Abstract Imaginary time evolution is a powerful tool for studying quantum systems. While it is possible to simulate with a classical computer, the time and memory requirements generally scale exponentially with the system size. Conversely, quantum computers can efficiently simulate quantum systems, but not non-unitary imaginary time evolution. We propose a variational algorithm for simulating imaginary time evolution on a hybrid quantum computer. We use this algorithm to find the ground-state energy of many-particle systems; specifically molecular hydrogen and lithium hydride, finding the ground state with high probability. Our method can also be applied to general optimisation problems and quantum machine learning. As our algorithm is hybrid, suitable for error mitigation and can exploit shallow quantum circuits, it can be implemented with current quantum computers.
Abstract Imaginary time evolution is a powerful tool for studying quantum systems. While it is possible to simulate with a classical computer, the time and memory requirements generally scale exponentially with the system size. Conversely, quantum computers can efficiently simulate quantum systems, but not non-unitary imaginary time evolution. We propose a variational algorithm for simulating imaginary time evolution on a hybrid quantum computer. We use this algorithm to find the ground-state energy of many-particle systems; specifically molecular hydrogen and lithium hydride, finding the ground state with high probability. Our method can also be applied to general optimisation problems and quantum machine learning. As our algorithm is hybrid, suitable for error mitigation and can exploit shallow quantum circuits, it can be implemented with current quantum computers.
Being able to predict molecular properties and interactions is of utmost interest for academia as well as industry. But the vast complexity of strongly correlated molecular systems limits the performance of classical algorithms. In contrast, quantum computation has the potential to be a game changer in the field of molecular simulations. Despite the hope in quantum computation, the capabilities of current quantum computers are still insufficient for handling molecular systems of interest. In this paper, we propose a variational ansatz for today's noisy quantum computers to calculate the ground state with the help of imaginary time evolution. Although the imaginary time evolution operator is not unitary, it can be implemented on a quantum computer by a linear decomposition and subsequent Taylor series expansion. This has the advantage that only a set of shallow circuits needs to be computed on a quantum computer. The parallel nature of this algorithm can be exploited to speed-up simulations even further, if a privileged access to quantum computers is granted.
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.
Imaginary time evolution is a powerful tool applied in quantum physics, while existing classical algorithms for simulating imaginary time evolution suffer high computational complexity as the quantum systems become larger and more complex. In this work, we propose a probabilistic algorithm for implementing imaginary time evolution based on non-unitary quantum circuit. We demonstrate the feasibility of this method by solving the ground state energy of several quantum many-body systems, including H2, LiH molecules and the quantum Ising chain. Moreover, we perform experiments on superconducting and trapped ion cloud platforms respectively to find the ground state energy of H2 and its most stable molecular structure. We also analyze the successful probability of the algorithm, which is a polynomial of the output error and introduce an approach to increase the success probability by rearranging the terms of Hamiltonian.
Calculating the energy spectrum of a quantum system is an important task, for example to analyse reaction rates in drug discovery and catalysis. There has been significant progress in developing algorithms to calculate the ground state energy of molecules on near-term quantum computers. However, calculating excited state energies has attracted comparatively less attention, and it is currently unclear what the optimal method is. We introduce a low depth, variational quantum algorithm to sequentially calculate the excited states of general Hamiltonians. Incorporating a recently proposed technique, we employ the low depth swap test to energetically penalise the ground state, and transform excited states into ground states of modified Hamiltonians. We use variational imaginary time evolution as a subroutine, which deterministically propagates towards the target eigenstate. We discuss how symmetry measurements can mitigate errors in the swap test step. We numerically test our algorithm on Hamiltonians which encode 3SAT optimisation problems of up to 18 qubits, and the electronic structure of the Lithium Hydride molecule. As our algorithm uses only low depth circuits and variational algorithms, it is suitable for use on near-term quantum hardware.
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.
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.