$XY$-mixers: analytical and numerical results for QAOA

Abstract: The Quantum Alternating Operator Ansatz (QAOA) is a promising gate-model meta-heuristic for combinatorial optimization. Applying the algorithm to problems with constraints presents an implementation challenge for near-term quantum resources. This work explores strategies for enforcing hard constraints by using $XY$-Hamiltonians as mixing operators (mixers). Despite the complexity of simulating the $XY$ model, we demonstrate that for problems represented through one-hot-encoding, certain classes of the mixer Hamiltonian can be implemented without Trotter error in depth $O(κ)$ where $κ$ is the number of assignable colors. We also specify general strategies for implementing QAOA circuits on all-to-all connected hardware graphs and linearly connected hardware graphs inspired by fermionic simulation techniques. Performance is validated on graph coloring problems that are known to be challenging for a given classical algorithm. The general strategy of using $XY$-mixers is borne out numerically, demonstrating a significant improvement over the general $X$-mixer, and moreover the generalized $W$-state yields better performance than easier-to-generate classical initial states when $XY$ mixers are used.

• The paper introduces XY-mixers in QAOA to preserve constraints and minimize Trotter error using O(κ)-depth circuits.
• It shows that XY-mixers outperform traditional X-mixers by achieving higher approximation ratios in graph coloring experiments.
• The study outlines strategies for implementing XY-mixers on all-to-all and linear architectures using fermionic simulation techniques.

Analytical and Numerical Evaluation of $XY$-Mixers in QAOA

Introduction to QAOA and $XY$-Mixers

The Quantum Alternating Operator Ansatz (QAOA) is a powerful quantum algorithm designed for addressing combinatorial optimization problems, particularly on near-term quantum devices. A key challenge in applying QAOA is the implementation of constraints, which this paper addresses using $XY$-Hamiltonians as mixing operators (mixers). This approach preserves constraint satisfaction while minimizing Trotter error, demonstrated effectively in encoding discrete values into a quantum circuit with depth $O(\kappa)$, where $\kappa$ represents the number of discrete values.

Given the practicality of implementing QAOA circuits on hardware with limited connectivity, the authors develop strategies for both all-to-all and linearly connected architectures, leveraging techniques from fermionic simulations. The numerical validation focuses on the graph coloring problem, illustrating that $XY$-mixers outperform traditional $X$-mixers and enhance performance when initialized in a generalized $W$-state.

QAOA Framework

QAOA operates by alternating between applying a problem-specific, phase-separating Hamiltonian, denoted as $H_{PS}$, and a mixing unitary, typically parameterized by angles optimized within the framework. The algorithm seeks the ground state of $H_{PS}$ in the presence of constraints by enforcing evolution within a feasible subspace of the solution space.

Central to the paper is the exploration of $XY$-mixers to maintain the computation within the feasible subspace, defined by Hamming-weight constraints. This approach negates the need for penalty terms commonly employed with $X$-mixers, resulting in more efficient optimization and increased proximity to optimal solutions.

Circuit Realization of $XY$-Mixers

Figure 1: Left: The original graph to-be colored. Right: The qubit-layout encoding the problem. Each vertex $v$ is represented by $\kappa$ qubits $x_{v,c}$.

In classical graph coloring translated into quantum circuits, vertices are color-encoded into qubits such that adjacent vertices differ in color. The $XY$-mixers enable direct manipulation of these encoded states by acting on linear superpositions effectively. Within this framework, simultaneous and partitioned mixers on a complete or ring graph topology are considered.

For a complete-graph mixer, the simulation demonstrates the capability to implement the necessary circuits in $O(\kappa)$ depth, assuming ideal connectivity. Conversely, the ring mixer capitalizes on the Jordan-Wigner transformation to convert spin interactions into quadratic fermionic couplings, suitable for systems with reduced connectivity to maintain desirable performance characteristics.

Empirical Evaluation

Figure 2: (a) 2-coloring and (b) 3-coloring of a triangle with level 1 QAOA, depicting the approximation ratio versus penalty weight $\alpha$.

The empirical results profile small hard-to-color graphs, such as the triangle, envelope, and prism graphs. Results indicate a consistent superiority of $XY$-mixers over $X$-mixers in scenarios necessitating high approximation ratios at modest depths.

In addition to performance metrics, the choice between simultaneous and partitioned mixers is explored, with the simultaneous mixer exhibiting a notable advantage across tested graph instances at lower QAOA levels.

Conclusion

The investigation into $XY$-mixers within QAOA indicates their benefit in efficiently enforcing constraints while improving solution quality for complex combinatorial problems. Numerically, they present a compelling case for preference over $X$-mixers, especially on near-term quantum devices where resources are limited.

These findings support continuing exploration into more varied application domains and contribute foundational insights essential for deploying quantum optimization in practical, real-world scenarios. The challenges remaining chiefly relate to scalability and tackling noisier environments, which will maintain their relevance as the field progresses toward more robust quantum architectures.