There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.
There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.
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.
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.
In the search for quantum advantage in real--world problems, one promising avenue is to use a quantum algorithm to improve on the solution found using an efficient classical algorithm. The quantum approximate optimization algorithm (QAOA) is particularly well adapted for such a "warm start" approach, and can be combined with the powerful classical Goemans-Williamson (GW) algorithms based on semi-definite programming. Nonetheless, the best way to leverage the power of the QAOA remains an open question. Here we propose a general model that describes a class of QAOA variants, and use it to explore routes to quantum advantages in a canonical optimization problem, MaxCut. For these algorithms we derive analytic expectation values of the cost Hamiltonian for the MaxCut problem in the level-1 case. Using these analytic results we obtain reliable averages over many instances for fairly large numbers of qubits. We find that the warm start adaptive-bias QAOA (WS-ab-QAOA) initialized by the GW algorithm outperforms previously proposed warm start variants on problems with 40 to 180 qubits. To assess whether a quantum advantage exists with this algorithm, we did numerical simulations with up to 1000 qubits to see whether the level-1 WS-ab-QAOA can improve the GW solution for 3-regular graphs. In fact the improvement in the 1000-qubit case even in level 1 can only be matched by the GW algorithm after about 105.5 random projections performed after the semi-definite program stage. This work gives evidence that the final stage of optimization after an efficient classical algorithm has produced an approximate solution may be a place where quantum advantages can be realized.
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.
This thesis is devoted to four modern quantum optimisation algorithms and their applications. In the first chapter, we provide a complete Quantum Approximate Optimisation Algorithm (QAOA) pipeline for the discrete global minimum-variance portfolio (DGMVP) problem. We introduce an ansatz with efficient initial-state preparation, new hard mixing operators satisfying the constraints of the DGMVP problem, and scalable cost operators utilising binary encoding of the qubits. Benchmarking the method with the most effective optimisation strategies---dual annealing and unfrozen layerwise optimisation---indicates a scaling advantage over classical methods in finding the global optimum of the DGMVP problem. In the second chapter, we show an end-to-end quantum/quantum-inspired Goemans-Williamson semidefinite programming (SDP) method for relaxing Quadratic Unconstrained Binary Optimisation (QUBO) problems and extracting their rounding energy. Our method identifies a class of QUBO instances that is effectively solvable in polylogarithmic time relative to the problem dimension, under the assumptions of Pauli sparsification of the QUBO cost matrix, reducibility of the diagonal constraints, and efficient evaluation of the corresponding Gibbs-state expectation values. Further, we identify the commuting 1D Hamiltonian and the Kronecker graph as practically important instances. Our numerical experiments show that our method solves the SDP and QUBO with up to 250 variables for 1D Hamiltonian instances and achieves results competitive with the best classical solver for Kronecker graphs, giving strong evidence of the power of our algorithm. In the third chapter, we improve the current QAOA method with an iterative warm-start initial-state preparation method. We implement the t-th order statistic state as the warm-start initial state generated from each QAOA optimisation loop. We achieve an improved approximation ratio when solving the 3-regular MaxCut problem, and find that the worst-case approximation ratio converges to that of the best classical algorithm for p=1 QAOA. Additionally, our method achieves more favourable scaling than standalone QAOA, single-string warm-started QAOA, and a classical constrained-sampling approach when solving the DGMVP problem. In the fourth chapter, we provide a hard-constrained Grover adaptive search (GAS) method for fixed-sum quadratic binary optimisation problems, and an extension of the alternating direction method of multipliers (ADMM) framework for fixed-sum quartic binary optimisation. Our method guarantees at least a quadratic speed-up over the classical method, and exponentially reduces the gate count for solving fixed-sum binary optimisation when compared with the soft-constrained GAS in the fault-tolerant quantum computing regime. In summary, our contributions include efficient algorithms and novel frameworks that substantially improve performance and broaden the applicability of these quantum optimisation methods. The proposed methods exhibit favourable scaling behaviour and indicate potential quantum advantages in solving challenging optimisation problems, thereby motivating further research at the interface of quantum computing and optimisation.
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.