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Quantum Science and Technology

Gate-based path integral quantum control applied to quantum chemistry

Peyman Najafi, Aarón Villanueva, Hilbert J. Kappen

Published Oct 13, 2026• 0 citations
Publisher

Abstract

Abstract The Path integral Quantum Control (PiQC) algorithm was recently introduced as a new approach for computing optimal controls in continuous-time open and closed quantum systems. In this work, we adapt the PiQC algorithm to optimize parametrized quantum circuits and refer to this version of PiQC as the Gate-Based PiQC (GB-PiQC) algorithm. We apply GB-PiQC and the original continuous-time version of PiQC for ground state preparation of electronic structure problems, and benchmark them against the Variational Quantum Eigensolver, optimized using the standard Simultaneous Perturbation Stochastic Approximation (SPSA) optimizer, on a set of standard benchmark molecular Hamiltonians. For a fair comparison, we optimize the hyperparameters of the algorithms for each molecule. We find that both PiQC algorithms exhibit greater robustness than SPSA to variations in the target Hamiltonian induced by changes in interatomic distance, when hyperparameters for both methods are fixed at values tuned near equilibrium. Furthermore, PiQC algorithms also achieve superior performance compared to SPSA in most instances under this fixed-hyperparameter setting, particularly at stretched interatomic distances, where the Hartree–Fock solution becomes less accurate and its error grows relative to equilibrium.

DOI
10.1088/2058-9565/aeb3b1

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Abstract The Path integral Quantum Control (PiQC) algorithm was recently introduced as a new approach for computing optimal controls in continuous-time open and closed quantum systems. In this work, we adapt the PiQC algorithm to optimize parametrized quantum circuits and refer to this version of PiQC as the Gate-Based PiQC (GB-PiQC) algorithm. We apply GB-PiQC and the original continuous-time version of PiQC for ground state preparation of electronic structure problems, and benchmark them against the Variational Quantum Eigensolver, optimized using the standard Simultaneous Perturbation Stochastic Approximation (SPSA) optimizer, on a set of standard benchmark molecular Hamiltonians. For a fair comparison, we optimize the hyperparameters of the algorithms for each molecule. We find that both PiQC algorithms exhibit greater robustness than SPSA to variations in the target Hamiltonian induced by changes in interatomic distance, when hyperparameters for both methods are fixed at values tuned near equilibrium. Furthermore, PiQC algorithms also achieve superior performance compared to SPSA in most instances under this fixed-hyperparameter setting, particularly at stretched interatomic distances, where the Hartree–Fock solution becomes less accurate and its error grows relative to equilibrium.
We investigate open-loop quantum state preparation for a class of open quantum systems whose dynamics follow a Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation that admits a trajectory-based stochastic representation. The deterministic control objective is reformulated as a stochastic optimal control problem -- interpreting stochasticity as a methodological tool akin to stochastic Schrödinger equation unravelings -- which situates the problem within the path integral control framework. For the class of GKLS generators under consideration, this reformulation leads to an explicit expression for the optimal control as a weighted average over stochastic quantum trajectories, thereby eliminating the need for gradient evaluations. Building on this theoretical result, we derive a control update rule for piecewise-constant control pulses and demonstrate that adaptive importance sampling progressively enhances the control estimator during optimization, culminating in the algorithm we term Path integral Quantum Control (PiQC). We further introduce an annealed variant of PiQC, wherein a synthetic noise schedule gradually steers open-system trajectories toward closed-system dynamics, enabling high-fidelity unitary state preparation. Numerical studies on a dissipative single-qubit system and a multi-qubit Nuclear Magnetic Resonance model verify that PiQC yields precise open-loop controls and displays robustness to Hamiltonian perturbations. We propose PiQC as a trajectory-based alternative to gradient-based approaches, which might offer a viable solution in quantum control problems where gradient computation is infeasible or computationally demanding.
The variational quantum eigensolver (VQE) is currently the flagship algorithm for solving electronic structure problems on near-term quantum computers. This hybrid quantum/classical algorithm involves implementing a sequence of parameterized gates on quantum hardware to generate a target quantum state, and then measuring the expectation value of the molecular Hamiltonian. Due to finite coherence times and frequent gate errors, the number of gates that can be implemented remains limited on current quantum devices, preventing accurate applications to systems with significant entanglement, such as strongly correlated molecules. In this work, we propose an alternative algorithm (which we refer to as ctrl-VQE) where the quantum circuit used for state preparation is removed entirely and replaced by a quantum control routine which variationally shapes a pulse to drive the initial Hartree-Fock state to the full CI target state. As with VQE, the objective function optimized is the expectation value of the qubit-mapped molecular Hamiltonian. However, by removing the quantum circuit, the coherence times required for state preparation can be drastically reduced by directly optimizing the pulses. We demonstrate the potential of this method numerically by directly optimizing pulse shapes which accurately model the dissociation curves of the hydrogen molecule (covalent bond) and helium hydride ion (ionic bond), and we compute the single point energy for LiH with four transmons.
This work studies pulse based variational quantum algorithms (VQAs), which are designed to determine the ground state of a quantum mechanical system by combining classical and quantum hardware. In contrast to more standard gate based methods, pulse based methods aim to directly optimize the laser pulses interacting with the qubits, instead of using some parametrized gate based circuit. Using the mathematical formalism of optimal control, these laser pulses are optimized. This method has been used in quantum computing to optimize pulses for quantum gate implementations, but has only recently been proposed for full optimization in VQAs. Pulse based methods have several advantages over gate based methods such as faster state preparation, simpler implementation and more freedom in moving through the state space. Based on these ideas, we present the development of a novel adjoint based variational method. This method can be tailored towards and applied in neutral atom quantum computers. This method of pulse based variational quantum optimal control is able to approximate molecular ground states of simple molecules up to chemical accuracy and is able to compete with the gate based variational quantum eigensolver in terms of total number of quantum evaluations. The total evolution time T and the form of the control Hamiltonian Hc are important factors in the convergence behavior to the ground state energy, both having influence on the quantum speed limit and the controllability of the system.
In this work we investigate methods to improve the efficiency and scalability of quantum algorithms for quantum chemistry applications. We propose a transformation of the electronic structure Hamiltonian in the second quantization framework into the particle-hole picture, which offers a better starting point for the expansion of the system wave function. The state of the molecular system at study is parametrized so as to constrain the sampling of the corresponding wave function within the sector of the molecular Fock space that contains the desired solution. To this end, we explore different mapping schemes to encode the molecular ground state wave function in a quantum register. Taking advantage of known post-Hartree-Fock quantum chemistry approaches and heuristic Hilbert space search quantum algorithms, we propose a new family of quantum circuits based on exchange-type gates that enable accurate calculations while keeping the gate count (i.e., the circuit depth) low. The particle-hole implementation of the unitary coupled-cluster (UCC) method within the variational quantum eigensolver approach gives rise to an efficient quantum algorithm, named q-UCC, with important advantages compared to the straightforward translation of the classical coupled-cluster counterpart. In particular, we show how a single Trotter step in the expansion of the system wave function can accurately and efficiently reproduce the ground-state energy of simple molecular systems.
In the noisy near-term quantum computing era variational quantum algorithms are promising to explore the boundaries of quantum advantage. One prevalent instance is the variational quantum eigensolver (VQE) employed in quantum chemistry to approximate ground-state energies of molecules. The performance of VQE depends on the design of the parametrized quantum circuit, as well as the optimization algorithm used to optimize those parameters. Circuit optimization methods have been developed to generate quantum circuits to reach accurate approximations to the ground-states. In this work, we conduct an empirical comparison between two prominent adaptive quantum circuit optimization algorithms, namely qubit-ADAPT-VQE and a reinforcement learning (RL) based method, on the problem of finding the ground state energy of Lithium Hydride (LiH). In our experiment, we focus on the circuit depth generated by each circuit optimization method and the number of two-qubit quantum gates in the best-found circuit.
Quantum computing promises to efficiently and accurately solve many important problems in quantum chemistry which elude classical solvers, such as the electronic structure problem of highly correlated materials. Two leading methods in solving the ground state problem are the Variational Quantum Eigensolver (VQE) and Adiabatic Quantum Computing (AQC) algorithms. VQE often struggles with convergence due to the energy landscape being highly non-convex and the existence of barren plateaux, and implementing AQC is beyond the capabilities of current quantum devices as it requires deep circuits. Adiabatically-inspired algorithms aim to fill this gap. In this paper, we first present a unifying framework for these algorithms and then benchmark the following methods: the Adiabatically Assisted VQE (AAVQE) (Garcia-Saez and Latorre (2018)), the Variational Adiabatic Quantum Computing (VAQC) (Harwood et al (2022)), and the Adiabatic Quantum Computing with Parametrized Quantum Circuits (AQC-PQC) (Kolotouros et al (2025)) algorithms. Second, we introduce a novel hybrid approach termed G-AQC-PQC, which generalizes the AQC-PQC method, and combines adiabatic-inspired initialization with the low-memory BFGS optimizer, reducing the quantum computational cost of the method. Third, we compare the accuracy of the methods for chemistry applications using the beryllium hydride molecule (BeH2_22​). We compare the approaches across a number of different choices (ansätze types, depth, discretization steps, initial Hamiltonian, adiabatic schedules and method used). Our results show that the G-AQC-PQC outperforms conventional VQE. We further discuss limitations such as the zero-gradient problem and identify regimes where adiabatically-inspired methods offer a tangible advantage for near-term quantum chemistry applications.