Quantum simulation for time-dependent Hamiltonians -- with applications to non-autonomous ordinary and partial differential equations

Abstract

Non-autonomous dynamical systems appear in a very wide range of interesting applications, both in classical and quantum dynamics, where in the latter case it corresponds to having a time-dependent Hamiltonian. However, the quantum simulation of these systems often needs to appeal to rather complicated procedures involving the Dyson series, considerations of time-ordering, requirement of time steps to be discrete and/or requiring multiple measurements and postselection. These procedures are generally much more complicated than the quantum simulation of time-independent Hamiltonians. Here we propose an alternative formalism that turns any non-autonomous unitary dynamical system into an autonomous unitary system, i.e., quantum system with a time-independent Hamiltonian, in one higher dimension, while keeping time continuous. This makes the simulation with time-dependent Hamiltonians not much more difficult than that of time-independent Hamiltonians, and can also be framed in terms of an analogue quantum system evolving continuously in time. We show how our new quantum protocol for time-dependent Hamiltonians can be performed in a resource-efficient way and without measurements, and can be made possible on either continuous-variable, qubit or hybrid systems. Combined with a technique called Schrodingerisation, this dilation technique can be applied to the quantum simulation of any linear ODEs and PDEs, and nonlinear ODEs and certain nonlinear PDEs, with time-dependent coefficients.

References in corpus (13)

• Adiabatic Quantum Computing
• Hamiltonian Simulation by Qubitization
• Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space
• Higher Order Decompositions of Ordered Operator Exponentials
• Quantum Time
• Time evolution of non-Hermitian Hamiltonian systems
• Time-dependent Hamiltonian simulation with $L^1$-norm scaling
• Quantum simulation of partial differential equations via Schrodingerisation: technical details
• Time-dependent unbounded Hamiltonian simulation with vector norm scaling
• Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations
• Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation
• Optimal Hamiltonian simulation for time-periodic systems
• Control of quantum noise: on the role of dilations

Cited by in corpus (3)

• Efficient explicit gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations
• Unifying framework for quantum simulation algorithms for time-dependent Hamiltonian dynamics
• Quantum Framework for Simulating Linear PDEs with Robin Boundary Conditions