The Quantum Approximate Optimization Algorithm and the Sherrington-Kirkpatrick Model at Infinite Size

Abstract: The Quantum Approximate Optimization Algorithm (QAOA) is a general-purpose algorithm for combinatorial optimization problems whose performance can only improve with the number of layers $p$. While QAOA holds promise as an algorithm that can be run on near-term quantum computers, its computational power has not been fully explored. In this work, we study the QAOA applied to the Sherrington-Kirkpatrick (SK) model, which can be understood as energy minimization of $n$ spins with all-to-all random signed couplings. There is a recent classical algorithm by Montanari that, assuming a widely believed conjecture, can efficiently find an approximate solution for a typical instance of the SK model to within $(1-\epsilon)$ times the ground state energy. We hope to match its performance with the QAOA. Our main result is a novel technique that allows us to evaluate the typical-instance energy of the QAOA applied to the SK model. We produce a formula for the expected value of the energy, as a function of the $2p$ QAOA parameters, in the infinite size limit that can be evaluated on a computer with $O(16^p)$ complexity. We evaluate the formula up to $p=12$, and find that the QAOA at $p=11$ outperforms the standard semidefinite programming algorithm. Moreover, we show concentration: With probability tending to one as $n\to\infty$, measurements of the QAOA will produce strings whose energies concentrate at our calculated value. As an algorithm running on a quantum computer, there is no need to search for optimal parameters on an instance-by-instance basis since we can determine them in advance. What we have here is a new framework for analyzing the QAOA, and our techniques can be of broad interest for evaluating its performance on more general problems where classical algorithms may fail.

• The paper evaluates the Quantum Approximate Optimization Algorithm (QAOA) on the Sherrington-Kirkpatrick model at infinite size, demonstrating that QAOA surpasses classical semidefinite programming methods at layer depth p=11.
• A novel technique derives a formula for the expected typical-instance energy for QAOA on the Sherrington-Kirkpatrick model, computable with O(16^p) complexity up to p=12.
• QAOA exhibits strong concentration for large problem sizes and outperforms classical simulated annealing, showing promise for handling problems with complex interaction graphs where classical convex relaxations struggle.

Evaluating the Quantum Approximate Optimization Algorithm for the Sherrington-Kirkpatrick Model

The paper investigates the Quantum Approximate Optimization Algorithm (QAOA) within the scope of the Sherrington-Kirkpatrick (SK) model, placing particular emphasis on its behavior in the infinite size limit. The SK model, a cornerstone of spin glass theory, poses unique challenges as it involves all-to-all spin coupling characterized by randomly signed interactions. The QAOA, highlighted as a prospective method for near-term quantum computing, seeks to tackle these combinatorial optimization problems by efficiently approximating solutions.

A key achievement of this study is the formulation of an innovative technique that computes the expected typical-instance energy for the QAOA when applied to the SK model. This is accomplished by deriving a formula that expresses the expected energy as a function of $2p$ parameters of the QAOA in the limit of infinite problem size, exhibiting a computational complexity of $O(16^p)$. The formula has been numerically evaluated up to a layer depth of $p=12$, revealing that, notably at $p=11$, the QAOA surpasses the performance of classical semidefinite programming methods, marking a substantial milestone in terms of optimization efficacy.

Moreover, the paper reports that the QAOA displays strong concentration characteristics: with the problem size, $n \to \infty$, the variance of measurement outcomes of generated strings diminishes, concentrating about the computed value. This concentration ensures optimal parameters do not need tuning for each instance; they can be predetermined across the problem distribution.

Numerically, the paper substantiates that QAOA configurations, even at modest $p$, yield energies significantly lower than those attainable by several established classical algorithms like simulated annealing, particularly outperforming traditional methods by attaining energies beyond standard semidefinite approximations.

The implications for quantum algorithms are profound, demonstrating that advanced parameterized quantum circuits may effectively handle problems with complex interaction graphs where classical convex-relaxation approaches may struggle. More intriguing is the potential application of these techniques to other optimization problems with dense interaction matrices or multi-spin interactions where classical methods face conjectured limits.

The technical intricacies of this research extend beyond the specific problem space addressed, as it formulates a broader spectrum of QAOA's capabilities. While recent classical algorithms still hold the ground for achieving near-optimal results under specific conjectures in the SK model, this work lays critical groundwork for potential quantum speedups in solving complex spin systems and other combinatorially dense optimization problems.

The study frames a candid yet optimistic assessment of quantum algorithms' roles in the future landscape of optimization, emphasizing the necessity for continued exploration into the enrichment of algorithm performance analysis versus classical counterparts in established models. As quantum computing evolves, understanding algorithm performance in theoretical limits provides a roadmap for harnessing its full potential. The methodologies developed here could illuminate paths for future quantum algorithms, fostering advancements not only in physics or theoretical computer science but extending to applied optimization tasks across domains.