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arXiv:2012.07171 · Physical Review Letters

Variational Quantum Eigensolvers for Sparse Hamiltonians

William M. Kirby, Peter J. Love

Published Sep 9, 2021• 43 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.127.110503

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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.
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
We describe the contextual subspace variational quantum eigensolver (CS-VQE), a hybrid quantum-classical algorithm for approximating the ground state energy of a Hamiltonian. The approximation to the ground state energy is obtained as the sum of two contributions. The first contribution comes from a noncontextual approximation to the Hamiltonian, and is computed classically. The second contribution is obtained by using the variational quantum eigensolver (VQE) technique to compute a contextual correction on a quantum processor. In general the VQE computation of the contextual correction uses fewer qubits and measurements than the VQE computation of the original problem. Varying the number of qubits used for the contextual correction adjusts the quality of the approximation. We simulate CS-VQE on tapered Hamiltonians for small molecules, and find that the number of qubits required to reach chemical accuracy can be reduced by more than a factor of two. The number of terms required to compute the contextual correction can be reduced by more than a factor of ten, without the use of other measurement reduction schemes. This indicates that CS-VQE is a promising approach for eigenvalue computations on noisy intermediate-scale quantum devices.
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The ability of near-term quantum computers to represent classically-intractable quantum states has brought much interest in using such devices for estimating the ground and excited state energies of fermionic Hamiltonians. The usefulness of such near-term techniques, generally based on the Variational Quantum Eigensolver (VQE), however, is limited by device noise and the need to perform many circuit repetitions. This paper addresses these challenges by generalizing VQE to consider wavefunctions in a subspace spanned by classically tractable states and states that can be prepared on a quantum computer. The manuscript shows how the ground and excited state energies can be estimated using such "classical-boosting" and how this approach can be combined with VQE Hamiltonian decomposition techniques. Unlike existing VQE approaches, the sensitivity to sampling error and device noise approaches zero in the limit where the classically tractable states are able to describe an eigenstate. A detailed analysis of the measurement requirements in the simplest case, where a single computational basis state is used to boost conventional VQE, shows that the ground-state energy estimation of several closed-shell homonuclear diatomic molecules can be accelerated by a factor of approximately 10-1000. The analysis also shows that the measurement reduction of such single basis state boosting, relative to conventional VQE, can be estimated using only the overlap between the ground state and the computational basis state used for boosting.
The Variational Quantum Algorithms (VQAs) are hybrid quantum-classical algorithms and they can be used in the Nosiy Intermadiate Scale Quantum (NISQ) devises. The Variational Quantum Eigensolver (VQE) was suggested as a first VQA. VQE is based on the variational method of quantum mechanics and it is used to find the ground state energy of a quantum system. In this study, VQE is used for the analysis of NMR spectra for the AB and AB2 spin systems. The frequencies and the spin coupling values are obtained from the sample spectra for these spin systems. Then the Hamiltonians are written in terms of pauli spin operators and transformed into a suitable forms for quantum computer. By employing VQE the ground state energies are obtained for the related spin systems. They are found to be in good agreement with the results obtained from the known variation method.
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.