Trotter and linear combination of unitary (LCU) operations are two popular Hamiltonian simulation methods. The Trotter method is easy to implement and enjoys good system-size dependence endowed by commutator scaling, while the LCU method admits high-accuracy simulation with a smaller gate cost. We propose Hamiltonian simulation algorithms using LCU to compensate Trotter error, which enjoy both of their advantages. By adding few gates after the K th -order Trotter formula, we realize a better time scaling than 2 K th -order Trotter. Our first algorithm exponentially improves the accuracy scaling of the K th -order Trotter formula. For a generic Hamiltonian, the estimated gate counts of the first algorithm can be 2 orders of magnitude smaller than the best analytical bound of fourth-order Trotter formula. In the second algorithm, we consider the detailed structure of Hamiltonians and construct LCU for Trotter errors with commutator scaling. Consequently, for lattice Hamiltonians, the algorithm enjoys almost linear system-size dependence and quadratically improves the accuracy of the K th -order Trotter. For the lattice system, the second algorithm can achieve 3 to 4 orders of magnitude higher accuracy with the same gate costs as the optimal Trotter algorithm. These algorithms provide an easy-to-implement approach to achieve a low-cost and high-precision Hamiltonian simulation.
Trotter and linear combination of unitary (LCU) operations are two popular Hamiltonian simulation methods. The Trotter method is easy to implement and enjoys good system-size dependence endowed by commutator scaling, while the LCU method admits high-accuracy simulation with a smaller gate cost. We propose Hamiltonian simulation algorithms using LCU to compensate Trotter error, which enjoy both of their advantages. By adding few gates after the K th -order Trotter formula, we realize a better time scaling than 2 K th -order Trotter. Our first algorithm exponentially improves the accuracy scaling of the K th -order Trotter formula. For a generic Hamiltonian, the estimated gate counts of the first algorithm can be 2 orders of magnitude smaller than the best analytical bound of fourth-order Trotter formula. In the second algorithm, we consider the detailed structure of Hamiltonians and construct LCU for Trotter errors with commutator scaling. Consequently, for lattice Hamiltonians, the algorithm enjoys almost linear system-size dependence and quadratically improves the accuracy of the K th -order Trotter. For the lattice system, the second algorithm can achieve 3 to 4 orders of magnitude higher accuracy with the same gate costs as the optimal Trotter algorithm. These algorithms provide an easy-to-implement approach to achieve a low-cost and high-precision Hamiltonian simulation.
Product formulas are among the most practical approaches to Hamiltonian simulation, requiring no ancillary qubits and exhibiting error bounds governed by nested commutators rather than only by Hamiltonian norms. Their circuit size, however, scales polynomially with the inverse precision. We develop a high-order nested-commutator compensation (HNCC) algorithm that preserves the main advantages of product formulas while achieving polylogarithmic precision dependence in the circuit size and the standard O(ε−2) sampling cost. HNCC uses a truncated Baker--Campbell--Hausdorff expansion to represent high-order Trotter errors by products of nested commutators and compensates these errors at the channel level through randomly sampled Pauli-rotation channels, avoiding Hadamard tests and ancillary qubits. For a fixed K-th order product formula applied to a k-local Hamiltonian on N qubits with Γ Pauli terms and local interaction strength g0, HNCC estimates tr[Oe−itHρeitH] to additive precision ε∥O∥ using O(ε−2) repetitions. Its maximum gate count per circuit is O(kN2K+11Γ1−2K+11max{Γ,Nlog(1/ε)}2K+11(kg0tlog(1/ε))1+2K+11). Finite-size resource estimates for the periodic Heisenberg chain indicate that HNCC has the lowest estimated T-gate count per circuit among the product-formula-based methods considered.
Trotter product formulas are a leading approach for Hamiltonian simulation on quantum computers, yet their practical performance has remained difficult to assess due to the challenge of accurately estimating the Trotter error. In this work, we develop new theoretical results, algorithms, and software tools that advance the state-of-the-art in Trotter error estimation by orders of magnitude in both scale and accuracy. On the theoretical side, we prove that in the asymptotic limit the error of a product formula depends on the diagonal elements of the Baker-Campbell-Hausdorff (BCH) error operator in the eigenbasis of the Hamiltonian, rather than its full spectral norm -- yielding an improved scaling for Hamiltonian simulation using product formulas. On the algorithmic side, we introduce a compact representation of the BCH expansion that reduces the number of commutators from O(n3) to O(n) for second-order, and from O(n5) to O(n2) for fourth-order formulas on n fragments, complemented by an importance sampling scheme to further reduce the computational cost. We provide implementations of these techniques in software and demonstrate their power on two applications: (i) X-ray absorption spectroscopy of an electronic Hamiltonian (Li4Mn2O) at up to 56 qubits using tensor networks; and (ii) vibronic dynamics of naphthalene at over 100 qubits using ML-MCTDH, where we find that naive analytical bounds overestimate the required number of Trotter steps by nearly five orders of magnitude. Our framework enables, for the first time, the accurate estimation of Trotter error at practically relevant system sizes, providing a foundation for fair algorithmic comparisons and rational design of product formulas.
Owing to their simplicity and low overhead, Suzuki-Trotter formulas remain the de facto Hamiltonian simulation methods on current quantum computing platforms. Systematic Trotter errors, however, will quickly become limiting when scaling to larger problems and aiming for higher accuracy. We present a mechanism that removes the systematic error of any k-th order Suzuki-Trotter simulation, at the cost of a constant sampling overhead. The key insight is that the Trotter error is itself a coherent dynamics to be reversed, rather than a deviation to be bounded. By identifying the structure of this error in closed form, we carry out that reversal through quasi-probabilistic decompositions. The resulting algorithm, called Probabilistic Trotter Error Reversal (PTER), is unbiased and still improves the gate-count scaling of Suzuki-Trotter formulas while retaining their simplicity. Numerical simulations of a Heisenberg spin chain support the predicted resource advantage already at modest system sizes.
Product formulas are a basic tool for simulating quantum many-body dynamics, but a fundamental degree of freedom in their application---the ordering of Hamiltonian terms---has remained largely unexplored. Here we systematically study such ordering effects, quantifying their impact on simulation accuracy and exploiting them to design more accurate product-formula simulations. We develop an efficiently computable, ordering-sensitive bound on the finite-step-size Trotter error for geometrically local Hamiltonians for arbitrarily high-order product formulas. For translation-invariant one-dimensional chains, our bounds are close to exact second-, fourth-, and sixth-order Trotter errors in small-system tests, while on large-scale systems they improve on previous higher-order analytical bounds by orders of magnitude. We further use the local structure of the error bounds and estimates to optimize ordering efficiently, reducing the long-chain problem to a minimum-mean-weight-cycle problem on a weighted de Bruijn graph. For random translation-invariant chains, sequential (staircase-like) orderings consistently outperform standard brickwall circuits; when circuit depth is restricted, the optimizer instead favors wave-like parallel-sequential orderings, which preserve much of the accuracy advantage over brickwall circuits. These results establish ordering as a useful degree of freedom for both understanding and improving product-formula simulation.
A multi-product formula (MPF) is a promising approach for Hamiltonian simulation efficiently both in the system size N and the inverse allowable error 1 / ε by combining Trotterization and the linear combination of unitaries (LCU). It achieves poly-logarithmic cost in 1 / ε like LCU [G. H. Low, V. Kliuchnikov, N. Wiebe, (2019)]. The efficiency in N is expected to come from the commutator scaling in Trotterization, and this appears to be confirmed by the error bound of MPF expressed by nested commutators [J. Aftab, D. An, K. Trivisa, (2024)]. However, we point out that the efficiency of MPF in the system size N is not exactly resolved yet in that the present error bound expressed by nested commutators is incompatible with the size-efficient complexity reflecting the commutator scaling. The problem is that q -fold nested commutators with arbitrarily large q are involved in their requirement and error bound. The benefit of commutator scaling by locality is absent, and the cost efficient in N becomes prohibited in general. In this paper, we show an alternative commutator-scaling error of MPF and derive its size-efficient cost properly inheriting the advantage in Trotterization. The requirement and the error bound in our analysis, derived by techniques from the Floquet-Magnus expansion, have a certain truncation order in the nested commutators and can fully exploit the locality. We prove that Hamiltonian simulation by MPF certainly achieves the cost whose system-size dependence is as large as Trotterization while keeping the p o l y l o g ( 1 / ε ) -scaling like the LCU. Our results will provide improved or accurate error and cost also for various algorithms using interpolation or extrapolation of Trotterization.
Trotterization is the most common and convenient approximation method for Hamiltonian simulations on digital quantum computers, but estimating its error accurately is computationally difficult for large quantum systems. Here, we develop a method for measuring the Trotter error without ancillary qubits on quantum circuits by combining the m th - and n th -order ( m < n ) Trotterizations rather than consulting with mathematical error bounds. Using this method, we make Trotterization precision guaranteed, developing an algorithm named Trotter ( m , n ) , in which the Trotter error at each time step is within an error tolerance ε preset for our purpose. Trotter ( m , n ) is applicable to both time-independent and -dependent Hamiltonians, and it adaptively chooses almost the largest step size δ t , which keeps quantum circuits shallowest, within the error tolerance. Benchmarking it in a quantum spin chain, we find the adaptively chosen δ t to be about 10 times larger than that inferred from known upper bounds of Trotter errors. Published by the American Physical Society 2024