In analog and digital simulations of practically relevant quantum systems, the target dynamics can only be implemented approximately. The Trotter product formula is the most common approximation scheme as it is a generic method which allows tuning accuracy. The Trotter simulation precision will always be inexact for noncommuting operators, but it is currently unknown what the minimum possible error is. This is an important quantity because upper bounds for the Trotter error are known to often be vast overestimates. Here we present explicit lower bounds on the error, in norm and on states, allowing to derive minimum resource requirements. Numerical comparison with the true error shows that our bounds offer accurate and tight estimates.
In analog and digital simulations of practically relevant quantum systems, the target dynamics can only be implemented approximately. The Trotter product formula is the most common approximation scheme as it is a generic method which allows tuning accuracy. The Trotter simulation precision will always be inexact for noncommuting operators, but it is currently unknown what the minimum possible error is. This is an important quantity because upper bounds for the Trotter error are known to often be vast overestimates. Here we present explicit lower bounds on the error, in norm and on states, allowing to derive minimum resource requirements. Numerical comparison with the true error shows that our bounds offer accurate and tight estimates.
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.
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
We establish tight connections between entanglement entropy and the approximation error in Trotter-Suzuki product formulas for Hamiltonian simulation. Product formulas remain the workhorse of quantum simulation on near-term devices, yet standard error analyses yield worst-case bounds that can vastly overestimate the resources required for structured problems. For systems governed by geometrically local Hamiltonians with maximum entanglement entropy Smax across all bipartitions, we prove that the first-order Trotter error scales as O(t2Smaxpolylog(n)/r) rather than the worst-case O(t2n/r), where n is the system size and r is the number of Trotter steps. This yields improvements of Ω~(n2) for one-dimensional area-law systems and Ω~(n3/2) for two-dimensional systems. We extend these bounds to higher-order Suzuki formulas, where the improvement factor involves 2pS∗/2 for the p-th order formula. We further establish a separation result demonstrating that volume-law entangled systems fundamentally require Ω~(n) more Trotter steps than area-law systems to achieve the same precision. This separation is tight up to logarithmic factors. Our analysis combines Lieb-Robinson bounds for locality, tensor network representations for entanglement structure, and novel commutator-entropy inequalities that bound the expectation value of nested commutators by the Schmidt rank of the state. These results have immediate applications to quantum chemistry, condensed matter simulation, and resource estimation for fault-tolerant quantum computing.
Efficient error estimates for the Trotter product formula are central in quantum computing, mathematical physics, and numerical simulations. However, the Trotter error's dependency on the input state and its application to unbounded operators remains unclear. Here, we present a general theory for error estimation, including higher-order product formulas, with explicit input state dependency. Our approach overcomes two limitations of the existing operator-norm estimates in the literature. First, previous bounds are too pessimistic as they quantify the worst-case scenario. Second, previous bounds become trivial for unbounded operators and cannot be applied to a wide class of Trotter scenarios, including atomic and molecular Hamiltonians. Our method enables analytical treatment of Trotter errors in chemistry simulations, illustrated through a case study on the hydrogen atom. Our findings reveal the following: (i) for states with fat-tailed energy distribution, such as low-angular-momentum states of the hydrogen atom, the Trotter error scales worse than expected (sublinearly) in the number of Trotter steps; (ii) certain states do not admit an advantage in the scaling from higher-order Trotterization and, thus, the higher-order Trotter hierarchy breaks down for these states, including the hydrogen atom's ground state; (iii) the scaling of higher-order Trotter bounds might depend on the order of the Hamiltonians in the Trotter product for states with fat-tailed energy distribution. Physically, the enlarged Trotter error is caused by the atom's ionization due to the Trotter dynamics. Mathematically, we find that certain domain conditions are not satisfied by some states so higher moments of the potential and kinetic energies diverge. Our analytical error analysis agrees with numerical simulations, indicating that we can estimate the state-dependent Trotter error scaling genuinely. Published by the American Physical Society 2024
Multi-product formulas (MPFs) are linear combinations of Trotter circuits offering high-quality simulation of Hamiltonian time evolution with fewer Trotter steps. Here we report two contributions aimed at making multi-product formulas more viable for near-term quantum simulations. First, we extend the theory of Trotter error with commutator scaling developed by Childs [A. M. Childs , ] to multi-product formulas. Our result implies that multi-product formulas can achieve a quadratic reduction of Trotter error in 1-norm (nuclear norm) on arbitrary time intervals compared with the regular product formulas without increasing the required circuit depth or qubit connectivity. The number of circuit repetitions grows only by a constant factor. Second, we introduce dynamic multi-product formulas with time-dependent coefficients chosen to minimize a certain efficiently computable proxy for the Trotter error. We use a minimax estimation method to make dynamic multi-product formulas robust to uncertainty from algorithmic errors, sampling, and hardware noise. We call this method the minimax MPF and we provide a rigorous bound on its error. Published by the American Physical Society 2024