Displaying items by tag: NISQ algorithms

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In this paper, the meta-VQE algorithm is presented, which is an adaptation of the Variational Quantum Eigensolver (VQE). VQE is a variational algorithm that was originally proposed for finding the ground state energy of a given Hamiltonian by variationally minimizing its expectation value with a parametrized quantum circuit. The cost function of VQE is the expected value of the model Hamiltonian. The variational principle states that this value is an upper bound of the ground state energy, so everything reduces to minimize this value by fine-tuning the parameters of the circuit.

The meta-VQE algorithm is inspired by quantum machine learning algorithms (QML) and is able to learn the ground state energy profile of a parametrized Hamiltonian. First, this circuit is trained with a set of Hamiltonian parameters, which are encoded in the gates of an encoding unitary. By designing a cost function with the expected values of all these Hamiltonian training points, the algorithm extracts the optimal values of the variational parameters. Then, one can compute the energy for other Hamiltonian values by just running the meta-VQE circuit with the parameters obtained in the minimization. In addition, one can also use the result of a meta-VQE training as a starting point of a standard VQE algorithm, the opt-meta-VQE, instead of random initialization.

This technique can make the algorithm converge to the correct solution, while previous variants of VQE often suffer from convergence issues due to the exponential increase of the Hilbert space paired with a random initialized ansatz, typically resulting in a limited probability of finding the ground state as the quantum system increases. The characteristic trait of the meta-VQE is that it can be used to first explore the ground state energies of Hamiltonian parameter space with only a few training points and then use the result as an initial state for a precise VQE, resulting in high precision. Furthermore, meta-VQE can easily be adapted to be a part of other VQE strategies to increase performance. Moreover, the meta-VQE is able to capture global correlation with a few training points alleviating the refined optimization of the individual points in a later step. The authors conclude that the meta-VQE can find the general energy shape but not provide an accurate value, in contrast to standard VQE. However, the opt-meta-VQE proves valuable, achieving better results than standard VQE with random initialization.

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Quantum computational supremacy (QCS) arguments have been provided to demonstrate that quantum computers will soon outperform classical computers in a variety of algorithms. However, in order to truly prove supremacy, several strict measures need to be taken: an appropriate algorithm must be selected, a quantum device to run the algorithm must be designed, the ability to verify the results of the calculations must be considered and a complexity theory that supports the claim that a classical computer would be unable to run such an algorithm should be provided. Quantum circuits running on quantum computing chips that are currently experimentally realized might still be able to be simulated in highly parallelized state-of-the-art classical supercomputers, therefore one can only make conjectures about QCS at the moment. Typically, classical simulation of certain families of quantum circuits require scaling that is worse than any polynomial in the size of the circuit, which prevents us from calculating exactly the number of qubits these quantum circuits must have for their classical simulation to be intractable on modern classical supercomputers.

In this paper, three refined fine-grained conjectures about quantum supremacy are provided and it is calculated that 208 qubits and 500 gates for Instantaneous Quantum Polynomial-Time (IQP) circuits, 420 qubits and 500 constraints for Approximate Optimization Algorithm (QAOA) circuits and 98 photons and 500 optical elements are sufficient. Although noise in current quantum devices cannot be fully approximated, a lower bound on the runtime of all three algorithms for any multiplicative-error classical simulation is provided.

This paper provides a concrete estimation on the number of qubits required for three algorithms that have gained a lot of attention during the NISQ era. While the orginal work stems from 2018,  the number of qubits required has been recalculated in the newest version of this paper, which provides a good indication of how fidelity of quantum chips has been improved in the last two years, as well as the latest understanding in complexity and the on-going evolution in classical competition.

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In this article, John Preskill provides his view on the near-term (5-10 years ahead) societal and commercial impact of quantum-computers. Specifically, the author focuses on what he calls Noisy Intermediate-Scale Quantum (NISQ) technology: quantum computers with 50-100 qubits for which noise in quantum gates will limit the size of quantum circuits that can be executed reliably. Such NISQ devices may be able to perform tasks which surpass the capabilities of today’s classical digital computers and will be useful tools for exploring e.g. many-body quantum physics, and may have other useful applications, but, the author states, the 100-qubit quantum computer will not change the world right away and we should regard them as a significant step toward the more powerful quantum technologies of the future.

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Quantum Approximate Optimization Algorithm (QAOA) is a variational quantum algorithm that has been heavily investigated due to its potential during the NISQ era. It is designed to find approximate solutions to combinatorial search problems and was first applied to the Max-Cut problem for d-regular graphs. The system is initially prepared in a product state and then p layers of unitaries U(C,γ) and U(B, β) are alternately applied; this can be seen as a Trotterized version of (non-adiabatic) quantum annealing with parametrized annealing pathway.

In these 2 papers by Farhi, Gamarnik & Gutmann posted on April 20 and May 18, the focus problem is to find a large independent set in a random graph of fixed average degree d for the problem of Maximum Independent Set (MIS) on random graphs. Generally, the performance of the QAOA can only improve with depth p, but it is shown that for MIS the algorithm will fail to pass a certain performance barrier if 2p is less than w*log(n)/log(d/ln(2)) for any w <1 with d big enough. The quantum algorithm consists of p unitaries that each respect the locality of the underlying graph. With a fixed average degree of d this means that each qubit typically has an influence sphere of roughly dp other qubits. For qubits further than 2p apart on the graph these influence spheres do not intersect and we can show that measurements of these qubits are uncorrelated, however if p is large enough that dp exceeds n our arguments do not apply and we have no indication that the QAOA will fail.

Overlap Gap Property states that for a given random graph the intersection of any two large independent sets is either big or small, there is no middle ground. Using OGP and the locality of the QAOA, it is shown that if p is less than a d-dependent constant times log n, the QAOA cannot do better than finding an independent set of size .854 times the optimal for d large. Because the logarithm is slowly growing, even at one million qubits we can only show that the algorithm is blocked if p is in single digits. At higher p the algorithm “sees” the whole graph and we have no indication that performance is limited.

The worst case performance circumstances can be easily created to exploit QAOA’s weaknesses. Through construction of operators C and B, QAOA is inherently a local quantum algorithm since when conjugating a single qubit operator produces an operator only involving that qubit and those connected to it on the graph, creating a shallow circuit. This can be exploited to construct examples where the QAOA’s performance is provably below optimal. For example, for Max-Cut when p is a small enough constant times log(n) we show that the approximation ratio is no better than ½ for large d, and for MIS the approximation ratio goes to 0 at large d.

This is an important result for the problem of MIS on random graphs that, although not directly generalizable to other problems, is still valuable for creating bounds for QAOA-applied problems. Knowing that on the average case for MIS, QAOA requires to see the whole graph, therefore require a large p, will change the way QAOA and its potential is viewed for certain problems, and these papers illustrate methods to quantify that.

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In recent years many academics and corporates have focus on solving combinatorial optimization problems on quantum-annealing devices like those offered by D-Wave. Now that noisy intermediate scale (NISQ) gate-based quantum-processers (like those of Google, IBM, Rigetti and Intel) are nearing the moment of quantum-supremacy, it is interesting to learn what gate-based quantum-computers can bring to combinatorial optimization problems. In this work, In this paper, Zahedinejad et al. provide a survey of the approaches to solving different types of combinatorial optimization problems, in particular quadratic unconstrained binary optimization (QUBO) problems on a gate model quantum computer. They focus on four different approaches including digitizing the adiabatic quantum computing, global quantum optimization algorithms, the quantum algorithms that approximate the ground state of a general QUBO problem, and quantum sampling. 

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In this paper, Matthew Hastings presents a quantum algorithm to exactly solve certain problems in combinatorial optimization, including weighted MAX-2-SAT.  While the time required is still exponential, the algorithm provably outperforms Grover's algorithm assuming a mild condition on the number of low energy states of the target Hamiltonian.

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Quantum computers are envisioned to have significantly higher computational capabilities compared to their classical counterparts, especially for optimization and machine learning problems that involve a large classical data set. However, existing quantum algorithms use the trivial methods of turning large classical datasets into either quantum oracles or quantum states which are so expensive that negate any possible quantum advantage. Such quantum algorithms focus at problems in which classical runtime scales rapidly with the input size, perhaps exponentially. To be able to achieve quantum speedup with algorithms like Grover search, a “quantum RAM” is proposed, which is a large classical memory that can be queried in superposition. Although quantum RAMs do not yet exist and creating one might encounter the same challenges that quantum computer hardware faces, it has the potential to provide significant speedup to applications like the k-means clustering, logistical regression, zero-sum games and boosting.

This paper introduces hybrid classical-quantum algorithms for problems involving a large classical data set X and a space of models Y such that a quantum computer has superposition access to Y but not X. Then a data reduction technique is used to construct a weighted subset of X called a coreset that yields approximately the same loss for each model. The coreset can be constructed either by a classical computer or by the combination of classical – quantum computer by utilization of quantum measurements.

The main message of this work is that in order to avoid losing quantum speedup for ‘big-data’ applications, techniques such as data reduction are required, so that the time for loading and storing the data set is limited. Also, non-fault tolerant quantum algorithms should be designed in a way that does not require an excessive amount of gates, so that the algorithm can run before qubits lose their coherence and invalidate the result. The goal of the paper is to draw attention to problems that arise from such actions like testing for quantum advantage when data reduction is used, explore general data reduction techniques and investigate new hybrid classical-quantum algorithms.

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Clustering is a form of unsupervised machine learning, where instances are organized into groups whose members share similarities. The assignments are, in contrast to classification, not known a priori, but generated by the algorithm. In this paper, Neukart et al.  present an algorithm for quantum-assisted cluster analysis (QACA) that makes use of the topological properties of a D-Wave 2000Q quantum processing unit (QPU). They explain how the problem can be expressed as a quadratic unconstrained binary optimization (QUBO) problem, and show that the introduced quantum-assisted clustering algorithm is, regarding accuracy, equivalent to commonly used classical clustering algorithms.

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02 March 2018

Quantum circuit learning

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Quantum machine learning (QML) algorithms based on the Harrow-Hassidim- Lloyd (HHL) algorithm rely on quantum phase estimation which requires high circuit-depth. To allow QML on current noisy intermediate scale quantum (NISQ) devices classical-quantum hybrid algorithms have been suggested applying low-depth circuits like quantum variational eigensolvers and quantum approximate optimization. Such hybrid algorithms typically divide the ML problem into two parts, each part to be performed either classically or on a quantum-computer. In this paper, Mitarai et al. present a new hybrid framework, called quantum circuit learning (QCL), which is easily realizable on current NISQ devices. Under QCL a circuit learns by providing input data, while iteratively tuning the circuit parameters to give the desired output. They show that QCL is able to learn nonlinear functions and perform simple classification tasks. They also show that a 6-qubit circuit is capable of learning dynamics of a 10-spin system with a fully connected Ising Hamiltonian, implying that QCL could be well suited for learning complex many-body systems.

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The perceptron algorithm dates back to the late 1950s and is an algorithm for supervised learning of binary classifiers. In a 2016 paper, Wiebe et al. proposed a quantum algorithm (based on Grover’s quantum-search approach), which can quadratically speed-up the training of a perceptron. In this paper, Zheng et al. describe their design for a quantum-circuit to implement the training-algorithm of Wiebe et al. They also analyze the resource requirements (qubits and gates) and demonstrate the feasibility of their quantum-circuit by testing it on the ibmqx5 (a 16 qubit universal gate quantum processor developed by IBM)

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