Grover adaptive search (GAS) is a quantum exhaustive search algorithm designed to solve binary optimization problems. In this article, we propose higher order binary formulations that can simultaneously reduce the numbers of qubits and gates required for GAS. Specifically, we consider two novel strategies: one that reduces the number of gates through polynomial factorization, and the other that halves the order of the objective function, subsequently decreasing circuit runtime and implementation cost. Our analysis demonstrates that the proposed higher order formulations improve the convergence performance of GAS by reducing both the search space size and the number of quantum gates. Our strategies are also beneficial for general combinatorial optimization problems using one-hot encoding.
Grover adaptive search (GAS) is a quantum exhaustive search algorithm designed to solve binary optimization problems. In this article, we propose higher order binary formulations that can simultaneously reduce the numbers of qubits and gates required for GAS. Specifically, we consider two novel strategies: one that reduces the number of gates through polynomial factorization, and the other that halves the order of the objective function, subsequently decreasing circuit runtime and implementation cost. Our analysis demonstrates that the proposed higher order formulations improve the convergence performance of GAS by reducing both the search space size and the number of quantum gates. Our strategies are also beneficial for general combinatorial optimization problems using one-hot encoding.
This paper presents a novel approach to Grover adaptive search (GAS) for a combinatorial optimization problem whose objective function involves spin variables. While the GAS algorithm with a conventional design of a quantum dictionary subroutine handles a problem associated with an objective function with binary variables{0,1}, we reformulate the problem using spin variables{+1,−1}to simplify the algorithm. Specifically, we introduce a novel quantum dictionary subroutine that is designed for this spin-based formulation. A key benefit of this approach is the substantial reduction in the number of CNOT gates required to construct the quantum circuit. We theoretically demonstrate that, for certain problems, our proposed approach can reduce the gate complexity from an exponential order to a polynomial order, compared to the conventional binary-based approach. This improvement has the potential to enhance the scalability and efficiency of GAS, particularly in larger quantum computations.
We demonstrate that the search space of the quadratic assignment problem (QAP), known as an NP-hard combinatorial optimization problem, can be reduced using Grover adaptive search (GAS) with permutation preparation operator (PPO). To that end, we first revise the traditional quadratic unconstrained binary optimization (QUBO) formulation of the QAP into a higher-order unconstrained binary optimization (HUBO) formulation, introducing a binary encoding method. Algebraic analyses in terms of the number of qubits, quantum gates, circuit depth, and query complexity are performed, which indicate that our proposed approach significantly reduces the search space size, improving convergence performance to the optimal solution compared to the conventional one. Furthermore, although the PPO for HUBO has a greater circuit depth than the PPO for QUBO, when the analysis is extended to the entire state preparation operator, both HUBO and QUBO exhibit comparable depths. Therefore, owing to its smaller number of variables, HUBO can be concluded to be more effective.
In this paper, we propose a novel method of formulating an NP-hard wireless channel assignment problem as a higher-order unconstrained binary optimization (HUBO), where the Grover adaptive search (GAS) is used to provide a quadratic speedup for solving the problem. The conventional method relies on a one-hot encoding of the channel indices, resulting in a quadratic formulation. By contrast, we conceive ascending and descending binary encodings of the channel indices, construct a specific quantum circuit, and derive the exact numbers of qubits and gates required by GAS. Our analysis clarifies that the proposed HUBO formulation significantly reduces the number of qubits and the query complexity compared with the conventional quadratic formulation. This advantage is achieved at the cost of an increased number of quantum gates, which we demonstrate can be reduced by our proposed descending binary encoding.
In this paper we discuss Grover Adaptive Search (GAS) for Constrained Polynomial Binary Optimization (CPBO) problems, and in particular, Quadratic Unconstrained Binary Optimization (QUBO) problems, as a special case. GAS can provide a quadratic speed-up for combinatorial optimization problems compared to brute force search. However, this requires the development of efficient oracles to represent problems and flag states that satisfy certain search criteria. In general, this can be achieved using quantum arithmetic, however, this is expensive in terms of Toffoli gates as well as required ancilla qubits, which can be prohibitive in the near-term. Within this work, we develop a way to construct efficient oracles to solve CPBO problems using GAS algorithms. We demonstrate this approach and the potential speed-up for the portfolio optimization problem, i.e. a QUBO, using simulation and experimental results obtained on real quantum hardware. However, our approach applies to higher-degree polynomial objective functions as well as constrained optimization problems.
High-order binary optimization (HUBO) formulation can represent many real world problems. This mathematical approach is particularly relevant within the quantum computation domain because quantum hardware allows to directly map and solve these problems. The thesis explores the implementation of Grover Adaptive Search (GAS) by varying the different free parameters to evaluate the quality of the solution and the success rate of the algorithm, along with a possible optimization procedure. In particular, GAS is an algorithm that is compliant with HUBO and theoretically offer a quadratic speed up over classical computation. This thesis implements a quantum circuit for GAS algorithm to solve a 4-city traveling salesman problem along with two strategies to reduce the gate counts. The results include the output quality of the value of the objective function and the success rate of reaching the true minimum of 7, where the optimal configuration achieves an average value of 9.4 and a success rate of 36%. This configuration also maintains a 7% lower average value and a 10% higher success rate compared to randomly generated parameters.
Maximum-likelihood multiuser detection incurs a large computational complexity, and its low-complexity detection scheme suffers from a performance loss, where this tradeoff is inevitable and inherent in a classical computer. In this paper, we use the Grover adaptive search (GAS) to break the tradeoff, which is a quantum exhaustive search algorithm guaranteed to obtain the optimal solution, achieving a quadratic speedup. Specifically, we design two specific parameters of GAS to achieve the optimal performance with a reduced complexity: the initial threshold and the number of Grover rotations. The initial threshold of GAS can be optimized using a solution of semi-definite programming, and it is possible to calculate the distribution of the number of solutions smaller than the initial threshold in advance, which depends on instantaneous channel coefficients. In addition, we analyze the number of quantum gates required for GAS and show that the gate count can be reduced by bypassing the higher-order terms in the objective function, leading to a reduced circuit runtime. Our analysis and simulation results demonstrate that the proposed approach achieves the same performance as the optimal maximum-likelihood detection while reducing the query complexity of GAS, implying that the large constant overhead of quadratic speedup can be further reduced.