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Quantum simulation of Liouville equation in geometrical optics with partial transmission and reflection via Schrödingerization

Shi Jin, Shuyi Zhang

Abstract

This paper investigates quantum simulation algorithms for the Liouville equation in geometrical optics with partial transmission and reflection at sharp interfaces, based on the Schrödingerization method. By means of a warped phase transformation in one higher dimension, the Schrödingerization method converts linear partial differential equations into a system of Schrödinger-type equations with unitary evolution, thereby rendering them suitable for quantum simulation. In this work, the Schrödingerization method is combined with a Hamiltonian-preserving scheme that incorporates partial transmission and reflection into the numerical flux. A main difficulty is that the interface treatment in the classical scheme relies on threshold-dependent "if/else" procedures, making it highly nontrivial to reformulate the method in a matrix form suitable for quantum simulation. To overcome this difficulty, we encode the interface conditions into a partial transmission and reflection matrix prepared a priori, rather than during the time evolution. We present detailed constructions of the resulting quantum algorithms and show through complexity analysis that the proposed methods achieve polynomial quantum advantage in the precision parameter $ε$ over their classical counterparts.

Quantum simulation of Liouville equation in geometrical optics with partial transmission and reflection via Schrödingerization

Abstract

This paper investigates quantum simulation algorithms for the Liouville equation in geometrical optics with partial transmission and reflection at sharp interfaces, based on the Schrödingerization method. By means of a warped phase transformation in one higher dimension, the Schrödingerization method converts linear partial differential equations into a system of Schrödinger-type equations with unitary evolution, thereby rendering them suitable for quantum simulation. In this work, the Schrödingerization method is combined with a Hamiltonian-preserving scheme that incorporates partial transmission and reflection into the numerical flux. A main difficulty is that the interface treatment in the classical scheme relies on threshold-dependent "if/else" procedures, making it highly nontrivial to reformulate the method in a matrix form suitable for quantum simulation. To overcome this difficulty, we encode the interface conditions into a partial transmission and reflection matrix prepared a priori, rather than during the time evolution. We present detailed constructions of the resulting quantum algorithms and show through complexity analysis that the proposed methods achieve polynomial quantum advantage in the precision parameter over their classical counterparts.
Paper Structure (17 sections, 3 theorems, 125 equations, 8 figures, 2 algorithms)

This paper contains 17 sections, 3 theorems, 125 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Liouville equation in geometrical optics
  3. The behavior of waves at an interface
  4. Overview of the Schrödingerization method
  5. The Schrödingerization method
  6. Complexity analysis
  7. Example: the linear advection with interface
  8. Quantum simulation of Liouville equation with interface condition \ref{['equ:ddimLiouvilleOpticsinterfacecondition']} in one space dimension
  9. The original algorithm
  10. The ODE form
  11. Schrödingerization
  12. Quantum simulation of Liouville equation with interface condition \ref{['equ:ddimLiouvilleOpticsinterfaceconditionPartial']} in two space dimension
  13. The original algorithm
  14. The ODE form
  15. Schrödingerization
  16. ...and 2 more sections

Key Result

Lemma 3.1

A $d$-sparse Hamiltonian $H$ on $n$ qubits with matrix elements specified to $m$ bits of precision can be simulated for time interval $[0,t]$, error $\epsilon$, and success probability at least $1 - 2\epsilon$ with $\mathcal{O}[td\|H\|_{\max}+\log{(1/\epsilon)}/\log{\log{(1/\epsilon)}}]$ queries to

Figures (8)

  • Figure 1: Wave transmission and reflection at an interface.
  • Figure 2: Example \ref{['ex:I2ex1']}, the density distribution function $f(x,\xi,t)$ at $t=1$. First row: 3D plot; second row: contour plot. First column: the exact solution; second column: the classical numerical solution; third column: the Schrödingerization solution.
  • Figure 3: Example \ref{['ex:I2ex1']}, the density $\rho$ and averaged slowness $u$ at $t = 1$. Solid blue line: the exact solution; green "o": the classical numerical solution; red solid line with "$\times$": the Schrödingerization solution. Left: the density $\rho$; Right: the averaged slowness $u$.
  • Figure 4: Example \ref{['ex:I2ex2']}, the density distribution function $f(x,\xi,t)$ at $t=1$. First row: 3D plot; second row: contour plot. First column: the exact solution; second column: the classical numerical solution; third column: the Schrödingerization solution.
  • Figure 5: Example \ref{['ex:I2ex2']}, the density $\rho$ and averaged slowness $u$ at $t = 1$. Solid blue line: the exact solution; green "o": the classical numerical solution; red solid line with "$\times$": the Schrödingerization solution. Left: the density $\rho$; Right: the averaged slowness $u$.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Lemma 3.1
  • Definition 4.1
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof