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arXiv:2410.03130

A variational quantum algorithm by Bayesian Inference with von Mises-Fisher distribution

Huynh, Trung, Gary C. An, Minsu Kim, Jeon, Yu-Seong, Jinhyoung Lee

Published Oct 4, 2024• 0 citations
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Abstract

The variational quantum eigensolver algorithm has gained attentions due to its capability of locating the ground state and ground energy of a Hamiltonian, which is a fundamental task in many physical and chemical problems. Although it has demonstrated promising results, the use of various types of measurements remains a significant obstacle. Recently, a quantum phase estimation algorithm inspired measurement scheme has been proposed to overcome this issue by introducing an additional ancilla system that is coupled to the primary system. Based on this measurement scheme, we present a novel approach that employs Bayesian inference principles together with von Mises-Fisher distribution and theoretically demonstrates the new algorithm's capability in identifying the ground state with certain for various random Hamiltonian matrices. This also opens a new way for exploring the von Mises-Fisher distribution potential in other quantum information science problems.

DOI
10.48550/arxiv.2410.03130

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The variational quantum eigensolver algorithm has gained attentions due to its capability of locating the ground state and ground energy of a Hamiltonian, which is a fundamental task in many physical and chemical problems. Although it has demonstrated promising results, the use of various types of measurements remains a significant obstacle. Recently, a quantum phase estimation algorithm inspired measurement scheme has been proposed to overcome this issue by introducing an additional ancilla system that is coupled to the primary system. Based on this measurement scheme, we present a novel approach that employs Bayesian inference principles together with von Mises-Fisher distribution and theoretically demonstrates the new algorithm's capability in identifying the ground state with certain for various random Hamiltonian matrices. This also opens a new way for exploring the von Mises-Fisher distribution potential in other quantum information science problems.
We develop an iterative Bayesian variational quantum eigensolver in which the unknown ground state, parametrised as a real unit vector, is inferred through a von Mises--Fisher distribution sequentially updated from Hadamard-test measurements on a single ancilla qubit. Building on a companion paper that treated a single success outcome via exact moment matching, we generalise the scheme to condition on mmm success outcomes for the likelihood function per iteration. Since exact moment matching becomes impractical for m>1m>1m>1, we exploit the large-concentration regime, in which a first-order expansion of the log-likelihood yields a closed-form von Mises--Fisher posterior with mmm entering only through a multiplicative weight. We prove that the two convergence properties of the single-outcome scheme---non-decreasing overlap and non-decreasing concentration---are preserved provided mmm lies within an explicitly characterised range. Based on the theory, we propose a measurement-based algorithm and illustrate the convergence on randomly generated three- and four-qubit Hamiltonian matrices, on HeH+\mathrm{HeH}^+HeH+, and on H2\mathrm{H}_2H2​.
The variational quantum eigensolver (or VQE), first developed by Peruzzo et al. (2014), has received significant attention from the research community in recent years. It uses the variational principle to compute the ground state energy of a Hamiltonian, a problem that is central to quantum chemistry and condensed matter physics. Conventional computing methods are constrained in their accuracy due to the computational limits facing exact modeling of the exponentially growing electronic wavefunction for these many-electron systems. The VQE may be used to model these complex wavefunctions in polynomial time, making it one of the most promising near-term applications for quantum computing. One important advantage is that variational algorithms have been shown to present some degree of resilience to the noise in the quantum hardware. Finding a path to navigate the relevant literature has rapidly become an overwhelming task, with many methods promising to improve different parts of the algorithm, but without clear descriptions of how the diverse parts fit together. The potential practical advantages of the algorithm are also widely discussed in the literature, but with varying conclusions. Despite strong theoretical underpinnings suggesting excellent scaling of individual VQE components, studies have pointed out that their various pre-factors could be too large to reach a quantum computing advantage over conventional methods. This review aims at disentangling the relevant literature to provide a comprehensive overview of the progress that has been made on the different parts of the algorithm, and to discuss future areas of research that are fundamental for the VQE to deliver on its promises. All the different components of the algorithm are reviewed in detail. These include the representation of Hamiltonians and wavefunctions on a quantum computer, the optimization process to find ground state energies, the post processing mitigation of quantum errors, and suggested best practices. We identify four main areas of future research: (1) optimal measurement schemes for reduction of circuit repetitions required; (2) large scale parallelization across many quantum computers; (3) ways to overcome the potential appearance of vanishing gradients in the optimization process for large systems, and how the number of iterations required for the optimization scales with system size; (4) the extent to which VQE suffers for quantum noise, and whether this noise can be mitigated in a tractable manner. The answers to these open research questions will determine the routes for the VQE to achieve quantum advantage as the quantum computing hardware scales up and as the noise levels are reduced.
Background Quantum phase estimation (QPE) is a foundational subroutine in quantum algorithms that ranges from Shor’s factoring scheme to variational quantum eigensolvers and quantum metrology protocols. Classical post-processing replacements for the inverse quantum Fourier transform (iQFT) —specifically Kitaev-style iterative estimation and Bayesian update schemes—substantially reduce circuit depth, making them practical for noisy intermediate-scale quantum (NISQ) hardware. Despite this promise, the statistical performance of these methods under the conditions that actually arise in practice—non-eigenstate inputs characterised by eigenstate overlap p ≡ O = | ψ | ϕ | 2 < 1 and mixed-state purity γ ≡ P = Tr ( ρ 2 ) < 1 —remains uncharacterised in the literature. Methods We develop a rigorous information-theoretic framework for Bayesian QPE (BQPE) beyond the eigenstate assumption. Starting from the binary measurement likelihood attenuated by the joint factor O P , we derive exact classical Fisher information expressions and apply the Cramér–Rao inequality to obtain tight variance lower bounds. We then construct an adaptive measurement protocol based on von Mises posterior updates, prove its asymptotic efficiency in the frequentist sense, and establish matching upper and lower bounds on sample complexity. All theoretical results are validated against 2,000 -trial Monte Carlo simulations and cross-checked against published nitrogen-vacancy (NV) centre and photonic experimental data. Results We prove that the fundamental variance floor for any unbiased frequentist estimator is ( 1 − p γ ) / ( N p γ n ̄ 2 ) 1 / ( N ( p γ ) 2 n ̄ 2 ) , where N
The problem of finding the ground state energy of a Hamiltonian using a quantum computer is currently solved using either the quantum phase estimation (QPE) or variational quantum eigensolver (VQE) algorithms. For precision $\ensuremath{\epsilon}$, QPE requires O(1)O(1)O(1) repetitions of circuits with depth $O(1/\ensuremath{\epsilon})$, whereas each expectation estimation subroutine within VQE requires $O(1/{\ensuremath{\epsilon}}^{2})$ samples from circuits with depth O(1)O(1)O(1). We propose a generalized VQE algorithm that interpolates between these two regimes via a free parameter $\ensuremath{\alpha}\ensuremath{\in}[0,1]$, which can exploit quantum coherence over a circuit depth of $O(1/{\ensuremath{\epsilon}}^{\ensuremath{\alpha}})$ to reduce the number of samples to $O(1/{\ensuremath{\epsilon}}^{2(1\ensuremath{-}\ensuremath{\alpha})})$. Along the way, we give a new routine for expectation estimation under limited quantum resources that is of independent interest.
Variational algorithms for strongly correlated chemical and materials systems are one of the most promising applications of near-term quantum computers. We present an extension to the variational quantum eigensolver that approximates the ground state of a system by solving a generalized eigenvalue problem in a subspace spanned by a collection of parametrized quantum states. This allows for the systematic improvement of a logical wavefunction ansatz without a significant increase in circuit complexity. To minimize the circuit complexity of this approach, we propose a strategy for efficiently measuring the Hamiltonian and overlap matrix elements between states parametrized by circuits that commute with the total particle number operator. This strategy doubles the size of the state preparation circuits but not their depth, while adding a small number of additional two-qubit gates relative to standard variational quantum eigensolver. We also propose a classical Monte Carlo scheme to estimate the uncertainty in the ground state energy caused by a finite number of measurements of the matrix elements. We explain how this Monte Carlo procedure can be extended to adaptively schedule the required measurements, reducing the number of circuit executions necessary for a given accuracy. We apply these ideas to two model strongly correlated systems, a square configuration of H4 and the π-system of hexatriene (C6H8).
Variational algorithms for strongly correlated chemical and materials systems are one of the most promising applications of near-term quantum computers. We present an extension to the variational quantum eigensolver that approximates the ground state of a system by solving a generalized eigenvalue problem in a subspace spanned by a collection of parametrized quantum states. This allows for the systematic improvement of a logical wavefunction ansatz without a significant increase in circuit complexity. To minimize the circuit complexity of this approach, we propose a strategy for efficiently measuring the Hamiltonian and overlap matrix elements between states parametrized by circuits that commute with the total particle number operator. This strategy doubles the size of the state preparation circuits but not their depth, while adding a small number of additional two-qubit gates relative to standard variational quantum eigensolver. We also propose a classical Monte Carlo scheme to estimate the uncertainty in the ground state energy caused by a finite number of measurements of the matrix elements. We explain how this Monte Carlo procedure can be extended to adaptively schedule the required measurements, reducing the number of circuit executions necessary for a given accuracy. We apply these ideas to two model strongly correlated systems, a square configuration of H 4 and the π -system of hexatriene (C 6 H 8 ).