Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2101.07808 · Quantum

Randomizing multi-product formulas for Hamiltonian simulation

Paul K. Faehrmann, Mark Steudtner, Richard Kueng, Mária Kieferová, Jens Eisert

Published Sep 19, 2022• 40 citations
arXivHTMLPDF

Abstract

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.

DOI
10.22331/q-2022-09-19-806

Related papers

Randomizing multi-product formulas for improved Hamiltonian simulation.

Paul K. Faehrmann, Mark Steudtner, Richard Kueng +2 more

Quantum simulation 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 is 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 and higher-order multi-product formulas as they are used in linear-combination-of-unitaries (LCU) algorithms. In doing so, we propose a framework of randomized sampling that could prove useful for programmable quantum simulators and present two new algorithms tailored to it. Our framework greatly reduces the circuit depth by circumventing the need for oblivious amplitude amplification required by standard LCU methods, rendering it especially useful for near-term quantum computing. 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 also discuss examples at hand of a Hamiltonian comprising pairwise anticommuting terms.

•Jan 19, 2021•7 citations

Doubling the order of approximation via the randomized product formula

Chien‐Hung Cho, Dominic William Berry, Min-Hsiu Hsieh

Hamiltonian simulation is a major application of quantum computing, for example, enabling prediction of the properties of molecules. Prior work has used product formulas with randomization to improve performance, but has only yielded modest improvements over the excellent performance provided by deterministic high-order product formulas. In this work, we provide a randomized scheme that greatly increases the order of product formulas, thereby providing a large advantage over the best-performing deterministic schemes. Our scheme is based on applying randomly chosen corrections to a high-order symmetric product formula. If the original product formula is of order $2k$ (so the error is of order $2k+1$), then the corrected formula is of order $4k+1$, corresponding to a doubling of the order of the error. In practice, applying the corrections in a quantum algorithm requires some structure to the Hamiltonian, for example, the Pauli strings as are commonly used in the simulation of quantum chemistry.

arXiv:2210.11281•Physical Review A•Jun 24, 2024•14 citations

Optimized Randomized Hamiltonian Simulation via Average-Error Analysis

Hayata Morisaki, Keisuke Fujii

Hamiltonian simulation is a central application of quantum computing. Randomized Hamiltonian simulation approximates the target dynamics by sampling quantum circuits and often allows simpler circuit implementations. We develop a framework for randomized Hamiltonian simulation in which Hamiltonian terms are sampled with arbitrary probabilities and the corresponding short-time evolutions are implemented sequentially. The leading channel error relative to ideal time evolution is governed by the variance of the sampled generators. Minimizing the variance-based upper bound on the worst-case error recovers qDRIFT, a leading randomized Hamiltonian simulation algorithm that samples terms in proportion to their operator norms. For typical input states, however, this sampling distribution need not be optimal. Minimizing Haar-averaged error bounds instead yields sampling probabilities proportional to the Hilbert-Schmidt norms of the terms. This choice can provide smaller average-error bounds than conventional qDRIFT. For Hamiltonians expressed as sums of Pauli strings, we combine this framework with commuting Pauli grouping to obtain smaller error bounds than conventional qDRIFT at fixed $R_z$ depth. Numerical benchmarks for the Sachdev-Ye-Kitaev model show error-bound improvement factors consistent with linear scaling in the number of qubits. For the 108-qubit FeMoco Hamiltonian, the error bounds are reduced by a factor of about 5.65. These results establish average-error optimization as a practical design principle for randomized Hamiltonian simulation.

quant-pharXiv:2609.36694•Sep 30, 2026•0 citations

Randomized product formulas beyond optimal deterministic scaling

Leeseok Kim, Luis Pedro García-Pintos

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 $\mathcal O(α^2)$ 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 $A+αB_\ell$ for $B = \sum_{\ell}B_\ell$, 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 $\mathcal O(α^3)$ error scaling with only constant-factor expected gate overhead. We also establish an $Ω(α^2)$ lower bound for deterministic product formulas in this access model. Numerical simulations confirm gate-count reductions for simulating physically motivated systems.

quant-pharXiv:2608.07720•Aug 11, 2026•0 citations

Phase Estimation with Partially Randomized Time Evolution

Jakob Günther, Freek Witteveen, Alexander Schmidhuber +3 more

Quantum phase estimation combined with Hamiltonian simulation is the most promising algorithmic framework to computing ground-state energies on quantum computers. Its main computational overhead derives from the Hamiltonian simulation subroutine. In this paper we use randomization to speed up product formulas, one of the standard approaches to Hamiltonian simulation. We propose partially randomized Hamiltonian simulation methods in which some terms are kept deterministically and others are randomly sampled. We perform a detailed resource estimate for single-ancilla phase estimation using partially randomized product formulas for benchmark systems in quantum chemistry and obtain orders-of-magnitude improvements compared to other simulations based on product formulas. When applied to the hydrogen chain, we have numerical evidence that our methods exhibit asymptotic scaling with the system size that is competitive with the best known qubitization approaches.

arXiv:2503.05647•PRX Quantum•Mar 31, 2026•4 citations

Well-conditioned multi-product formulas for hardware-friendly Hamiltonian simulation

Almudena Carrera Vazquez, Daniel J. Egger, David Ochsner +1 more

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.

arXiv:2207.11268•Quantum•Jul 25, 2023•30 citations
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.
Quantum simulation 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 is 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 and higher-order multi-product formulas as they are used in linear-combination-of-unitaries (LCU) algorithms. In doing so, we propose a framework of randomized sampling that could prove useful for programmable quantum simulators and present two new algorithms tailored to it. Our framework greatly reduces the circuit depth by circumventing the need for oblivious amplitude amplification required by standard LCU methods, rendering it especially useful for near-term quantum computing. 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 also discuss examples at hand of a Hamiltonian comprising pairwise anticommuting terms.
Hamiltonian simulation is a major application of quantum computing, for example, enabling prediction of the properties of molecules. Prior work has used product formulas with randomization to improve performance, but has only yielded modest improvements over the excellent performance provided by deterministic high-order product formulas. In this work, we provide a randomized scheme that greatly increases the order of product formulas, thereby providing a large advantage over the best-performing deterministic schemes. Our scheme is based on applying randomly chosen corrections to a high-order symmetric product formula. If the original product formula is of order 2k2k2k (so the error is of order 2k+12k+12k+1), then the corrected formula is of order 4k+14k+14k+1, corresponding to a doubling of the order of the error. In practice, applying the corrections in a quantum algorithm requires some structure to the Hamiltonian, for example, the Pauli strings as are commonly used in the simulation of quantum chemistry.
Hamiltonian simulation is a central application of quantum computing. Randomized Hamiltonian simulation approximates the target dynamics by sampling quantum circuits and often allows simpler circuit implementations. We develop a framework for randomized Hamiltonian simulation in which Hamiltonian terms are sampled with arbitrary probabilities and the corresponding short-time evolutions are implemented sequentially. The leading channel error relative to ideal time evolution is governed by the variance of the sampled generators. Minimizing the variance-based upper bound on the worst-case error recovers qDRIFT, a leading randomized Hamiltonian simulation algorithm that samples terms in proportion to their operator norms. For typical input states, however, this sampling distribution need not be optimal. Minimizing Haar-averaged error bounds instead yields sampling probabilities proportional to the Hilbert-Schmidt norms of the terms. This choice can provide smaller average-error bounds than conventional qDRIFT. For Hamiltonians expressed as sums of Pauli strings, we combine this framework with commuting Pauli grouping to obtain smaller error bounds than conventional qDRIFT at fixed RzR_zRz​ depth. Numerical benchmarks for the Sachdev-Ye-Kitaev model show error-bound improvement factors consistent with linear scaling in the number of qubits. For the 108-qubit FeMoco Hamiltonian, the error bounds are reduced by a factor of about 5.65. These results establish average-error optimization as a practical design principle for randomized Hamiltonian simulation.
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+αBH=A+αBH=A+αB, where ααα is small. In the standard access model, where one can implement exponentials of AAA and BBB separately, our randomized formulas achieve O(α2)\mathcal O(α^2)O(α2) error scaling at the cost of only doubling the gate depth of the corresponding deterministic formula. We further prove an Ω(α)Ω(α)Ω(α) lower bound for deterministic product formulas. In a stronger access model, allowing exponentials of A+αBℓA+αB_\ellA+αBℓ​ for B=∑ℓBℓB = \sum_{\ell}B_\ellB=∑ℓ​Bℓ​, 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)\mathcal O(α^3)O(α3) error scaling with only constant-factor expected gate overhead. We also establish an Ω(α2)Ω(α^2)Ω(α2) lower bound for deterministic product formulas in this access model. Numerical simulations confirm gate-count reductions for simulating physically motivated systems.
Quantum phase estimation combined with Hamiltonian simulation is the most promising algorithmic framework to computing ground-state energies on quantum computers. Its main computational overhead derives from the Hamiltonian simulation subroutine. In this paper we use randomization to speed up product formulas, one of the standard approaches to Hamiltonian simulation. We propose partially randomized Hamiltonian simulation methods in which some terms are kept deterministically and others are randomly sampled. We perform a detailed resource estimate for single-ancilla phase estimation using partially randomized product formulas for benchmark systems in quantum chemistry and obtain orders-of-magnitude improvements compared to other simulations based on product formulas. When applied to the hydrogen chain, we have numerical evidence that our methods exhibit asymptotic scaling with the system size that is competitive with the best known qubitization approaches.
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.