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arXiv:2008.03661 · PRX Quantum

Quantum Power Method by a Superposition of Time-Evolved States

Kazuhiro Seki, Seiji Yunoki

Published Feb 26, 2021• 106 citations
arXivHTMLPDF

Abstract

We propose a quantum-classical hybrid algorithm of the power method, here dubbed as the quantum power method, to evaluate Ĥn |ψ with quantum computers, where n is a non-negative integer, Ĥ is a time-independent Hamiltonian of interest, and |ψ is a quantum state.We show that the number of gates required for approximating Ĥn scales linearly in the power and the number of qubits, making it a promising application for near-term quantum computers.Using numerical simulation, we show that the power method can control systematic errors in approximating the Hamiltonian power Ĥn for n as large as 100.As an application, we combine our method with a multireference Krylov-subspace-diagonalization scheme to show how one can improve the estimation of ground-state energies and the ground-state fidelities found using a variational-quantum-eigensolver scheme.Finally, we outline other applications of the quantum power method, including several moment-based methods.We numerically demonstrate the connectedmoment expansion for the imaginary-time evolution and compare the results with the multireference Krylov-subspace diagonalization.

DOI
10.1103/prxquantum.2.010333

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We propose a quantum-classical hybrid algorithm of the power method, here dubbed as the quantum power method, to evaluate Ĥn |ψ with quantum computers, where n is a non-negative integer, Ĥ is a time-independent Hamiltonian of interest, and |ψ is a quantum state.We show that the number of gates required for approximating Ĥn scales linearly in the power and the number of qubits, making it a promising application for near-term quantum computers.Using numerical simulation, we show that the power method can control systematic errors in approximating the Hamiltonian power Ĥn for n as large as 100.As an application, we combine our method with a multireference Krylov-subspace-diagonalization scheme to show how one can improve the estimation of ground-state energies and the ground-state fidelities found using a variational-quantum-eigensolver scheme.Finally, we outline other applications of the quantum power method, including several moment-based methods.We numerically demonstrate the connectedmoment expansion for the imaginary-time evolution and compare the results with the multireference Krylov-subspace diagonalization.
The task of estimating the ground state of Hamiltonians is an important problem in physics with numerous applications ranging from solid-state physics to combinatorial optimization. We provide a hybrid quantum-classical algorithm for approximating the ground state of a Hamiltonian that builds on the powerful Krylov subspace method in a way that is suitable for current quantum computers. Our algorithm systematically constructs the ansatz by using any given choice of the initial state and the unitaries describing the Hamiltonian. The only task of the quantum computer is to measure overlaps and no feedback loops are required. The measurements can be performed efficiently on current quantum hardware without requiring any complicated measurements such as the Hadamard test. Finally, a classical computer solves a well-characterized quadratically constrained optimization program. Our algorithm can reuse previous measurements to calculate the ground state of a wide range of Hamiltonians without requiring additional quantum resources. Further, we demonstrate our algorithm for solving a class of problems with thousands of qubits. The algorithm works for almost every random choice of the initial state and circumvents the barren plateau problem.
The generating function of a Hamiltonian HHH is defined as F(t)=⟨e−itH⟩F(t)=\langle e^{-itH}\rangleF(t)=⟨e−itH⟩, where ttt is the time and where the expectation value is taken on a given initial quantum state. This function gives access to the different moments of the Hamiltonian ⟨HK⟩\langle H^{K}\rangle⟨HK⟩ at various orders KKK. With the constraint of minimizing the quantum resources needed on near-term quantum computers, we show that the evaluation of the generating function can be made with only one extra ancillary qubit. Due to the limited stability of actual quantum devices, the function F(t)F(t)F(t) can a priori be estimated on a restricted time interval limiting automatically the orders of the moments that could be computed. Despite of this current limitation and even with a small number of moments, a post-processing on classical computers can be used to predict approximate ground or excited state energies and/or approximate long-time evolutions. This post-processing can be achieved using methods based on the Krylov space and/or on the ttt-expansion approach that is based on imaginary time evolution. Hybrid quantum-classical calculations are illustrated in many-body interacting systems using the pairing and Fermi-Hubbard models. Possible extensions using symmetry breaking and restoration as well as a multi-reference framework are discussed.
Abstract Quantum algorithms on noisy intermediate‐scale quantum (NISQ) devices are expected to soon simulate quantum systems that are classically intractable. However, the non‐negligible gate error present on NISQ devices impedes the implementation of many purely quantum algorithms, necessitating the use of hybrid quantum‐classical algorithms. One such hybrid quantum‐classical algorithm, is based upon quantum computed Hamiltonian moments , with respect to quantum state . In this tutorial review, we will give a brief review of these quantum algorithms with focuses on the typical ways of computing Hamiltonian moments using quantum hardware and improving the accuracy of the estimated state energies based on the quantum computed moments. We also present a computation of the Hamiltonian moments of a four‐site Heisenberg model and compute the energy and magnetization of the model utilizing the imaginary time evolution on current IBM‐Q hardware. Finally, we discuss some possible developments and applications of Hamiltonian moment methods.
We present a quantum-classical hybrid random power method that approximates a ground state of a Hamiltonian. The quantum part of our method computes a fixed number of elements of a Hamiltonian-matrix polynomial via quantum polynomial filtering techniques with either Hamiltonian simulation or block encoding. The use of the techniques provides a computational advantage that may not be achieved classically in terms of the degree of the polynomial. The classical part of our method is a randomized iterative algorithm that takes as input the matrix elements computed from the quantum part and outputs an approximation of ground state of the Hamiltonian. We prove that with probability one, our method converges to an approximation of a ground state of the Hamiltonian, requiring a constant scaling of the per-iteration classical complexity. The required quantum circuit depth is independent of the initial overlap and has no or a square-root dependence on the spectral gap. The iteration complexity scales linearly as the dimension of the Hilbert space when the quantum polynomial filtering corresponds to a sparse matrix. We numerically validate this sparsity condition for well-known model Hamiltonians. We also present a lower bound of the fidelity, which depends on the magnitude of noise occurring from quantum computation regardless of its charateristics, if it is smaller than a critical value. Several numerical experiments demonstrate that our method provides a good approximation of ground state in the presence of systematic and/or sampling noise.
Abstract Imaginary time evolution is a powerful tool for studying quantum systems. While it is possible to simulate with a classical computer, the time and memory requirements generally scale exponentially with the system size. Conversely, quantum computers can efficiently simulate quantum systems, but not non-unitary imaginary time evolution. We propose a variational algorithm for simulating imaginary time evolution on a hybrid quantum computer. We use this algorithm to find the ground-state energy of many-particle systems; specifically molecular hydrogen and lithium hydride, finding the ground state with high probability. Our method can also be applied to general optimisation problems and quantum machine learning. As our algorithm is hybrid, suitable for error mitigation and can exploit shallow quantum circuits, it can be implemented with current quantum computers.
The estimation of low energies of many-body systems is a cornerstone of the computational quantum sciences. Variational quantum algorithms can be used to prepare ground states on pre-fault-tolerant quantum processors, but their lack of convergence guarantees and impractical number of cost function estimations prevent systematic scaling of experiments to large systems. Alternatives to variational approaches are needed for large-scale experiments on pre-fault-tolerant devices. Here, we use a superconducting quantum processor to compute eigenenergies of quantum many-body systems on two-dimensional lattices of up to 56 sites, using the Krylov quantum diagonalization algorithm, an analog of the well-known classical diagonalization technique. We construct subspaces of the many-body Hilbert space using Trotterized unitary evolutions executed on the quantum processor, and classically diagonalize many-body interacting Hamiltonians within those subspaces. These experiments demonstrate exponential convergence towards an estimate of the ground state energy, and show that quantum diagonalization algorithms are poised to complement their classical counterparts at the foundation of computational methods for quantum systems. The estimation of low energies of many-body systems is a cornerstone of the computational quantum sciences. This paper demonstrates on a superconducting quantum processor that the Krylov quantum diagonalization algorithm is poised to complement its classical counterparts at the foundation of computational methods for quantum systems.