Fair sampling with temperature-targeted QAOA based on quantum-classical correspondence theory

Abstract

In combinatorial optimization problems with degenerate ground states, fair sampling of degenerate solutions is essential. However, the quantum approximate optimization algorithm (QAOA) with a standard transverse-field mixer induces biases among degenerate states as circuit depth increases. Based on quantum-classical correspondence theory, we propose SBO-QAOA, which employs a temperature-dependent Hamiltonian encoding a Gibbs distribution as its ground state. Numerical simulations show that, unlike standard QAOA, SBO-QAOA yields ground-state probabilities converging to finite-temperature values with uniform distribution among degenerate states. These fairness and temperature-targeting properties are preserved even with only four variational parameters under a linear schedule.

Figures (3)

• Figure 1: (Color online) The structure of the 5-spin toy model used in the numerical simulation. The nodes represent spins, and the edges represent interactions. Blue solid lines indicate ferromagnetic interactions ($J_{ij} =+1$), and red dashed lines indicate antiferromagnetic interactions ($J_{ij} = -1$). The competition between these interactions induces frustration, resulting in a six-fold ground-state degeneracy.
• Figure 2: (Color online) Dependence of the ground-state probability on the circuit depth $p$. (a) full-parameter QAOA, (b) linearized QAOA, (c) full-parameter SBO-QAOA, and (d) linearized SBO-QAOA. The probability of each state-pair (state-pair 1–3) and the total ground-state probability $P_{\mathrm{GS}}$ are plotted.
• Figure 3: (Color online) Dependence of the total variation distance $D_{\mathrm{TVD}}$ between the final distribution $P_p(\sigma)$ and the target Gibbs distribution $P_{\mathrm{Gibbs}}(\sigma)$ on the circuit depth $p$. (a) Full-parameter SBO-QAOA and (b) linearized SBO-QAOA. Different curves correspond to different temperatures $T$.