The computation of electronic excited states and real-time quantum dynamics of many-fermion systems is among the most promising applications of near-term quantum computing. In this work, we generalize the reinforcement learning contracted quantum eigensolver (RL-CQE), previously developed for ground-state problems, to electronic excited states and real-time quantum dynamics, in which a deep Q-network agent adaptively selects the two-body operators at each iteration, yielding more compact ansätze and improved robustness with respect to critical hyperparameters. A key feature of the algorithm is a scalable state representation based on the ACSE residuals, whose dimension grows with the one-particle basis but remains independent of the number of targeted excited states. We also verify the equivalence of sign-free qubit operators in the excited-state setting, extending a result previously established for ground-state problems. Our RL-CQE for time evolution derives from a constant-scaling ansatz that represents the wave function with a fixed number of unitary transformations independent of simulation time $t$, enabled by the shared unitary structure of the purified ensemble treatment of excited states. Benchmarks on chemical systems demonstrate chemical accuracy with minimal operator counts across a range of bond lengths.
The computation of electronic excited states and real-time quantum dynamics of many-fermion systems is among the most promising applications of near-term quantum computing. In this work, we generalize the reinforcement learning contracted quantum eigensolver (RL-CQE), previously developed for ground-state problems, to electronic excited states and real-time quantum dynamics, in which a deep Q-network agent adaptively selects the two-body operators at each iteration, yielding more compact ansätze and improved robustness with respect to critical hyperparameters. A key feature of the algorithm is a scalable state representation based on the ACSE residuals, whose dimension grows with the one-particle basis but remains independent of the number of targeted excited states. We also verify the equivalence of sign-free qubit operators in the excited-state setting, extending a result previously established for ground-state problems. Our RL-CQE for time evolution derives from a constant-scaling ansatz that represents the wave function with a fixed number of unitary transformations independent of simulation time t, enabled by the shared unitary structure of the purified ensemble treatment of excited states. Benchmarks on chemical systems demonstrate chemical accuracy with minimal operator counts across a range of bond lengths.
Solving for the many-body wavefunction represents a significant challenge on both classical and quantum devices because of the exponential scaling of the Hilbert space with system size. While the complexity of the wavefunction can be reduced through conventional ansätze (e.g., the coupled cluster ansatz), it can still grow rapidly with system size even on quantum devices. An exact, universal two-body exponential ansatz for the many-body wavefunction has been shown to be generated from the solution of the contracted Schrödinger equation (CSE), and recently, this ansatz has been implemented without classical approximation on quantum simulators and devices for the scalable simulation of many-body quantum systems. Here we combine the solution of the CSE with a form of artificial intelligence known as reinforcement learning (RL) to generate highly compact circuits that implement this ansatz without sacrificing accuracy. As a natural extension of CSE, we reformulate the wavefunction update as a Markovian decision process and train the agent to select the optimal actions at each iteration based upon only the current CSE residual. Compact circuits with high accuracy are achieved for H3 and H4 molecules over a range of molecular geometries.
Abstract Computing excited-state properties of molecules and solids is considered one of the most important near-term applications of quantum computers. While many of the current excited-state quantum algorithms differ in circuit architecture, specific exploitation of quantum advantage, or result quality, one common feature is their rooting in the Schrödinger equation. However, through contracting (or projecting) the eigenvalue equation, more efficient strategies can be designed for near-term quantum devices. Here we demonstrate that when combined with the Rayleigh–Ritz variational principle for mixed quantum states, the ground-state contracted quantum eigensolver (CQE) can be generalized to compute any number of quantum eigenstates simultaneously. We introduce two excited-state (anti-Hermitian) CQEs that perform the excited-state calculation while inheriting many of the remarkable features of the original ground-state version of the algorithm, such as its scalability. To showcase our approach, we study several model and chemical Hamiltonians and investigate the performance of different implementations.
Calculating ground and excited states is an exciting prospect for near-term quantum computing applications, and accurate and efficient algorithms are needed to assess viable directions. We develop an excited-state approach based on the contracted quantum eigensolver (ES-CQE), which iteratively attempts to find a solution to a contraction of the Schrödinger equation projected onto a subspace and does not require a priori information on the system. We focus on the anti-Hermitian portion of the equation, leading to a two-body unitary ansatz. We investigate the role of symmetries, initial states, constraints, and overall performance within the context of the model strongly correlated rectangular H 4 system. We show that the ES-CQE achieves near-exact accuracy across the majority of states, covering regions of strong and weak electron correlation, while also elucidating challenging instances for two-body unitary ansatz.
Abstract Quantum many-body control is a central milestone en route to harnessing quantum technologies. However, the exponential growth of the Hilbert space dimension with the number of qubits makes it challenging to classically simulate quantum many-body systems and, consequently, to devise reliable and robust optimal control protocols. Here we present a framework for efficiently controlling quantum many-body systems based on reinforcement learning (RL). We tackle the quantum-control problem by leveraging matrix product states (1) for representing the many-body state and (2) as part of the trainable machine learning architecture for our RL agent. The framework is applied to prepare ground states of the quantum Ising chain, including states in the critical region. It allows us to control systems far larger than neural-network-only architectures permit, while retaining the advantages of deep learning algorithms, such as generalizability and trainable robustness to noise. In particular, we demonstrate that RL agents are capable of finding universal controls, of learning how to optimally steer previously unseen many-body states and of adapting control protocols on the fly when the quantum dynamics is subject to stochastic perturbations. Furthermore, we map our RL framework to a hybrid quantum–classical algorithm that can be performed on noisy intermediate-scale quantum devices and test it under the presence of experimentally relevant sources of noise.
A contracted quantum eigensolver (CQE) finds a solution to the many-electron Schrödinger equation by solving its integration (or contraction) to the two-electron space─a contracted Schrödinger equation (CSE)─on a quantum computer. When applied to the anti-Hermitian part of the CSE (ACSE), the CQE iterations optimize the wave function, with respect to a general product ansatz of two-body exponential unitary transformations that can exactly solve the Schrödinger equation. In this work, we accelerate the convergence of the CQE and its wave function ansatz via tools from classical optimization theory. By treating the CQE algorithm as an optimization in a local parameter space, we can apply quasi-second-order optimization techniques, such as quasi-Newton approaches or nonlinear conjugate gradient approaches. Practically, these algorithms result in superlinear convergence of the wave function to a solution of the ACSE. Convergence acceleration is important because it can both minimize the accumulation of noise on near-term intermediate-scale quantum (NISQ) computers and achieve highly accurate solutions on future fault-tolerant quantum devices. We demonstrate the algorithm, as well as some heuristic implementations relevant for cost-reduction considerations, comparisons with other common methods such as variational quantum eigensolvers, and a Fermionic-encoding-free form of the CQE.
The field of quantum chemistry has long been defined by the central computational challenge of the quantum many-body fermionic system problem, main the electronic Schrödinger equation. This exponential scaling makes it computationally intractable to solve, the exact governing electronic Schrödinger equation. Approximation methods have been developed, ranging from the widely used Density Functional Theory (DFT) to Coupled Cluster (CC) theory. While powerful, these conventional approaches often involve controlled truncations which caused a trade-off between accuracy, leaving many systems—particularly those with strong electron correlation—beyond the reach of ab initio simulation. Furthermore, the famous fermionic sign oscillation problem is particularly severe in stochastic methods like Quantum Monte Carlo (QMC). 1 Traditional Ansätze are also frequently engineered specifically for particular systems, which results in a fragmentation of results and fundamentally limits the transferability. Recent research demonstrates that with carefully designed architectures, ANNs can not only represent ground states of model systems with high precision but also tackle the sign problem, offering a promising path forward for strongly correlated electronic structure calculations.2,3 This review charts the paradigm shift driven by Neural Network Quantum States (NNQS), which leverage the representational power of deep learning to overcome these barriers. We detail the architectural evolution from Restricted Boltzmann Machines to autoregressive models like RNNs and Transformers, which enable exact, uncorrelated sampling and bypass critical bottlenecks.2,4,5Furthermore, we explore the geometric foundation of high-efficiency optimization via the neural networks, which decouples optimization cost from model complexity.6 The frontier of the field is marked by the integration of operator learning and hybrid quantum-classical frameworks, such as Reinforcement Learning for contracted quantum eigen-solvers and Quantum-Enhanced Neural Networks, which leverage quantum processors to enhance expressivity for hardware-efficient Ansätze.7,8 By framing the problem through the lens of computational resource allocation—specifically, trading exponential memory demands for polynomialcomplexity optimization and sampling—this review elucidates how recent advances at the intersection of artificial neural networks and quantum computing have evolved. The emergence of those methods not merely progress into a powerful simulation-tools, but also a hybrid computational interface.9 This convergence offers a versatile and systematically improvable framework for potentially addressing some of the most challenging problems in quantum physics and chemistry.