Faster Algorithmic Quantum and Classical Simulations by Corrected Product Formulas
Mohsen Bagherimehrab, Luis Mantilla Calderón, Dominic William Berry, Philipp Schleich, Mohammad Ghazi Vakili, Abdulrahman Aldossary, Jorge A. Campos-Gonzalez-Angulo, Christoph Gorgulla, Alán Aspuru‐Guzik
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.
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.
Simulating the time-evolution of a Hamiltonian is one of the most promising applications of quantum computers. Multi-Product Formulas (MPFs) are well suited to replace standard product formulas since they scale better with respect to time and approximation errors. Hamiltonian simulation with MPFs was first proposed in a fully quantum setting using a linear combination of unitaries. Here, we analyze and demonstrate a hybrid quantum-classical approach to MPFs that classically combines expectation values evaluated with a quantum computer. This has the same approximation bounds as the fully quantum MPFs, but, in contrast, requires no additional qubits, no controlled operations, and is not probabilistic. We show how to design MPFs that do not amplify the hardware and sampling errors, and demonstrate their performance. In particular, we illustrate the potential of our work by theoretically analyzing the benefits when applied to a classically intractable spin-boson model, and by computing the dynamics of the transverse field Ising model using a classical simulator as well as quantum hardware. We observe an error reduction of up to an order of magnitude when compared to a product formula approach by suppressing hardware noise with Pauli Twirling, pulse efficient transpilation, and a novel zero-noise extrapolation based on scaled cross-resonance pulses. The MPF methodology reduces the circuit depth and may therefore represent an important step towards quantum advantage for Hamiltonian simulation on noisy hardware.
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.
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
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.
Product formulas, also known as Trotter formulas, are among the most widely used and practical methods for simulating quantum systems on quantum computers. Here we introduce two new classes of randomized product formulas for simulating Hamiltonians with separated energy scales, H=A+αB, where α is small. In the standard access model, where one can implement exponentials of A and B separately, our randomized formulas achieve O(α2) error scaling at the cost of only doubling the gate depth of the corresponding deterministic formula. We further prove an Ω(α) lower bound for deterministic product formulas. In a stronger access model, allowing exponentials of A+αBℓ for B=∑ℓBℓ, our randomized formula, based on Trotter Heuristic Resource Improved Formulas for Time-dynamics (THRIFT)~[J. L. Bosse et al., Nat. Commun. 16, 2673 (2025)], achieves O(α3) error scaling with only constant-factor expected gate overhead. We also establish an Ω(α2) lower bound for deterministic product formulas in this access model. Numerical simulations confirm gate-count reductions for simulating physically motivated systems.
Recent study has shown that it can be advantageous to implement a composite channel that partitions the Hamiltonian H for a given simulation problem into subsets A and B such that H=A+B , where the terms in A are simulated with a Trotter-Suzuki channel and the B terms are randomly sampled via the Qdrift algorithm. Here we extend Qdrift and composite product formulas to imaginary time, formulating candidate classical algorithms for quantum Monte Carlo calculations. We upper bound the induced Schatten- 1→1 norm on both imaginary-time Qdrift and composite channels. Another recent result demonstrated that simulations of lattice Hamiltonians containing geometrically local interactions can be improved using a Lieb-Robinson argument to decompose H into subsets that contain only terms supported on that subset of the lattice. Here, we provide a quantum algorithm by unifying this result with the composite approach into “local composite channels” and we upper bound the diamond distance. We provide exact numerical simulations of algorithmic cost by counting the number of gates of the form e−iHjt and e−Hjβ to meet a certain error tolerance ε . In doing so, we optimize the partitioning into sets A and B using gradient boosted tree models from machine learning. These numerical studies are important given that product formulas have been historically known to outperform analytic upper bounds. We show constant factor advantages for a variety of interesting Hamiltonians, the maximum of which is a ≈20 -fold speedup that occurs in the simulation of Jellium. Published by the American Physical Society 2024