A new methodology is introduced to solve classical Boolean problems as Hamiltonians, using the quantum approximate optimization algorithm (QAOA). This methodology is termed the "Boolean-Hamiltonians Transform for QAOA" (BHT-QAOA). Because a great deal of research and studies are mainly focused on solving combinatorial optimization problems using QAOA, the BHT-QAOA adds an additional capability to QAOA to find all optimized approximated solutions for Boolean problems, by transforming such problems from Boolean oracles (in different structures) into Phase oracles, and then into the Hamiltonians of QAOA. From such a transformation, we noticed that the total utilized numbers of qubits and quantum gates are dramatically minimized for the generated Hamiltonians of QAOA. In this article, arbitrary Boolean problems are examined by successfully solving them with our BHT-QAOA, using different structures based on various logic synthesis methods, an IBM quantum computer, and a classical optimization minimizer. Accordingly, the BHT-QAOA will provide broad opportunities to solve many classical Boolean-based problems as Hamiltonians, for the practical engineering applications of several algorithms, digital synthesizers, robotics, and machine learning, just to name a few, in the hybrid classical-quantum domain.
A new methodology is introduced to solve classical Boolean problems as Hamiltonians, using the quantum approximate optimization algorithm (QAOA). This methodology is termed the "Boolean-Hamiltonians Transform for QAOA" (BHT-QAOA). Because a great deal of research and studies are mainly focused on solving combinatorial optimization problems using QAOA, the BHT-QAOA adds an additional capability to QAOA to find all optimized approximated solutions for Boolean problems, by transforming such problems from Boolean oracles (in different structures) into Phase oracles, and then into the Hamiltonians of QAOA. From such a transformation, we noticed that the total utilized numbers of qubits and quantum gates are dramatically minimized for the generated Hamiltonians of QAOA. In this article, arbitrary Boolean problems are examined by successfully solving them with our BHT-QAOA, using different structures based on various logic synthesis methods, an IBM quantum computer, and a classical optimization minimizer. Accordingly, the BHT-QAOA will provide broad opportunities to solve many classical Boolean-based problems as Hamiltonians, for the practical engineering applications of several algorithms, digital synthesizers, robotics, and machine learning, just to name a few, in the hybrid classical-quantum domain.
A new methodology is proposed to solve classical Boolean problems as Hamiltonians, using the quantum approximate optimization algorithm (QAOA). Our methodology successfully finds all optimized approximated solutions for Boolean problems, after converting them from Boolean oracles (in different structures) into Phase oracles, and then into the Hamiltonians of QAOA. From such a conversion, we noticed that the total utilized numbers of qubits and quantum gates are dramatically minimized for the final quantum circuits of Hamiltonians. In this paper, arbitrary classical Boolean problems are examined by successfully solving them with our proposed methodology, using structures based on various logic synthesis methods, an IBM quantum computer, and a classical optimization minimizer. Accordingly, this methodology will provide broad opportunities to solve many classical Boolean problems as Hamiltonians, for the practical engineering applications of several algorithms, robotics, machine learning, just to name a few, in the hybrid classical-quantum domain.
The BHT-QAOA is a hybrid classical-quantum algorithm that solves arbitrary classical Boolean problems as Hamiltonians in the quantum domain, using the quantum approximate optimization algorithm (QAOA) [1]. The BHT-QAOA stands for the "Boolean-Hamiltonians Transform for QAOA" [2]. Research and studies are mainly focused on solving combinatorial optimization problems using QAOA, e.g., the MaxCut problem [1]. However, the BHT-QAOA adds an additional capability to QAOA to find all optimized approximated solutions for classical Boolean problems, as expressed in the following steps and demonstrated in the figure below. Design classical Boolean problems into quantum Boolean oracles in different logical structures, such as POS, SOP, ESOP, CSP-SAT, XOR-SAT, just to name a few. Convert these quantum Boolean oracles into quantum Phase oracles. Transform these quantum Phase oracles into the Hamiltonians (HC and HM) of QAOA. Please observe that, from the aforementioned steps, the total utilized numbers of qubits and quantum gates are significantly reduced for the final generated Hamiltonians (HC and HM) of QAOA. Accordingly, the BHT-QAOA will provide broad opportunities to solve many classical Boolean-based problems as Hamiltonians, for the practical engineering applications of several algorithms, digital synthesizers, robotics, machine learning, just to name a few, in the hybrid classical-quantum domain.
This article investigates the performance impact of five classical optimization approximation algorithms on our previously introduced quantum search algorithm, termed the Boolean–Hamiltonians Transform for Quantum Approximate Optimization Algorithm (BHT-QAOA), to effectively search for all best-approximated solutions for Boolean-based problems. These optimization approximation algorithms are BFGS, L-BFGS-B, SLSQP, COBYLA, and COBYQA. Their performance impact is evaluated and compared using two proposed performance metrics—(i) the final number of function evaluations (the lower numbers denote the best optimization approximation algorithms) and (ii) the final quality of qubit measurements (the higher values indicate all best-approximated solutions were found for a problem). Arbitrary classical Boolean problems in various logical structures were examined and evaluated using the BHT-QAOA, these five optimization approximation algorithms, and a simulated noisy model of an IBM quantum computer. Broadly, the BHT-QAOA, with these five algorithms, successfully finds all optimized approximated solutions for these problems. Specifically, both the BFGS and SLSQP algorithms successfully find all best-approximated solutions for these problems, in the context of fewer function evaluations and higher quality of qubit measurements.
The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum algorithm described as ansatzes that represent both the problem and the mixer Hamiltonians. Both are parameterizable unitary transformations executed on a quantum machine/simulator and whose parameters are iteratively optimized using a classical device to optimize the problem's expectation value. To do so, in each QAOA iteration, most of the literature uses a quantum machine/simulator to measure the QAOA outcomes. However, this poses a severe bottleneck considering that quantum machines are hardly constrained (e.g. long queuing, limited qubits, etc.), likewise, quantum simulation also induces exponentially-increasing memory usage when dealing with large problems requiring more qubits. These limitations make today's QAOA implementation impractical since it is hard to obtain good solutions with a reasonably-acceptable time/resources. Considering these facts, this work presents a new approach with two main contributions, including (I) removing the need for accessing quantum devices or large-sized classical machines during the QAOA optimization phase, and (II) ensuring that when dealing with some k-bounded pseudo-Boolean problems, optimizing the exact problem's expectation value can be done in polynomial time using a classical computer.
The Quantum Approximate Optimization Algorithm (QAOA) is an algorithmic framework for finding approximate solutions to combinatorial optimization problems. It consists of interleaved unitary transformations induced by two operators labelled the mixing and problem Hamiltonians. To fit this framework, one needs to transform the original problem into a suitable form and embed it into these two Hamiltonians. In this paper, for the well-known NP-hard Traveling Salesman Problem (TSP), we encode its constraints into the mixing Hamiltonian rather than the conventional approach of adding penalty terms to the problem Hamiltonian. Moreover, we map edges (routes) connecting each pair of cities to qubits, which decreases the search space significantly in comparison to other approaches. As a result, our method can achieve a higher probability for the shortest round-trip route with only half the number of qubits consumed compared to IBM Q’s approach. We argue the formalization approach presented in this paper would lead to a generalized framework for finding, in the context of QAOA, high-quality approximate solutions to NP optimization problems.
Quantum computing is an emerging field on the multidisciplinary interface between physics, engineering, and computer science with the potential to make a large impact on computational intelligence (CI). The aim of this paper is to introduce quantum approximate optimization methods to the CI community because of direct relevance to solving combinatorial problems. We introduce quantum computing and variational quantum algorithms (VQAs). VQAs are an effective method for the near-term implementation of quantum solutions on noisy intermediate-scale quantum (NISQ) devices with less reliable qubits and early-stage error correction. Then, we explain Farhi et al.'s quantum approximate optimization algorithm (Farhi's QAOA, to prevent confusion). This VQA is generalized by Hadfield et al. to the quantum alternating operator ansatz (QAOA), which is a nature-inspired (particularly, adiabatic) quantum metaheuristic for approximately solving combinatorial optimization problems on gate-based quantum computers. We discuss connections of QAOA to relevant domains, such as computational learning theory and genetic algorithms, discussing current techniques and known results regarding hybrid quantum-classical intelligence systems. We present a schematic of how QAOA is constructed, and also discuss how CI techniques can be used to improve QAOA. We conclude with QAOA implementations for the well-known maximum cut, maximum bisection, and traveling salesperson problems, which can serve as templates for CI practitioners interested in using QAOA.