Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.10191

Regularized Warm-Started Quantum Approximate Optimization and Conditions for Surpassing Classical Solvers on the Max-Cut Problem

Zichang He, Anuj Apte, Brandon Augustino, Arman Babakhani, Abid Khan, Sivaprasad Omanakuttan, Ruslan Shaydulin

quant-phSubmitted Mar 10, 2026• Announced Mar 12, 2026• 0 citations
arXivHTMLPDF

Abstract

Demonstrating quantum heuristics that outperform strong classical solvers on large-scale optimization remains an open challenge. Here we introduce Regularized Warm-Started QAOA (RWS-QAOA), which initializes qubits by minimizing expected energy with a regularizer that penalizes near-bitstring states, preventing QAOA from stalling. We further propose a protocol that yields fixed, instance-independent parameters, enabling RWS-QAOA to operate as a non-variational algorithm in which the quantum circuit parameters are fixed and only a classical warm starting step is instance-dependent. We evaluate RWS-QAOA on the Max-Cut problem for random regular graphs, where this protocol yields a constant-depth quantum circuit, across three complementary settings. First, on Quantinuum's trapped-ion processor, RWS-QAOA outperforms the classical algorithms with the best provable guarantees for Max-Cut on $3$-regular graphs, namely Goemans-Williamson and Halperin-Livnat-Zwick, on $96$-node instances. Second, tensor-network simulations on graphs with up to $N{=}10{,}000$ nodes show that depth-$6$ RWS-QAOA, achieving an average cut fraction of $0.9167$, surpasses the best classical heuristics under matched restrictions (no local-search post-processing and no iterative refinement). Third, we remove these restrictions and benchmark against the strongest unrestricted classical heuristics, including an optimized parallel Burer-Monteiro solver that improves upon the MQLib implementation. Even against this stronger baseline, we project that surface-code RWS-QAOA reaches a quantum-classical runtime crossover below $0.2$ seconds on $3{,}000$-node graphs with fewer than $1.3$ million physical qubits. Our results show that constant-depth quantum circuits combined with a classical warm start have a credible potential to surpass classical solvers on the Max-Cut problem when executed on future quantum computers.

Comments
24 pages, 13 figures

Related papers

Warm-Started QAOA with Custom Mixers Provably Converges and Computationally Beats Goemans-Williamson's Max-Cut at Low Circuit Depths

Reuben Tate, Jai Moondra, Bryan Gard +2 more

We generalize the Quantum Approximate Optimization Algorithm (QAOA) of Farhi et al. (2014) to allow for arbitrary separable initial states with corresponding mixers such that the starting state is the most excited state of the mixing Hamiltonian. We demonstrate this version of QAOA, which we call Q A O A − w a r m e s t , by simulating Max-Cut on weighted graphs. We initialize the starting state as a w a r m − s t a r t using 2 and 3 -dimensional approximations obtained using randomized projections of solutions to Max-Cut's semi-definite program, and define a warm-start dependent c u s t o m m i x e r . We show that these warm-starts initialize the QAOA circuit with constant-factor approximations of 0.658 for 2 -dimensional and 0.585 for 3 -dimensional warm-starts for graphs with non-negative edge weights, improving upon previously known trivial (i.e., 0.5 for standard initialization) worst-case bounds at p = 0 . These factors in fact lower bound the approximation achieved for Max-Cut at higher circuit depths, since we also show that QAOA-warmest with any separable initial state converges to Max-Cut under the adiabatic limit as p → ∞ . However, the choice of warm-starts significantly impacts the rate of convergence to Max-Cut, and we show empirically that our warm-starts achieve a faster convergence compared to existing approaches. Additionally, our numerical simulations show higher quality cuts compared to standard QAOA, the classical Goemans-Williamson algorithm, and a warm-started QAOA without custom mixers for an instance library of 1148 graphs (upto 11 nodes) and depth p = 8 . We further show that QAOA-warmest outperforms the standard QAOA of Farhi et al. in experiments on current IBM-Q and Quantinuum hardware.

arXiv:2112.11354•Quantum•Sep 26, 2023•43 citations

Warm-Starting MaxCut Relaxation via Low-Depth Quantum Approximate Optimization Algorithm

Bao G. Bach, Ilya Safro, Filip B. Maciejewski

Quantum optimization has attracted growing interest as quantum hardware continues to improve, yet state-of-the-art classical solvers remain a formidable benchmark for practical utility. Rather than seeking a fully quantum replacement for classical optimization, we propose a hybrid strategy that uses quantum information to enhance leading classical heuristics. Specifically, we introduce a warm-start method based on local correlators obtained from the Quantum Approximate Optimization Algorithm (QAOA), and use this information to initialize the Burer-Monteiro (BM) rank-two relaxation. We demonstrate numerically that, compared to a random, multi-start initialization baseline (a standard strategy used for BM), this quantum-informed initialization offers a significant head start, i.e., high-quality solutions with very small number of iterations, for two problem classes -- random Erdős Rényi graphs with edge density of $10\%$ (ER-10) and fully-connected Sherrington Kirkpatrick (SK) spin glass models, at $n=500$ and $n=1000$ qubits. At the same time, given enough iterations, the random baseline often eventually catches up and slightly outperforms the warm-start strategy on average, an effect visibly stronger for $n=500$ than for $n=1000$. The results demonstrate an exploitation/exploration tradeoff of using WS to quickly arrive at very good solutions vs exploring slightly better solutions with a larger iterations budget via a standard strategy. Our results highlight how low-depth quantum circuits can provide useful structural information for classical optimization and suggest a promising route toward near-term quantum utility through quantum-assisted initialization.

quant-pharXiv:2608.06212•Aug 7, 2026•0 citations

Bridging Classical and Quantum with SDP initialized warm-starts for QAOA

Reuben Tate, Majid Farhadi, Creston D. Herold +2 more

We study the Quantum Approximate Optimization Algorithm ( QAOA ) in the context of the Max-Cut problem. Noisy quantum devices are only able to accurately execute QAOA at low circuit depths, while classically-challenging problem instances may call for a relatively high circuit-depth. This is due to the need to build correlations between reachable pairs of vertices in potentially large graphs [ 16 ]. To enhance the solving power of low-depth QAOA, we introduce a classical pre-processing step that initializes QAOA with a biased superposition of possible cuts in the graph, referred to as a warm-start . In particular, we initialize QAOA with a solution to a low-rank semidefinite programming relaxation of the Max-Cut problem. Our experimental results show that this variant of QAOA , called QAOA-warm , is able to outperform standard QAOA on lower circuit depths in solution quality and training time. While this improvement is partly due to the classical warm-start, we find strong evidence of further improvement using QAOA circuit at small depth. We provide experimental evidence of improved performance as well as theoretical properties of the proposed framework.

arXiv:2010.14021•ACM Transactions on Quantum Computing•Jul 19, 2022•61 citations

Selection and Optimization of Hyperparameters in Warm-Started Quantum Optimization for the MaxCut Problem

Felix Truger, Martin Beisel, Johanna Barzen +2 more

Today’s quantum computers are limited in their capabilities, e.g., the size of executable quantum circuits. The Quantum Approximate Optimization Algorithm (QAOA) addresses these limitations and is, therefore, a promising candidate for achieving a near-term quantum advantage. Warm-starting can further improve QAOA by utilizing classically pre-computed approximations to achieve better solutions at a small circuit depth. However, warm-starting requirements often depend on the quantum algorithm and problem at hand. Warm-started QAOA (WS-QAOA) requires developers to understand how to select approach-specific hyperparameter values that tune the embedding of classically pre-computed approximations. In this paper, we address the problem of hyperparameter selection in WS-QAOA for the maximum cut problem using the classical Goemans–Williamson algorithm for pre-computations. The contributions of this work are as follows: We implement and run a set of experiments to determine how different hyperparameter settings influence the solution quality. In particular, we (i) analyze how the regularization parameter that tunes the bias of the warm-started quantum algorithm towards the pre-computed solution can be selected and optimized, (ii) compare three distinct optimization strategies, and (iii) evaluate five objective functions for the classical optimization, two of which we introduce specifically for our scenario. The experimental results provide insights on efficient selection of the regularization parameter, optimization strategy, and objective function and, thus, support developers in setting up one of the central algorithms of contemporary and near-term quantum computing.

Electronics•Mar 25, 2022•15 citations

Theoretical Approximation Ratios for Warm-Started QAOA on 3-Regular Max-Cut Instances at Depth $p=1$

Reuben Tate, Stephan Eidenbenz

We generalize Farhi et al.'s 0.6924-approximation result technique of the Max-Cut Quantum Approximate Optimization Algorithm (QAOA) on 3-regular graphs to obtain provable lower bounds on the approximation ratio for warm-started QAOA. Given an initialization angle $θ$, we consider warm-starts where the initial state is a product state where each qubit position is angle $θ$ away from either the north or south pole of the Bloch sphere; of the two possible qubit positions the position of each qubit is decided by some classically obtained cut encoded as a bitstring $b$. We illustrate through plots how the properties of $b$ and the initialization angle $θ$ influence the bound on the approximation ratios of warm-started QAOA. We consider various classical algorithms (and the cuts they produce which we use to generate the warm-start). Our results strongly suggest that there does not exist any choice of initialization angle that yields a (worst-case) approximation ratio that simultaneously beats standard QAOA and the classical algorithm used to create the warm-start. Additionally, we show that at $θ=60^\circ$, warm-started QAOA is able to (effectively) recover the cut used to generate the warm-start, thus suggesting that in practice, this value could be a promising starting angle to explore alternate solutions in a heuristic fashion.

arXiv:2402.12631•Feb 20, 2024•0 citations

Solving General QUBOs with Warm-Start QAOA via a Reduction to Max-Cut

Bikrant Bhattacharyya, Michael Capriotti, Reuben Tate

The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm that finds approximate solutions to problems in combinatorial optimization, especially those that can be formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem. In prior work, researchers have considered various ways of "warm-starting" QAOA by constructing an initial quantum state using classically-obtained solutions or information; these warm-starts typically cause QAOA to yield better approximation ratios at much lower circuit depths. For the Max-Cut problem, one warm-start approaches constructs the initial state using the high-dimensional vectors that are output from an SDP relaxation of the corresponding Max-Cut problem. This work leverages these semidefinite warmstarts for a broader class of problem instances by using a standard reduction that transforms any QUBO instance into a Max-Cut instance. We empirically compare this approach to a "QUBO-relaxation" approach that relaxes the QUBO directly. Our results consider a variety of QUBO instances ranging from randomly generated QUBOs to QUBOs corresponding to specific problems such as the traveling salesman problem, maximum independent set, and portfolio optimization. We find that the best choice of warmstart approach is strongly dependent on the problem type.

arXiv:2504.06253•Apr 8, 2025•1 citation
Demonstrating quantum heuristics that outperform strong classical solvers on large-scale optimization remains an open challenge. Here we introduce Regularized Warm-Started QAOA (RWS-QAOA), which initializes qubits by minimizing expected energy with a regularizer that penalizes near-bitstring states, preventing QAOA from stalling. We further propose a protocol that yields fixed, instance-independent parameters, enabling RWS-QAOA to operate as a non-variational algorithm in which the quantum circuit parameters are fixed and only a classical warm starting step is instance-dependent. We evaluate RWS-QAOA on the Max-Cut problem for random regular graphs, where this protocol yields a constant-depth quantum circuit, across three complementary settings. First, on Quantinuum's trapped-ion processor, RWS-QAOA outperforms the classical algorithms with the best provable guarantees for Max-Cut on 333-regular graphs, namely Goemans-Williamson and Halperin-Livnat-Zwick, on 969696-node instances. Second, tensor-network simulations on graphs with up to N=10,000N{=}10{,}000N=10,000 nodes show that depth-666 RWS-QAOA, achieving an average cut fraction of 0.91670.91670.9167, surpasses the best classical heuristics under matched restrictions (no local-search post-processing and no iterative refinement). Third, we remove these restrictions and benchmark against the strongest unrestricted classical heuristics, including an optimized parallel Burer-Monteiro solver that improves upon the MQLib implementation. Even against this stronger baseline, we project that surface-code RWS-QAOA reaches a quantum-classical runtime crossover below 0.20.20.2 seconds on 3,0003{,}0003,000-node graphs with fewer than 1.31.31.3 million physical qubits. Our results show that constant-depth quantum circuits combined with a classical warm start have a credible potential to surpass classical solvers on the Max-Cut problem when executed on future quantum computers.
We generalize the Quantum Approximate Optimization Algorithm (QAOA) of Farhi et al. (2014) to allow for arbitrary separable initial states with corresponding mixers such that the starting state is the most excited state of the mixing Hamiltonian. We demonstrate this version of QAOA, which we call Q A O A − w a r m e s t , by simulating Max-Cut on weighted graphs. We initialize the starting state as a w a r m − s t a r t using 2 and 3 -dimensional approximations obtained using randomized projections of solutions to Max-Cut's semi-definite program, and define a warm-start dependent c u s t o m m i x e r . We show that these warm-starts initialize the QAOA circuit with constant-factor approximations of 0.658 for 2 -dimensional and 0.585 for 3 -dimensional warm-starts for graphs with non-negative edge weights, improving upon previously known trivial (i.e., 0.5 for standard initialization) worst-case bounds at p = 0 . These factors in fact lower bound the approximation achieved for Max-Cut at higher circuit depths, since we also show that QAOA-warmest with any separable initial state converges to Max-Cut under the adiabatic limit as p → ∞ . However, the choice of warm-starts significantly impacts the rate of convergence to Max-Cut, and we show empirically that our warm-starts achieve a faster convergence compared to existing approaches. Additionally, our numerical simulations show higher quality cuts compared to standard QAOA, the classical Goemans-Williamson algorithm, and a warm-started QAOA without custom mixers for an instance library of 1148 graphs (upto 11 nodes) and depth p = 8 . We further show that QAOA-warmest outperforms the standard QAOA of Farhi et al. in experiments on current IBM-Q and Quantinuum hardware.
Quantum optimization has attracted growing interest as quantum hardware continues to improve, yet state-of-the-art classical solvers remain a formidable benchmark for practical utility. Rather than seeking a fully quantum replacement for classical optimization, we propose a hybrid strategy that uses quantum information to enhance leading classical heuristics. Specifically, we introduce a warm-start method based on local correlators obtained from the Quantum Approximate Optimization Algorithm (QAOA), and use this information to initialize the Burer-Monteiro (BM) rank-two relaxation. We demonstrate numerically that, compared to a random, multi-start initialization baseline (a standard strategy used for BM), this quantum-informed initialization offers a significant head start, i.e., high-quality solutions with very small number of iterations, for two problem classes -- random Erdős Rényi graphs with edge density of 10%10\%10% (ER-10) and fully-connected Sherrington Kirkpatrick (SK) spin glass models, at n=500n=500n=500 and n=1000n=1000n=1000 qubits. At the same time, given enough iterations, the random baseline often eventually catches up and slightly outperforms the warm-start strategy on average, an effect visibly stronger for n=500n=500n=500 than for n=1000n=1000n=1000. The results demonstrate an exploitation/exploration tradeoff of using WS to quickly arrive at very good solutions vs exploring slightly better solutions with a larger iterations budget via a standard strategy. Our results highlight how low-depth quantum circuits can provide useful structural information for classical optimization and suggest a promising route toward near-term quantum utility through quantum-assisted initialization.
We study the Quantum Approximate Optimization Algorithm ( QAOA ) in the context of the Max-Cut problem. Noisy quantum devices are only able to accurately execute QAOA at low circuit depths, while classically-challenging problem instances may call for a relatively high circuit-depth. This is due to the need to build correlations between reachable pairs of vertices in potentially large graphs [ 16 ]. To enhance the solving power of low-depth QAOA, we introduce a classical pre-processing step that initializes QAOA with a biased superposition of possible cuts in the graph, referred to as a warm-start . In particular, we initialize QAOA with a solution to a low-rank semidefinite programming relaxation of the Max-Cut problem. Our experimental results show that this variant of QAOA , called QAOA-warm , is able to outperform standard QAOA on lower circuit depths in solution quality and training time. While this improvement is partly due to the classical warm-start, we find strong evidence of further improvement using QAOA circuit at small depth. We provide experimental evidence of improved performance as well as theoretical properties of the proposed framework.
Today’s quantum computers are limited in their capabilities, e.g., the size of executable quantum circuits. The Quantum Approximate Optimization Algorithm (QAOA) addresses these limitations and is, therefore, a promising candidate for achieving a near-term quantum advantage. Warm-starting can further improve QAOA by utilizing classically pre-computed approximations to achieve better solutions at a small circuit depth. However, warm-starting requirements often depend on the quantum algorithm and problem at hand. Warm-started QAOA (WS-QAOA) requires developers to understand how to select approach-specific hyperparameter values that tune the embedding of classically pre-computed approximations. In this paper, we address the problem of hyperparameter selection in WS-QAOA for the maximum cut problem using the classical Goemans–Williamson algorithm for pre-computations. The contributions of this work are as follows: We implement and run a set of experiments to determine how different hyperparameter settings influence the solution quality. In particular, we (i) analyze how the regularization parameter that tunes the bias of the warm-started quantum algorithm towards the pre-computed solution can be selected and optimized, (ii) compare three distinct optimization strategies, and (iii) evaluate five objective functions for the classical optimization, two of which we introduce specifically for our scenario. The experimental results provide insights on efficient selection of the regularization parameter, optimization strategy, and objective function and, thus, support developers in setting up one of the central algorithms of contemporary and near-term quantum computing.
Theoretical Approximation Ratios for Warm-Started QAOA on 3-Regular Max-Cut Instances at Depth p=1p=1p=1
We generalize Farhi et al.'s 0.6924-approximation result technique of the Max-Cut Quantum Approximate Optimization Algorithm (QAOA) on 3-regular graphs to obtain provable lower bounds on the approximation ratio for warm-started QAOA. Given an initialization angle θθθ, we consider warm-starts where the initial state is a product state where each qubit position is angle θθθ away from either the north or south pole of the Bloch sphere; of the two possible qubit positions the position of each qubit is decided by some classically obtained cut encoded as a bitstring bbb. We illustrate through plots how the properties of bbb and the initialization angle θθθ influence the bound on the approximation ratios of warm-started QAOA. We consider various classical algorithms (and the cuts they produce which we use to generate the warm-start). Our results strongly suggest that there does not exist any choice of initialization angle that yields a (worst-case) approximation ratio that simultaneously beats standard QAOA and the classical algorithm used to create the warm-start. Additionally, we show that at θ=60∘θ=60^\circθ=60∘, warm-started QAOA is able to (effectively) recover the cut used to generate the warm-start, thus suggesting that in practice, this value could be a promising starting angle to explore alternate solutions in a heuristic fashion.
The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm that finds approximate solutions to problems in combinatorial optimization, especially those that can be formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem. In prior work, researchers have considered various ways of "warm-starting" QAOA by constructing an initial quantum state using classically-obtained solutions or information; these warm-starts typically cause QAOA to yield better approximation ratios at much lower circuit depths. For the Max-Cut problem, one warm-start approaches constructs the initial state using the high-dimensional vectors that are output from an SDP relaxation of the corresponding Max-Cut problem. This work leverages these semidefinite warmstarts for a broader class of problem instances by using a standard reduction that transforms any QUBO instance into a Max-Cut instance. We empirically compare this approach to a "QUBO-relaxation" approach that relaxes the QUBO directly. Our results consider a variety of QUBO instances ranging from randomly generated QUBOs to QUBOs corresponding to specific problems such as the traveling salesman problem, maximum independent set, and portfolio optimization. We find that the best choice of warmstart approach is strongly dependent on the problem type.