We prove that the Quantum Approximate Optimization Algorithm (QAOA) approaches the ground-state energy per spin of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity first and circuit depth then increases. For every accuracy, some finite depth and deterministic angles, independent of system size and disorder, achieve that accuracy in the limiting expected energy per spin using the standard cost Hamiltonian and transverse-field mixer. This proves the eventual Parisi-optimality conjecture of Basso, Farhi, Marwaha, Villalonga, and Zhou in its fixed-parameter thermodynamic formulation. We give no quantitative bound on the required depth or efficient angle-selection procedure.
We prove that the Quantum Approximate Optimization Algorithm (QAOA) approaches the ground-state energy per spin of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity first and circuit depth then increases. For every accuracy, some finite depth and deterministic angles, independent of system size and disorder, achieve that accuracy in the limiting expected energy per spin using the standard cost Hamiltonian and transverse-field mixer. This proves the eventual Parisi-optimality conjecture of Basso, Farhi, Marwaha, Villalonga, and Zhou in its fixed-parameter thermodynamic formulation. We give no quantitative bound on the required depth or efficient angle-selection procedure.
The quantum approximate optimization algorithm (QAOA) achieves monotonically improving performance with circuit depth p, yet the study of the high-depth regime has been obstructed by the exponential in p cost of existing exact evaluation techniques. In this Letter, we prove that, in the infinite-size limit, the depth-p QAOA state for the Sherrington-Kirkpatrick (SK) model converges to the state of a spin coupled to p bosonic modes. We simulate the spi-boson system using matrix product states and provide numerical evidence that QAOA obtains a (1-ε) approximation to the optimal energy of the SK model with circuit depth O(n/ε^{1.13}) in the average case. The modest computational cost of our approach allows us to optimize QAOA parameters and observe that QAOA achieves ϵ≲2.2% at p=160 in the infinite-size limit, extending far beyond p≤20 accessible to prior exact methods. Our mapping provides a many-body route to study and optimize high-depth QAOA in regimes previously inaccessible to exact evaluation.
We perform a semiclassical, large-spin S, analysis of the quantum approximate optimization algorithm (QAOA) on the Sherrington-Kirkpatrick (SK) model, using the truncated Wigner approximation. Fixing the QAOA angles to their previously determined optimal S=1/2 values, we observe a non-monotonic dependence of the final energy on the spin. At small S the semiclassics is dominated by noise, while the large-S limit is constrained by the exponential growth of the initial fluctuations. For a depth-p QAOA one achieves the optimal balance at S of order p, resulting in a convergence of the final energy to the Parisi value like log(p)/p. We find that the semiclassics slightly outperforms the true spin-1/2 QAOA, and thus suggest they both converge to the Parisi value in the same way. Finally, removing all the initial noise, and re-optimizing the parameters to account for that change, results in superior performance with 1/p convergence.
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 − ϵ ) 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 2 p 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 → ∞ , 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 Quantum Approximate Optimization Algorithm (QAOA) finds approximate solutions to combinatorial optimization problems. Its performance monotonically improves with its depth p. We apply the QAOA to MaxCut on large-girth D-regular graphs. We give an iterative formula to evaluate performance for any D at any depth p. Looking at random D-regular graphs, at optimal parameters and as D goes to infinity, we find that the p=11 QAOA beats all classical algorithms (known to the authors) that are free of unproven conjectures. While the iterative formula for these D-regular graphs is derived by looking at a single tree subgraph, we prove that it also gives the ensemble-averaged performance of the QAOA on the Sherrington-Kirkpatrick (SK) model defined on the complete graph. We also generalize our formula to Max-q-XORSAT on large-girth regular hypergraphs. Our iteration is a compact procedure, but its computational complexity grows as O(p24p). This iteration is more efficient than the previous procedure for analyzing QAOA performance on the SK model, and we are able to numerically go to p=20. Encouraged by our findings, we make the optimistic conjecture that the QAOA, as p goes to infinity, will achieve the Parisi value. We analyze the performance of the quantum algorithm, but one needs to run it on a quantum computer to produce a string with the guaranteed performance.
Combinatorial optimization is regarded as a potentially promising application of near and long-term quantum computers. The best-known heuristic quantum algorithm for combinatorial optimization on gate-based devices, the Quantum Approximate Optimization Algorithm (QAOA), has been the subject of many theoretical and empirical studies. Unfortunately, its application to specific combinatorial optimization problems poses several difficulties: among these, few performance guarantees are known, and the variational nature of the algorithm makes it necessary to classically optimize a number of parameters. In this work, we partially address these issues for a specific combinatorial optimization problem: diluted spin models, with MAX-CUT as a notable special case. Specifically, generalizing the analysis of the Sherrington-Kirkpatrick model by Farhi et al., we establish an explicit algorithm to evaluate the performance of QAOA on MAX-CUT applied to random Erdos-Renyi graphs of expected degree d for an arbitrary constant number of layers p and as the problem size tends to infinity. This analysis yields an explicit mapping between QAOA parameters for MAX-CUT on Erdos-Renyi graphs of expected degree d, in the limit d→∞, and the Sherrington-Kirkpatrick model, and gives good QAOA variational parameters for MAX-CUT applied to Erdos-Renyi graphs. We then partially generalize the latter analysis to graphs with a degree distribution rather than a single degree d, and finally to diluted spin-models with D-body interactions (D≥3). We validate our results with numerical experiments suggesting they may have a larger reach than rigorously established; among other things, our algorithms provided good initial, if not nearly optimal, variational parameters for very small problem instances where the infinite-size limit assumption is clearly violated.
Quantum Approximate Optimization Algorithm (QAOA) is a promising quantum heuristic with empirical evidence of speedup over classical state-of-the-art for some problems. QAOA uses a parameterized circuit with p layers, where higher p yields better solutions, but requires optimizing 2p independent parameters, which is challenging at large p. We present an iterative interpolation method that exploits the smoothness of optimal parameter schedules by expressing them in a basis of orthogonal functions, generalizing the work of Zhou et al. By optimizing a small number of basis coefficients and iteratively increasing both circuit depth and coefficient count until convergence, our method constructs high-quality schedules for large p. We provide theoretical justification using Jackson's theorem and Lipschitz continuity to bound the required number of basis coefficients for a given accuracy. Our approach achieves better performance with fewer optimization steps than existing methods across three benchmark problems: the Sherrington-Kirkpatrick (SK) model, portfolio optimization, and Low Autocorrelation Binary Sequences (LABS). For the largest LABS instance, we achieve near-optimal merit factors with schedules exceeding 1000 layers, an order of magnitude beyond previous methods. Additionally, we observe that a mild growth in QAOA depth suffices to solve the SK model exactly, a result of independent theoretical interest.