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arXiv:2412.08719 · Physical Review A

Short-time simulation of quantum dynamics by Pauli measurements

Paul K. Faehrmann, Jens Eisert, Mária Kieferová, Richard Kueng

Published Jun 12, 2025• 4 citations
arXivHTMLPDF

Abstract

Simulating the dynamics of complex quantum systems is a central application of quantum devices. Here we propose leveraging the power of measurements to simulate short-time quantum dynamics of physically prepared quantum states in classical postprocessing using a truncated Taylor series approach. While limited to short simulation times, our hybrid quantum-classical method is equipped with rigorous error bounds. It is extendable to estimate low-order Taylor approximations of smooth time-dependent functions of tractable linear combinations of measurable operators. These insights can be made use of in the context of Hamiltonian learning and device verification, short-time imaginary-time evolution, or the application of intractable operations to subuniversal quantum simulators in classical postprocessing.

DOI
10.1103/6ggs-y392

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Simulating the dynamics of complex quantum systems is a central application of quantum devices. Here we propose leveraging the power of measurements to simulate short-time quantum dynamics of physically prepared quantum states in classical postprocessing using a truncated Taylor series approach. While limited to short simulation times, our hybrid quantum-classical method is equipped with rigorous error bounds. It is extendable to estimate low-order Taylor approximations of smooth time-dependent functions of tractable linear combinations of measurable operators. These insights can be made use of in the context of Hamiltonian learning and device verification, short-time imaginary-time evolution, or the application of intractable operations to subuniversal quantum simulators in classical postprocessing.
We describe a simple, efficient method for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator. Our method can simulate the time evolution of a wide variety of physical systems. As in another recent algorithm, the cost of our method depends only logarithmically on the inverse of the desired precision, which is optimal. However, we simplify the algorithm and its analysis by using a method for implementing linear combinations of unitary operations together with a robust form of oblivious amplitude amplification.
We propose a hybrid approach to simulate quantum many-body dynamics by combining Trotter-based quantum evolution with classical dynamic mode decomposition. In quantum simulation, the interest often lies in estimating observables rather than explicitly obtaining the wave function. Instead of characterizing a quantum state, our method predicts observables of the quantum state at later times by using data from a set of short-time measurements from a quantum computer. For a fixed set of measurements, the global error bound for our method scales at most as O(t3/2)\mathcal{O}(t^{3/2})O(t3/2), with constants determined by the DMD approximation error and the conditioning of the learned modes. We apply our method to quench dynamics in the Hubbard model and nearest neighbor spin systems and show that the observable properties can be predicted up to a reasonable error by controlling the number of data points obtained from the quantum measurements.
The efficient simulation of quantum dynamics and ground states is a central challenge in physics and a key frontier for quantum advantage. While short-time evolution in one-dimensional systems can often be simulated classically, extending this to higher dimensions remains difficult. Here, we introduce an efficient classical algorithm for simulating the short-time dynamics of arbitrary local quantum systems. For any local Hamiltonian HHH and constant evolution time ttt, our method estimates expectation values of the form ⟨ψ∣eiHtOe−iHt∣ψ⟩\langleψ|e^{iHt}Oe^{-iHt}|ψ\rangle⟨ψ∣eiHtOe−iHt∣ψ⟩ for global Pauli observables OOO and stabilizer states ∣ψ⟩|ψ\rangle∣ψ⟩, with high precision and exponentially small failure probability. Furthermore, we present a classical dequantization of a tailored quantum algorithm that efficiently solves the guided local Hamiltonian (GLH) problem to constant additive error - previously considered classically hard and hence a promising candidate for quantum computational advantage. These results reveal unexpected classical tractability in constant-time quantum dynamics and fundamental connections between Hamiltonian dynamics mean value and the GLH problem. Our work refines the boundary between classical and quantum computational power, identifying sharper criteria for regimes where quantum advantage may genuinely emerge.
As established in the seminal work by Berry et al.[1], expanding the time evolution operator using truncated Taylor series (up to some order KKK) makes a good candidate for simulating Hamiltonian dynamics. Here, we adapt the method but present an alternative quantum circuit that maintains an equivalent asymptotic elementary gate cost but has an exponentially reduced number of ancilla qubits. This is realized by utilizing mid-circuit measurements (with early abort-and-restart of circuit execution), and transforming a series of multi-controlled(Hk)(H^k)(Hk) to a series of singly-controlled(Hk′)(H^{k'})(Hk′), where HHH is a linear combination of unitaries and k,k′k, k'k,k′ are integers. The proposed circuit utilizes a total of ⌈log⁡(K)⌉+⌈log⁡(L)⌉+n\lceil \log(K) \rceil + \lceil \log(L) \rceil +n⌈log(K)⌉+⌈log(L)⌉+n qubits, where LLL is the number of terms in the Hamiltonian and nnn is the system qubit size. Our shorter width circuit with mid-measurements protocol is implemented and evaluated using the programming language Guppy[2,3].
Recent progress in the development of quantum technologies has enabled the direct investigation of the dynamics of increasingly complex quantum many-body systems.This motivates the study of the complexity of classical algorithms for this problem in order to benchmark quantum simulators and to delineate the regime of quantum advantage.Here, we present classical algorithms for approximating the dynamics of local observables and nonlocal quantities such as the Loschmidt echo, where the evolution is governed by a local Hamiltonian.For short times, their computational cost scales polynomially with the system size and the inverse of the approximation error.In the case of local observables, the proposed algorithm has a better dependence on the approximation error than algorithms based on the Lieb-Robinson bound.Our results use cluster-expansion techniques adapted to the dynamical setting, for which we give a novel proof of their convergence.This has important physical consequences besides our efficient algorithms.In particular, we establish a novel quantum speed limit, a bound on dynamical phase transitions, and a concentration bound for product states evolved for short times.
Entanglement is not only the origin of exotic quantum phenomena, but also widely regarded as the fundamental barrier to classical simulation of quantum dynamics. We overturn this intuition: for predicting local observables under Hamiltonian evolution, entanglement in the quantum state actually heals the classical simulation error in the Heisenberg picture. Classical algorithms that propagate observables in the Pauli basis with proper truncation have shown remarkable empirical success in simulating noiseless quantum dynamics, yet all prior rigorous guarantees required noise or randomness---leaving the physically relevant noiseless regime without theoretical foundation. To close this gap, we prove that the truncation error of Low-weight Pauli Dynamics (LPD) admits an average-case bound without assuming randomness, provided the state is sufficiently entangled. Since tensor-network methods efficiently simulate low-entanglement states while LPD thrives precisely in the complementary regime, together they extend rigorous classical simulation to longer times, sharpening the boundary between classically simulable and genuinely quantum dynamics.