Combinatorial optimization is one of the promising applications of quantum computation. Variational Quantum Algorithm (VQA) is a classical-quantum hybrid algorithm framework for noisy quantum devices. However, the quality of the solution is readily affected by statistical fluctuations and physical noise, making it challenging to maintain applicability for largescale problems. In contrast, sampling-based quantum algorithms have recently been successfully applied to large-scale quantum chemistry problems. In the sampling-based framework, the quantum device is used only for sampling, while the ground state and its energy are estimated on the classical device. In this study, we propose the Sampling-based Quantum Optimization Algorithm (SQOA) that leverages these advantages for combinatorial optimization. There are two challenges in constructing SQOA. The first challenge is that we need to encode the optimization problem in a non-diagonal Hamiltonian, even though many VQAs encode it into the Ising Hamiltonian, which is diagonal. The second challenge is that we need a method for efficient preparation of the input state to be sampled. To address the first challenge, we utilize the Quantum Relaxation (QR) method for encoding an optimization problem within a Hamiltonian. This method achieves a higher encoding density by mapping multiple variables to a single qubit, thus requiring fewer qubits for larger problems compared to the Ising Hamiltonian approach. Moreover, to address the second challenge, we investigate the parameter transferability within the Quantum Alternating Operator Ansatz for QR Hamiltonians. We show that restricting parameters to a linear form exhibits moderate transferability for 3-regular Max-Cut problems, similar to transferability observed in the Quantum Approximate Optimization Algorithm. Therefore, this property allows us to efficiently prepare the input state for a large problem instance using the parameters obtained from a small problem instance. We leveraged transferability to create input states and applied SQOA with QR to the Max-Cut instances. By transferring parameters from a 20-node problem, we demonstrate that SQOA with QR provides high-quality solutions for 40-node problems without variational parameter optimization.
Combinatorial optimization is one of the promising applications of quantum computation. Variational Quantum Algorithm (VQA) is a classical-quantum hybrid algorithm framework for noisy quantum devices. However, the quality of the solution is readily affected by statistical fluctuations and physical noise, making it challenging to maintain applicability for largescale problems. In contrast, sampling-based quantum algorithms have recently been successfully applied to large-scale quantum chemistry problems. In the sampling-based framework, the quantum device is used only for sampling, while the ground state and its energy are estimated on the classical device. In this study, we propose the Sampling-based Quantum Optimization Algorithm (SQOA) that leverages these advantages for combinatorial optimization. There are two challenges in constructing SQOA. The first challenge is that we need to encode the optimization problem in a non-diagonal Hamiltonian, even though many VQAs encode it into the Ising Hamiltonian, which is diagonal. The second challenge is that we need a method for efficient preparation of the input state to be sampled. To address the first challenge, we utilize the Quantum Relaxation (QR) method for encoding an optimization problem within a Hamiltonian. This method achieves a higher encoding density by mapping multiple variables to a single qubit, thus requiring fewer qubits for larger problems compared to the Ising Hamiltonian approach. Moreover, to address the second challenge, we investigate the parameter transferability within the Quantum Alternating Operator Ansatz for QR Hamiltonians. We show that restricting parameters to a linear form exhibits moderate transferability for 3-regular Max-Cut problems, similar to transferability observed in the Quantum Approximate Optimization Algorithm. Therefore, this property allows us to efficiently prepare the input state for a large problem instance using the parameters obtained from a small problem instance. We leveraged transferability to create input states and applied SQOA with QR to the Max-Cut instances. By transferring parameters from a 20-node problem, we demonstrate that SQOA with QR provides high-quality solutions for 40-node problems without variational parameter optimization.
Mathematical optimization is a field where the potential of quantum algorithms is actively being explored. The Quantum Approximate Optimization Algorithm (QAOA) serves as a notable quantum-classical hybrid approach aimed at solving combinatorial optimization challenges on Noisy Intermediate-Scale Quantum (NISQ) devices. In QAOA, classical variables find representation in qubits via a one-to-one mapping. A newer entrant, Quantum Random Access Optimization (QRAO), encodes problems using fewer qubits than QAOA by leveraging the Quantum Random Access Code (QRAC) to structure a problem into its Hamiltonian form. Notably, the QRAO Hamiltonian, dubbed the Quantum Relaxation Hamiltonian, boasts of a ground state energy inferior to that of the QAOA Hamiltonian. Nonetheless, the QRAO's ground state exhibits entanglement, making it starkly different from QAOA's ground state. As a workaround, preceding studies formulated solutions to the original problem using a rounding algorithm. Our research introduces the Quantum Relaxation based Branch-and-Bound (QR-BnB) algorithm. This novel method marries Quantum Relaxation and the classical exact solution strategy, Branch-and-Bound. Within the QR-BnB framework, a classical device shoulders the responsibility of the branch-and-bound operation, while its quantum counterpart dedicates itself to determining each ground state energy for the subproblem's Quantum Relaxation Hamiltonian. This fusion eliminates the prerequisite of the rounding algorithm by crafting the optimal solution through branching operations. Preliminary simulations of the QR-BnB approach attest to its capability in accurately solving a MaxCut problem. As future work, exploring its applicability on diverse combinatorial optimization problems holds promise. Our algorithm signals a step forward in harnessing NISQ devices for tackling expansive optimization challenges.
The promise of quantum computing to address complex problems requiring high computational resources has long been hindered by the intrinsic and demanding requirements of quantum hardware development. Nonetheless, the current state of quantum computing, denominated Noisy Intermediate-Scale Quantum (NISQ) era, has introduced algorithms and methods that are able to harness the computational power of current quantum computers with advantages over classical computers (referred to as quantum advantage). Achieving quantum advantage is of particular relevance for the combinatorial optimization domain, since it often implies solving an NP-Hard optimization problem. Moreover, combinatorial problems are highly relevant for practical application areas, such as operations research, or resource allocation problems. Among quantum computing methods, Variational Quantum Algorithms (VQA) have emerged as one of the strongest candidates towards reaching practical applicability of NISQ systems. This paper explores the current state and recent developments of VQAs, emphasizing their applicability to combinatorial optimization. We identify the Quantum Approximate Optimization Algorithm (QAOA) as the leading candidate for these problems. Furthermore, we implement QAOA circuits with varying depths to solve the MaxCut problem on graphs with 10 and 20 nodes, demonstrating the potential and challenges of using VQAs in practical optimization tasks. We release our code, dataset and optimized circuit parameters under https://github.com/DanielFPerez/VQA-for-MaxCut.
Abstract MaxCut is a key NP-hard combinatorial optimization problem. Quantum computing offers methods to solve such problems potentially better than classical counterparts, with the Quantum Approximate Optimization Algorithm (QAOA) being a state-of-the-art example. However, the performance of quantum methods is currently hindered by hardware noise and limited qubit volumes. We present a variational Qubit-Efficient MaxCut (QEMC) algorithm that requires only O(logN) O ( log N ) qubits to tackle graphs of size N , an exponential reduction compared to QAOA. We demonstrate cutting-edge performance for 32-node graph instances (5 qubits) on real superconducting hardware, and for graphs with up to 2048 nodes (11 qubits) via classical simulations. The QEMC algorithm is based on an innovative encoding scheme, with potentially broad applicability, that empowers it with strong noise resilience, but also enables its efficient classical simulation. As such, the QEMC algorithm provides a challenging benchmark for QAOA on noisy devices and offers a novel quantum-inspired approach.
Variational quantum algorithms have emerged as a cornerstone of contemporary quantum algorithms research. While they have demonstrated considerable promise in solving problems of practical interest, efficiently determining the minimal quantum resources necessary to obtain such a solution remains an open question. In this work, inspired by concepts from classical machine learning, we investigate the impact of overparameterization on the performance of variational algorithms. Our study focuses on the quantum approximate optimization algorithm (QAOA) -- a prominent variational quantum algorithm designed to solve combinatorial optimization problems. We investigate if circuit overparametrization is necessary and sufficient to solve such problems in QAOA, considering two representative problems -- MAX-CUT and MAX-2-SAT. For MAX-CUT we observe that overparametriation is both sufficient and (statistically) necessary for attaining exact solutions, as confirmed numerically for up to 20 qubits. In fact, for MAX-CUT on 2-regular graphs we show the necessity to be exact, based on the analytically found optimal depth. In sharp contrast, for MAX-2-SAT, underparametrized circuits suffice to solve most instances. This result highlights the potential of QAOA in the underparametrized regime, supporting its utility for current noisy devices.
The emergent practical applicability of the Quantum Approximate Optimization Algorithm (QAOA) for approximate combinatorial optimization is a subject of considerable interest. One of the primary limitations of QAOA is the task of finding a set of good parameters, which is usually done using a variational optimization loop. Parameter transfer, or parameter concentration, is a phenomenon where QAOA angles trained on problem instances that are self-similar tend to perform well for other problem instances from that similar class. This suggests a potentially highly efficient and scalable nonvariational learning method for QAOA angle finding. In this work, we systematically study QAOA parameter transferability from small problem sizes (16 and 27 decision variables) onto large problem instances (up to 156 qubits) for heavy-hex graph Ising models with geometrically local higher-order terms using the Julia based QAOA simulation tool to perform classical angle finding for up to 49 QAOA layers ( p ) . Parameter transfer of the fixed angles is validated using a combination of full state vector, projected entangled pair states, matrix product state, and LOWESA numerical simulations. We find that the QAOA parameter transfer from single instances applied to other (unseen) problem instances does not in general provide monotonically improving performance as a function of p —there are many cases where the performance temporarily decreases as a function of p —but despite this the transferred angles have a general trend of improved expectation value as the QAOA depth increases, in many cases converging close to the true ground-state energy of the 100 + qubit instances. We also sample the hardware-compatible Ising models using the ensemble of transfer-learned QAOA parameters on several superconducting qubit IBM quantum processors with 127, 133, and 156 qubits. We find continuous solution quality improvement of the hardware-compatible QAOA circuits run on the IBM noisy intermediate-scale quantum processors up to p = 5 on , up to p = 9 on , and up to p = 10 on .
Solving combinatorial optimization problems using variational quantum algorithms (VQAs) has emerged as a promising research direction. Since the introduction of the Quantum Approximate Optimization Algorithm (QAOA), numerous variants have been proposed to enhance its performance. QAOA was later extended to the Quantum Alternating Operator Ansatz (QAOA+), which generalizes the initial state, phase-separation operator, and mixer to address constrained problems without relying on the standard Quadratic Unconstrained Binary Optimization (QUBO) formulation. However, QAOA+ often requires additional ancilla qubits and a large number of multi-controlled Toffoli gates to prepare the superposition of feasible states, resulting in deep circuits that are challenging for near-term quantum devices. Furthermore, VQAs are generally hindered by issues such as barren plateaus and suboptimal local minima. Recently, Quantum Imaginary Time Evolution (QITE), a groundstate preparation algorithm, has been explored as an alternative to QAOA and its variants. QITE has demonstrated improved performance in quantum chemistry problems and has been applied to unconstrained combinatorial problems such as MaxCut. In this work, we apply the variational form of QITE (VarQITE) to solve the Multiple Knapsack Problem (MKP), a constrained problem, using a Max-Cut-tailored ansatz. To the best of our knowledge, this is the first attempt to address constrained optimization using VarQITE. We show that VarQITE achieves significantly lower mean optimality gaps compared to QAOA and other conventional methods. Moreover, we demonstrate that scaling the Hamiltonian coefficients can further reduce optimization costs and accelerate convergence.