Simulation of continuous-time evolution requires time discretization on both classical and quantum computers.A finer time step improves simulation precision but it inevitably leads to increased computational efforts.This is particularly costly for today's noisy intermediate-scale quantum computers, where notable gate imperfections limit the circuit depth that can be executed at a given accuracy.Classical adaptive solvers are well developed to save numerical computation times.However, it remains an outstanding challenge to make optimal usage of the available quantum resources by means of adaptive time steps.Here, we introduce a quantum algorithm to solve this problem, providing a controlled solution of the quantum many-body dynamics of local observables.The key conceptual element of our algorithm is a feedback loop that self-corrects the simulation errors by adapting time steps, thereby significantly outperforming conventional Trotter schemes on a fundamental level and reducing the circuit depth.It even allows for a controlled asymptotic long-time error, where the usual Trotterized dynamics faces difficulties.Another key advantage of our quantum algorithm is that any desired conservation law can be included in the self-correcting feedback loop, which has a potentially wide range of applicability.We demonstrate the capabilities by enforcing gauge invariance, which is crucial for a faithful and long-sought-after quantum simulation of lattice gauge theories.Our algorithm can potentially be useful on a more general level whenever time discretization is involved also concerning, e.g., numerical approaches based on time-evolving block-decimation methods.
Simulation of continuous-time evolution requires time discretization on both classical and quantum computers.A finer time step improves simulation precision but it inevitably leads to increased computational efforts.This is particularly costly for today's noisy intermediate-scale quantum computers, where notable gate imperfections limit the circuit depth that can be executed at a given accuracy.Classical adaptive solvers are well developed to save numerical computation times.However, it remains an outstanding challenge to make optimal usage of the available quantum resources by means of adaptive time steps.Here, we introduce a quantum algorithm to solve this problem, providing a controlled solution of the quantum many-body dynamics of local observables.The key conceptual element of our algorithm is a feedback loop that self-corrects the simulation errors by adapting time steps, thereby significantly outperforming conventional Trotter schemes on a fundamental level and reducing the circuit depth.It even allows for a controlled asymptotic long-time error, where the usual Trotterized dynamics faces difficulties.Another key advantage of our quantum algorithm is that any desired conservation law can be included in the self-correcting feedback loop, which has a potentially wide range of applicability.We demonstrate the capabilities by enforcing gauge invariance, which is crucial for a faithful and long-sought-after quantum simulation of lattice gauge theories.Our algorithm can potentially be useful on a more general level whenever time discretization is involved also concerning, e.g., numerical approaches based on time-evolving block-decimation methods.
Digital quantum simulation relies on Trotterization to discretize time evolution into elementary quantum gates. On current quantum processors with notable gate imperfections, there is a critical trade-off between improved accuracy for finer time steps, and increased error rate on account of the larger circuit depth. We present an adaptive Trotterization algorithm to cope with time dependent Hamiltonians, where we propose a concept of piecewise "conserved" quantities to estimate errors in the time evolution between two (nearby) points in time; these allow us to bound the errors accumulated over the full simulation period. They reduce to standard conservation laws in the case of time independent Hamiltonians, for which we first developed an adaptive Trotterization scheme [H. Zhao et al., Making Trotterization adaptive and energy-self-correcting for NISQ devices and beyond, PRX Quantum 4, 030319 (2023).2691-339910.1103/PRXQuantum.4.030319]. We validate the algorithm for a time dependent quantum spin chain, demonstrating that it can outperform the conventional Trotter algorithm with a fixed step size at a controlled error.
Accurate digital quantum simulation at long times is limited by the accumulation of errors inherent to approximate simulation. Here we introduce RL-Trotter, a reinforcement-learning framework that treats unavoidable approximation errors as resources for error correction rather than merely imperfections to suppress. We show that low-dimensional information from conservation laws, such as the energy and energy variance, provides a sufficient learning signal to guide the agent, which learns to adapt a single scalar---the next Trotter step size---without access to the target wave function. By optimizing the entire long-time evolution rather than individual steps, RL-Trotter discovers self-correcting sequences in which later errors compensate for those accumulated earlier, increasing the accuracy of the long-time dynamics. The learned policies are intrinsically robust to measurement noise, substantially reducing measurement overhead. They also generalize to previously unseen, physically similar initial states and transfer from small, classically simulable systems to systems an order of magnitude larger. This enables a practical protocol based on classical pretraining followed by direct deployment or limited fine-tuning on quantum hardware. Our results establish a broader perspective for quantum algorithms: errors in approximate evolution can be orchestrated into resources for accurate and resource-efficient quantum dynamics.
Noisy Intermediate-Scale Quantum (NISQ) devices are promising platforms for practical quantum computing, but their performance is significantly affected by limited qubit availability, gate errors, decoherence, circuit depth, and hardware-dependent noise. Conventional quantum circuits are generally designed and optimized before execution and may not respond effectively to changing hardware conditions during computation. This research proposes a Self-Adaptive Quantum Circuit framework that dynamically modifies quantum circuits according to the current computational resources and noise characteristics of NISQ devices. The proposed framework monitors parameters such as qubit fidelity, gate error rates, circuit depth, execution time, and resource availability, and automatically selects suitable circuit configurations to improve computational performance. Adaptive mechanisms can be employed to reduce unnecessary gates, optimize qubit utilization, and mitigate the impact of hardware noise while maintaining the required computational accuracy. The framework will be evaluated using simulation platforms and available NISQ quantum hardware, comparing adaptive circuits with conventional static circuits using metrics such as accuracy, fidelity, circuit depth, gate count, execution time, and error rate. The expected outcome is a resource-efficient and noise-aware quantum circuit optimization approach capable of improving the reliability and performance of quantum applications on near-term quantum hardware. The proposed research aims to contribute toward more practical, flexible, and hardware-aware quantum computing systems for NISQ-era applications. Keywords: NISQ Computing, Self-Adaptive Quantum Circuits, Quantum Circuit Optimization, Quantum Error Mitigation, Quantum Noise, Quantum Computing.
Understanding the dynamics of quantum systems is crucial in many areas of physics, but simulating many-body systems presents significant challenges due to the large Hilbert space to navigate and the exponential growth of computational overhead. Quantum computers offer a promising platform to overcome these challenges, particularly for simulating the time evolution with Hamiltonians. Trotterization is a widely used approach among available algorithms in this regard, and well suited for near-term quantum devices. However, it introduces algorithmic Trotter errors due to the non-commutativity of Hamiltonian components. Several techniques such as multi-product formulas have been developed to mitigate Trotter errors, but often require deep quantum circuits, which can introduce additional physical errors. In this work, we propose a resource-efficient scheme to reduce the algorithmic Trotter error with relatively shallow circuit depth. We develop a profiling method by introducing an auxiliary parameter to estimate the error effects in expectation values, enabling significant error suppression with a fixed number of Trotter steps. Our approach offers an efficient way of quantum simulation on near-term quantum processors with shallow circuits.
We propose a variational alternative to the Trotter-Suzuki decomposition that provides greater control over errors while preserving the unitary structure of time evolution. The variational parameters in our ansatz are derived from a global action principle, where Euler-Lagrange equations govern their optimal dynamics. Unlike conventional wavefunction-based variational methods, our approach specifically targets the time evolution operation and this allows a single set of optimized parameters to be applied to any initial state for a fixed Hamiltonian avoiding costly optimization procedures. Our method outperforms the standard Trotter-Suzuki formulas, typically achieving higher accuracy than higher-order Suzuki schemes. This translates directly to quantum computing applications, where it enables the design of quantum circuits with fewer gates which reduces noise and improves precision. Although we focus on quantum dynamics, the method is broadly applicable to problems involving general time-evolution operators. Applied to various model Hamiltonians, our approach reduces errors by factors of 2 to 5 compared to Trotter-Suzuki decompositions, demonstrating its promise for accurate quantum simulation with improved efficiency. In certain cases, the variational ansatz achieves higher accuracy than more complex higher-order Suzuki formulas while reducing the gate count by nearly half within a single circuit layer. Furthermore, we derive approximate analytical expressions for the variational parameters up to cubic order in time, valid for generic Hamiltonians. These approximations enable long-time quantum simulations with improved accuracy over equivalent Suzuki decompositions, providing ready-to-use evolution formulas that match Suzuki's gate complexity while delivering better performance.
Various Hamiltonian simulation algorithms have been proposed to efficiently study the dynamics of quantum systems on a quantum computer. The existing algorithms generally approximate the time evolution operators, which may need a deep quantum circuit that is beyond the capability of near-term noisy quantum devices. Here, focusing on the time evolution of a fixed input quantum state, we propose an adaptive approach to construct a low-depth time evolution circuit. By introducing a measurable quantifier that characterizes the simulation error, we use an adaptive strategy to learn the shallow quantum circuit that minimizes that error. We numerically test the adaptive method with electronic Hamiltonians of the H_{2}O and H_{4} molecules, and the transverse field Ising model with random coefficients. Compared to the first-order Suzuki-Trotter product formula, our method can significantly reduce the circuit depth (specifically the number of two-qubit gates) by around two orders while maintaining the simulation accuracy. We show applications of the method in simulating many-body dynamics and solving energy spectra with the quantum Krylov algorithm. Our work sheds light on practical Hamiltonian simulation with noisy-intermediate-scale-quantum devices.