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Warm-Starting the VQE with Approximate Complex Amplitude Encoding

Felix Truger, Johanna Barzen, Frank Leymann, Julian Obst

TL;DR

The evaluation of the approach shows that the warm-started VQE reaches higher quality solutions earlier than the original VQE, and opens the path to fruitful combinations of classical approximation algorithms and quantum algorithms.

Abstract

The Variational Quantum Eigensolver (VQE) is a Variational Quantum Algorithm (VQA) to determine the ground state of quantum-mechanical systems. As a VQA, it makes use of a classical computer to optimize parameter values for its quantum circuit. However, each iteration of the VQE requires a multitude of measurements, and the optimization is subject to obstructions, such as barren plateaus, local minima, and subsequently slow convergence. We propose a warm-starting technique, that utilizes an approximation to generate beneficial initial parameter values for the VQE aiming to mitigate these effects. The warm-start is based on Approximate Complex Amplitude Encoding, a VQA using fidelity estimations from classical shadows to encode complex amplitude vectors into quantum states. Such warm-starts open the path to fruitful combinations of classical approximation algorithms and quantum algorithms. In particular, the evaluation of our approach shows that the warm-started VQE reaches higher quality solutions earlier than the original VQE.

Warm-Starting the VQE with Approximate Complex Amplitude Encoding

TL;DR

The evaluation of the approach shows that the warm-started VQE reaches higher quality solutions earlier than the original VQE, and opens the path to fruitful combinations of classical approximation algorithms and quantum algorithms.

Abstract

The Variational Quantum Eigensolver (VQE) is a Variational Quantum Algorithm (VQA) to determine the ground state of quantum-mechanical systems. As a VQA, it makes use of a classical computer to optimize parameter values for its quantum circuit. However, each iteration of the VQE requires a multitude of measurements, and the optimization is subject to obstructions, such as barren plateaus, local minima, and subsequently slow convergence. We propose a warm-starting technique, that utilizes an approximation to generate beneficial initial parameter values for the VQE aiming to mitigate these effects. The warm-start is based on Approximate Complex Amplitude Encoding, a VQA using fidelity estimations from classical shadows to encode complex amplitude vectors into quantum states. Such warm-starts open the path to fruitful combinations of classical approximation algorithms and quantum algorithms. In particular, the evaluation of our approach shows that the warm-started VQE reaches higher quality solutions earlier than the original VQE.
Paper Structure (29 sections, 7 equations, 5 figures)

This paper contains 29 sections, 7 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background and Fundamentals
  3. Variational Quantum Eigensolver
  4. Approximate Complex Amplitude Encoding
  5. Related Work
  6. Warm-Starting the VQE
  7. Biased Initial State
  8. Parameter Initialization
  9. Evaluation
  10. Setup and Configurations
  11. Problem Instances
  12. Classical Approximation
  13. Ansatz and VQE Implementation
  14. Shots and Snapshots
  15. Parameter Initialization
  16. ...and 14 more sections

Figures (5)

  • Figure 1: Shaping WS-VQE: a) The standard VQE without warm-start. b) A simple warm-start for the VQE using amplitude encoding to prepend a biased initial state based on an approximate eigenvector $\vec{v}$ to the VQE Ansatz. c) A warm-start for the VQE that prepends a pretrained ACAE Ansatz instead of amplitude encoding. d) Our approach, a warm-start for VQE based on ACAE pretraining of the VQE Ansatz to approximately encode a given approximate eigenvector.
  • Figure 2: Qiskit's hardware efficient SU(2) Ansatz (EfficientSU2) with two repetitions as used in the evaluation qiskit.
  • Figure 3: Median approximation ratio (horizontal markers) reached after $80$ iterations of VQE on $1\,000$ random problem instances per number of shots.
  • Figure 4: Progress of the (WS-)VQE optimization: Median approximation ratio after each iteration $i\in\{20, \dots, 100\}$ for the $500$ problem instances (see \ref{['subsec:problem-instances']}) with different initial step sizes $\varrho$, as declared in \ref{['subsec:optimizer-config']}.
  • Figure 5: Slices of the parameter space of VQE and ACAE (left) for three different problem instances as compared to the actual fidelity of the quantum states prepared by the Ansatz to the approximate and an optimal eigenvector of the instance, $\vec{v}_\text{approx.}$ and $\vec{v}_\text{opt.}$, (right). Each line corresponds to a problem instance as described in \ref{['subsec:problem-instances']} for which a random two-dimensional slice of the full 18-dimensional parameter space of the Ansatz depicted in \ref{['fig:circuit']} is plotted. The remaining parameter values were selected uniformly at random from $[-\pi, \pi]$. Labels on the color bars indicate minimum and maximum values observed.