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arXiv:2511.06280 · Physica A Statistical Mechanics and its Applications

Hybrid Real-Imaginary Time Evolution for Low-Depth Hamiltonian Simulation in Quantum Optimization

Fei Li, Xiao-Wei Li

quant-phSubmitted Nov 9, 2025• Announced Nov 11, 2025• 0 citations
v1 · Nov 9, 2025v2 · Apr 14, 2026
arXivHTMLPDF

Abstract

Counterdiabatic (CD) driving is a powerful technique for accelerating adiabatic quantum computing. However, it becomes self-limiting in complex optimizations like the Sherrington-Kirkpatrick model: long evolution times $T$ needed to traverse crossings force the CD strength to scale as $1/T$, causing it to vanish before convergence and wasting the quantum resources invested in its implementation. We break this trade-off with a Hybrid adaptive variational quantum dynamics simulation (HAVQDS). HAVQDS combines adaptive real-time evolution for circuit compression with imaginary-time steps that suppress excitations at no extra gate cost. For the SK model (6--14 qubits), HAVQDS achieves higher approximation ratios than adiabatic or CD approaches, while reducing CNOT counts by 1--2 orders of magnitude, enabling high-fidelity quantum optimization.

Journal reference
Physica A 658 (2026) 131539
DOI
10.1016/j.physa.2026.131539

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Counterdiabatic (CD) driving is a powerful technique for accelerating adiabatic quantum computing. However, it becomes self-limiting in complex optimizations like the Sherrington-Kirkpatrick model: long evolution times TTT needed to traverse crossings force the CD strength to scale as 1/T1/T1/T, causing it to vanish before convergence and wasting the quantum resources invested in its implementation. We break this trade-off with a Hybrid adaptive variational quantum dynamics simulation (HAVQDS). HAVQDS combines adaptive real-time evolution for circuit compression with imaginary-time steps that suppress excitations at no extra gate cost. For the SK model (6--14 qubits), HAVQDS achieves higher approximation ratios than adiabatic or CD approaches, while reducing CNOT counts by 1--2 orders of magnitude, enabling high-fidelity quantum optimization.
Preparing ground states of correlated molecules on near-term quantum computers requires algorithms that are both accurate and resource-efficient. We introduce the qubit-adaptive variational quantum imaginary time evolution (QAITE) algorithm, which synthesizes the systematic convergence of imaginary-time evolution with hardware-efficient, adaptive ansatz construction. By integrating the linear-scaling qubit operator pool into the adaptive variational quantum imaginary time evolution framework, QAITE deterministically constructs a compact ansatz without costly classical optimization loops. Numerical simulations for molecular systems (H4, LiH, H2O, BeH2), benchmarked against our own implementations of established adaptive algorithms under identical conditions, show that QAITE achieves chemical accuracy with a 15-45% reduction in CNOT gates compared to the standard adaptive variational quantum imaginary time evolution (AVQITE) method. For hydrogen chains (RHH = 1.5 Å, N = 6-16 qubits), QAITE's circuit complexity grows approximately quadratically and remains the lowest of the four methods at every system size, reaching 31% fewer CNOT gates than AVQITE at 16 qubits─a saving of 1322 gates. This efficiency extends to heterogeneous (LiH)n and (HBe)n chains, where QAITE maintains a CNOT gate ratio below 1 for all sizes studied (e.g., 0.81 for (LiH)2). We further show that the deterministic parameter update is numerically robust: the results are insensitive to the regularization of the quantum geometric tensor over at least 4 orders of magnitude, and chemical accuracy is retained under realistic shot noise applied to both the quantum geometric tensor and the force vector. By outperforming established adaptive algorithms in both absolute gate count and scaling, QAITE establishes a promising, scalable pathway for quantum molecular simulations on noisy intermediate-scale quantum devices.
Counterdiabatic (CD) driving has the potential to speed up adiabatic quantum state preparation by suppressing unwanted excitations. However, existing approaches either require intractable classical computations or are based on approximations that do not have performance guarantees. We propose and analyze a nonvariational, system-agnostic CD expansion method and analytically show that it converges exponentially quickly in the expansion order. In finite systems, the required resources scale inversely with the spectral gap, which we argue is asymptotically optimal. To extend our method to the thermodynamic limit and suppress errors stemming from high-frequency transitions, we leverage finite-time adiabatic protocols. In particular, we show that a time determined by the quantum speed limit is sufficient to prepare the desired ground state, without the need to optimize the adiabatic trajectory. Numerical tests of our method on the quantum Ising chain show that our method can outperform state-of-the-art variational CD approaches.
Quantum simulation has begun to penetrate the field of quantum chemistry in hopes of efficiently calculating ground state energies and approximating real-time evolution. With modern research highlighting nonadiabatic dynamics, tunably approximating deep circuits representing potential landscapes becomes crucial for simulating real quantum systems. Variationally approximating unitaries allows for shallower circuits and accuracy tunable to hardware fidelity, so long as the observable quantities are preserved. We show the variational compression of Trotter terms preserve reaction rate coefficients via classical emulation of a hybrid quantum-classical optimization method, as well as fast-forwarded adiabatic dynamics on quantum hardware. Compressed circuits can be incorporated with product-formula-based time evolution to approximate dynamics of a particle in two coupled harmonic potentials, allowing tunability when removing high-cost qubit interactions. Approximate rate coefficients are recovered after substituting terms in a nonadiabatic dynamic process, giving proof-of-principle for observable preservation under variational optimization. Attention is paid to minimizing qubit and gate-count resources.
Among variational quantum algorithms designed for NISQ devices, ADAPT-VQE stands out for its robustness against barren plateaus, particularly in estimating molecular ground states. On the other hand, counterdiabatic algorithms have shown advantages in both performance and circuit depth when compared to standard adiabatic approaches. In this work, we propose a hybrid method that integrates the ADAPT-VQE framework with counterdiabatic driving within an adiabatic evolution scheme. Specifically, we map the molecular Hamiltonian to a qubit representation and construct an adiabatic Hamiltonian, from which an approximate adiabatic gauge potential is computed using nested commutators. The resulting operator terms define the operator pool, and the ADAPT-VQE algorithm is applied to iteratively select the most relevant elements for the ansatz. Our results demonstrate improvements in performance and reductions in circuit depth compared to using either counterdiabatic algorithms or ADAPT-VQE with fermionic excitation operators, thus supporting the effectiveness of combining both paradigms in molecular simulations.
Using counterdiabatic (CD) driving-aiming at suppression of diabatic transition-in digitized adiabatic evolution has garnered immense interest in quantum protocols and algorithms. However, improving the approximate CD terms with a nested commutator ansatz is a challenging task. In this work, we propose a technique of finding optimal coefficients of the CD terms using a variational quantum circuit. By classical optimization routines, the parameters of this circuit are optimized to provide the coefficients corresponding to the CD terms. Then their improved performance is exemplified in Greenberger-Horne-Zeilinger state preparation on the nearest-neighbour Ising model. Finally, we also show the advantage over the usual quantum approximation optimization algorithm, in terms of fidelity with bounded time. This article is part of the theme issue 'Shortcuts to adiabaticity: theoretical, experimental and interdisciplinary perspectives'.
Shortcuts to adiabaticity are well-known methods for controlling the quantum dynamics beyond the adiabatic criteria, where counterdiabatic (CD) driving provides a promising means to speed up quantum many-body systems. In this work, we show the applicability of CD driving to enhance the digitized adiabatic quantum computing paradigm in terms of fidelity and total simulation time. We study the state evolution of an Ising spin chain using the digitized version of the standard CD driving and its variants derived from the variational approach. We apply this technique in the preparation of Bell and Greenberger-Horne-Zeilinger states with high fidelity using a very shallow quantum circuit. We implement this proposal on the IBM quantum computer, proving its usefulness for the speed up of adiabatic quantum computing in noisy intermediate-scale quantum devices.