wu, J. Yao, P. Zhang, H. Zhai, “Expressivity of Quantum Neural Networks” arXiv:2101.04273 (Jan. 2021)

[3] P. Braccia, F. Caruso, L. Banchi, “How to enhance quantum generative adversarial learning of noisy information” arXiv:2012.05996 (Dec. 2020)

[4] Xun Gao, Eric R. Anschuetz, Sheng-Tao Wang, J. Ignacio Cirac, Mikhail D. Lukin, “Enhancing Generative Models via Quantum Correlations” arXiv:2101.08354 (Jan. 2021)

[5] Z.Holmes, K. Sharma, M. Cerezo, P.J. Coles, “Connecting ansatz expressibility to gradient magnitudes and barren plateaus” arXiv:2101.02138 (Jan. 2021)

[6] Maria Schuld, “Quantum Machine Learning Models are kernel methods” arXiv:2101.11020 (Jan. 2021)

Tagged under

07 January 2021
###
The cost of universality

The authors of this work provides a comparative study of the overhead of state distillation and code-switching, and enables a deeper understanding of these two approaches to fault-tolerant universal quantum computation (FTQC). FTQC requires a gate set that involves at least one non-Clifford gate, which is typically generated through such protocols. State distillation relies on first producing many noisy encoded copies of a T state (magic state) and then processing them using Clifford operations to output a high fidelity encoded version of the state. The high-fidelity T state can be then used to implement the non-Clifford T gate. State generation and distillation is still a resource intensive process, despite significant recent improvements. A compelling alternative is ‘code-switching’, for example via gauge fixing to a 3D topological code. However, moving to 3D architectures is experimentally difficult, even if it significantly reduces the overhead compared to state distillation.

This work estimates the resources needed to prepare high-fidelity T states encoded in the 2D color code. This is accomplished via either state distillation or code-switching assuming that both approaches are implemented using noisy quantum-local operations in 3D. In particular, these two approaches were compared by simulating noisy circuits built from single-qubit state preparations, unitaries and measurements, and two-qubit unitaries between nearby qubits.

It is reported that the circuit noise threshold achieved for the code-switching implementation in this work is the highest presented to date and is equivalent to the circuit-noise threshold for the state distillation scheme with the 2D color code.

A direct way to switch between the 2D color code and the 3D subsystem color code is explored. The method exploits a particular gauge-fixing of the 3D subsystem color code for which the code state admits a local tensor product structure in the bulk and can therefore be prepared in constant time. The restriction decoder was adapted for the 3D color code with a boundary and produced a high error threshold. However, the failure probability of implementing the T gate with code-switching and the estimated T gate threshold is found to be lower than the one calculated for the state distillation.

The results suggest that code-switching does not offer substantial savings over state distillation in terms of both space overhead, i.e., the number of physical qubits required, and space-time overhead (the space overhead multiplied by the number of physical time units required) for most circuit noise error rates. Code switching holds a lot of promise, but is dependent on the clever design of codes that have transversal implementations of non-Clifford gates, and such a protocol is at the moment inferior to the state distillation. There exist other protocols that rival state distillation, like the pieceable fault-tolerant implementation of the CCZ gate, which in general shows the importance of having a protocol for non-Clifford gate implementation with a relatively small number of qubits and operations.

This work estimates the resources needed to prepare high-fidelity T states encoded in the 2D color code. This is accomplished via either state distillation or code-switching assuming that both approaches are implemented using noisy quantum-local operations in 3D. In particular, these two approaches were compared by simulating noisy circuits built from single-qubit state preparations, unitaries and measurements, and two-qubit unitaries between nearby qubits.

It is reported that the circuit noise threshold achieved for the code-switching implementation in this work is the highest presented to date and is equivalent to the circuit-noise threshold for the state distillation scheme with the 2D color code.

A direct way to switch between the 2D color code and the 3D subsystem color code is explored. The method exploits a particular gauge-fixing of the 3D subsystem color code for which the code state admits a local tensor product structure in the bulk and can therefore be prepared in constant time. The restriction decoder was adapted for the 3D color code with a boundary and produced a high error threshold. However, the failure probability of implementing the T gate with code-switching and the estimated T gate threshold is found to be lower than the one calculated for the state distillation.

The results suggest that code-switching does not offer substantial savings over state distillation in terms of both space overhead, i.e., the number of physical qubits required, and space-time overhead (the space overhead multiplied by the number of physical time units required) for most circuit noise error rates. Code switching holds a lot of promise, but is dependent on the clever design of codes that have transversal implementations of non-Clifford gates, and such a protocol is at the moment inferior to the state distillation. There exist other protocols that rival state distillation, like the pieceable fault-tolerant implementation of the CCZ gate, which in general shows the importance of having a protocol for non-Clifford gate implementation with a relatively small number of qubits and operations.

Tagged under

- Entangling power and quantum circuit complexity 11 April 2021 in Blog
- Modular photonic quantum computing 31 March 2021 in Blog
- Experimental Quantum Reinforcement Learning 28 March 2021 in Blog
- Variational Quantum Phase Estimation for Hamiltonian eigenstates 21 March 2021 in Blog
- Strongly Universal Hamiltonian Simulators 14 March 2021 in Blog
- Nanophotonic Quantum Computing 05 March 2021 in Blog

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