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arXiv:2101.07755 · IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings

Quantum Permutation Synchronization

Tolga Birdal, Vladislav Golyanik, Christian Theobalt, Leonidas J. Guibas

Published Jun 1, 2021• 37 citations
arXivHTMLPDF

Abstract

We present QuantumSync, the first quantum algorithm for solving a synchronization problem in the context of computer vision. In particular, we focus on permutation synchronization which involves solving a non-convex optimization problem in discrete variables. We start by formulating synchronization into a quadratic unconstrained binary optimization problem (QUBO). While such formulation respects the binary nature of the problem, ensuring that the result is a set of permutations requires extra care. Hence, we: (i) show how to insert permutation constraints into a QUBO problem and (ii) solve the constrained QUBO problem on the current generation of the adiabatic quantum computers D-Wave. Thanks to the quantum annealing, we guarantee global optimality with high probability while sampling the energy landscape to yield confidence estimates. Our proof-of-concepts realization on the adiabatic D-Wave computer demonstrates that quantum machines offer a promising way to solve the prevalent yet difficult synchronization problems.

DOI
10.1109/cvpr46437.2021.01292

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We present QuantumSync, the first quantum algorithm for solving a synchronization problem in the context of computer vision. In particular, we focus on permutation synchronization which involves solving a non-convex optimization problem in discrete variables. We start by formulating synchronization into a quadratic unconstrained binary optimization problem (QUBO). While such formulation respects the binary nature of the problem, ensuring that the result is a set of permutations requires extra care. Hence, we: (i) show how to insert permutation constraints into a QUBO problem and (ii) solve the constrained QUBO problem on the current generation of the adiabatic quantum computers D-Wave. Thanks to the quantum annealing, we guarantee global optimality with high probability while sampling the energy landscape to yield confidence estimates. Our proof-of-concepts realization on the adiabatic D-Wave computer demonstrates that quantum machines offer a promising way to solve the prevalent yet difficult synchronization problems.
Matching problems on 3D shapes and images are challenging as they are frequently formulated as combinatorial quadratic assignment problems (QAPs) with permutation matrix constraints, which are NP-hard. In this work, we address such problems with emerging quantum computing technology and propose several reformulations of QAPs as unconstrained problems suitable for efficient execution on quantum hardware. We investigate several ways to inject permutation matrix constraints in a quadratic unconstrained binary optimization problem which can be mapped to quantum hardware. We focus on obtaining a sufficient spectral gap, which further increases the probability to measure optimal solutions and valid permutation matrices in a single run. We perform our experiments on the quantum computer D-Wave 2000Q (2^11 qubits, adiabatic). Despite the observed discrepancy between simulated adiabatic quantum computing and execution on real quantum hardware, our reformulation of permutation matrix constraints increases the robustness of the numerical computations over other penalty approaches in our experiments. The proposed algorithm has the potential to scale to higher dimensions on future quantum computing architectures, which opens up multiple new directions for solving matching problems in 3D computer vision and graphics.
Matching problems on 3D shapes and images are challenging as they are frequently formulated as combinatorial quadratic assignment problems (QAPs) with permutation matrix constraints, which are NP-hard. In this work, we address such problems with emerging quantum computing technology and propose several reformulations of QAPs as unconstrained problems suitable for efficient execution on quantum hardware. We investigate several ways to inject permutation matrix constraints in a quadratic unconstrained binary optimization problem which can be mapped to quantum hardware. We focus on obtaining a sufficient spectral gap, which further increases the probability to measure optimal solutions and valid permutation matrices in a single run. We perform our experiments on the quantum computer D-Wave 2000Q(211qubits, adiabatic). Despite the observed discrepancy between simulated adiabatic quantum computing and execution on real quantum hardware, our reformulation of permutation matrix constraints increases the robustness of the numerical computations over other penalty approaches in our experiments. The proposed algorithm has the potential to scale to higher dimensions on future quantum computing architectures, which opens up multiple new directions for solving matching problems in 3D computer vision and graphics.
Quadratic Unconstrained Binary Optimization (QUBO) is a standard NP-hard optimization problem. Recently, it has gained renewed interest through quantum computing, as QUBOs directly reduce to the Ising model, on which quantum annealing devices are based. We introduce a QUBO formulation for permutations using compare-exchange networks, with only O(nlog⁡2n)O(n \log^2 n)O(nlog2n) binary variables. This is a substantial improvement over the standard permutation matrix encoding, which requires n2n^2n2 variables and has a much denser interaction graph. A central feature of our approach is uniformity: each permutation corresponds to a unique variable assignment, enabling unbiased sampling. Our construction also allows additional constraints, including fixed points and parity. Moreover, it provides a representation of permutations that supports the operations multiplication and inversion, and also makes it possible to check the order of a permutation. This can be used to uniformly generate permutations of a given order or, for example, permutations that commute with a specified permutation. To our knowledge, this is the first result linking oblivious compare-exchange networks with QUBO encodings. While similar functionality can be achieved using permutation matrices, our method yields QUBOs that are both smaller and sparser. We expect this method to be practically useful in areas where unbiased sampling of constrained permutations is important, including cryptography and combinatorial design.
Many combinatorial optimization problems can be formulated as a problem to determine the order of sequence or to find a corresponding mapping of the objects. We call such problems permutation-based optimization problems. Many such problems can be formulated as a quadratic unconstrained binary optimization (QUBO) or Ising model by introducing a penalty coefficient to the permutation constraint terms. While classical and quantum annealing approaches have been proposed to solve QUBOs to date, they face issues with optimality and feasibility. Here we treat a given QUBO solver as a black box and propose techniques to enhance its performance. First, to ensure an effective search for good quality solutions, a smooth energy landscape is needed; we propose a data scaling approach that reduces the amplitudes of the input without compromising optimality. Second, we need to tune the penalty coefficient. In this paper, we illustrate that for certain problems, it suffices to tune the parameter by performing random sampling on the penalty coefficients to achieve good performance. Finally, to handle possible infeasibility of the solution, we introduce a polynomial-time projection algorithm. We apply these techniques along with a divide-and-conquer strategy to solve some large-scale permutation-based problems and present results for TSP and QAP.
We consider the problem of transforming a given symmetric matrix into a nearly block diagonal form by permutation of its rows and columns. Such a transformation is useful as preconditioning to accelerate the convergence of an eigenvalue solver, but the problem of finding an optimal permutation that maximizes the Frobenius norms of the diagonal blocks is NP-complete. We formulate this problem as QUBO (Quadratic Unconstrained Binary Optimization) and solve it using D-Wave Advantage quantum annealing machine. Experimental results on small problems show that the true minimum can be obtained with high probability. We also discuss how to improve the mapping of the problem onto the physical qubit network to increase the size of the problems that can be solved.
Quantum annealing is a method based on simulated annealing where temperature variations are replaced by quantum fluctuations that cause qubit state transitions. ISING problems and Quadratic Unconstrained Binary Optimization problems (QUBO) can be tackled by quantum annealing-based machines. We can link the interconnected qubits of the machine and the binaries of our model: qubits are binary variables and each pair of them linked by a coupler has a strong impact on the equality or the non-equality of the two binaries of the associated pair. Quantum annealing is based on the fact that any system tends to seek its minimum energy state. Starting from qubits in a state of superposition where all the solutions to the problem are fairly represented, the machine will apply a magnetic field by targeting the qubits and couplers in such a way as to make their value energetically favorable in the direction of optimization (minimization). For the coupled qubits, it is the quadratic products of the binary variables that are considered here and the physical system will make it energetically favorable for them to take (or not) the same values. To work on these machines based on quantum annealing in order to solve optimization problems with constraints, the latter must be relaxed while penalizing the Objective function. After presenting the case of the Max-Cut problem, we will look at such constrained problems.