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arXiv:quant-ph/9809009

Higher-Order Methods for Quantum Simulations

Andrew T. Sornborger, Ewan D. Stewart

Published Sep 4, 1998• 4 citations
arXivHTMLPDF

Abstract

To efficiently implement many-particle quantum simulations on quantum computers we develop and present methods for inverting the Campbell-Baker-Hausdorff lemma to 3rd and 4th order in the commutator. That is, we reexpress exp{-i(H_1 + H_2 + ...)dt} as a product of factors exp(-i H_1 dt), exp(-i H_2 dt), ... which is accurate to 3rd or 4th order in dt.

DOI
10.48550/arxiv.quant-ph/9809009

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To efficiently implement many-particle quantum simulations on quantum computers we develop and present methods for inverting the Campbell-Baker-Hausdorff lemma to 3rd and 4th order in the commutator. That is, we reexpress exp{-i(H_1 + H_2 + ...)dt} as a product of factors exp(-i H_1 dt), exp(-i H_2 dt), ... which is accurate to 3rd or 4th order in dt.
To implement many-qubit gates for use in quantum simulations on quantum computers efficiently, we develop and present methods reexpressing $\mathrm{exp}[\ensuremath{-}{i(H}_{1}{+H}_{2}+\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot})\ensuremath{\Delta}t]$ as a product of factors $\mathrm{exp}[\ensuremath{-}{\mathrm{iH}}_{1}\ensuremath{\Delta}t],$ $\mathrm{exp}[\ensuremath{-}{\mathrm{iH}}_{2}\ensuremath{\Delta}t],\dots{}$, which is accurate to third or fourth order in $\ensuremath{\Delta}t$. The methods we derive are an extended form of the symplectic method, and can also be used for an integration of classical Hamiltonians on classical computers. We derive both integral and irrational methods, and find the most efficient methods in both cases.
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)\mathcal{O}(n^3)O(n3) to O(n)\mathcal{O}(n)O(n) for second-order, and from O(n5)\mathcal{O}(n^5)O(n5) to O(n2)\mathcal{O}(n^2)O(n2) for fourth-order formulas on nnn 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 (Li4_44​Mn2_22​O) 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.
The Baker--Campbell--Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics. It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved. The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators. In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest. In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.
We solve the solution of the Time-Dependent Schrodinger Equation (TDSE) using the higher-order Trotter-Suzuki Decomposition. We use The Gaussian function as a stationary state in a linear potential system. The TDSE solution using Baker - Campbell - Hausdorff was used to validate the results and to measure the accuracy of the Trotter - Suzuki decomposition. So that the difference between the Baker - Campbell - Hausdorff result and the Suzuki - Trotter decomposition is considered an error. The error of the TDSE solution by the Trotter - Suzuki second-order decomposition was lower than the first order. Meanwhile, the error of the TDSE solution by second-order hybrid will be lower than the second-order Trotter - Suzuki decomposition when the value of dx=0.1 and 0.05 with dt < 0.0001. The error comparison of these three methods is only valid when time t<1.