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arXiv:2607.09599

Generalized Nonlinear Imaginary-Time Evolution

Chenyu Shi, Hao Wang, Jin-Fu Chen

quant-phSubmitted Jul 10, 2026• Announced Jul 13, 2026• 0 citations
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Abstract

Imaginary-time evolution (ITE) is a powerful method for ground-state preparation of a given Hamiltonian. The normalized ITE can be viewed as a gradient flow of the energy expectation value with respect to the Fubini--Study metric. In this work, we propose a generalized nonlinear imaginary-time evolution (NITE) for more general quantum state-preparation tasks. We further present a hardware-efficient variational implementation of NITE and reveal its connection to quantum natural gradient descent. NITE is applied to several subroutine tasks, including variance minimization in variational eigensolvers, probe-state preparation in variational quantum sensing, and excited-state preparation using penalty terms. We prove that NITE achieves a local exponential convergence rate under reasonable assumptions. Our results show that NITE outperforms standard gradient descent and can serve as an efficient optimization method for variational tasks beyond ground-state preparation.

Comments
6 + 7 pages, 2 + 4 figures

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Imaginary-time evolution (ITE) is a powerful method for ground-state preparation of a given Hamiltonian. The normalized ITE can be viewed as a gradient flow of the energy expectation value with respect to the Fubini--Study metric. In this work, we propose a generalized nonlinear imaginary-time evolution (NITE) for more general quantum state-preparation tasks. We further present a hardware-efficient variational implementation of NITE and reveal its connection to quantum natural gradient descent. NITE is applied to several subroutine tasks, including variance minimization in variational eigensolvers, probe-state preparation in variational quantum sensing, and excited-state preparation using penalty terms. We prove that NITE achieves a local exponential convergence rate under reasonable assumptions. Our results show that NITE outperforms standard gradient descent and can serve as an efficient optimization method for variational tasks beyond ground-state preparation.
Imaginary-time evolution (ITE) underpins a broad family of algorithms for ground-state preparation in quantum simulation and quantum many-body physics. In these methods, convergence is governed by the energy variance of the instantaneous state, causing the flow to approach the ground state only asymptotically. We introduce an augmented imaginary-time evolution (AITE) framework that replaces the standard gradient flow on the energy landscape with a geometrically informed descent along locally optimal directions, which are identified by exploiting the higher-order statistical structure of the instantaneous energy distribution. The resulting flow strictly outperforms standard ITE throughout the entire evolution and exhibits two qualitatively distinct regimes: a superlinear convergence regime, followed by an extinction regime in which the energy error vanishes exactly at a finite value of the imaginary-time flow parameter, in contrast to the asymptotic exponential decay of ITE. Standard ITE is recovered in the zero-skewness limit of AITE, implying that the acceleration extends naturally across the broader ITE algorithmic family.
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.
The imaginary-time evolution of quantum states is integral to various fields, ranging from natural sciences to classical optimization or machine learning. Since simulating quantum imaginary-time evolution generally requires storing an exponentially large wave function, quantum computers are emerging as a promising platform for this task. However, variational approaches, suitable for near-term quantum computers, struggle with a prohibitive number of measurements and impractical runtimes for relevant system sizes. Here, we suggest a stochastic approach to variational quantum imaginary-time evolution, which allows a significant reduction in runtimes. Our approach allows trading off invested resources and accuracy, which makes it also suitable for ground state preparation, where simulating the exact dynamics is not required. We demonstrate the efficiency of our algorithm in simulations and show a hardware experiment performing the imaginary-time evolution of the transverse field Ising model on 27 qubits.
Numerous methodologies have been proposed to implement imaginary-time evolution (ITE) on quantum computers. Among these, variational ITE (VITE) methods for noisy intermediate-scale quantum (NISQ) computers have attracted much attention, as they use parameterized quantum circuits to mimic non-unitary dynamics. Although widely studied, conventional variational quantum algorithms face challenges in achieving high accuracy due to their strong dependence on the choice of ansatz quantum circuits. Recently, the variational quantum-neural hybrid eigensolver (VQNHE), which combines neural networks (NN) with a variational quantum eigensolver, has been proposed. This approach enhances the expressive power of variational states and improves the estimation of expectation values. Motivated by this idea, we explore the hybridization of VITE with an NN-based non-unitary operator. In this study, we propose a method named variational quantum-neural hybrid ITE (VQNHITE). By combining the NN with a parameterized quantum circuit, our proposal enhances the expressive power compared to conventional approaches, enabling more accurate tracking of imaginary-time dynamics. In addition, to mitigate the instability arising from randomly initialized NN parameters, we introduce an initial-parameter optimization procedure at a small imaginary-time step, which stabilizes the subsequent variational evolution. We tested our approach with numerical simulations on Heisenberg spin chains under both nearest-neighbor and all-to-all circuit connectivities. The results demonstrate that VQNHITE consistently achieves higher fidelity with the exact ITE state compared to VITE.
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.