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arXiv:2112.07416 · Physical Review Research

Quantum expectation-value estimation by computational basis sampling

Masaya Kohda, Ryosuke Imai, Keita Kanno, Kosuke Mitarai, Wataru Mizukami, Yuya O. Nakagawa

Published Sep 2, 2022• 31 citations
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Abstract

Measuring expectation values of observables is an essential ingredient in variational quantum algorithms. A practical obstacle is the necessity of a large number of measurements for statistical convergence to meet requirements of precision, such as chemical accuracy in the application to quantum chemistry computations. Here we propose an algorithm to estimate the expectation value based on its approximate expression as a weighted sum of classically tractable matrix elements with some modulation, where the weight and modulation factors are evaluated by sampling appropriately prepared quantum states in the computational basis on quantum computers. Each of those states is prepared by applying a unitary transformation consisting of at most $N$ CNOT gates, where $N$ is the number of qubits, to a target quantum state whose expectation value is evaluated. Our algorithm is expected to require fewer measurements than conventional methods for a required statistical precision of the expectation value when the target quantum state is concentrated in particular computational basis states. We provide numerical comparisons of our method with existing ones for measuring electronic ground-state energies (expectation values of electronic Hamiltonians for the lowest-energy states) of various small molecules. Numerical results show that our method can reduce the numbers of measurements to obtain the ground-state energies for a targeted precision by several orders of magnitudes for molecules whose ground states are concentrated. Our results provide another route to measure expectation values of observables, which could accelerate the variational quantum algorithms.

DOI
10.1103/physrevresearch.4.033173

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Measuring expectation values of observables is an essential ingredient in variational quantum algorithms. A practical obstacle is the necessity of a large number of measurements for statistical convergence to meet requirements of precision, such as chemical accuracy in the application to quantum chemistry computations. Here we propose an algorithm to estimate the expectation value based on its approximate expression as a weighted sum of classically tractable matrix elements with some modulation, where the weight and modulation factors are evaluated by sampling appropriately prepared quantum states in the computational basis on quantum computers. Each of those states is prepared by applying a unitary transformation consisting of at most NNN CNOT gates, where NNN is the number of qubits, to a target quantum state whose expectation value is evaluated. Our algorithm is expected to require fewer measurements than conventional methods for a required statistical precision of the expectation value when the target quantum state is concentrated in particular computational basis states. We provide numerical comparisons of our method with existing ones for measuring electronic ground-state energies (expectation values of electronic Hamiltonians for the lowest-energy states) of various small molecules. Numerical results show that our method can reduce the numbers of measurements to obtain the ground-state energies for a targeted precision by several orders of magnitudes for molecules whose ground states are concentrated. Our results provide another route to measure expectation values of observables, which could accelerate the variational quantum algorithms.
Expectation value estimation is ubiquitous in quantum algorithms. The expectation value of a Hamiltonian, which is essential in various practical applications, is often estimated by measuring a large number of Pauli strings on quantum computers and performing classical post-processing. In the case of nnn-qubit molecular Hamiltonians in quantum chemistry calculations, it is necessary to evaluate O(n4)O(n^4)O(n4) Pauli strings, requiring a large number of measurements for accurate estimation. To reduce the measurement cost, we assess an existing idea that uses two copies of an nnn-qubit quantum state of interest and coherently measures them in the Bell basis, which enables the simultaneous estimation of the absolute values of expectation values of all the nnn-qubit Pauli strings. We numerically investigate the efficiency of energy estimation for molecular Hamiltonians of up to 12 qubits. The results show that, when the target precision is no smaller than tens of milli-Hartree, this method requires fewer measurements than conventional sampling methods. This suggests that the method may be useful for many applications that rely on expectation value estimation of Hamiltonians and other observables as well when moderate precision is sufficient.
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Iterative phase estimation has long been used in quantum computing to estimate Hamiltonian eigenvalues. This is done by applying many repetitions of the same fundamental simulation circuit to an initial state, and using statistical inference to glean estimates of the eigenvalues from the resulting data. Here, we show a generalization of this framework where each of the steps in the simulation uses a different Hamiltonian. This allows the precision of the Hamiltonian to be changed as the phase estimation precision increases. Additionally, through the use of importance sampling, we can exploit knowledge about the ground state to decide how frequently each Hamiltonian term should appear in the evolution, and minimize the variance of our estimate. We rigorously show, if the Hamiltonian is gapped and the sample variance in the ground state expectation values of the Hamiltonian terms sufficiently small, that this process has a negligible impact on the resultant estimate and the success probability for phase estimation. We demonstrate this process numerically for two chemical Hamiltonians, and observe substantial reductions in the number of terms in the Hamiltonian; in one case, we even observe a reduction in the number of qubits needed for the simulation. Our results are agnostic to the particular simulation algorithm, and we expect these methods to be applicable to a range of approaches.
We propose an efficient and almost optimal scheme for measuring molecular Hamiltonians in quantum chemistry on quantum computers, which requires 2N2 distinct measurements in the leading order with N being the number of molecular orbitals. It achieves the state-of-the-art by improving a previous proposal by Bonet-Monroig [], which exhibits 103N2 scaling in the leading order. We develop a method based on a finite projective plane to construct sets of simultaneously measurable operators contained in molecular Hamiltonians. Each measurement only requires a depth- O(N) circuit consisting of O(N2) one- and two-qubit gates under the Jordan-Wigner and parity mapping, assuming the linear connectivity of qubits on quantum hardwares. We perform numerical simulation of our method for molecular Hamiltonians of hydrogen chains. We count the number of sets of simultaneously measurable operators generated by our method and estimate the number of measurement shots required to achieve the small standard deviation of the energy expectation value for precise quantum chemistry calculations. Because evaluating expectation values of molecular Hamiltonians is one of the major bottlenecks in the applications of quantum devices to quantum chemistry, our finding is expected to accelerate such applications. Published by the American Physical Society 2024
Quantum chemistry calculations are important applications of quantum annealing. For practical applications in quantum chemistry, it is essential to estimate a ground state energy of the Hamiltonian with chemical accuracy. However, there are no known methods to guarantee the accuracy of the estimation of the energy calculated by quantum annealing. Here, we propose a way to improve the accuracy of the estimate of the ground state energy by combining quantum annealing with classical computation. In our scheme, before running the QA, we need a pre-estimation of the energies of the ground state and first excited state with some error bars (corresponding to possible estimation error) by performing classical computation with some approximations. We show that, if an expectation value and variance of the energy of the state after the QA are smaller than certain threshold values (that we can calculate from the pre-estimation), the QA provides us with a better estimate of the ground state energy than that of the pre-estimation. Since the expectation value and variance of the energy can be experimentally measurable by the QA, our results pave the way for accurate estimation of the ground state energy with the QA.
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.