Abstract Quantum algorithms for simulation of Hamiltonian evolution are often based on product formulae. The fractal methods give a systematic way to find arbitrarily high-order product formulae, but result in a large number of exponentials. On the other hand, product formulae with fewer exponentials can be found by numerical solution of simultaneous non-linear equations. It is also possible to reduce the cost of long-time simulations by processing, where a kernel is repeated and a processor need only be applied at the beginning and end of the simulation. In this work, we found thousands of new product formulae, and numerically tested these formulae, together with many formulae from prior literature. We provide methods to fairly compare product formulae of different lengths and different orders. For the case of 8th order, we have found new product formulae with exceptional performance, about two orders of magnitude better accuracy than prior work, both in the processed and non-processed cases. The processed product formula provides the best performance due to being shorter than the non-processed product formula. It outperforms all other tested product formulae over a range of many orders of magnitude in system parameters T (time) and ε (allowable error). That includes reasonable combinations of parameters to be used in quantum algorithms, where the size of the simulation is large enough to be classically intractable, but not so large it takes an impractically long time on a quantum computer.
Abstract Quantum algorithms for simulation of Hamiltonian evolution are often based on product formulae. The fractal methods give a systematic way to find arbitrarily high-order product formulae, but result in a large number of exponentials. On the other hand, product formulae with fewer exponentials can be found by numerical solution of simultaneous non-linear equations. It is also possible to reduce the cost of long-time simulations by processing, where a kernel is repeated and a processor need only be applied at the beginning and end of the simulation. In this work, we found thousands of new product formulae, and numerically tested these formulae, together with many formulae from prior literature. We provide methods to fairly compare product formulae of different lengths and different orders. For the case of 8th order, we have found new product formulae with exceptional performance, about two orders of magnitude better accuracy than prior work, both in the processed and non-processed cases. The processed product formula provides the best performance due to being shorter than the non-processed product formula. It outperforms all other tested product formulae over a range of many orders of magnitude in system parameters T (time) and ε (allowable error). That includes reasonable combinations of parameters to be used in quantum algorithms, where the size of the simulation is large enough to be classically intractable, but not so large it takes an impractically long time on a quantum computer.
Product formula approximations of the time-evolution operator on quantum computers are of great interest due to their simplicity, and good scaling with system size by exploiting commutativity between Hamiltonian terms. However, product formulas exhibit poor scaling with the time t and error ε of simulation as the gate cost of a single step scales exponentially with the order m of accuracy. We introduce well-conditioned multiproduct formulas, which are a linear combination of product formulas, where a single step has polynomial cost O(m2log(m)) and succeeds with probability Ω(1/log2(m)). Our multiproduct formulas imply a simple and generic simulation algorithm that simultaneously exploits commutativity in arbitrary systems and has a worst-case cost O(tlog2(t/ε)) which is optimal up to poly-logarithmic factors. In contrast, prior Trotter and post-Trotter Hamiltonian simulation algorithms realize only one of these two desirable features. A key technical result of independent interest is our solution to a conditioning problem in previous multiproduct formulas that amplified numerical errors by eΩ(m) in the classical setting, and led to a vanishing success probability e−Ω(m) in the quantum setting.
This thesis describes quantum algorithms for Hamiltonian simulation, ordinary differential equations (ODEs), and partial differential equations (PDEs). Product formulas are used to simulate Hamiltonians which can be expressed as a sum of terms which can each be simulated individually. By simulating each of these terms in sequence, the net effect approximately simulates the total Hamiltonian. We find that the error of product formulas can be improved by randomizing over the order in which the Hamiltonian terms are simulated. We prove that this approach is asymptotically better than ordinary product formulas and present numerical comparisons for small numbers of qubits. The ODE algorithm applies to the initial value problem for time-independent first order linear ODEs. We approximate the propagator of the ODE by a truncated Taylor series, and we encode the initial value problem in a large linear system. We solve this linear system with a quantum linear system algorithm (QLSA) whose output we perform a post-selective measurement on. The resulting state encodes the solution to the initial value problem. We prove that our algorithm is asymptotically optimal with respect to several system parameters. The PDE algorithms apply the finite difference method (FDM) to Poisson's equation, the wave equation, and the Klein-Gordon equation. We use high order FDM approximations of the Laplacian operator to develop linear systems for Poisson's equation in cubic volumes under periodic, Neumann, and Dirichlet boundary conditions. Using QLSAs, we output states encoding solutions to Poisson's equation. We prove that our algorithm is exponentially faster with respect to the spatial dimension than analogous classical algorithms. We also consider how high order Laplacian approximations can be used for simulating the wave and Klein-Gordon equations. We consider under what conditions it suffices to use Hamiltonian simulation for time evolution, and we propose an algorithm for these cases that uses QLSAs for state preparation and post-processing.
In this work we propose an approach for implementing time-evolution of a quantum system using product formulas. The quantum algorithms we develop have provably better scaling (in terms of gate complexity and circuit depth) than a naive application of well-known Trotter formulas, for systems where the evolution is determined by a Hamiltonian with different energy scales (i.e., one part is "large" and another part is "small"). Our algorithms generate a decomposition of the evolution operator into a product of simple unitaries that are directly implementable on a quantum computer. Although the theoretical scaling is suboptimal compared with state-of-the-art algorithms (e.g., quantum signal processing), the performance of the algorithms we propose is highly competitive in practice. We illustrate this via extensive numerical simulations for several models. For instance, in the strong-field regime of the 1D transverse-field Ising model, our algorithms achieve an improvement of one order of magnitude in both the system size and evolution time that can be simulated with a fixed budget of 1000 arbitrary 2-qubit gates, compared with standard Trotter formulas.
Hamiltonian simulation using product formulas is arguably the most straightforward and practical approach for algorithmic simulation of a quantum system’s dynamics on a quantum computer. Here we present corrected product formulas (CPFs), a variation of product formulas achieved by injecting auxiliary terms called correctors into standard product formulas. We establish several correctors that improve the accuracy of standard product formulas by orders of magnitude when simulating Hamiltonians comprised of two exactly simulatable partitions, a common structure of lattice Hamiltonians. Importantly, injecting these correctors increases the overall simulation cost by only a small additive or multiplicative factor. We show that correctors are particularly advantageous for perturbed systems, where one partition has a relatively small norm compared to the other, as they allow the small norm to be utilized as an additional parameter for controlling the simulation error. We demonstrate the performance of CPFs by numerical simulations for several lattice Hamiltonians. Numerical results show that our theoretical error bounds for CPFs match or outperform the empirical error scaling of standard product formulas for these systems. We also demonstrate improvements offered by CPFs by implementing small-size systems on actual quantum hardware, as well as on noisy and noiseless quantum simulators. CPFs could be a valuable algorithmic tool for early fault-tolerant quantum computers with limited computing resources. As for standard product formulas, CPFs could also be used for simulations on a classical computer.
Exponential-in-Nc2 cost reduction of product-formula-based quantum simulations of quantum chromodynamics
Quantum algorithms for simulating quantum chromodynamics (QCD) have matured steadily since the pioneering work of Byrnes and Yamamoto [PRA 73, 022328 (2006)]. The most popular strategies for Hamiltonian simulation involve product-formula decompositions. However, the application of product-formula methods to SU(Nc) lattice gauge theories by Byrnes and Yamamoto leads to O(Λ8(Nc2−1)) gate complexity per Trotter step, where Λ is the bosonic cutoff in the electric (i.e., irreducible-representation) basis. A seminal work by Kan and Nam [arXiv:2107.12769 (2021)] significantly improves over such an undesirable cost and reports an O(Λpolylog(Λ)) scaling, yet it still calls for an unrealistically large number of quantum gates. Here, we illuminate one of the reasons behind this high cost estimate and show that a factor of size O(24(Nc2−1)) can be removed from the per-Trotter-step cost estimate by Kan and Nam. We specifically show that, by using methods developed in our past works [PRD 112, 014508 (2025); Quantum 7, 1213 (2023)], exponentiated-Hamiltonian decomposition---a necessary step in the application of product-formula algorithms---can be performed far more efficiently than previously thought. Our method reduces the T-gate cost estimate of QCD simulations using a second-order product formula by a factor of nearly 1014, independent of simulation parameters and sizes. Focusing on simulations in the electric basis, we further contrast our results with other methods: the local-multiplet basis approach of Ciavarella, Klco, and Savage [PRD 103, 094501 (2021)] and the near-optimal algorithm of Rhodes, Kreshchuk, and Pathak [PRX Quantum 5, 040347 (2024)]. This work highlights the importance of continued algorithmic improvement to bringing the quantum-simulation cost of QCD within reach of realistic quantum computers.
Quantum simulation, the simulation of quantum processes on quantum computers, suggests a path forward for the efficient simulation of problems in condensed-matter physics, quantum chemistry, and materials science. While the majority of quantum simulation algorithms are deterministic, a recent surge of ideas has shown that randomization can greatly benefit algorithmic performance. In this work, we introduce a scheme for quantum simulation that unites the advantages of randomized compiling on the one hand and higher-order multi-product formulas, as they are used for example in linear-combination-of-unitaries (LCU) algorithms or quantum error mitigation, on the other hand. In doing so, we propose a framework of randomized sampling that is expected to be useful for programmable quantum simulators and present two new multi-product formula algorithms tailored to it. Our framework reduces the circuit depth by circumventing the need for oblivious amplitude amplification required by the implementation of multi-product formulas using standard LCU methods, rendering it especially useful for early quantum computers used to estimate the dynamics of quantum systems instead of performing full-fledged quantum phase estimation. Our algorithms achieve a simulation error that shrinks exponentially with the circuit depth. To corroborate their functioning, we prove rigorous performance bounds as well as the concentration of the randomized sampling procedure. We demonstrate the functioning of the approach for several physically meaningful examples of Hamiltonians, including fermionic systems and the Sachdev–Ye–Kitaev model, for which the method provides a favorable scaling in the effort.