The accuracy of quantum dynamics simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on certain norm of the Hamiltonian. For unbounded operators, after suitable discretization, the norm of the Hamiltonian can be very large, which significantly increases the simulation cost. However, the operator norm measures the worst-case error of the quantum simulation, while practical simulation concerns the error with respect to a given initial vector at hand. We demonstrate that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the Hamiltonian increases using Trotter type methods. In this sense, our result outperforms all previous error bounds in the quantum simulation literature. Our result extends that of [Jahnke, Lubich, BIT Numer. Math. 2000] to the time-dependent setting. We also clarify the existence and the importance of commutator scalings of Trotter and generalized Trotter methods for time-dependent Hamiltonian simulations.
The accuracy of quantum dynamics simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on certain norm of the Hamiltonian. For unbounded operators, after suitable discretization, the norm of the Hamiltonian can be very large, which significantly increases the simulation cost. However, the operator norm measures the worst-case error of the quantum simulation, while practical simulation concerns the error with respect to a given initial vector at hand. We demonstrate that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the Hamiltonian increases using Trotter type methods. In this sense, our result outperforms all previous error bounds in the quantum simulation literature. Our result extends that of [Jahnke, Lubich, BIT Numer. Math. 2000] to the time-dependent setting. We also clarify the existence and the importance of commutator scalings of Trotter and generalized Trotter methods for time-dependent Hamiltonian simulations.
Abstract Large-scale classical simulation of quantum computers is crucial for benchmarking quantum algorithms, establishing boundaries of quantum advantage and exploring heuristic quantum algorithms. We present a full-state vector simulation algorithm and software implementation designed to perform HPC simulation of layers of rotations around a string of Pauli operators. We demonstrate robust scalability of the simulation method on large distributed CPU and GPU systems. Our distributed computation harnessed up to 16384 CPU cores and 512 NVIDIA H100 GPUs, using 32 TB of memory. The simulator significantly outperforms other high-performance libraries, showing a typical speedup of 10-100 for large-scale multi-GPU workloads. As a first application of our approach, we report a numerical experiment aimed at simulating exactly Hamiltonian dynamics of up to 40 qubits to investigate the Trotter error for a quantum chemistry problem. Bounding the Trotter error is important for evaluating the cost of quantum algorithms for chemistry, and rigorous bounds are often conservative, as our simulations confirm. Our software, specifically designed for quantum time evolution applications, is also well equipped to manage circuits that utilize standard gate sets.
We present a quantum algorithm for the dynamical simulation of time-dependent Hamiltonians.Our method involves expanding the interaction-picture Hamiltonian as a sum of generalized permutations, which leads to an integral-free Dyson series of the time-evolution operator.Under this representation, we perform a quantum simulation for the time-evolution operator by means of the linear combination of unitaries technique.We optimize the time steps of the evolution based on the Hamiltonian's dynamical characteristics, leading to a gate count that scales with an L 1 -norm-like scaling with respect only to the norm of the interaction Hamiltonian, rather than that of the total Hamiltonian.We demonstrate that the cost of the algorithm is independent of the Hamiltonian's frequencies, implying its advantage for systems with highly oscillating components, and for time-decaying systems the cost does not scale with the total evolution time asymptotically.In addition, our algorithm retains the near optimal log(1/ )/ log log(1/ ) scaling with simulation error .
Abstract. Known as the no fast-forwarding theorem in quantum computing (see, e.g., Theorem 3 in [D. W. Berry et al., Comm. Math. Phys., 270 (2007), pp. 359–371], Theorem 5 in [A. M. Childs, Comm. Math. Phys., 294 (2010), pp. 581–603], and [R. Kothari, Efficient Simulation of Hamiltonians, M.S. thesis, University of Waterloo, 2010]), the simulation time for the Hamiltonian evolution is typically [Formula: see text], which essentially states that one cannot go across the multiple scales as the simulation time for the Hamiltonian evolution needs to be strictly greater than the physical time. We demonstrated in the context of the semiclassical Schrödinger equation that the computational cost for a class of observables can be much lower than the state-of-the-art bounds. In the semiclassical regime (the effective Planck constant [Formula: see text]), the operator norm of the Hamiltonian is [Formula: see text]. We show that the number of Trotter steps used for the observable evolution can be [Formula: see text], that is, to simulate some observables of the Schrödinger equation on a quantum scale only takes the simulation time comparable to the classical scale. In terms of error analysis, we improve the additive observable error bounds in [C. Lasser and C. Lubich, Acta Numer., 29 (2020), pp. 229–401] to uniform-in-[Formula: see text] observable error bounds. This is, to our knowledge, the first uniform observable error bound for the semiclassical Schrödinger equation without sacrificing the convergence order of the numerical method. Based on semiclassical calculus and discrete microlocal analysis, our result showcases the potential improvements taking advantage of multiscale properties, such as the smallness of the effective Planck constant, and of the underlying dynamics, and sheds light on going across the scale for quantum dynamics simulation.
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.
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
In this thesis, we present three innovations that can be used to simulate quantum dynamics on both quantum and classical computers. Our primary innovation is an efficient quantum algorithm for simulating time-dependent Hamiltonian evolution of general input states on a quantum computer. Given conditions on the smoothness of the Hamiltonian, the complexity of the algorithm is close to linear in the evolution time, and therefore is comparable to algorithms for time-independent Hamiltonians. In addition, we show how the complexity can be reduced by optimizing the time steps. The complexity of the algorithm is quantified by calls to an oracle, which yields information about the Hamiltonian. In contrast to previous work, which allowed an oracle query to yield an arbitrary number of bits or qubits, we assign a cost for each bit or qubit accessed, revealing hitherto unnoticed simulation costs. We also account for discretization errors in the time and the representation of the Hamiltonian. This work is enabled by our development of a decomposition scheme based on Lie-Trotter- Suzuki product formulae for ordered operator exponentials. We show, by counterexample, that Lie-Trotter-Suzuki approximations may be lower order than expected when applied to problems that have singularities or discontinuous derivatives of appropriate order. We address this problem by presenting a set of criteria that is sufficient for the validity of these approximations, prove convergence and provide upper bounds for the approximation error. We also provide a new analysis of the evolution of quantum adiabatic systems in this thesis. Our work contains the tightest upper bounds that are currently known, and the first tight lower bounds, for the error in the adiabatic approximation applied to fixed Hamiltonians in the limit of slow evolution. We use our results to address the Marzlin- Sanders counterexample Hamiltonian. Specifically, we provide sufficiency conditions for the validity of the adiabatic approximation and demonstrate that the Marzlin-Sanders counterexample fails to satisfy them. We also find a new class of time-dependent Hamiltonians that are related to the Marzlin0Sanders counterexample Hamiltonian, but nonetheless obey the adiabatic approximation for sufficiently slow evolutions.