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Mitigating Precision Errors in Quantum Annealing via Coefficient Reduction of Embedded Hamiltonians

Kentaro Ohno, Nozomu Togawa

Abstract

Quantum annealing is a quantum algorithm to solve combinatorial optimization problems. In the current quantum annealing devices, the dynamic range of the input Ising Hamiltonian, defined as the ratio of the largest to the smallest coefficient, significantly affects the quality of the output solution due to limited hardware precision. Several methods have been proposed to reduce the dynamic range by reducing large coefficients in the Ising Hamiltonian. However, existing studies do not take into account minor-embedding, which is an essential process in current quantum annealers. In this study, we revisit three existing coefficient-reduction methods under the constraints of minor-embedding. We evaluate to what extent these methods reduce the dynamic range of the minor-embedded Hamiltonian and improve the sample quality obtained from the D-Wave Advantage quantum annealer. The results show that, on the set of problems tested in this study, the interaction-extension method effectively improves the sample quality by reducing the dynamic range, while the bounded-coefficient integer encoding and the augmented Lagrangian method have only limited effects. Furthermore, we empirically show that reducing external field coefficients at the logical Hamiltonian level is not required in practice, since minor-embedding automatically has the role of reducing them. These findings suggest future directions for enhancing the sample quality of quantum annealers by suppressing hardware errors through preprocessing of the input problem.

Mitigating Precision Errors in Quantum Annealing via Coefficient Reduction of Embedded Hamiltonians

Abstract

Quantum annealing is a quantum algorithm to solve combinatorial optimization problems. In the current quantum annealing devices, the dynamic range of the input Ising Hamiltonian, defined as the ratio of the largest to the smallest coefficient, significantly affects the quality of the output solution due to limited hardware precision. Several methods have been proposed to reduce the dynamic range by reducing large coefficients in the Ising Hamiltonian. However, existing studies do not take into account minor-embedding, which is an essential process in current quantum annealers. In this study, we revisit three existing coefficient-reduction methods under the constraints of minor-embedding. We evaluate to what extent these methods reduce the dynamic range of the minor-embedded Hamiltonian and improve the sample quality obtained from the D-Wave Advantage quantum annealer. The results show that, on the set of problems tested in this study, the interaction-extension method effectively improves the sample quality by reducing the dynamic range, while the bounded-coefficient integer encoding and the augmented Lagrangian method have only limited effects. Furthermore, we empirically show that reducing external field coefficients at the logical Hamiltonian level is not required in practice, since minor-embedding automatically has the role of reducing them. These findings suggest future directions for enhancing the sample quality of quantum annealers by suppressing hardware errors through preprocessing of the input problem.

Paper Structure

This paper contains 30 sections, 29 equations, 29 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Quantum Annealers
  3. Numerical Precision in Quantum Annealers
  4. Minor-Embedding
  5. Coefficient Reduction Methods
  6. Interaction-extension Method
  7. Bounded-coefficient Integer Encoding
  8. Augmented Lagrangian Method
  9. Coefficients in Embedded Hamiltonian
  10. Experiments
  11. Experimental Setup
  12. Problem Datasets
  13. Evaluation Workflow
  14. Computational Setup
  15. Reduction of External Fields through Minor-embedding
  16. ...and 15 more sections

Figures (29)

  • Figure 1: Example of minor-embedding of triangle graph onto square lattice. The chain $C(3) = \{3,4\}$ of the logical node $3$ is represented by the dashed box.
  • Figure 2: Procedure of interaction-extension method oku2020reduce.
  • Figure 3: Example of bounded-coefficient encoding karimi2019practical. The coefficient-reduction effect on logical couplings is assessed through $\max_{i\ne j} a_i a_j$, assuming $z^2$ appears in the objective function.
  • Figure 4: Scaling factors on QUBO instances in MQLIB.
  • Figure 5: Probability $P_\mathrm{opt}$ to obtain ground states for the trivial problem versus coupling strength. Baseline probability $0.5$ obtained by uniform sampling is labeled as 'random'.
  • ...and 24 more figures