In Search of a Time-Independent Reset Hamiltonian

Abstract

In this work, we consider the problem of simulating time-independent Lindblad evolution in continuous time using time-independent Schrödinger evolution. The task is distinct from the usual computational problem of outputting the final state of a time evolution, as we seek to have our simulation be close to the original evolution at all times between the start and end. This goal is broken down into two parts, this work focusing only on the first: approximating an instantaneous reset of a qubit with time-independent Schrödinger evolution and combining multiple time-independent approximations into a single time-independent evolution that inherits characteristics of its constituents.

We propose and analyze a family of time-independent Hamiltonians and initial ancilla states that approximate in continuous time an instantaneous reset of a single qubit with four parameters: state error, time to perform the reset, time before the reset in which the initial state is unchanged, and time after the reset in which the state remains reset. The specific time-independent Hamiltonians are constructed using the idea of interpolating a unitary that exchanges the information between the input register and a register initialized to the |0⟩ state, but only after a set number of applications and then leaves the input register untouched for a set number of additional applications.