Time-Dependent Hamiltonian Simulation Using Discrete-Clock Constructions

Compared with time-independent Hamiltonians, the dynamics of generic quantum Hamiltonians <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mi>H</a:mi><a:mo stretchy="false">(</a:mo><a:mi>t</a:mi><a:mo stretchy="false">)</a:mo></a:math> are complicated by the presence of time ordering in the evolution operator. In the context of digital quantum simulation, this difficulty prevents a direct adaptation of many time-independent simulation algorithms for time-dependent simulation. However, there exists a framework within the theory of dynamical systems that eliminates time ordering by adding a “clock” degree of freedom. In this work, we provide a computational framework, based on this reduction, for encoding time-dependent dynamics as time-independent systems. As a result, we make two advances in digital Hamiltonian simulation. First, we create a time-dependent simulation algorithm based on performing qubitization on the augmented clock system and, in doing so, provide the first qubitization-based approach to time-dependent Hamiltonians that goes beyond Trotterization of the ordered exponential. Second, we define a natural generalization of multiproduct formulas for time-ordered exponentials and then propose and analyze an algorithm based on these formulas. Unlike other algorithms of similar accuracy, the multiproduct approach achieves commutator scaling, meaning that this method outperforms existing methods for physically local time-dependent Hamiltonians with sufficient smoothness. Our work reduces the disparity between time-dependent and time-independent simulation and indicates a step toward optimal quantum simulation of time-dependent Hamiltonians. Published by the American Physical Society 2024