Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2307.00766 · Physical Review Research

Accelerated variational quantum eigensolver with joint Bell measurement

Chenfeng Cao, Hiroshi Yano, Yuya O. Nakagawa

Published Feb 26, 2024• 17 citations
arXivHTMLPDF

Abstract

The variational quantum eigensolver (VQE) stands as a prominent quantum-classical hybrid algorithm for near-term quantum computers to obtain the ground states of molecular Hamiltonians in quantum chemistry. However, due to the noncommutativity of the Pauli operators in the Hamiltonian, the number of measurements required on quantum computers increases significantly as the system size grows, which may hinder practical applications of VQE. In this work, we present a protocol termed (JBM-VQE) to reduce the number of measurements and speed up the VQE algorithm. Our method employs joint Bell measurements, enabling the simultaneous measurement of the absolute values of all expectation values of Pauli operators present in the Hamiltonian. In the course of the optimization, JBM-VQE estimates the absolute values of the expectation values of the Pauli operators for each iteration by the joint Bell measurement, while the signs of them are measured less frequently by the conventional method to measure the expectation values. Our approach is based on the empirical observation that the signs do not often change during optimization. We illustrate the speed-up of JBM-VQE compared to conventional VQE by numerical simulations for finding the ground states of molecular Hamiltonians of small molecules, and the speed-up of JBM-VQE at the early stage of the optimization becomes increasingly pronounced in larger systems. Our approach based on the joint Bell measurement is not limited to VQE and can be utilized in various quantum algorithms whose cost functions are expectation values of many Pauli operators. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.013205

Related papers

Truncation technique for variational quantum eigensolver for Molecular Hamiltonians

Xu, Qidong, Kanav Setia

The variational quantum eigensolver (VQE) is one of the most promising quantum algorithms for the near-term noisy intermediate-scale quantum (NISQ) devices. The VQE typically involves finding the minimum energy of a quantum Hamiltonian through classical optimization of a parametrized quantum ansatz. One of the bottlenecks in VQEs is the number of quantum circuits to be measured. In this work, we propose a physically intuitive truncation technique that starts the optimization procedure with a truncated Hamiltonian and then gradually transitions to the optimization for the original Hamiltonian via an operator classification method. This strategy allows us to reduce the required number of evaluations for the expectation value of Hamiltonian on a quantum computer. The reduction in required quantum resources for our strategy is substantial and likely scales with the system size. With numerical simulations, we demonstrate our method for various molecular systems.

arXiv:2402.01630•Feb 2, 2024•0 citations

Efficient measurement schemes for the Monte Carlo projective quantum eigensolver

Divye Baid, Maria-Andreea Filip

The Monte Carlo Projective Quantum Eigensolver (MC-PQE) was recently introduced as an alternative to hybrid algorithms like the variational quantum eigensolver (VQE). By using a quantum Monte Carlo-inspired scheme for energy estimation, MC\nobreakdash-PQE was found to decrease the required measurement cost relative to comparable conventional approaches. However, in the context of VQE, numerous techniques have been developed to reduce the measurement overhead by joint measurement of multiple observables. In this work, we extend these approaches to the asymmetric expectation values required in MC-PQE and assess the performance of different techniques for various molecular systems of up to 12 qubits. We find full commuting Pauli term grouping combined with tailored measurement allocation techniques leads to a 5-10$\times$ reduction in standard error for the same total number of quantum measurements. Conventional Hamiltonian grouped measurement outperforms tractable classical shadows tomography based techniques for the considered systems.

quant-pharXiv:2608.30612•Sep 1, 2026•0 citations

Variational denoising for variational quantum eigensolver

Quoc Hoan Tran, Shinji Kikuchi, Hirotaka Oshima

The variational quantum eigensolver (VQE) is a hybrid algorithm that has the potential to provide a quantum advantage in practical chemistry problems that are currently intractable on classical computers. VQE trains parameterized quantum circuits using a classical optimizer to approximate the eigenvalues and eigenstates of a given Hamiltonian. However, VQE faces challenges in task-specific design and machine-specific architecture, particularly when running on noisy quantum devices. This can have a negative impact on its trainability, accuracy, and efficiency, resulting in noisy quantum data. We propose variational denoising, an unsupervised learning method that employs a parameterized quantum neural network to improve the solution of VQE by learning from noisy VQE outputs. Our approach can significantly decrease energy estimation errors and increase fidelities with ground states compared to noisy input data for the H2 , LiH, and BeH2 molecular Hamiltonians and the transverse field Ising model. Surprisingly, it only requires noisy data for training. Variational denoising can be integrated into quantum hardware, increasing its versatility as an end-to-end quantum processing for quantum data. Published by the American Physical Society 2024

arXiv:2304.00549•Physical Review Research•May 17, 2024•4 citations

Variational Quantum Eigensolvers for Sparse Hamiltonians

William M. Kirby, Peter J. Love

Hybrid quantum-classical variational algorithms such as the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) are promising applications for noisy, intermediate-scale quantum computers. Both VQE and QAOA variationally extremize the expectation value of a Hamiltonian. All work to date on VQE and QAOA has been limited to Pauli representations of Hamiltonians. However, many cases exist in which a sparse representation of the Hamiltonian is known but there is no efficient Pauli representation. We extend VQE to general sparse Hamiltonians. We provide a decomposition of a fermionic second-quantized Hamiltonian into a number of one-sparse, self-inverse, Hermitian terms linear in the number of ladder operator monomials in the second-quantized representation. We provide a decomposition of a general d-sparse Hamiltonian into O(d^{2}) such terms. In both cases, a single sample of any term can be obtained using two ansatz state preparations and at most six oracle queries. The number of samples required to estimate the expectation value to precision ε scales as ε^{-2} as for Pauli-based VQE. This widens the domain of applicability of VQE to systems whose Hamiltonian and other observables are most efficiently described in terms of sparse matrices.

arXiv:2012.07171•Physical Review Letters•Sep 9, 2021•43 citations

Measurement optimization in the variational quantum eigensolver using a minimum clique cover

Vladyslav Verteletskyi, Tzu-Ching Yen, Artur F. Izmaylov

Solving the electronic structure problem using the Variational Quantum Eigensolver (VQE) technique involves the measurement of the Hamiltonian expectation value. The current hardware can perform only projective single-qubit measurements, and thus, the Hamiltonian expectation value is obtained by measuring parts of the Hamiltonian rather than the full Hamiltonian. This restriction makes the measurement process inefficient because the number of terms in the Hamiltonian grows as O(N4) with the size of the system, N. To optimize the VQE measurement, one can try to group as many Hamiltonian terms as possible for their simultaneous measurement. Single-qubit measurements allow one to group only the terms commuting within the corresponding single-qubit subspaces or qubit-wise commuting. We found that the qubit-wise commutativity between the Hamiltonian terms can be expressed as a graph and the problem of the optimal grouping is equivalent to finding a minimum clique cover (MCC) for the Hamiltonian graph. The MCC problem is NP-hard, but there exist several polynomial heuristic algorithms to solve it approximately. Several of these heuristics were tested in this work for a set of molecular electronic Hamiltonians. On average, grouping qubit-wise commuting terms reduced the number of operators to measure three times less compared to the total number of terms in the considered Hamiltonians.

arXiv:1907.03358•The Journal of Chemical Physics•Mar 27, 2020•269 citations

The Variational Quantum Eigensolver: A review of methods and best practices

Jules Tilly, Hongxiang Chen, Shuxiang Cao +8 more

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.

arXiv:2111.05176•Physics Reports•Sep 23, 2022•1,031 citations
The variational quantum eigensolver (VQE) stands as a prominent quantum-classical hybrid algorithm for near-term quantum computers to obtain the ground states of molecular Hamiltonians in quantum chemistry. However, due to the noncommutativity of the Pauli operators in the Hamiltonian, the number of measurements required on quantum computers increases significantly as the system size grows, which may hinder practical applications of VQE. In this work, we present a protocol termed (JBM-VQE) to reduce the number of measurements and speed up the VQE algorithm. Our method employs joint Bell measurements, enabling the simultaneous measurement of the absolute values of all expectation values of Pauli operators present in the Hamiltonian. In the course of the optimization, JBM-VQE estimates the absolute values of the expectation values of the Pauli operators for each iteration by the joint Bell measurement, while the signs of them are measured less frequently by the conventional method to measure the expectation values. Our approach is based on the empirical observation that the signs do not often change during optimization. We illustrate the speed-up of JBM-VQE compared to conventional VQE by numerical simulations for finding the ground states of molecular Hamiltonians of small molecules, and the speed-up of JBM-VQE at the early stage of the optimization becomes increasingly pronounced in larger systems. Our approach based on the joint Bell measurement is not limited to VQE and can be utilized in various quantum algorithms whose cost functions are expectation values of many Pauli operators. Published by the American Physical Society 2024
The variational quantum eigensolver (VQE) is one of the most promising quantum algorithms for the near-term noisy intermediate-scale quantum (NISQ) devices. The VQE typically involves finding the minimum energy of a quantum Hamiltonian through classical optimization of a parametrized quantum ansatz. One of the bottlenecks in VQEs is the number of quantum circuits to be measured. In this work, we propose a physically intuitive truncation technique that starts the optimization procedure with a truncated Hamiltonian and then gradually transitions to the optimization for the original Hamiltonian via an operator classification method. This strategy allows us to reduce the required number of evaluations for the expectation value of Hamiltonian on a quantum computer. The reduction in required quantum resources for our strategy is substantial and likely scales with the system size. With numerical simulations, we demonstrate our method for various molecular systems.
The Monte Carlo Projective Quantum Eigensolver (MC-PQE) was recently introduced as an alternative to hybrid algorithms like the variational quantum eigensolver (VQE). By using a quantum Monte Carlo-inspired scheme for energy estimation, MC\nobreakdash-PQE was found to decrease the required measurement cost relative to comparable conventional approaches. However, in the context of VQE, numerous techniques have been developed to reduce the measurement overhead by joint measurement of multiple observables. In this work, we extend these approaches to the asymmetric expectation values required in MC-PQE and assess the performance of different techniques for various molecular systems of up to 12 qubits. We find full commuting Pauli term grouping combined with tailored measurement allocation techniques leads to a 5-10×\times× reduction in standard error for the same total number of quantum measurements. Conventional Hamiltonian grouped measurement outperforms tractable classical shadows tomography based techniques for the considered systems.
The variational quantum eigensolver (VQE) is a hybrid algorithm that has the potential to provide a quantum advantage in practical chemistry problems that are currently intractable on classical computers. VQE trains parameterized quantum circuits using a classical optimizer to approximate the eigenvalues and eigenstates of a given Hamiltonian. However, VQE faces challenges in task-specific design and machine-specific architecture, particularly when running on noisy quantum devices. This can have a negative impact on its trainability, accuracy, and efficiency, resulting in noisy quantum data. We propose variational denoising, an unsupervised learning method that employs a parameterized quantum neural network to improve the solution of VQE by learning from noisy VQE outputs. Our approach can significantly decrease energy estimation errors and increase fidelities with ground states compared to noisy input data for the H2 , LiH, and BeH2 molecular Hamiltonians and the transverse field Ising model. Surprisingly, it only requires noisy data for training. Variational denoising can be integrated into quantum hardware, increasing its versatility as an end-to-end quantum processing for quantum data. Published by the American Physical Society 2024
Hybrid quantum-classical variational algorithms such as the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) are promising applications for noisy, intermediate-scale quantum computers. Both VQE and QAOA variationally extremize the expectation value of a Hamiltonian. All work to date on VQE and QAOA has been limited to Pauli representations of Hamiltonians. However, many cases exist in which a sparse representation of the Hamiltonian is known but there is no efficient Pauli representation. We extend VQE to general sparse Hamiltonians. We provide a decomposition of a fermionic second-quantized Hamiltonian into a number of one-sparse, self-inverse, Hermitian terms linear in the number of ladder operator monomials in the second-quantized representation. We provide a decomposition of a general d-sparse Hamiltonian into O(d^{2}) such terms. In both cases, a single sample of any term can be obtained using two ansatz state preparations and at most six oracle queries. The number of samples required to estimate the expectation value to precision ε scales as ε^{-2} as for Pauli-based VQE. This widens the domain of applicability of VQE to systems whose Hamiltonian and other observables are most efficiently described in terms of sparse matrices.
Solving the electronic structure problem using the Variational Quantum Eigensolver (VQE) technique involves the measurement of the Hamiltonian expectation value. The current hardware can perform only projective single-qubit measurements, and thus, the Hamiltonian expectation value is obtained by measuring parts of the Hamiltonian rather than the full Hamiltonian. This restriction makes the measurement process inefficient because the number of terms in the Hamiltonian grows as O(N4) with the size of the system, N. To optimize the VQE measurement, one can try to group as many Hamiltonian terms as possible for their simultaneous measurement. Single-qubit measurements allow one to group only the terms commuting within the corresponding single-qubit subspaces or qubit-wise commuting. We found that the qubit-wise commutativity between the Hamiltonian terms can be expressed as a graph and the problem of the optimal grouping is equivalent to finding a minimum clique cover (MCC) for the Hamiltonian graph. The MCC problem is NP-hard, but there exist several polynomial heuristic algorithms to solve it approximately. Several of these heuristics were tested in this work for a set of molecular electronic Hamiltonians. On average, grouping qubit-wise commuting terms reduced the number of operators to measure three times less compared to the total number of terms in the considered Hamiltonians.
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.