Applying splitting methods with complex coefficients to the numerical integration of unitary problems

Blanes, Sergio; Casas, Fernando; Escorihuela-Tomàs, Alejandro

doi:10.3934/jcd.2021022

Cited by 9 publications

(28 citation statements)

References 21 publications

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“…The schemes presented in eqs. (33) and (39) have therefore, to our knowledge, not been derived before and outperform all methods of the same order we found in the literature by at least a factor two, including other non-unitary schemes.…”

Section: Using Complex Coefficientssupporting

confidence: 64%

“…We therefore highly recommend using these novel decompositions when possible. Empirically we find that this scheme (33) is well behaved at long times and therefore more performant in practice than its theoretically more efficient q = 5 counterpart (39). Somewhat surprisingly, we find that the gap between our non-unitary schemes to the next best unitary ones is even larger than suggested by the theoretical efficiency in the case of more than two stages.…”

Section: 26mentioning

confidence: 51%

“…Especially the results in [33] are noteworthy nonetheless. There the differences between symmetric ("palindromic") and so-called "symmetric-conjugate" splitting methods are discussed.…”

Section: Using Complex Coefficientsmentioning

confidence: 88%

“…Just this year Casas, Escorihuela-Tomàs and different collaborators have published several works on higher order symplectic integrators with complex coefficients [12,[31][32][33]. Again, as Omelyan [1] did with real coefficients, they do not optimise for general Trotter decompositions.…”

Section: Using Complex Coefficientsmentioning

confidence: 99%

“…3.2.9 Non-Unitary (order n = 4, q = 5 cycles) Similarly to the decomposition scheme (33) this choice of complex coefficients…”

Section: 26mentioning

confidence: 99%

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Optimised Trotter Decompositions for Classical and Quantum Computing

Ostmeyer¹

2022

Preprint

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Suzuki-Trotter decompositions of exponential operators like exp(Ht) are required in almost every branch of numerical physics. Often the exponent under consideration has to be split into more than two operators H = k A k , for instance as local gates on quantum computers. We demonstrate how highly optimised schemes originally derived for exactly two operators A1,2 can be applied to such generic Suzuki-Trotter decompositions, providing a formal proof of correctness as well as numerical evidence of efficiency. A comprehensive review of existing symmetric decomposition schemes up to order n ≤ 4 is presented and complemented by a number of novel schemes, including both real and complex coefficients. We derive the theoretically most efficient unitary and non-unitary 4th order decompositions. The list is augmented by several exceptionally efficient schemes of higher order n ≤ 8. Furthermore we show how Taylor expansions can be used on classical devices to reach machine precision at a computational effort at which state of the art Trotterization schemes do not surpass a relative precision of 10 −4 . Finally, a short and easily understandable summary explains how to choose the optimal decomposition in any given scenario.1 Symplectic integrators rely on the property that a large number of high order commutators vanishes [1]. This means that using the coefficients of a symplectic integrator for a general splitting method leads to a reduced efficiency starting from order n ≥ 4 and might not even give the expected order at all for n ≥ 6.

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Section: Using Complex Coefficientssupporting

confidence: 64%

Section: 26mentioning

confidence: 51%

“…Especially the results in [33] are noteworthy nonetheless. There the differences between symmetric ("palindromic") and so-called "symmetric-conjugate" splitting methods are discussed.…”

Section: Using Complex Coefficientsmentioning

confidence: 88%

Section: Using Complex Coefficientsmentioning

confidence: 99%

“…3.2.9 Non-Unitary (order n = 4, q = 5 cycles) Similarly to the decomposition scheme (33) this choice of complex coefficients…”

Section: 26mentioning

confidence: 99%

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Optimised Trotter Decompositions for Classical and Quantum Computing

Ostmeyer¹

2022

Preprint

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Symmetric-conjugate splitting methods for linear unitary problems

Bernier,

Blanes,

Casas

et al. 2023

Bit Numer Math

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We analyze the preservation properties of a family of reversible splitting methods when they are applied to the numerical time integration of linear differential equations defined in the unitary group. The schemes involve complex coefficients and are conjugated to unitary transformations for sufficiently small values of the time step-size. New and efficient methods up to order six are constructed and tested on the linear Schrödinger equation.

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