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Variational Quantum Algorithms for Many-Body Systems

Mirko Consiglio

TL;DR

This work develops and assesses variational quantum algorithms (VQAs) tailored for challenging many-body problems on near-term quantum hardware. It introduces three complementary approaches: a VQE for SU($N$) fermions in a number-preserving Hubbard framework, a Variational Separability Verifier (VSV) to quantify entanglement via the Hilbert–Schmidt distance, and a variational Gibbs-state-preparation scheme that estimates the von Neumann entropy through modular ancilla–system PQCs while minimising a generalized Helmholtz free energy. The methods address core NISQ challenges, including BP, measurement overhead, and circuit depth, by leveraging Pauli-string decompositions, efficient grouping, and symmetry-preserving hardware-efficient designs. Results on SU($N$) Hubbard models (SU(3), SU(4)) and thermal-state benchmarks (Ising, XY, XXZ) show high state fidelities in statevector simulations, with degraded but informative performance under realistic noise and hardware constraints, illustrating the approach as a viable pathway for quantum thermodynamics and many-body physics on NISQ devices. The work provides modular, extensible frameworks with open-source code, and outlines clear directions for improving trainability, error mitigation, and scalability, aiming to broaden the practical impact of VQAs in quantum simulation.

Abstract

Variational quantum algorithms (VQAs) incorporate hybrid quantum-classical computation aimed at harnessing the power of noisy intermediate-scale quantum (NISQ) computers to solve challenging computational problems. In this thesis, three main VQAs are presented, each tackling a different facet of many-body physics.

Variational Quantum Algorithms for Many-Body Systems

TL;DR

This work develops and assesses variational quantum algorithms (VQAs) tailored for challenging many-body problems on near-term quantum hardware. It introduces three complementary approaches: a VQE for SU() fermions in a number-preserving Hubbard framework, a Variational Separability Verifier (VSV) to quantify entanglement via the Hilbert–Schmidt distance, and a variational Gibbs-state-preparation scheme that estimates the von Neumann entropy through modular ancilla–system PQCs while minimising a generalized Helmholtz free energy. The methods address core NISQ challenges, including BP, measurement overhead, and circuit depth, by leveraging Pauli-string decompositions, efficient grouping, and symmetry-preserving hardware-efficient designs. Results on SU() Hubbard models (SU(3), SU(4)) and thermal-state benchmarks (Ising, XY, XXZ) show high state fidelities in statevector simulations, with degraded but informative performance under realistic noise and hardware constraints, illustrating the approach as a viable pathway for quantum thermodynamics and many-body physics on NISQ devices. The work provides modular, extensible frameworks with open-source code, and outlines clear directions for improving trainability, error mitigation, and scalability, aiming to broaden the practical impact of VQAs in quantum simulation.

Abstract

Variational quantum algorithms (VQAs) incorporate hybrid quantum-classical computation aimed at harnessing the power of noisy intermediate-scale quantum (NISQ) computers to solve challenging computational problems. In this thesis, three main VQAs are presented, each tackling a different facet of many-body physics.

Paper Structure

This paper contains 92 sections, 214 equations, 43 figures, 9 tables.

Table of Contents

  1. List of Publications
  2. List of Abbreviations
  3. Introduction
  4. (Quantum) Computational Complexity Theory
  5. The Era of Noise. Is there a Solution?
  6. Overview of the Thesis
  7. Building a Variational Quantum Algorithm
  8. Objective Function
  9. Parametrised Quantum Circuit
  10. Problem-Inspired Ansätze
  11. Hardware-Efficient Ansätze
  12. Other Ansätze
  13. Measurement Technique
  14. Classical Optimiser
  15. Gradient-Based Approach
  16. ...and 77 more sections

Figures (43)

  • Figure 1: Illustration of computational complexity classes along with a few examples of problems contained within some of the classes. Note that the containers are suggestive, and have not been mathematically proven for many of the classes in the figure. One important conjecture is in fact whether P $=$NP. Figure adapted from Bharti2022.
  • Figure 2: Applications of VQA. Figure adapted from Cerezo2021b.
  • Figure 3: A diagram showcasing the setup of a VQA, with the four main modules: a) The objective function $f$, which encodes the problem to be solved; b) the PQC, in which the parameters $\bm{\theta}$ are variationally updated to minimise the objective function; c) the measurement technique, which involves basis changes and measurements needed to compute the objective function; and d) the classical optimiser that minimises the objective function while proposing a new set of variational parameters. Figure adapted from Bharti2022.
  • Figure 4: Example of an $n$-qubit hardware-efficient PQC with $m$ layers, where $R_y$ gates are used to hold the parameters with CNOT gates used as the entangling operators.
  • Figure 5: A schematic representation of gradient descent and stochastic gradient descent.
  • ...and 38 more figures