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

TL;DR

This work introduces a dilation framework that converts any linear non-autonomous dynamical system into an autonomous, time-independent quantum dynamics on an enlarged space by introducing a clock mode, enabling continuous-time quantum simulation without time-ordering or intermediate measurements. Through Schrödingerisation, the method extends to non-unitary dynamics and open quantum systems, unifying the treatment of a broad class of linear ODEs and PDEs, including certain nonlinear cases, with explicit constructions of time-independent Hamiltonians $\bar{\boldsymbol{H}}$. The approach applies across continuous-variable, qubit, and hybrid platforms, and is demonstrated via numerical experiments on Hamiltonian dynamics, open quantum systems, and time-dependent Fokker–Planck equations, confirming theoretical error scalings and resource estimates. The framework significantly broadens the utility of quantum simulation for time-dependent problems, enabling resource-efficient, measurement-free implementations and paving the way for analogue quantum simulations of time-dependent phenomena in physics, chemistry, and beyond.

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.

Figures (7)

• Figure 1: Time-independent Hamiltonian simulation for non-autonomous quantum dynamics in qubit or hybrid setting. Here we use the simpler first protocol in Theorem \ref{['thm:two']}, where we have a time-dependent Hamiltonian $\boldsymbol{H}(t)$. This is in principle suitable for fully continuous-variable systems, but for illustration we assume that we use $\log_2(N)$ qubits for the clock mode. Here we can begin with a mixed quantum initial state $\rho_0$ which is simple to prepare and we are assumed we are given $|y_0\rangle$. Then the system can evolve under a time-independent Hamiltonian $\boldsymbol{\bar{H}}$. Retrieval of an approximation to $|y(t)\rangle$ is very simple, where we simply throw away the clock mode at the end.
• Figure 2: Time-independent Hamiltonian simulation for general non-autonomous linear PDEs. Here we use the simpler first protocol in Theorem \ref{['thm:three']}. In the ideal initial state preparation, we have the ancillary initial state as the $s=0$ eigenstate $\rho_0=|s=0\rangle \langle s=0|$. In this scenario, after the evolution of the initial state with respect to the time-independent Hamiltonian $\boldsymbol{\bar{H}}$, we perform the measurement $\hat{P}_{>0}$ postselected on $\xi>0$, and we retrieve the state $|\bar{w}(t)\rangle=\boldsymbol{\bar{w}}(t)/\|\boldsymbol{\bar{w}}(t)\|$ with probability $\|\boldsymbol{\bar{w}}(t)\|^2/\|\boldsymbol{\bar{w}}(0)\|^2$ where $\boldsymbol{\bar{w}}(t)=\exp(-i\boldsymbol{\bar{H}}t)\boldsymbol{\bar{w}}(0)$, with $\boldsymbol{\bar{w}}(0)=\boldsymbol{y}_0|s=0\rangle$. Then $|\bar{w}(t)\rangle=|s=t\rangle |u(t)\rangle$ and by tracing out the ancilla state $|s=t\rangle$, we retrieve $|u(t)\rangle$ exactly. However, in general $|s=0\rangle$ is an ideal state preparation that cannot be achieved in reality and we can use an approximation. We can also prepare more general and possibly mixed state $\rho_0=\iint g(s,s')|s\rangle \langle s'| ds ds'$, where the output state $\gamma(t)=\text{Tr}_s(\sigma(t)) \approx |y(t)\rangle \langle y(t)|$ when $g(s,s)$ is close enough to $\delta(s)$.
• Figure 3: The error quantified by $1-\text{Fid}$ with respect to $\omega$ for various time functions $g$ and magnitude $a$. The discrete data points are errors using the quantum algorithm in Theorem \ref{['thm:three']}, whereas dashed lines represent the theoretical prediction of the error by Lemma \ref{['lemma::error_quantum']}, namely, we draw the line $\mathsf{C}\omega^2$.
• Figure 4: We consider a two-dimensional ODE in \ref{['eg::ode']} and visualise observables $\expval{\sigma_Z}$ along the time for the normalised state $\ket{\boldsymbol{u(t)}}$ for various time-dependence $g$ and $\omega$.
• Figure 5: We visualise observables and their corresponding errors for quantum algorithms with various imperfect clock modes ($s$-mode). This picture corresponds to the third case in Eq. \ref{['eqn::fp_eg']}, namely, $g(t) = 0.5 s^3, \beta(t) = 0.3 s$.

Theorems & Definitions (22)

• Lemma 1
• Theorem 2
• proof : Proof of Theorem \ref{['thm:one']}
• Remark 3
• Remark 4
• Remark 5
• Lemma 6
• proof
• Remark 7
• Example 8: Hamiltonian commuting at different times