Average-Case Speedup for Product Formulas

Quantum simulation is a promising application of future quantum computers. Product formulas, or Trotterization, are the oldest and still remain an appealing method to simulate quantum systems. For an accurate product formula approximation, the state-of-the-art gate complexity depends on the number o...

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Vydané v:Communications in mathematical physics Ročník 405; číslo 2; s. 32
Hlavní autori: Chen, Chi-Fang, Brandão, Fernando G. S. L.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2024
Springer Nature B.V
Predmet:
Algorithms
Classical and Quantum Gravitation
Complex Systems
Complexity
Estimates
Gate counting
Martingales
Mathematical and Computational Physics
Mathematical Physics
Operators (mathematics)
Physics
Physics and Astronomy
Quantum computers
Quantum computing
Quantum Physics
Relativity Theory
Simulation
Smoothness
Theoretical
ISSN:0010-3616, 1432-0916
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Shrnutí:Quantum simulation is a promising application of future quantum computers. Product formulas, or Trotterization, are the oldest and still remain an appealing method to simulate quantum systems. For an accurate product formula approximation, the state-of-the-art gate complexity depends on the number of terms in the Hamiltonian and a local energy estimate. In this work, we give evidence that product formulas, in practice, may work much better than expected. We prove that the Trotter error exhibits a qualitatively better scaling for the vast majority of input states, while the existing estimate is for the worst states. For general k -local Hamiltonians and higher-order product formulas, we obtain gate count estimates for input states drawn from any orthogonal basis. The gate complexity significantly improves over the worst case for systems with large connectivity. Our typical-case results generalize to Hamiltonians with Fermionic terms, with input states drawn from a fixed-particle number subspace, and with Gaussian coefficients (e.g., the SYK models). Technically, we employ a family of simple but versatile inequalities from non-commutative martingales called uniform smoothness , which leads to Hypercontractivity , namely p -norm estimates for k -local operators. This delivers concentration bounds via Markov’s inequality. For optimality, we give analytic and numerical examples that simultaneously match our typical-case estimates and the existing worst-case estimates. Therefore, our improvement is due to asking a qualitatively different question, and our results open doors to the study of quantum algorithms in the average case.
Bibliografia:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-023-04912-5