Quantum approximate optimization for hard problems in linear algebra

Abstract

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.

Quantum computing for smart grid applications

“…The combinatorial optimization problems are typically NP‐hard and involve finding an optimal object out of a finite set of objects. Although QAOA was originally designed to solve MaxCut problems, it is possible to implement QAOA for QUBOs, and Ising‐type Hamiltonians [105, 106]. The hybrid quantum‐classical approach QAOA requires a quantum processor to prepare a quantum state $$|\psi (\beta , \gamma )\rangle$$ using a unitary $$U(\beta , \gamma )$$ characterized by the set of variational parameters (β, γ).…”

“…The worst‐case AR of QAOA in the MaxCut problem with $$p=1$$ was computed as 0.6924 [104]. Along with the parameter optimization challenge, it was reported in recent research that the performance of QAOA may also be affected by the quantum volume and qubit connectivity of the quantum processors [105]. Besides combinatorial optimizations, the potential applications of QAOA can be machine learning [109], and linear algebra [105].…”

Quantum Model-Discovery

“…Recent algorithm developments have also focused on compatibility with current noisy intermediate scale quantum (NISQ) devices [27]. Such algorithms are typically variational in nature, and are being applied to fields such as quantum chemistry [20,28] and heuristic optimization [29,30].…”

Deterministic Ansätze for the measurement-based variational quantum eigensolver

“…Hybrid quantum-classical algorithms and specifically the variational methods, which embed the problems into parameterized short-depth quantum circuits and employ the classical optimization routines to find the quantum circuits that best solve the problem at hand, have attracted significant interest [6][7][8][9]. They address the problem of estimating the ground state energy of a quantum many-body Hamiltonian and have applications in quantum chemistry [9][10][11], high-energy physics [12][13][14], materials science [15], and classical optimization [16,17]. Alongside the success, this approach suffers from a few challenges: in general, training variational quantum algorithms is NP-hard [18] and the error-mitigation might require a superpolynomial number of samples even for logarithmically shallow circuits, threatening any possible quantum advantage [19].…”