[Paper Review] Improving the Accuracy of an Adiabatic Quantum Computer

John E. Dorband|arXiv (Cornell University)|May 4, 2017

Quantum Computing Algorithms and Architecture2 references3 citations

TL;DR

This paper proposes a post-processing method to improve the accuracy of results from D-Wave adiabatic quantum computers by iteratively flipping qubit values to minimize the Ising model objective function. The method checks each qubit's influence (Ii) on the energy function and flips it if doing so reduces the total energy, guaranteeing a local minimum near the original sample, though not necessarily the global minimum. The approach significantly improves sample quality across multiple test cases, bringing results closer to theoretical expectations.

ABSTRACT

The purpose of the D-Wave adiabatic quantum computer is to find a set of qubit values that minimize its objective function. For various reasons, the set of qubit values returned by the D-Wave has errors. This paper presents a method of improving the results returned by the D-Wave. The method individually modifies the qubit values returned by the D-Wave to find a set of values which is a minimum of the objective function. That set however is not necessarily guaranteed to be a global minimum. The method is simple and easily incorporated into any algorithm that has direct access to the sets of values returned by the D-Wave. Examples are also presented that demonstrate the merit of using such a sample improvement method.

Motivation & Objective

• To address the issue of D-Wave quantum computers returning non-minimum energy states due to quantum noise and near-degenerate excited states.
• To develop a simple, general-purpose post-processing technique that improves the quality of samples returned by the D-Wave without requiring changes to the quantum hardware.
• To demonstrate that correcting samples can significantly improve performance in practical applications like Boltzmann machine training and Ising model simulations.
• To quantify the improvement of corrected samples over uncorrected ones in terms of energy minimization and statistical fidelity to theoretical models.

Proposed method

• For each qubit i, compute its influence Ii = ai + Σ(j,i)∈C bijqj, where C is the set of active couplers connected to qubit i.
• If Ii and qi have the same sign, flipping qi to −qi will reduce the objective function F by 2|Ii|, indicating a potential energy decrease.
• Iteratively apply qubit flips based on sign mismatch between Ii and qi until no further reduction in F is possible, resulting in a local minimum.
• The method operates solely on classical post-processing of raw D-Wave samples, requiring only access to the returned qubit values and the problem's coefficients.
• The algorithm is guaranteed to converge to a local minimum of F for the given coefficient set, though not necessarily the global minimum.
• The method is easily integrated into any algorithm that uses D-Wave samples, such as optimization solvers or machine learning training pipelines.

Experimental results

Research questions

• RQ1 Can a simple classical post-processing method significantly improve the quality of samples returned by a D-Wave quantum computer?
• RQ2 How does correcting D-Wave samples affect the statistical behavior of Ising model systems, particularly in terms of qubit and virtual qubit probabilities?
• RQ3 To what extent do corrected samples reduce the discrepancy between experimental D-Wave results and theoretical predictions for the Ising model?
• RQ4 Does post-processing improve the performance of machine learning models trained using D-Wave samples, such as Chimera Boltzmann machines?
• RQ5 How does the correction method affect the distribution of objective function values across multiple samples?

Key findings

• The correction method successfully shifts the distribution of objective function values F toward lower energies, with corrected samples consistently achieving lower F than uncorrected ones across all tested configurations.
• In the random coefficient test, corrected samples were always at least local minima, and the corrected F distribution was shifted slightly toward smaller values compared to uncorrected samples.
• For the Chimera Boltzmann machine, training using corrected D-Wave samples led to higher test accuracy and faster convergence, with corrected training achieving 95% accuracy compared to 85% with uncorrected samples.
• The probability distributions of physical and virtual qubits from corrected samples were significantly closer to the theoretically ideal distributions than those from uncorrected samples, especially in the presence of noise.
• In the 12-qubit chain study, corrected samples showed markedly improved agreement with theoretical Ising model predictions, particularly in the P(q=1) vs. Cc and P(q=1) vs. Cq plots.
• The use of voting over physical qubits in virtual qubits further enhanced fidelity, and corrected samples with voting closely matched theoretical curves, demonstrating robust error mitigation.