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.
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.
The Linear Combination of Hamiltonian Simulation (LCHS) framework simulates dissipative linear dynamics by representing time evolution as an integral over unitary operators, which is discretized by quadrature and implemented via Hamiltonian simulation. While existing analyses achieve near-optimal scaling in time and precision using norm-based quantities of the dissipative generator, we show that implementing the Hamiltonian simulation steps with Multi-Product Formulas (MPFs) yields commutator-sensitive error and complexity bounds. We demonstrate that the quadrature rule affects not only discretization error but also commutator structure and query complexity. This dependence is quantified through post-quadrature analysis for abstract MPF error profiles and for general time-independent and local Hamiltonians using known commutator-sensitive MPF error estimates. We compare uniform trapezoidal and free-scale sinh--sinh quadrature, showing improved quadrature-cardinality scaling for the latter, and illustrate the framework with applications to fractional diffusion, advection--diffusion, and open quantum systems.
The well-conditioned multi-product formula (MPF), proposed by Low-Kliuchnikov-Wiebe (2019), is a high-order, time-independent Hamiltonian simulation algorithm that implements a linear combination of low-order product formulas. Prior work established its well-conditioned algorithmic construction and near-optimal time and precision dependence, but did not simultaneously provide an explicit error bound in terms of nested commutators, which has left its practical advantage uncertain. In this work, we provide a rigorous complexity analysis of the well-conditioned MPF, explicitly establishing both commutator scaling and near-optimal dependence on time and precision through a rigorous Baker-Campbell-Hausdorff (BCH) and variation-of-parameters-based error analysis. Using our improved complexity estimates, we present several physically relevant applications where the second-order-based MPF asymptotically outperforms the second-order product formula in all three aspects of system size, time, and precision. We further demonstrate that, while MPF yields worse scalings than the best post-Trotter methods in most applications, there exist certain parameter regimes in the nonlocal fast-transform Bogoliubov-de Gennes model where MPF can outperform both leading post-Trotter methods and product formulas with fixed or adaptively optimized order.
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.
Product formula (PF), which approximates the time evolution under a many-body Hamiltonian by the product of local time-evolution operators, is one of the central approaches for simulating quantum dynamics by quantum computers. It has been of great interest whether the errors of PFs can be expressed by commutators among local terms, since it brings the substantial suppression of the computational cost in the system size reflecting the locality of Hamiltonians. Although recent studies have revealed such error bounds for generic time-independent systems, it remains to be a difficult problem for time-dependent Hamiltonians except for low-order PFs. In this paper, we derive an error bound of generic PFs for smooth time-dependent Hamiltonians, which is expressed explicitly by nested commutators and their time derivatives. It indicates the substantial error suppression in the system size for generic local Hamiltonians, thereby giving a much better estimate of gate counts. Our derivation employs Floquet theory originally developed for time-periodic systems, with which we analyze the time-dependent PF errors by mapping them to those for time-independent PFs defined on an infinite-dimensional space. This approach is also applicable to a family of time-dependent PFs, such as the one for non-unitary dynamics. In particular, we clarify the error bound of a time-dependent multi-product formula, with which we can achieve exponentially better scaling with respect to the target accuracy than the PF. Our results will shed light on applications of near-term and future quantum computers, ranging from quantum simulation of nonequilibrium materials to faster algorithms exploiting time-dependent Hamiltonians such as adiabatic state preparation.
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
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.