With the advent of quantum computers, researchers are exploring if quantum mechanics can be leveraged to solve important problems in ways that may provide advantages not possible with conventional or classical methods. A previous work by O'Malley and Vesselinov in 2016 briefly explored using a quantum annealing machine for solving linear least squares problems for real numbers. They suggested that it is best suited for binary and sparse versions of the problem. In our work, we propose a more compact way to represent variables using two's and one's complement on a quantum annealer. We then do an in-depth theoretical analysis of this approach, showing the conditions for which this method may be able to outperform the traditional classical methods for solving general linear least squares problems. Finally, based on our analysis and observations, we discuss potentially promising areas of further research where quantum annealing can be especially beneficial.
The Quantum Approximate Optimization Algorithm (QAOA) by Farhi et al. is a framework for hybrid quantum/classical optimization. In this paper, we explore using QAOA for binary linear least squares; a problem that can serve as a building block of several other hard problems in linear algebra. Most of the previous efforts in quantum computing for solving these problems were done using the quantum annealing paradigm. For the scope of this work, our experiments were done on the QISKIT simulator and an IBM Q 5 qubit machine. We highlight the possibilities of using QAOA and QAOA-like variational algorithms for solving such problems, where the result outputs produced are classical. We find promising numerical results, and point out some of the challenges involved in currentday experimental implementations of this technique on a cloudbased quantum computer.
The use of quantum computing for applications involving optimization has been regarded as one of the areas it may prove to be advantageous (against classical computation). To further improve the quality of the solutions, post-processing techniques are often used on the results of quantum optimization. One such recent approach is the Multi Qubit Correction (MQC) algorithm by Dorband. In this paper, we will discuss and analyze the strengths and weaknesses of this technique. Then based on our discussion, we perform an experiment on how pairing heuristics on the input of MQC can affect the results of a quantum optimizer and a comparison between MQC and the built-in optimization method that D-wave Systems offers. Among our results, we are able to show that the built-in post-processing rarely beats MQC in our tests. We hope that by using the ideas and insights presented in this paper, researchers and developers will be able to make a more informed decision on what kind of post-processing methods to use for their quantum optimization needs.
The quantum approximate optimization algorithm (QAOA) by Farhi et al. is a quantum computational framework for solving quantum or classical optimization tasks. Here, we explore using QAOA for binary linear least squares (BLLS); a problem that can serve as a building block of several other hard problems in linear algebra, such as the non-negative binary matrix factorization (NBMF) and other variants of the non-negative matrix factorization (NMF) problem. Most of the previous efforts in quantum computing for solving these problems were done using the quantum annealing paradigm. For the scope of this work, our experiments were done on noiseless quantum simulators, a simulator including a device-realistic noise-model, and two IBM Q 5-qubit machines. We highlight the possibilities of using QAOA and QAOA-like variational algorithms for solving such problems, where trial solutions can be obtained directly as samples, rather than being amplitude-encoded in the quantum wavefunction. Our numerics show that even for a small number of steps, simulated annealing can outperform QAOA for BLLS at a QAOA depth of p\leq3p≤3 for the probability of sampling the ground state. Finally, we point out some of the challenges involved in current-day experimental implementations of this technique on cloud-based quantum computers.
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