Randomizing multi-product formulas for Hamiltonian simulation

Paul K. Faehrmann1, Mark Steudtner1, Richard Kueng2, Mária Kieferová3, and Jens Eisert1,4

1Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
2Institute for Integrated Circuits, Johannes Kepler University Linz, Austria
3Centre for Quantum Computation and Communication Technology, Centre for Quantum Software and Information, University of Technology Sydney, NSW 2007, Australia
4Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany

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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 multiproduct 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.

Simulating the dynamics of interacting quantum systems is one of the most eagerly anticipated use cases for quantum computing. However, most algorithms require large quantum computers with precise control and will not be implementable on near-term devices. Implementing state-of-the-art algorithms on an actual device needs a lot of resources. Unfortunately, these resource costs are prohibitive in the near and intermediate term, constituting a roadblock.

But there is a new key ingredient that enters here that renders the task of simulating quantum many-body systems easier: This is randomness. It is too much to ask of the algorithm to lead to the correct result in every run. Instead, being exact only on average is much more resource-efficient.

Consequently, we propose randomly applying gates, generating the desired superpositions required for higher-order schemes on average, giving rise to more precise implementations. We find that this random compilation avoids the need for complex quantum circuits while maintaining the benefits of more accurate, higher-order schemes.

This work introduces new techniques that make quantum simulators feasible already in the intermediate regime of programmable quantum devices. It is thus more suited for near- and intermediate-term devices. Due to its comparative simplicity, our scheme could also apply to programmable quantum simulators. Within the developed framework, there is a lot of potential for new methods, for example, more efficient ways of determining ground states.

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Cited by

[1] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu, "A Theory of Trotter Error", arXiv:1912.08854.

[2] Natalie Klco, Alessandro Roggero, and Martin J. Savage, "Standard model physics and the digital quantum revolution: thoughts about the interface", Reports on Progress in Physics 85 6, 064301 (2022).

[3] Troy J. Sewell and Christopher David White, "Mana and thermalization: Probing the feasibility of near-Clifford Hamiltonian simulation", Physical Review B 106 12, 125130 (2022).

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[6] Chi-Fang Chen, Hsin-Yuan Huang, Richard Kueng, and Joel A. Tropp, "Concentration for Random Product Formulas", PRX Quantum 2 4, 040305 (2021).

[7] Xiantao Li, "Some error analysis for the quantum phase estimation algorithms", Journal of Physics A Mathematical General 55 32, 325303 (2022).

[8] Jacob Watkins, Nathan Wiebe, Alessandro Roggero, and Dean Lee, "Time-dependent Hamiltonian Simulation Using Discrete Clock Constructions", arXiv:2203.11353.

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