Quantum Permutation Synchronization
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(I) QUBO Preparation (II) Quantum Annealing (III) Global Synchronization Quantum Permutation Synchronization 1;? 2;? 2 1 Tolga Birdal Vladislav Golyanik Christian Theobalt Leonidas Guibas Unembedding 1Stanford University 2Max Planck Institute for Informatics, SIC QUBO Problem Logical Abstract Formulation We present QuantumSync, the first quantum algorithm for solving a synchronization problem in the context of com- puter vision. In particular, we focus on permutation syn- Embedding chronization which involves solving a non-convex optimiza- Quantum Annealing tion problem in discrete variables. We start by formulating Solution synchronization into a quadratic unconstrained binary opti- mization problem (QUBO). While such formulation respects Unembedding the binary nature of the problem, ensuring that the result is a set of permutations requires extra care. Hence, we: (i) Figure 1. Overview of QuantumSync. QuantumSync formulates show how to insert permutation constraints into a QUBO permutation synchronization as a QUBO and embeds its logical instance on a quantum computer. After running multiple anneals, problem and (ii) solve the constrained QUBO problem on it selects the lowest energy solution as the global optimum. the current generation of the adiabatic quantum computers D-Wave. Thanks to the quantum annealing, we guarantee reconstruction and multi-shape analysis pipelines [86, 23, global optimality with high probability while sampling the 25] because it heavy-lifts the global constraint satisfaction energy landscape to yield confidence estimates. Our proof- while respecting the geometry of the parameters. In fact, of-concepts realization on the adiabatic D-Wave computer most of the multiview-consistent inference problems can be demonstrates that quantum machines offer a promising way expressed as some form of a synchronization [108, 15]. to solve the prevalent yet difficult synchronization problems. In this paper, our focus is permutation synchronization, where the edges of the graph are labeled by permutation 1. Introduction matrices denoting the correspondences either between two 2D images or two 3D shapes. Specifically, we seek to find Computer vision literature accommodates a myriad of an absolute ordering for each point of each frame which efficient and effective methods for processing rich informa- sorts all the corresponding points into the same bin. Un- tion from widely available 2D and 3D cameras. Many algo- fortunately, this problem by definition involves a combi- rithms at our disposal excel at processing single frames or natorial non-convex optimization, for which attaining the sequence of images such as videos [83, 34, 13]. Oftentimes, global minimum is intractable under standard formulations a scene can be observed by multiple cameras, from differ- targeting classical von Neumann computers. This difficulty ent, usually unknown viewpoints. To this date it remains an arXiv:2101.07755v1 [quant-ph] 19 Jan 2021 have encouraged scholars to seek continuous relaxations for open question how to consistently combine the multi-view which either a good local optimum [14, 16] or a closed form cues acquired without respecting any particular order [44]. solution [6, 58,5] could be found. The solution to this Synchronization [92, 42, 93] is one of the proposed solu- approximately equivalent relaxed problem should then be tions to the aforementioned problem. On an abstract note, it rounded back to the permutation matrices to report a valid involves distributing the discrepancies over the graph con- result. Ideally, we would like to avoid such approximations necting multiple viewpoints such that the estimates are con- and work with the original set of constraints at hand. This sistent across all considered nodes. To this end, synchro- is exactly what we propose to do in this work. Because on nization simultaneously averages the pairwise local infor- a classical computer we cannot speak of a global optimality mation into a global one [49, 55,4, 12]. This procedure guarantee of our discrete problem, we turn our attention to is a fundamental piece of most state-of-the-art multi-view a new family of processors, i.e., quantum computers. ? both authors contributed equally to this work A quantum computer is a computing machine which 1 takes advantage of quantum effects such as quantum su- of algorithms. Primarily, there exists a plethora of works perposition, entanglement, tunnelling and contextuality to on group structures arising in different applications [49, 48, solve problems notoriously difficult (i.e., NP-hard) on a 16,5,4, 57, 55,4, 101, 28, 97, 99,6,9]. Some of the pro- classical computer. In fact, numerous quantum algorithms posed solutions are closed form [6,5, 73,4] or minimize have demonstrated improved computational complexity fi- robust losses [54, 27]. Others address certifiability [84] nally reaching the desired supremacy [7]. At this point of and global optimality [21]. Bayesian treatment or uncer- time, we see the first practical uses of quantum comput- tainty quantification is also considered [98, 14, 12, 16] as ers thanks to the programmable quantum processing units well as low rank matrix factorizations [10]. Recent algo- (QPU) available to the research community such as D-Wave rithms tend to incorporate synchronization into deep learn- 2000Q/Advantage system 1:1 [2] or IBM [1]. Because these ing [59, 43]. In this work, we are concerned with syn- new generation of computers offer a completely different chronizing correspondence sets, otherwise known as per- computing paradigm, directly porting classical problem for- mutation synchronization (PS) [82]. This sub-field also at- mulations is far from being trivial. In fact, oftentimes we are tracted a descent amount of attention: low-rank formula- required to revise the problem at hand altogether [50]. tions [105, 102], convex programming [56], multi-graph In this paper, we show how to formulate the classical per- matching[87], distributed optimization [56] or Riemannian mutation synchronization problem in terms of a quadratic optimization [16]. unconstrained binary optimization (QUBO) that is quantum All of these approaches try to cope with the intrinsic computer friendly. Unfortunately, a QUBO optimizes for non-convexity of the synchronization problem one way or binary variables and not permutations. Hence, we are re- another. Unfortunately, solving our problem on a classical quired to ensure that permutation constraints are respected computer is notoriously difficult. To the best of our knowl- during optimization. To this end, we turn all constraints into edge, we are the firsts to address this problem through the linear ones and incorporate them into QUBO. Finally, we lens of a new paradigm, adiabatic quantum computing. embed our problem to a real quantum computer and show Quantum computing. Since its motivation in the 1980s that it is highly likely to achieve the lowest energy solution, [72, 39], quantum computing has become an active research the global optimum (see Fig.1). Our contributions are: area, both from the hardware [30, 62, 36, 103, 74, 112, 66, 1. We formulate the classical synchronization in a 69, 67] and algorithmic side [90, 91, 17, 50, 38, 52, 100, 53, form consumable by an adiabatic quantum computer 31,3, 71, 88, 76, 37, 109, 95]. Quantum methods offering (AQC). speedup compared to the classical counterparts have been 2. We show how to introduce permutations as linear con- demonstrated for domains such as applied number theory straints into the QUBO problem. [90, 17, 100, 37], linear algebra [52, 53, 68, 96, 95], ma- 3. We numerically verify the validity of our formulation chine learning [3, 71, 109] and simulation of physical sys- in simulated experiments as well as on a real AQC (for tems [107, 80, 11, 63, 60], among others. the first time on D-Wave Advantage 1:1). Quantum annealing (QA) [61, 22] and the adiabatic 4. We provide extensive ablations studies giving insights quantum evolution algorithm by Farhi et al. [38] have trig- into this new way of computing. gered the development of adiabatic quantum annealers. In We obtain highly probable global optima up to either of: the last decade, the technology has matured and became ac- (i) eight views and three points per view; (ii) seven views cessible remotely for test and research purposes with the with four points; or (iii) five points with three views. We help of D-Wave [32]. A recent benchmarking of D-Wave experiment for the first time on an AQC with 5k qubits and AQC [33] has shown that for energy landscapes with large perform extensive evaluations and comparisons to classical and tall barriers, quantum annealing can achieve speed-ups methods. Our approach can also be classified as the first of up to eight orders of magnitude compared to simulated method for quantum matching of multiple point sets. While annealing [64] running on a single core. AQC has also been we are the first to implement synchronization on quantum successfully applied to traffic flow optimization while out- hardware, our evaluations and tests are proof of concepts as performing classical methods [76]. truly practical quantum computing is still a leap away. Quantum computer vision. Several quantum methods for computer vision problems have been proposed in the litera- 2. Related Work ture including algorithms for image recognition [77], clas- sification [20, 79] and facial feature learning by low-rank Synchronization. The art of consistently recovering abso- matrix factorization [81]. Recently, D-Wave has been ap- lute quantities from a