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 (2209.07714v5)

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.

• The paper demonstrates a VQA framework that approximates solutions to diverse nonlinear PDEs with hybrid quantum-classical optimization.
• It details the application of gradient-based and stochastic optimizers to refine quantum circuit simulations using Cirq and QSimCirq platforms.
• The study highlights improved computational efficiency and accuracy compared to classical methods, suggesting promising applications in fluid dynamics and electrodynamics.

Analyzing the Efficacy of Variational Quantum Algorithms for Solving Partial Differential Equations

The paper presented explores the application of Variational Quantum Algorithms (VQAs) to effectively approximate solutions for a variety of Partial Differential Equations (PDEs) fundamental in modeling complex physical systems. This comprehensive paper examines the efficiency and viability of VQAs in processing a spectrum of PDEs, ranging from well-researched equations like the Navier-Stokes and Maxwell equations to more specialized ones like the KdV-Burgers, Drinfeld-Sokolov-Wilson, and Camassa-Holm equations.

Within the quantum computing paradigm, VQAs are acknowledged for their proficiency in tackling nonlinear PDEs by optimizing quantum circuits through a hybrid approach of quantum-classical computations. The research builds on earlier work by Lubasch et al., highlighting the advantages of utilizing quantum hardware over classical methods for certain computational tasks, particularly in scenarios involving intricate cost functions.

Key Contributions and Findings

1. Varied Equation Analysis: The research articulates quantum strategies for tackling each PDE, addressing their specific non-linear characteristics and computational challenges. Among the diverse equations studied, generalized KdV and Camassa-Holm equations receive particular focus due to their extensive applications in fluid dynamics and wave theory.
2. Algorithmic Framework: The paper outlines the generalized framework of the time evolution of quantum states using the VQA, integrating classical optimization routines for refining the quantum solutions. These routines include gradient-based and stochastic optimizers tested over numerous simulation iterations.
3. Simulation and Measurement: Through the use of the open-source Cirq and QSimCirq platforms, accurate quantum simulations are achieved, enabling the approximation of ground states for the PDEs under analysis. This work critically examines the quantum circuit models' depth and their implications on the viability of extractable solution states.
4. Optimization Challenges: The paper addresses an essential aspect of optimizing quantum circuits - the occurrence of barren plateaus. By evaluating the efficacy of various optimization strategies, including particle swarm and nevergrad optimizers, the paper delineates methodologies to mitigate optimization pitfalls across different PDEs.
5. Quantum Circuit Implementations: For each class of PDEs, the research provides exemplary quantum circuit diagrams outlining the gate models applied for the simulation, showcasing how variational parameters are iteratively refined through quantum-classical feedback loops.
6. Performance Analysis: The research emphasizes the VQA's proficiency in reducing computational complexity while maintaining solution accuracy, as evidenced by alignment with classically known solutions or previously computed approximations using numerical methods.

Implications and Future Directions

The implications of employing VQAs are significant in computational fluid dynamics, electrodynamics, and other fields utilizing PDEs. They offer a promising alternative to classical methodologies, particularly when handling systems requiring high non-linearity resolutions. The paper's findings suggest potential future exploration in scaling quantum algorithms to more complex PDEs and deploying these frameworks on near-term quantum devices, despite the noise challenges prevalent in existing technology.

The versatility demonstrated in adapting quantum algorithms to diverse PDEs underpins promising advancements in quantum simulations. Consequently, researchers in quantum computing and applied mathematics may find pathways for innovative developments, potentially enhancing the synergy between quantum technology and mathematical physics.

In conclusion, the research presented provides substantial insights into applying VQAs across a multitude of PDEs, advocating for the quantum computing landscape's role in addressing multifaceted scientific challenges. The outcomes strengthen the foundation towards developing more expressive quantum algorithms capable of addressing grand challenges in computational modeling and simulations.