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Schrödingerization based Quantum Circuits for Maxwell's Equation with time-dependent source terms

Chuwen Ma, Shi Jin, Nana Liu, Kezhen Wang, Lei Zhang

TL;DR

This paper explicitly constructs a quantum circuit for Maxwell's equations with perfect electric conductor boundary conditions and time-dependent source terms, based on Schr\"odingerization and autonomozation, with corresponding computational complexity analysis.

Abstract

The Schrödingerisation method combined with the autonomozation technique in \cite{cjL23} converts general non-autonomous linear differential equations with non-unitary dynamics into systems of autonomous Schrödinger-type equations, via the so-called warped phase transformation that maps the equation into two higher dimension. Despite the success of Schrödingerisation techniques, they typically require the black box of the sparse Hamiltonian simulation, suitable for continuous-variable based analog quantum simulation. For qubit-based general quantum computing one needs to design the quantum circuits for practical implementation. This paper explicitly constructs a quantum circuit for Maxwell's equations with perfect electric conductor (PEC) boundary conditions and time-dependent source terms, based on Schrödingerization and autonomozation, with corresponding computational complexity analysis. Through initial value smoothing and high-order approximation to the delta function, the increase in qubits from the extra dimensions only requires minor rise in computational complexity, almost $\log\log {1/\varepsilon}$ where $\varepsilon$ is the desired precision. Our analysis demonstrates that quantum algorithms constructed using Schrödingerisation exhibit polynomial acceleration in computational complexity compared to the classical Finite Difference Time Domain (FDTD) format.

Schrödingerization based Quantum Circuits for Maxwell's Equation with time-dependent source terms

TL;DR

This paper explicitly constructs a quantum circuit for Maxwell's equations with perfect electric conductor boundary conditions and time-dependent source terms, based on Schr\"odingerization and autonomozation, with corresponding computational complexity analysis.

Abstract

The Schrödingerisation method combined with the autonomozation technique in \cite{cjL23} converts general non-autonomous linear differential equations with non-unitary dynamics into systems of autonomous Schrödinger-type equations, via the so-called warped phase transformation that maps the equation into two higher dimension. Despite the success of Schrödingerisation techniques, they typically require the black box of the sparse Hamiltonian simulation, suitable for continuous-variable based analog quantum simulation. For qubit-based general quantum computing one needs to design the quantum circuits for practical implementation. This paper explicitly constructs a quantum circuit for Maxwell's equations with perfect electric conductor (PEC) boundary conditions and time-dependent source terms, based on Schrödingerization and autonomozation, with corresponding computational complexity analysis. Through initial value smoothing and high-order approximation to the delta function, the increase in qubits from the extra dimensions only requires minor rise in computational complexity, almost where is the desired precision. Our analysis demonstrates that quantum algorithms constructed using Schrödingerisation exhibit polynomial acceleration in computational complexity compared to the classical Finite Difference Time Domain (FDTD) format.

Paper Structure

This paper contains 18 sections, 9 theorems, 125 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. The matrix representation of Maxwell's equations
  3. Schrödingerization for Maxwell's equations
  4. Notations of finite difference operator
  5. Discretization of Maxwell's equations
  6. A review of general framework of Schrödingerisation
  7. Turning a non-autonomous system into an autonomous one
  8. Recovery from Schrödingerisation
  9. A higher order improvement in the extended domains
  10. Quantum circuit for Maxwell's equations
  11. Quantum circuit for $\mathcal{U}_1(\tau)$
  12. Quantum circuits for $\mathcal{U}_2(\tau)$
  13. Quantum circuit for $\mathcal{U}_3(\tau)$
  14. Complexity analysis
  15. Main results
  16. ...and 3 more sections

Key Result

Theorem 3.1

For the non-autonomous system in Equation eq:schro of tilde w, introduce the following initial-value problem of an autonomous PDE One can recover ${\boldsymbol{w}}_h(t)$ from ${\boldsymbol{v}}(t,s)$ using

Figures (8)

  • Figure 1: Quantum circuit for Schrödingerisation of Equation \ref{['eq:Hamiltonian v']}.
  • Figure 2: Quantum circuit for the operator $\tilde{\mathcal{U}}_1(\tau)$, where $I_{\theta} = R_z(-\theta) P(\theta)$.
  • Figure 3: Quantum circuit for $\mathcal{U}_{X,l_j}^{x}(2\theta)$ and $\mathcal{U}_{X,l}^{x}(2\theta)$ with $\mathcal{I}_{j,3m}^{1} = \{0\}$, $\mathcal{I}_{l,n_s}^1 =\{n_s-1\}$.
  • Figure 4: Quantum circuit for $\mathcal{U}_{X_p,l_j}^{x}$ with $\theta = \tau \theta_{l_p}^{\alpha,1}$.
  • Figure 5: Quantum circuit for $\mathcal{U}_{X_p,l}^{x}$ with $\theta = \tau \theta_{l}^{\alpha,1}$.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Lemma 5.1
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • Lemma 5.4
  • Lemma 5.5
  • ...and 3 more