Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Design and Analysis of an Improved Constrained Hypercube Mixer in Quantum Approximate Optimization Algorithm

Arkadiusz Wołk, Karol Capała, Katarzyna Rycerz

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is expected to offer advantages over classical approaches when solving combinatorial optimization problems in the Noisy Intermediate-Scale Quantum (NISQ) era. In its standard formulation, however, QAOA is not suited for constrained problems. One way to incorporate certain types of constraints is to restrict the mixing operator to the feasible subspace; however, this substantially increases circuit size, thereby reducing noise robustness. In this work, we refine an existing hypercube mixer method for enforcing hard constraints in QAOA. We present a modification that generates circuits with fewer gates for a broad class of constrained problems defined by linear functions. Furthermore, we calculate an analytical upper bound on the number of binary variables for which this reduction might not apply. Additionally, we present numerical experimental results demonstrating that the proposed approach improves robustness to noise. In summary, the method proposed in this paper allows for more accurate QAOA performance in noisy settings, bringing us closer to practical, real-world NISQ-era applications.

Design and Analysis of an Improved Constrained Hypercube Mixer in Quantum Approximate Optimization Algorithm

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is expected to offer advantages over classical approaches when solving combinatorial optimization problems in the Noisy Intermediate-Scale Quantum (NISQ) era. In its standard formulation, however, QAOA is not suited for constrained problems. One way to incorporate certain types of constraints is to restrict the mixing operator to the feasible subspace; however, this substantially increases circuit size, thereby reducing noise robustness. In this work, we refine an existing hypercube mixer method for enforcing hard constraints in QAOA. We present a modification that generates circuits with fewer gates for a broad class of constrained problems defined by linear functions. Furthermore, we calculate an analytical upper bound on the number of binary variables for which this reduction might not apply. Additionally, we present numerical experimental results demonstrating that the proposed approach improves robustness to noise. In summary, the method proposed in this paper allows for more accurate QAOA performance in noisy settings, bringing us closer to practical, real-world NISQ-era applications.
Paper Structure (22 sections, 92 equations, 4 figures, 2 tables)

This paper contains 22 sections, 92 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Quantum Approximate Optimization Algorithm
  5. Constrained hypercube mixer
  6. Quantum noise
  7. Standard implementation
  8. Modified circuit
  9. Extension for multiple linear functions
  10. Sequential
  11. Parallel
  12. Standard method
  13. Modified method
  14. Circuit size analysis
  15. Sequential approach to the standard method
  16. ...and 7 more sections

Figures (4)

  • Figure 1: (a) The overview of the circuit for constrained hypercube mixing operator $U_B$ (see Eq. \ref{['eq:general-ub']}) when $r = 1$ is used in approximation. (b) The standard implementation of operator $U_{B_j}$ (see Eq. \ref{['eq:standard-ubi']}), where $n$ is the number of binary variables, $n_l$ is the number of qubits to calculate linear function and $n_f$ is the number of flags used to control $RX_j$ operator.
  • Figure 2: (a) The overview of the circuit for the modified implementation of $U_{B'}$ (Eq. \ref{['eq:modified-ub']}) when $r = 1$ is used in approximation. (b) The modified implementation of operator $U_{B'_j}$ (Eq. \ref{['eq:modified-ubj']}), where $n$ is the number of binary variables, $n_l$ is the number of qubits to calculate linear function and $n_f$ is the number of flags used to control $RX_j$ operator.
  • Figure 3: Final state fidelity compared to the ideal result with 'depolarizing' noise model for problems with a single constraint (Table \ref{['tab:problems']})
  • Figure 4: Final state fidelity compared to the ideal result with 'depolarizing' noise model for problems with two constraints (Table \ref{['tab:problems']})