Optimised Trotter decompositions for classical and quantum computing

Abstract

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 Ak, 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.

Simple Ways to improve Discrete Time Evolution

“…We remark that sometimes "symplectic integrator" is used interchangeably with Trotterization, but this can be very misleading. For more details on the distinctive properties of symplectic integrators see Section 3. This proceeding expands on the results in our previous work [5] where all the technical background can be found that has been omitted here. In the next Section 2 we will reiterate the highly useful method to construct decompositions featuring arbitrarily many operators starting from a two-operator decomposition as in equation (1).…”

“…In these cases only less optimised schemes like those by Hatano and Suzuki [9] or Yoshida [10] have been applicable so far. Now recently, we have shown that every scheme of order 𝑛 applicable to two stages defines an order 𝑛 scheme for an arbitrary number of stages [5]. This allows to adapt the valuable work done on 2-stage decompositions so far and it furthermore provides a simple means to derive efficient new methods in the future since it is sufficient to analyse the method's properties in the simple case of two stages.…”

“…(3) through (5). Every ramp can now be populated with more stages in the last row without changing the validity of the decomposition.…”

“…More rigorously, the theorem proved in [5] reads as follows. Let the coefficients 𝑎 𝑖 and 𝑏 𝑖 as in equation ( 1) define a 2-stage decomposition of order 𝑛.…”

Time crystal embodies chimeralike state in periodically driven quantum spin system

“…During the T 1 −cycle, t ∈ [0, T/2], and the time-independent Hamiltonian Ĥ1 is applied on spin chain. Thus, exp We can simplify this expression further with the Suzuki-Trotter decomposition [93,94] that all operators with norms O(T 2 ) can be neglected in comparison to those with norm ∼ T for sufficiently large ω ≡ 2π/T. Thus, for instance, e T Â+T B = e T Â e T B e C2T 2 [ Â, B] e C3T 3 [ Â,[ Â, B]] • • • ≈ e T Âe T B, once all higher order operators are neglected after applying the Zassenhaus' formula [67].…”