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arXiv:2512.10875

Multiple-time Quantum Imaginary Time Evolution

Julio Del Castillo, Mats Granath, Evert van Nieuwenburg

quant-phSubmitted Dec 11, 2025• Announced Dec 12, 2025• 0 citations• Revised Jun 12, 2026 (v2)
v1 · Dec 11, 2025v2 · Jun 12, 2026
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Abstract

Quantum Imaginary-Time Evolution (QITE) is a powerful method for preparing ground states on quantum hardware. However, executing QITE has costly measurement budgets for general Hamiltonians. Both fidelity and computational cost are strongly dependent on the definition of suitable local domains and Hamiltonian partitions. In this work, we introduce the Multiple-Time QITE algorithm (MT-QITE). We show how using more than one imaginary time substantially improves the fidelity of the resulting ground state as well as the measurement overhead with respect to the previously published QITE algorithm, while preserving its deterministic character and its independence from ad hoc ansatze. Moreover, unlike QITE and other QITE-based algorithms, MT-QITE is parallelizable, and we show that even in Hamiltonians with non-local interactions, partitioning may entail a computational advantage.

Comments
13 pages, 8 figures. Reviewed figures

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Quantum Imaginary-Time Evolution (QITE) is a powerful method for preparing ground states on quantum hardware. However, executing QITE has costly measurement budgets for general Hamiltonians. Both fidelity and computational cost are strongly dependent on the definition of suitable local domains and Hamiltonian partitions. In this work, we introduce the Multiple-Time QITE algorithm (MT-QITE). We show how using more than one imaginary time substantially improves the fidelity of the resulting ground state as well as the measurement overhead with respect to the previously published QITE algorithm, while preserving its deterministic character and its independence from ad hoc ansatze. Moreover, unlike QITE and other QITE-based algorithms, MT-QITE is parallelizable, and we show that even in Hamiltonians with non-local interactions, partitioning may entail a computational advantage.
Quantum imaginary time evolution (QITE) is one of the promising candidates for finding the eigenvalues and eigenstates of a Hamiltonian on a quantum computer. However, the original proposal suffers from large circuit depth and measurements due to the size of the Pauli operator pool and Trotterization. To alleviate the requirement for deep circuits, we propose a time-dependent drifting scheme inspired by the qDRIFT algorithm [Campbell, E. Phys. Rev. Lett. 2019, 123, 070503]. We show that this drifting scheme removes the depth dependency on the size of the operator pool and converges inversely with respect to the number of steps. We further propose a deterministic algorithm that selects the dominant Pauli term to reduce the fluctuation for the ground state preparation. We also introduce an efficient measurement reduction scheme across Trotter steps that removes its cost dependence on the number of iterations. We analyze the main source of error for our scheme both theoretically and numerically. We numerically test the validity of depth reduction, convergence performance of our algorithms, and the faithfulness of the approximation for our measurement reduction scheme on several benchmark molecules. In particular, the results on the LiH molecule give circuit depths comparable to that of the advanced adaptive variational quantum eigensolver (VQE) methods while requiring much fewer measurements.
Many computationally hard problems can be encoded in quantum Hamiltonians. The solution to these problems is given by the ground states of these Hamiltonians. A state-of-the-art algorithm for finding the ground state of a Hamiltonian is the so-called Quantum Imaginary Time Evolution (QITE) which approximates imaginary time evolution by a unitary evolution that can be implemented in quantum hardware. In this paper, we review the original algorithm together with a comprehensive computer program, as well as, the variational version of it.
Abstract Most quantum algorithms designed to generate or probe properties of the ground state of a quantum many-body system require as input an initial state with a large overlap with the desired ground state. One approach for preparing such a ground state is Imaginary Time Evolution (ITE). Recent work by [Motta, M. , Sun, C. , Tan, A.T.K. et al (2020)] introduced an algorithm—which we will refer to as Quantum Imaginary Time Evolution (QITE)—that shows how ITE can be approximated by a sequence of unitary operators, making QITE potentially implementable on early fault-tolerant quantum computers. In this work, we provide a heuristic study of the capabilities of the QITE algorithm in approximating the ITE of lattice and molecular electronic structure Hamiltonians. We numerically study the performance of the QITE algorithm when provided with a good classical initial state for a large class of systems, some of which are of interest to industrial applications, and check if QITE is able to qualitatively replicate the ITE behavior and improve over a classical mean-field solution. The systems we consider in this work range from one- and two-dimensional lattice systems of various lattice geometries displaying short- and long-range interactions, to active spaces of molecular electronic structure Hamiltonians. In addition to the comparison of QITE and ITE, we explicitly show how imaginary time evolved fermionic Gaussian states can serve as initial states which can be efficiently computed on classical computers and efficiently implemented on quantum computers for generic spin Hamiltonians in arbitrary lattice geometries and dimensions, which can be of independent interest.
Energy shifting is usually trivial in imaginary-time evolution because it changes only the normalization of the evolved state. On a quantum computer, however, imaginary-time evolution can be implemented as a coherent or sampled superposition of real-time evolutions, in which energy shifts generate relative phases that can interfere. Here we introduce multi-shift quantum imaginary-time evolution (MS-QITE), which uses a distribution of energy shifts to engineer this interference and optimize different implementations. In a Monte Carlo realization, multi-shift reshapes the normalized sampling distribution and concentrates it within a shorter real-time window, thereby reducing the typical Hamiltonian-evolution time and improving the stability of ground-state-energy estimation. In a continuous-variable-assisted realization, it enables projection onto a state supported over a finite quadrature interval, substantially reducing the required squeezing over an intermediate temperature range while retaining accurate thermal-state preparation. Numerical results for transverse-field Ising models demonstrate that energy shifts provide an interference-based degree of freedom for optimizing quantum imaginary-time evolution.
Efficiently preparing approximate ground states of large, strongly correlated systems on quantum hardware is challenging, and yet, nature is innately adept at this. This has motivated the study of thermodynamically inspired approaches to ground-state preparation that aim to replicate cooling processes via imaginary-time evolution. However, synthesizing quantum circuits that efficiently implement imaginary-time evolution is itself difficult, with prior proposals generally adopting heuristic variational approaches or using deep block encodings. Here, we use the insight that quantum imaginary-time evolution is a solution of Brockett's double-bracket flow and synthesize circuits that implement double-bracket flows coherently on the quantum computer. We prove that our double-bracket quantum imaginary-time evolution (DB-QITE) algorithm inherits the cooling guarantees of imaginary-time evolution. Concretely, each step is guaranteed to (i) decrease the energy of an initial approximate ground state by an amount proportional to the energy fluctuations of the initial state and (ii) increase the fidelity with the ground state. We provide gate counts for DB-QITE through numerical simulations in qrisp that demonstrate scenarios where DB-QITE outperforms quantum phase estimation. Thus, DB-QITE provides a means to systematically improve the approximation of a ground state using shallow circuits.
Quantum imaginary-time evolution (QITE) provides a systematic route to ground-state preparation, but its cost grows rapidly with the locality DDD of the unitary approximation, for which the local Pauli basis scales as 4D4^D4D. We introduce Sparse-QITE, a regularized formulation that promotes compact generators, reducing the number of quantum operations and yielding circuits with fewer gates and lower depth. We derive rigorous bounds on the sparsification error and benchmark the method in one-dimensional transverse-field Ising and antiferromagnetic Heisenberg models. Sparse-QITE exhibits a controllable sparsity--accuracy trade-off while retaining strong compression even in the high-accuracy regime. In scalability experiments with respect to DDD, the active support grows much more slowly than in standard QITE and, in several regimes, approximately saturates over the domain sizes studied. After circuit construction and transpilation, some of the largest-domain cases require only about 1%1\%1% of the standard-QITE gate count while maintaining high accuracy. We further introduce adaptive and active-discovery variants: the former selects the regularization strength dynamically during the evolution, while the latter reduces the number of Pauli expectation values that need to be evaluated and, consequently, the measurement requirements. These results establish sparsity-promoting regularization as a route toward more resource-efficient QITE. More broadly, these results suggest that sparsity-promoting regularization may also benefit variational quantum algorithms.