Displaying items by tag: Quantum chemistry

In recent years, there have been significant developments in the field of quantum algorithms, especially in applications involving quantum chemistry, and representations of molecular Hamiltonians. These include variational quantum algorithms that aim to maximize the algorithmic performance despite the limited coherence times of currently available hardware. Unfortunately, that comes at the cost of introducing heuristic aspects, making it difficult to obtain rigorous performance guarantees. In contrast, quantum phase estimation approaches provide a route to accurately calculate eigenstates to small specifiable error, assuming sufficiently high overlap with the true eigenstates during the preparation of approximate eigenstates. Also, in fault-tolerant quantum computation, required resources will depend on the ability to bound errors in the algorithm which means that tighter error estimates will lead to fewer resource requirements.

Estimation of the resources required for phase estimation can be done using product-formula decomposition, also known as Trotterization. Previous works have shown that the number of fermions in the system can be used to offset the dependence of the error on the number of orbitals that can be applied to chemical systems in realistically sized basis sets. In this work, the authors present three factorized decompositions and use these in conjunction with the fermionic seminorm to obtain tighter Trotter error bounds in practice and apply it to the particular case of simulating a uniform electron gas (Jellium). Another objective of this approach is reducing the classical runtime and the gate complexity of quantum algorithms.

Each of the three factorized decompositions used in the work; namely spectral decomposition, cosine decomposition, and Cholesky decomposition, exhibits its own advantage. The spectral decomposition is generally applicable and extends beyond the plane wave dual basis. The cosine decomposition best exploits fermion number information and thus can be the most effective in the low-filling fraction regime. The Cholesky decomposition has the smallest constant factor overhead and therefore performs best in the medium and half-filling regimes.

The approach is applied to simulation of a uniform electron gas, finding substantial (over 100×) improvements in Trotter error for low-filling fraction and pushing towards much higher numbers of orbitals than is possible with existing methods. The approach was benchmarked against three prior art bounds: the analytic SHC bound, the fermionic commutator approach, and a similar Pauli commutator approach. A substantial classical runtime advantage for the calculation of the bounds was observed. In the case of fermionic and Pauli commutator approaches, the calculation of large spin-orbital number N>200 becomes intractable without access to > 100 GB of RAM. In contrast, the new bounds on a 512 spin-orbital can be calculated in less than 6 hours using < 16 GB of RAM. The work also calculates the T-bound for phase estimation of the ground state energy for Jellium, with significant improvements in gate complexity of over 10× as compared to the existing Trotter-based approaches. The gate counts in the results demonstrate that the trotter approach can comparatively perform better in specific regimes of interest than post-Trotter methods like qubitization. This holds true even while using fewer gates than qubitization for a large enough Wigner-Seitz radius.

The work focuses on second-order Trotter in the plane wave dual basis, but the techniques can be further generalized to higher-order trotter bounds. While the spectral and Cholesky decompositions are applicable in case of compact basis sets, an interesting potential lies in the cosine decomposition to obtain an even tighter bound in the low-filling fraction regime. Also, calculation of the higher-order trotter bounds would require time (scaling as O(N^7)), making it a costlier run. A potential alternative would be the post-Trotter methods, which exhibit superior asymptotic performance with respect to target error as well as being cost-effective due to the use of Hamiltonian factorizations. However, trotter methods possess good constant pre-factors in the runtime and few additional ancilla qubits requirements compared to post-Trotter methods, suggesting that trotter methods could perform comparatively better at specific tasks in the pre-asymptotic regime. It would be interesting to explore whether second quantized post-Trotter methods can similarly exploit low-filling fractions, where Trotter methods perform better.
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水, 26 5月 2021 12:00

Quantum Chemistry in First Quantization

In recent years, there have been significant developments in the field of quantum algorithms, especially in applications involving quantum chemistry and representations of molecular Hamiltonians. However, existing methods for simulating quantum chemistry (especially those leveraging simple basis functions like plane waves) are not feasible upon scaling to the continuum limit. The reason being that the majority of quantum computing algorithms for chemistry are based on second quantized simulations where the required number of qubits scales linearly with the number of spin-orbital basis functions.

To overcome this limitation, first quantized simulations of chemistry have been proposed where the idea is to track the basis state each particle is in, instead of storing the occupancy of each basis state. This is comparatively advantageous as the number of qubits needed to represent the state, scale logarithmically with the number of basis functions. Also, such simulations are highly adaptive in cases where the entanglement between the electronic and nuclear subsystems is non-negligible.

In this work, the authors analyze the finite resources required to implement two first quantized quantum algorithms for chemistry; block encodings for the qubitization and interaction picture frameworks. The objective is to compile and optimize these algorithms using a plane wave basis within an error-correctable gate set as well as developing new techniques to reduce the complexity. For qubitization of the Hamiltonian, control registers are used to select the momenta as well as the individual bits to multiply, which significantly decreases the multiplication cost. The results show that the qubitization algorithms requires much less surface code spacetime volume for simulating millions of plane waves as compared to the best second quantized algorithms require for simulating hundreds of Gaussian orbitals. In case of interaction picture based, the cost of computing the total kinetic energy is reduced by computing the sum of squares of momenta at the beginning.

The work shows that the total cost associated with the state preparation cost is reduced by a factor of 3, when one assumes that the state preparation is only needed for the kinetic energy operator, rather than using the standard amplitude amplification for the total energy. The number of bits needed for the discretization of the time intervals is also reduced by a factor of two as indicated by prior well-established methods. The main complexity is therefore focused on selecting the momenta registers. Unlike the previous approaches, only the dynamic degrees of freedom for electrons that participate in reactivity are employed by the algorithm, hence reducing encoding complexity. This approach is particularly more advantageous for a high number of electrons for the qubitization and despite the improved scaling of the interaction picture based method, the qubitization based method proves to be comparatively more practical. Also, the numerical experimentation reveals that these approaches require significantly fewer resources to reach comparable accuracy as compared to second quantized methods. Another interesting finding is that the first quantized approaches developed here, may give lower Toffoli complexities than previous work for realistic simulations of both material Hamiltonians and non-periodic molecules, suggesting more fault tolerant viability than second quantized methods.

This work provides the first explicit circuits and constant factors for simulation of any first quantized quantum algorithm, which can be a promising direction for simulating realistic materials Hamiltonians within quantum error-correction. But perhaps more impressively, the authors have also improved the constant factors; the results demonstrate reduction in circuit complexity by about a thousandfold as compared to naive implementations for modest sized systems. It also provides insights on the resources required to simulate various molecules and materials and gives the first impression of the first quantization-based algorithm for preparation of the eigenstates for phase estimation with required overlap with another eigenstate. This suggests many potential advantages in contrast to the second quantization-based algorithms, as the case in the continuum limit.

In addition to the progress made by the current work, many potential further improvements on the approach have already been proposed by the authors as well, such as modifying the algorithm to encode pseudopotential specific to the problem - relating it to bonds and chemical reactivity. Furthermore, using a non-orthogonal reciprocal lattice and efficient convergence in the thermodynamic limit might be a good direction for future research. The presented algorithms could also be adapted to more natively suit systems of reduced periodicity even thought the underlying representation is based on plane waves. The authors acknowledge relatively little is known about state-preparation in first-quantization as compared to second-quantization. Future work may investigate whether straightforward translations form the latter body of work to the current will be sufficiently efficient. Although this work focuses on preparing eigenstates for energy estimation, it can also be extended for simulation of (non-)Born-Oppenheimer dynamics.

The paper offers an interesting insight and significant progress in designing first-quantization picture quantum chemistry simulations on fault-tolerant hardware. Several strong advantages as compared to second-quantization methods are highlighted, and it will be interesting to see how this field evolves further.
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A prospective application of quantum computing is solving quantum chemistry problems, however, obtaining exact solutions is difficult due to the lack of general method of obtaining such solutions. Typically the solution lies in finding the ground state energy, even though the energy is not descriptive enough to fully characterize all desired properties of a system. In order to find such properties, many measurements of the wavefunction are required. These measurements are expensive, because the wavefunction cannot be copied and must often be re-prepared before a second measurement is performed. Finding the full wavefunction would require exponentially many measurements, so one option would be to encode many solutions into one measurement by using a machine learned (ML) model. Training of the ML model requires finding exact quantities at several different external potentials. Besides that, one can use density functional theory (DFT), to replace the wavefunction with the one-body density, n(r), which has fewer variables. When DFT is used, instead of the Hamiltonian, the universal functional, F[n], must be found. The quantities required for the classical user to find self-consistent solutions are the exact functional (determining the energy) and the functional derivative. So, in addition to finding F[n], one also must find some other quantity such as the density, n(r), or the Kohn-Sham(KS) potential, vs(r). With these components, one can fully characterize a quantum ground state and solve for other measurable quantities.

The authors propose an algorithm that finds the ML model for F[n] on the quantum computer if a ground-state wavefunction is available. The algorithm leaves the wavefunction largely undisturbed so it can be used as the starting state for another system, greatly reducing the pre-factor required to solve other systems through a quantum counting algorithm to extract descriptive quantities such as the density. Most of this algorithm is kept entirely on the quantum computer to motivate future improvements for speed, but the counting algorithm does allow for information to be output classically. Moreover, the authors demonstrate that the exact Kohn-Sham potential can be solved in a faster way with a gradient evaluated on a cost function for the Kohn-Sham system.

The novelty of the proposed algorithm suggested in this work is the limitation of the number of measurements and re-preparations of the wavefunction especially in the case of time-dependent quantities. Since there is no algorithm for the general case that is exponentially better than the proposed algorithm, limiting the number of measurements and re-preparations of the wavefunction is as best as one can achieve. The proposed algorithm is a combination of several known algorithms including quantum phase estimation, quantum amplitude estimation, and quantum gradient methods that are iteratively used to train a machine learned model.
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In this arXiv submission by Qu & Co and Covestro, a well-known approximation in classical computational methods for quantum chemistry is applied to a quantum computing scheme for simulating molecular chemistry efficiently on near-term quantum devices. The restricted mapping allows for a polynomial reduction in both the quantum circuit depth and the total number of measurements required, as compared to the conventional variational approaches based on near-term quantum simulation of molecular chemistry, such as UCCSD. This enables faster runtime convergence of the variational algorithm to a potentially higher accuracy by using a larger basis set allowed by the restricted mapping. The latter is shown via an example simulation of the disassociation curve of lithium hydride. These results open up a new direction for efficient near-term quantum chemistry simulation, as well as decreasing the effective quantum resource requirements for future fault-tolerant quantum computing schemes.

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Qu&Co comments on this publication:

In this article by McKinsey & Co, a strategy consulting firm, Florian Budde and Daniel Volz state that the chemical companies must act now to capture the benefits of quantum computing. Of course we at Qu & Co are a bit biased on this topic, but we do agree with the authors that the chemical sector is likely to be an early beneficiary of the vastly expanded modeling and computational capabilities, which is promised to be unlocked by quantum computing.

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火, 15 1月 2019 00:00

Quantum chemistry on quantum annealers

Qu&Co comments on this publication:

Thus far, quantum chemistry quantum algorithms have been experimentally demonstrated only on gate-based quantum computers. Efforts have been made to also map the chemistry problem Fermionic Hamiltonian to an Ising Hamiltonian in order to solve it on a quantum annealer.  However, the number of qubits required still scales exponentially with the problem size (the number of orbitals considered in the electronic structure problem). As an alternative, this paper presents a different approach exploiting the efficiency at which quantum annealers can solve discrete optimization problems, and mapping a qubit coupled cluster method to this form. They simulate their method on an ideal Ising machine and on a D-Wave 2000Q system, and find promising success rates for smaller molecules. However, further investigation would be necessary to investigate the usability for larger or more complex systems, as the scaling of their folding technique with the number of local minima is unknown. In addition, it is unclear from the experimental data whether the limitations of the D-Wave system  as compared to a perfect Ising machine could hinder expected performance gains for more complex systems.

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Qu&Co comments on this publication:

Most near-term quantum-computational chemistry experiments have so-far been implemented by applying the Variational Quantum Eigensolver (VQE) classical-quantum hybrid algorithm as an alternative to Quantum Phase Estimation (QPE). This is due to the fact that QPE requires many orders of magnitude more quantum gates than is feasible with typical coherence times of current and near-term quantum-processors. As an alternative, in this paper, Paesani et al. report experimental results of a recently proposed adaptive Bayesian approach to quantum phase estimation and use it to simulate molecular energies on a Silicon quantum photonic device. The approach is verified to be well suited for NISQ quantum-processors by investigating its superior robustness to noise and decoherence compared to the iterative phase estimation algorithm. There results shows a promising route to unlock the power of QPE much sooner than previously believed possible.

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Qu&Co comments on this publication:

Quantum computers can be used to address molecular structure, materials science and condensed matter physics problems, which currently stretch the limits of existing high-performance computing resources. Finding exact numerical solutions to these interacting fermion problems has exponential cost, while Monte Carlo methods are plagued by the fermionic sign problem. In Quantum Computational Chemistry solutions, the Variational Quantum Eigensolver (VQE) algorithm offers a hybrid classical-quantum, and thus low quantum circuit depth, alternative to the Phase Estimation algorithm used to measure the ground-state energy of a molecular Hamiltonian. In VQE the quantum computer is used to prepare variational trial states that depend on a set of parameters. Then, the expectation value of the energy is estimated and used by a classical optimizer to generate a new set of improved parameters. The advantage of VQE over classical simulation methods is that in VQE one can prepare trial states that are not amenable to efficient classical numerics. In this paper, Kandala et al. demonstrate the experimental results for determining the ground state energy for molecules of increasing size, up to BeH2 using the VQE algorithm.

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Quantum Computational Chemistry is one of the most promising applications for both near-term and large scale fault-tolerant quantum-computers. In this paper, McClean et al. present Open Fermion (www.openfermion.org), an open-source software library written largely in Python, aimed at enabling the simulation of fermionic models and quantum chemistry problems on quantum hardware. Without such a library, developing and studying algorithms for these problems is be difficult due to the prohibitive amount of domain knowledge required in both the area of chemistry and quantum algorithms. Beginning with an interface to common electronic structure packages, it simplifies the translation between a molecular specification and a quantum circuit for solving or studying the electronic structure problem on a quantum computer, minimizing the amount of domain expertise required to enter the field.

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Qu&Co comments on this publication:

Recently, promising experimental results have been shown for quantum-chemistry calculations using small, noisy quantum processors. As full scale fault-tolerant error correction is still many years away, near-term quantum computers will have a limited number of qubits, and each qubit will be noisy. Methods that reduce noise and correct errors without doing full error correction on every qubit will help extend the range of interesting problems that can be solved in the near-term. In this paper Otten et al. present a scheme for accounting (and removal) of errors in observables determined from quantum algorithms and apply this scheme to the variational quantum eigensolver algorithm, simulating the calculation of the ground state energy of equilibrium H2 and LiH in the presence of several noise sources, including amplitude damping, dephasing, thermal noise, and correlated noise. They show that their scheme provides a decrease in the needed quality of the qubits by up to two orders of magnitude.

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