Southwest Quantum Information and Technology

Simulating time-dependent Hamiltonians with finite-dimensional clocks

Presenting Author: Jacob Watkins, Michigan State University
Contributing Author(s): Nathan Wiebe, Alessandro Roggero, Dean Lee

To date, several simulation methods have been proposed that achieve optimal scaling for time-independent Hamiltonians. However, no such algorithm has been developed that saturates these lower bounds for a non-trivial time-dependent Hamiltonian. We solve this problem by providing a new approach for approximating an ordered operator exponential using an ordinary operator exponential acting on a larger, finite-dimensional Hilbert space, which we call a “clock space”. This approach allows us to translate results for simulating time-independent systems to the time-dependent case. Our result solves two open problems in simulation. First, we provide a rigorous way to generate time-dependent product and multiproduct formulas using translations on the clock, constructing a new family of multiproduct formulas for time-dependent Hamiltonians that yield both commutator scaling and poly-logarithmic error. Our construction outperforms existing methods for simulating physically local, time-dependent Hamiltonians. Second, we extend the application of qubitization to time-dependent Hamiltonians and achieve the current best computational scaling for linear time dependencies, matching the value for time-independent qubitization. We show that as the number of auxiliary qubits grows, the error in the ordered operator exponential vanishes, as well as the entanglement between the clock and the system of interest.

Read this article online: https://arxiv.org/abs/2203.11353

(Session 9b : Friday from 5:15 pm - 5:45 pm)