Quantum Hardware-Efficient Selection of Auxiliary Variables for QUBO Formulations

TL;DR

The paper tackles the problem that QAOA performance depends on efficiently embedding QUBO-encoded problems on hardware with limited connectivity. It introduces a hardware-aware auxiliary-variable selection that yields a regular interaction graph, specifically a chain of triangles, improving mapping to architecture with constrained qubit connectivity. Empirical results on an IBM device show an average circuit-depth improvement of $\approx 39.2\%$ for problems up to $N=16$, at the cost of increased circuit width, and the approach remains complementary to existing compilation techniques. The authors provide open-source implementations and discuss practical considerations for extending the method with qubit reuse and integration with QAOA-optimized compilers.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) requires considered optimization problems to be translated into a compatible format. A popular transformation step in this pipeline involves the quadratization of higher-order binary optimization problems, translating them into Quadratic Unconstrained Binary Optimization (QUBO) formulations through the introduction of auxiliary variables. Conventional algorithms for the selection of auxiliary variables often aim to minimize the total number of required variables without taking the constraints of the underlying quantum computer-in particular, the connectivity of its qubits-into consideration. This quickly results in interaction graphs that are incompatible with the target device, resulting in a substantial compilation overhead even with highly optimized compilers. To address this issue, this work presents a novel approach for the selection of auxiliary variables tailored for architectures with limited connectivity. By specifically constructing an interaction graph with a regular structure and a limited maximal degree of vertices, we find a way to construct QAOA circuits that can be mapped efficiently to a variety of architectures. We show that, compared to circuits constructed from a QUBO formulation using conventional auxiliary selection methods, the proposed approach reduces the circuit depth by almost 40%. An implementation of all proposed methods is publicly available at https://github.com/munich-quantum-toolkit/problemsolver.

Figures (8)

• Figure 1: The QAOA circuit for the optimization problem introduced in \ref{['ex:slack']} using one repetition layer.
• Figure 2: The coupling map of the 133-qubit ibm_torino superconducting quantum computer based on IBM's Heron processor type.
• Figure 3: The interaction graph constructed from the QUBO formulation proposed in \ref{['ex:slack']}, as well as its extension to 8 variables, following the strategy of repeatedly replacing all sub-products of two variables by a new auxiliary variable until only a quadratic cost function is left.
• Figure 4: The interaction graph constructed by selecting auxiliary variables to form chains of triangles. The red lines indicate the ordering in which the variables should be mapped to a path of qubits on the target device.
• Figure 5: Potential types of anomalies in the interaction graph if the variable selection algorithm is executed multiple times.

Theorems & Definitions (6)

• Example 1
• Example 2
• Example 3
• Example 4
• Example 5
• Example 6