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arXiv:2608.30363

Quantum Imaginary Time Evolution on an Infinite 1D Chain

Hao-Ti Hung, Tung Tsao, Ying-Jer Kao

quant-phSubmitted Aug 31, 2026• Announced Sep 1, 2026• 0 citations
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Abstract

We introduce a quantum-circuit algorithm for performing imaginary-time evolution on infinite one-dimensional lattice systems. The method uses a parameterized quantum circuit to represent a uniform matrix product state ansatz. We derive the ITE algorithm using the time-dependent variational principle and employ the quantum Lanczos algorithm to improve the ground-state energy estimate. As a benchmark, we simulate the transverse-field Ising model using both classical simulators and IBM Quantum devices. Our analysis includes a statistical study of the distributions of the cost function and energy density obtained from quantum measurements, illustrating the effects of finite-sampling noise on convergence.

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We introduce a quantum-circuit algorithm for performing imaginary-time evolution on infinite one-dimensional lattice systems. The method uses a parameterized quantum circuit to represent a uniform matrix product state ansatz. We derive the ITE algorithm using the time-dependent variational principle and employ the quantum Lanczos algorithm to improve the ground-state energy estimate. As a benchmark, we simulate the transverse-field Ising model using both classical simulators and IBM Quantum devices. Our analysis includes a statistical study of the distributions of the cost function and energy density obtained from quantum measurements, illustrating the effects of finite-sampling noise on convergence.
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.
Imaginary-time evolution (ITE) is a foundational method for ground state preparation on quantum computers. However, because ITE is non-unitary, existing implementations incur a sample complexity and/or classical cost that scales exponentially with the target imaginary time βββ. Moreover, the state itself converges slower than the energy, making accurate estimation of arbitrary ground state observables even more expensive. In this work, we improve upon standard ITE by introducing twirled imaginary-time evolution (TITE), which pairs ITE with real-time evolution applied for a random duration drawn from a carefully designed distribution. We prove that this randomization quadratically suppresses the trace distance to the ground state, and thus also the error of arbitrary observables, which allows roughly half of the imaginary time to be replaced with real-time evolution (β↦β/2β\mapsto β/2β↦β/2) while maintaining the same level of accuracy. Because real-time evolution is unitary and does not incur an overhead in sample complexity or classical computation, this affords a quadratic reduction in the cost of any black-box ITE implementation, including Trotterization and quantum imaginary-time evolution. We demonstrate the efficiency of our algorithm in noisy circuit-level simulations of a non-integrable Ising chain, showing substantial improvements over standard ITE.
Variational quantum time evolution is a fundamental issue in physics and engineering. It provides a strong framework for modeling the dynamics of quantum states, which is essential for efficiently running quantum networks. It presents an excellent toolkit that can improve the efficacy, reliability, resilience and performance of quantum networks, opening the door to more sophisticated distributed computer systems, quantum communication, and other quantum technologies. The imaginary time evolution of quantum systems is one useful tool that enables the generation of ground or thermal states. In this work, we address standard variational principles, such as the Dirac and Frenkel variational principles, the McLachlan variational principle and the time-dependent variational principle to model real-time dynamics, as well as the Rayleigh–Ritz approach to address static challenges. We also present the variational quantum time evolution VarQITE and VarQRTE formalism in the density matrix method, which is not influenced by the global phase of the quantum state. We calculate the ground-state energy of our quantum Ising spin chain, using VarQITE with different interaction energies J , where three cases are considered. In the non-interacting ( NI ) case ( J = 0 ), we show that the energy of the ground state accomplished with VarQITE converges to an energy of − 2 . 00 Hartree at t = 1 . 1 seconds. In the Ferromagnetic ( FM ) interaction case ( J > 0 ), four scenarios are used to obtain the energy of the Hamiltonian. We show that the convergence and the energy 〈 H 〉 are sensitive to the energy of interaction J . Furthermore, the influence of the negative values of J < 0 , which correspond to the antiferromagnetic ( AFM ) case, on the energy of the ground state is also studied. We find that VarQITE and the exact classical solution are matched. We also investigated the expectation values for magnetization 〈 m z 〉 , employing VarQRTE and the exact classical solution. The NI, FM, and AFM cases are accomplished here. We find that when the spins are not interacting, 〈 m z 〉 = 0 , while it oscillates over time t when J > 0 and J < 0 . We believe that our outcomes will improve our knowledge of the variational quantum time evolution and lead to a better understanding of real and imaginary time evolution of quantum systems (which form a quantum network) and therefore, to more information on the time in networks.
Variational quantum dynamics simulations (VQDS) provide a promising route to simulate real- and imaginary-time quantum dynamics on noisy intermediate-scale quantum devices using fixed-depth circuits. However, their practical performance is strongly limited by sampling noise arising from a finite number of circuit measurements. In this work, we systematically investigate the impact of sampling noise on VQDS, with a focus on ground-state preparation in one-dimensional Ising spin models using imaginary time evolution. We compare different regularization strategies for stabilizing the equations of motion and show that Tikhonov regularization provides robust performance in noisy imaginary-time evolution. We then benchmark measurement-distribution strategies that allocate shots by minimizing a cost function that characterizes the error in solving the equation of motion. Using noisy circuit simulations, we demonstrate that such optimized shot allocation can significantly improve state fidelity and reduce the total measurement cost by more than a factor of two compared to uniform shot distributions. We observe that the best results are found if a sufficiently large number of measurements is guaranteed for all circuits, suggesting that a finite fraction of shots should be distributed evenly. Our results provide practical guidelines for implementing measurement-efficient variational quantum dynamics and ground-state preparation on near-term quantum hardware.
Simulating time evolution of generic quantum many-body systems using classical numerical approaches has an exponentially growing cost either with evolution time or with the system size. In this work we present a polynomially scaling hybrid quantum-classical algorithm for time evolving a one-dimensional uniform system in the thermodynamic limit. This algorithm uses a layered uniform sequential quantum circuit as a variational Ansatz to represent infinite translation-invariant quantum states. We show numerically that this Ansatz requires a number of parameters polynomial in the simulation time for a given accuracy. Furthermore, this favorable scaling of the Ansatz is maintained during our variational evolution algorithm. All steps of the hybrid optimization are designed with near-term digital quantum computers in mind. After benchmarking the evolution algorithm on a classical computer, we demonstrate the measurement of observables of this uniform state using a finite number of qubits on a cloud-based quantum processing unit. With more efficient tensor contraction schemes, this algorithm may also offer improvements as a classical numerical algorithm.
Simulating time evolution of generic quantum many-body systems using classical numerical approaches has an exponentially growing cost either with evolution time or with the system size. In this work we present a polynomially scaling hybrid quantum-classical algorithm for time evolving a one-dimensional uniform system in the thermodynamic limit. This algorithm uses a layered uniform sequential quantum circuit as a variational Ansatz to represent infinite translation-invariant quantum states. We show numerically that this Ansatz requires a number of parameters polynomial in the simulation time for a given accuracy. Furthermore, this favorable scaling of the Ansatz is maintained during our variational evolution algorithm. All steps of the hybrid optimization are designed with near-term digital quantum computers in mind. After benchmarking the evolution algorithm on a classical computer, we demonstrate the measurement of observables of this uniform state using a finite number of qubits on a cloud-based quantum processing unit. With more efficient tensor contraction schemes, this algorithm may also offer improvements as a classical numerical algorithm.