A combinatorial optimization problem is to find an optimal combination under the constraints. This is one of the potential applications for quantum computers. Quantum Random Access Optimization (QRAO) is the quantum optimization algorithm that encodes multiple classical variables into a single qubit to construct a quantum Hamiltonian, thereby reducing the number of qubits required. The energy of the ground state of the QRAO Hamiltonian provides a lower bound on the original problem's optimal value before encoding. This property allows the QRAO Hamiltonian to be used as a relaxation of the original problem, and it is thus referred to as a quantum relaxed Hamiltonian. In the Branch-and-Bound method, solving the relaxation problem plays a significant role. In this study, we developed Quantum Relaxation based Branch-and-Bound (QR-BnB), a method incorporating quantum relaxation into the Branch-and-Bound framework. We solved the MaxCut Problem and the Travelling Salesman Problem in our experiments. In all instances in this study, we obtained the optimal solution whenever we successfully computed the exact lower bound through quantum relaxation. Internal strategies, such as relaxation methods and variable selection, influence the convergence of the Branch-and-Bound. Thus, we have further developed the internal strategies for QR-BnB and examined how these strategies influence its convergence. We show that our variable selection strategy via the expectation value of the Pauli operators gives better convergence than the naive random choice. QRAO deals with only unconstrained optimization problems, but QR-BnB can handle constraints more flexibly because of the Branch-and-Bound processes on the classical computing part. We demonstrate that in our experiments with the Travelling Salesman Problem, the convergence of QR-BnB became more than three times faster by using the information in the constraints.
A combinatorial optimization problem is to find an optimal combination under the constraints. This is one of the potential applications for quantum computers. Quantum Random Access Optimization (QRAO) is the quantum optimization algorithm that encodes multiple classical variables into a single qubit to construct a quantum Hamiltonian, thereby reducing the number of qubits required. The energy of the ground state of the QRAO Hamiltonian provides a lower bound on the original problem's optimal value before encoding. This property allows the QRAO Hamiltonian to be used as a relaxation of the original problem, and it is thus referred to as a quantum relaxed Hamiltonian. In the Branch-and-Bound method, solving the relaxation problem plays a significant role. In this study, we developed Quantum Relaxation based Branch-and-Bound (QR-BnB), a method incorporating quantum relaxation into the Branch-and-Bound framework. We solved the MaxCut Problem and the Travelling Salesman Problem in our experiments. In all instances in this study, we obtained the optimal solution whenever we successfully computed the exact lower bound through quantum relaxation. Internal strategies, such as relaxation methods and variable selection, influence the convergence of the Branch-and-Bound. Thus, we have further developed the internal strategies for QR-BnB and examined how these strategies influence its convergence. We show that our variable selection strategy via the expectation value of the Pauli operators gives better convergence than the naive random choice. QRAO deals with only unconstrained optimization problems, but QR-BnB can handle constraints more flexibly because of the Branch-and-Bound processes on the classical computing part. We demonstrate that in our experiments with the Travelling Salesman Problem, the convergence of QR-BnB became more than three times faster by using the information in the constraints.
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.
Relaxation is a common way for dealing with combinatorial optimization problems. QRAO (Quantum Random Access Optimization) is a quantum-relaxation based optimizer that uses fewer qubits than the number of bits in the original problem, by encoding multiple variables per qubit using QRAC (Quantum Random Access Code). Reducing the number of qubits will alleviate physical noise (typically, decoherence) and, as a result, the quality of the binary solution of QRAO may be robust against noise, which is however unknown. In this paper, we numerically demonstrate that the mean approximation ratio of the (3, 1)-QRAC Hamiltonian, i.e., the Hamiltonian utilizing the encoding of 3 bits into 1 qubit by QRAC, is less affected by noise compared to the conventional Ising Hamiltonian used in quantum annealer and QAOA (Quantum Approximate Optimization Algorithm). Based on this observation, we discuss a plausible mechanism behind the robustness of QRAO under depolarizing noise. Finally, we assess the number of shots required to estimate the values of binary variables correctly under depolarizing noise and show that the (3, 1)-QRAC Hamiltonian requires less shots to achieve the same accuracy compared to the Ising Hamiltonian.
Quantum Random Access Optimizer (QRAO) is a quantum-relaxation based optimization algorithm proposed by Fuller et al. that utilizes Quantum Random Access Code (QRAC) to encode multiple variables of binary optimization in a single qubit. The approximation ratio bound of QRAO for the maximum cut problem is 0.555 if the bit-to-qubit compression ratio is 3x, while it is 0.625 if the compression ratio is 2x, thus demonstrating a trade-off between space efficiency and approximability. In this research, we extend the quantum-relaxation by using another QRAC which encodes three classical bits into two qubits (the bit-to-qubit compression ratio is 1.5x) and obtain its approximation ratio for the maximum cut problem as 0.722. Also, we design a novel quantum relaxation that always guarantees a 2x bit-to-qubit compression ratio which is unlike the original quantum relaxation of Fuller~et~al. We analyze the condition when it has a non-trivial approximation ratio bound (>21). We hope that our results lead to the analysis of the quantum approximability and practical efficiency of the quantum-relaxation based approaches.
Compressed quantum encodings aim to overcome hardware limitations towards tackling challenging problems at scale, with many classical variables mapped onto noncommuting observables of fewer qubits. Classically, relaxations such as the semidefinite program formulation of MaxCut trade solution quality for computational efficiency. By contrast, quantum relaxations based on compression can amplify the worst-case complexity of the problem being solved. We study quantum random access optimization (QRAO), a special case of the Pauli correlation encoding (PCE) framework that assigns up to three binary variables to the Pauli X, Y, and Z observables of each qubit, with the packing choices determining the compressed Hamiltonian to be optimized. We identify explicit QRAO optimal energy promise problems complete for NP, StoqMA, and QMA, with inverse-polynomial promise gaps for the latter two. Our problem reductions preserve inverse-polynomial promise gaps without requiring gadgets or ancillas. For any prescribed packing, we show that weighted MaxCut instances compress, up to a known shift and rescaling, to arbitrary nonnegative-weight pairwise Pauli couplings allowed by the packing. For QRAO, using one aligned axis gives an NP-complete energy problem. Using two or three positive aligned Pauli axes generally gives QMA-complete problems, with bipartite restrictions in BQP ∩ StoqMA.
We show that this computational hardness survives compilation and is practically relevant. Notably, this result applies directly to the current QRAO compiler implementation in Qiskit Optimization 0.7.0, confirming our hardness results are not artifacts of artificial or contrived packing rules. Altogether our results identify worst-case complexity barriers arising from quantum compression, while making no broad claims about typical cases or the performance and trainability of algorithm pipelines that use it.
Quantum Random Access Optimizer (QRAO) is a quantum-relaxation based optimization algorithm proposed by Fuller et al. that utilizes Quantum Random Access Code (QRAC) to encode multiple variables of binary optimization in a single qubit. Differing from standard quantum optimizers such as QAOA, it utilizes the eigenstates of local quantum Hamiltonians that are not diagonal in the computational basis. There are indications that quantum entanglement may not be needed to solve binary optimization problems with standard quantum optimizers because their maximal eigenstates of diagonal Hamiltonians include classical states. In this study, while quantumness does not always improve the performance of quantum relaxations, we observed that there exist some instances in which quantum relaxation succeeds to find optimal solutions with the help of quantumness. Our results suggest that QRAO not only can scale the instances of binary optimization problems solvable with limited quantum computers but also can benefit from quantum entanglement,
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.