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arXiv:2111.12471

Probabilistic imaginary-time evolution by using forward and backward real-time evolution with a single ancilla: first-quantized eigensolver of quantum chemistry for ground states

Taichi Kosugi, Yusuke Nishiya, Nishi, Hirofumi, Yu‐ichiro Matsushita

Published Nov 24, 2021• 2 citations
arXivHTMLPDF

Abstract

Imaginary-time evolution (ITE) on a quantum computer is a promising formalism for obtaining the ground state of a quantum system. As a kind of it, the probabilistic ITE (PITE) takes advantage of measurements to implement the nonunitary operations. We propose a new approach of PITE which requires only a single ancillary qubit. Under a practical approximation, the circuit is constructed from the forward and backward real-time evolution (RTE) gates as black boxes, generated by the original many-qubit Hamiltonian. All the efficient unitary quantum algorithms for RTE proposed so far and those in the future can thus be transferred to ITE exactly as they are. Our approach can also be used for obtaining the Gibbs state at a finite temperature and the partition function. We apply the approach to several systems as illustrative examples to see its validity. We also discuss the application of our approach to quantum chemistry by focusing on the scaling of computational cost, leading to a novel framework denoted by first-quantized quantum eigensolver.

DOI
10.48550/arxiv.2111.12471

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Imaginary-time evolution (ITE) on a quantum computer is a promising formalism for obtaining the ground state of a quantum system. As a kind of it, the probabilistic ITE (PITE) takes advantage of measurements to implement the nonunitary operations. We propose a new approach of PITE which requires only a single ancillary qubit. Under a practical approximation, the circuit is constructed from the forward and backward real-time evolution (RTE) gates as black boxes, generated by the original many-qubit Hamiltonian. All the efficient unitary quantum algorithms for RTE proposed so far and those in the future can thus be transferred to ITE exactly as they are. Our approach can also be used for obtaining the Gibbs state at a finite temperature and the partition function. We apply the approach to several systems as illustrative examples to see its validity. We also discuss the application of our approach to quantum chemistry by focusing on the scaling of computational cost, leading to a novel framework denoted by first-quantized quantum eigensolver.
Imaginary-time evolution (ITE) on a quantum computer is a promising formalism for obtaining the ground state of a quantum system. The probabilistic ITE (PITE) exploits measurements to implement nonunitary operations, and it can avoid the restriction of dynamics to a low-dimensional subspace imposed by variational parameters unlike other types of ITE. In this paper, we propose a PITE approach that uses only one ancillary qubit. Unlike the existing PITE approaches, the one proposed here constructs, under a practical approximation, the circuit from forward and backward real-time evolution (RTE) gates as black boxes for the original Hamiltonian. Thus all efficient unitary algorithms for RTE can be transferred to the ITE without any modifications. Our approach can be used to obtain the Gibbs state at a finite temperature and partition function. We validate the approach via several illustrative systems where the trial states are found to converge rapidly to the ground states. In addition, we discuss its applicability to quantum chemistry by focusing on the scaling of computational cost; this leads to the development of a framework referred to as a first-quantized eigensolver. The nonvariational generic approach will expand the scope of practical quantum computation for versatile objectives.
A probabilistic imaginary-time evolution (PITE) method was proposed as a nonvariational method to obtain a ground state on a quantum computer. In this formalism, the success probability of obtaining all imaginary-time evolution operators acting on the initial state decreases as the imaginary time proceeds. To alleviate the undesirable nature, we propose quantum circuits for PITE combined with the quantum amplitude amplification (QAA) method. We reduce the circuit depth in the combined circuit with QAA by introducing a pre-amplification operator. We successfully demonstrated that the combination of PITE and QAA works efficiently and reported a case in which the quantum acceleration is achieved. Additionally, we have found that by optimizing a parameter of PITE, we can reduce the number of QAA operations and that deterministic imaginary-time evolution (deterministic ITE) can be achieved which avoids the probabilistic nature of PITE. We applied the deterministic ITE procedure to multiple imaginary-time steps and discussed the computational cost for the circuits. Finally, as an example, we demonstrate the numerical results of the PITE circuit combined with QAA in the first- and second-quantized Hamiltonians.
Imaginary-time evolution is a powerful tool for obtaining the ground state of a quantum system, but the complexity of classical algorithms designed for simulating imaginary-time evolution will increase significantly as the size of the quantum system becomes larger. Here, a probabilistic quantum algorithm based on Taylor expansion for implementing imaginary-time evolution is introduced. For Hamiltonians composed of Pauli product terms, the quantum circuit requires only a single ancillary qubit and is exclusively constructed using elementary single-qubit and two-qubit gates. Furthermore, similar principles are used to extend the algorithm to the case where the Hamiltonian takes a more general form. The algorithm only requires negligible precomputed numerical calculations, without the need for complex classical pre-mathematical calculations or optimization loops. We demonstrate the algorithm by solving the ground state energy of hydrogen molecules and Heisenberg Hamiltonians. Moreover, we conducted experiments on real quantum computers through the quantum cloud platform to find the ground state energy of Heisenberg Hamiltonians. Our work extends the methods for realizing imaginary-time evolution on quantum computers, and our algorithm exhibits potential for implementation on near-term quantum devices, particularly when the Hamiltonian consists of Pauli product terms.
Imaginary-time evolution (ITE) is a widely used technique for preparing the ground state of a target Hamiltonian. Building on the recently proposed framework of Double-Bracket quantum ITE (DB-QITE), we introduce a probabilistic variant (PDBQITE) that achieves an exponential reduction in circuit depth. Our algorithm requires only a single ancilla qubit and a quantum circuit that consists of one controlled real-time evolution and one mid-circuit measurement per iteration. The trade-off is an exponential decay of the total success probability with the number of iterations, for which we derive an analytical lower bound that is independent of system size. Our numerical simulations show that the per-step success probability remains close to unity, making it realistic to execute many iterations, thus providing a substantial improvement over prior constructions. We show that PDBQITE outperforms the near-term probabilistic algorithm PITE for ground-state preparation of molecular Hamiltonians, and that a hybrid QAOA-PDBQITE scheme achieves higher approximation ratios than QAOA at the same circuit depth on random regular graphs for the MaxCut problem.
Imaginary-time evolution is a powerful tool in the study of quantum physics. However, existing classical algorithms for simulating imaginary-time evolution suffer from high computational complexity as the dimension of the quantum system increases. In this study we propose a quantum algorithm for implementing imaginary-time evolution using nonunitary quantum circuits with one ancillary qubit. The success probability of our algorithm is a polynomial function of the output error and can be enhanced by reorganizing the terms of the Hamiltonian. To illustrate the practicality of our algorithm on current quantum devices, we conduct a demonstration on superconducting and trapped-ion quantum processors to calculate the ground-state energy and determine the most stable molecular structure of H2{\mathrm{H}}_{2}H2​. Additionally, we validate the feasibility of our algorithm by numerically simulating the ground-state energies of LiH molecules and the quantum Ising chain. In contrast to existing algorithms, our method provides a systematic approach to construct the required nonunitary circuits using universal quantum gates, making it suitable for experimental implementation. Our algorithm opens up possibilities for exploring other physical phenomena such as finite-temperature properties and non-Hermitian systems.
Ground-state preparation is an important task in quantum computation. The probabilistic imaginary-time evolution (PITE) method is a promising candidate for preparing the ground state of the Hamiltonian, which comprises a single ancilla qubit and forward- and backward-controlled real-time evolution operators. The ground state preparation is a challenging task even in the quantum computation, classified as complexity-class quantum Merlin-Arthur. However, optimal parameters for PITE could potentially enhance the computational efficiency to a certain degree. In this paper, we analyze the computational costs of the PITE method for both linear and exponential scheduling of the imaginary-time step size for reducing the computational cost. First, we analytically discuss an error defined as the closeness between the states acted on by exact and approximate imaginary-time evolution operators. The optimal imaginary-time step size and rate of change of imaginary time are also discussed. Subsequently, the analytical discussion is validated using numerical simulations for a one-dimensional Heisenberg chain. From the results, we find that linear scheduling works well in the case of unknown eigenvalues of the Hamiltonian. For a wide range of eigenstates, the linear scheduling returns smaller errors on average. However, the linearity of the scheduling causes problems for some specific energy regions of eigenstates. To avoid these problems, incorporating a certain level of nonlinearity into the scheduling, such as by inclusion of an exponential character, is preferable for reducing the computational costs of the PITE method. The findings of this paper can make a significant contribute to the field of ground-state preparation of many-body Hamiltonians on quantum computers.