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Variational quantum algorithm for measurement extraction from the Navier-Stokes, Einstein, Maxwell, B-type, Lin-Tsien, Camassa-Holm, DSW, H-S, KdV-B, non-homogeneous KdV, generalized KdV, KdV, translational KdV, sKdV, B-L and Airy equations

Pete Rigas

TL;DR

This work extends a variational quantum algorithm (VQA) framework to tackle nonlinear PDE IVPs by mapping time evolution to a quantum–classical optimization loop. It introduces a quantum nonlinear processing unit (QNPU) and a ZGR-QFT-based ansatz to encode and evolve PDE solutions, deriving explicit cost functions for the Boussinesq-type, Lin–Tsien, and Camassa–Holm equations in terms of operator-time evolutions. Across hundreds of ZGR-QFT ansätze, the approach yields readouts that approximate classical solutions under noiseless or controlled conditions, while discussing noise mitigation and hardware considerations for near-term devices. The study demonstrates the method’s potential for polynomial-time evolution on quantum hardware and illuminates the balance between circuit depth, parameter count, and optimization landscapes in complex nonlinear PDEs. These insights pave the way for quantum-assisted simulations of fluid dynamics, electromagnetism, gravitation, and wave propagation with structured, interpretable cost functions grounded in discretized operator algebra.

Abstract

Classical-quantum hybrid algorithms have recently garnered significant attention, which are characterized by combining quantum and classical computing protocols to obtain readout from quantum circuits of interest. Recent progress due to Lubasch et al in a 2019 paper provides readout for solutions to the Schrodinger and Inviscid Burgers equations, by making use of a new variational quantum algorithm (VQA) which determines the ground state of a cost function expressed with a superposition of expectation values and variational parameters. In the following, we analyze additional computational prospects in which the VQA can reliably produce solutions to other PDEs that are comparable to solutions that have been previously realized classically, which are characterized with noiseless quantum simulations. To determine the range of nonlinearities that the algorithm can process for other IVPs, we study several PDEs, first beginning with the Navier-Stokes equations and progressing to other equations underlying physical phenomena ranging from electromagnetism, gravitation, and wave propagation, from simulations of the Einstein, Boussniesq-type, Lin-Tsien, Camassa-Holm, Drinfeld-Sokolov-Wilson (DSW), and Hunter-Saxton equations. To formulate optimization routines that the VQA undergoes for numerical approximations of solutions that are obtained as readout from quantum circuits, cost functions corresponding to each PDE are provided in the supplementary section after which simulations results from hundreds of ZGR-QFT ansatzae are generated.

Variational quantum algorithm for measurement extraction from the Navier-Stokes, Einstein, Maxwell, B-type, Lin-Tsien, Camassa-Holm, DSW, H-S, KdV-B, non-homogeneous KdV, generalized KdV, KdV, translational KdV, sKdV, B-L and Airy equations

TL;DR

This work extends a variational quantum algorithm (VQA) framework to tackle nonlinear PDE IVPs by mapping time evolution to a quantum–classical optimization loop. It introduces a quantum nonlinear processing unit (QNPU) and a ZGR-QFT-based ansatz to encode and evolve PDE solutions, deriving explicit cost functions for the Boussinesq-type, Lin–Tsien, and Camassa–Holm equations in terms of operator-time evolutions. Across hundreds of ZGR-QFT ansätze, the approach yields readouts that approximate classical solutions under noiseless or controlled conditions, while discussing noise mitigation and hardware considerations for near-term devices. The study demonstrates the method’s potential for polynomial-time evolution on quantum hardware and illuminates the balance between circuit depth, parameter count, and optimization landscapes in complex nonlinear PDEs. These insights pave the way for quantum-assisted simulations of fluid dynamics, electromagnetism, gravitation, and wave propagation with structured, interpretable cost functions grounded in discretized operator algebra.

Abstract

Classical-quantum hybrid algorithms have recently garnered significant attention, which are characterized by combining quantum and classical computing protocols to obtain readout from quantum circuits of interest. Recent progress due to Lubasch et al in a 2019 paper provides readout for solutions to the Schrodinger and Inviscid Burgers equations, by making use of a new variational quantum algorithm (VQA) which determines the ground state of a cost function expressed with a superposition of expectation values and variational parameters. In the following, we analyze additional computational prospects in which the VQA can reliably produce solutions to other PDEs that are comparable to solutions that have been previously realized classically, which are characterized with noiseless quantum simulations. To determine the range of nonlinearities that the algorithm can process for other IVPs, we study several PDEs, first beginning with the Navier-Stokes equations and progressing to other equations underlying physical phenomena ranging from electromagnetism, gravitation, and wave propagation, from simulations of the Einstein, Boussniesq-type, Lin-Tsien, Camassa-Holm, Drinfeld-Sokolov-Wilson (DSW), and Hunter-Saxton equations. To formulate optimization routines that the VQA undergoes for numerical approximations of solutions that are obtained as readout from quantum circuits, cost functions corresponding to each PDE are provided in the supplementary section after which simulations results from hundreds of ZGR-QFT ansatzae are generated.
Paper Structure (22 sections, 100 equations, 5 figures)

This paper contains 22 sections, 100 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Variational quantum algorithmic objects
  4. PDE nonlineaerity library
  5. Paper organization
  6. Cost function derivations
  7. Boussinesq type equation
  8. Statement
  9. Derivation
  10. Lin-Tsien equation
  11. Statement
  12. Derivation
  13. Camassa-Holm equation
  14. Statement
  15. Derivation
  16. ...and 7 more sections

Figures (5)

  • Figure 1: Extracting measurements from the Boussinesq-type equation with $500$ time steps of evolution.
  • Figure 2: Extracting measurements from the Lin-Tsien equation with $400$ time steps of evolution.
  • Figure 3: A portion of the QNPU in quantum circuits for running the VQA for the Camassa-Holm PDE.
  • Figure 4: Extracting measurements from the Camassa-Holm equation with $8500$ time steps. Implementing the variational quantum algorithm for $8500$ time steps yields sinusoidal profiles of motion.
  • Figure 5: Extracting measurements from the Camassa-Holm equation. Implementing the variational quantum algorithm for $4500$ time steps yields sinusoidal profiles of motion.