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arXiv:2504.04958 · Physical Review Research

Probabilistic imaginary-time evolution in state-vector-based and shot-based simulations and on quantum devices

Satoshi Ejima, Kazuhiro Seki, Benedikt Fauseweh, Seiji Yunoki

Published Nov 19, 2025• 1 citation
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Abstract

Imaginary-time evolution, an important technique in tensor network and quantum Monte Carlo algorithms on classical computers, has recently been adapted to quantum computing. In this study, we focus on probabilistic imaginary-time evolution (PITE) algorithm and derive its formulation in the context of state-vector-based simulations, where quantum state vectors are directly used to compute observables without statistical errors. We compare the results with those of shot-based simulations, which estimate observables through repeated projective measurements. Applying the PITE algorithm to the Heisenberg chain, we investigate optimal initial conditions for convergence. We further demonstrate the method on the transverse-field Ising model using a state-of-the-art trapped-ion quantum device. Finally, we explore the potential of error mitigation in this framework, highlighting practical considerations for near-term digital quantum simulations.

DOI
10.1103/2s2p-kvcx

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Physical Review Research•Aug 11, 2022•55 citations
Imaginary-time evolution, an important technique in tensor network and quantum Monte Carlo algorithms on classical computers, has recently been adapted to quantum computing. In this study, we focus on probabilistic imaginary-time evolution (PITE) algorithm and derive its formulation in the context of state-vector-based simulations, where quantum state vectors are directly used to compute observables without statistical errors. We compare the results with those of shot-based simulations, which estimate observables through repeated projective measurements. Applying the PITE algorithm to the Heisenberg chain, we investigate optimal initial conditions for convergence. We further demonstrate the method on the transverse-field Ising model using a state-of-the-art trapped-ion quantum device. Finally, we explore the potential of error mitigation in this framework, highlighting practical considerations for near-term digital quantum simulations.
Imaginary-time evolution is an important technique in tensor network methods and quantum Monte Carlo methods on classical computers, and in recent years its application to quantum computers has also been progressing. In this work, we focus particularly on the Probabilistic Imaginary-Time Evolution (PITE) method. We first derive its formulation within the framework of state-vector-based simulations, compute physical quantities directly from quantum state vectors without statistical errors, and compare the results with those obtained from shot-based simulations. Next, we apply the PITE method to the Heisenberg chain and investigate the optimal initial state for convergence. We also apply it to the transverse-field Ising model on a state-of-the-art ion-trap quantum device to confirm its effectiveness. Finally, we examine the possibility of error mitigation in this model and explore its applicability to NISQ devices.
Abstract Imaginary-time evolution plays an important role in algorithms for computing ground-state and thermal equilibrium properties of quantum systems, but can be challenging to simulate on classical computers. Many quantum algorithms for imaginary-time evolution have resource requirements that are prohibitive for current quantum devices and face performance issues due to noise. Here, we propose a new algorithm for computing imaginary-time evolved expectation values on quantum computers, inspired by probabilistic error cancellation, an error-mitigation technique. Our algorithm works by decomposing a Trotterization of imaginary-time evolution into a probabilistic linear combination of operations, each of which is then implemented on a quantum computer. The measurement data is then classically post-processed to obtain the expectation value of the imaginary-time evolved state. Our algorithm requires no ancillary qubits and can be made noise-resilient without additional error-mitigation. It is well-suited for estimating thermal expectation values by making use of the notion of a thermal pure quantum state. We demonstrate our algorithm by performing numerical simulations of thermal pure quantum state preparation for the 1D Heisenberg Hamiltonian on 8 qubits, and by using an IBM quantum computer to estimate the energy of the same Hamiltonian on 2 qubits. We observe promising results compared to the exact values, illustrating the potential of our algorithm for probing the physics of quantum many-body systems on current hardware.
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.
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.
Simulating differential equations on classical computers becomes an intractable problem if the grid size is extremely large. Quantum computers are believed to achieve a possibly exponential speedup in the matrix operation. In this paper, we propose a quantum algorithm for solving the advection-diffusion-reaction equation by employing a novel approximate probabilistic imaginary-time evolution (PITE) operator. First, the effectiveness of the proposed approximate PITE operator is justified by the theoretical evaluation of the error. Next, we construct the explicit quantum circuit to realize the imaginary-time evolution of the Hamiltonian coming from the advection-diffusion equation, whose gate complexity is logarithmic regarding the size of the discretized Hamiltonian matrix. Compared to the existing algorithms for the quantum linear system problem, our algorithm achieves an exponential speedup regarding the matrix size at the cost of a worse dependence on the error bound. Moreover, numerical simulations using gate-based quantum emulator for 1D/2D examples are also provided to verify our algorithm. Finally, we extend our algorithm to the coupled system of advection-diffusion equations to show the prospects for practical applications.
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.