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A Quantum Computational Approach to Correspondence Problems on Point Sets

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A Quantum Computational Approach to Correspondence Problems on Point Sets A Quantum Computational Approach to Correspondence Problems on Point Sets Vladislav Golyanik Christian Theobalt Max Planck Institute for Informatics, Saarland Informatics Campus Abstract Modern adiabatic quantum computers (AQC) are al- ready used to solve difficult combinatorial optimisation problems in various domains of science. Currently, only a few applications of AQC in computer vision have been demonstrated. We review AQC and derive a new algorithm for correspondence problems on point sets suitable for ex- ecution on AQC. Our algorithm has a subquadratic com- putational complexity of the state preparation. Examples of successful transformation estimation and point set align- ment by simulated sampling are shown in the systematic ex- perimental evaluation. Finally, we analyse the differences Figure 1: Different 2D point sets — fish [47], qubit, kanji and composer — aligned with our QA approach. For every pair of point sets, the initial in the solutions and the corresponding energy values. misalignment is shown on the left, and the registration is shown on the right. QA is the first transformation estimation and point set alignment method which can be executed on adiabatic quantum computers. 1. Introduction Since their proposal in the early eighties [8, 43, 27], lems (QUBOP) defined as quantum computers have attracted much attention of physi- arg min qTPq; (1) cists and computer scientists. Impressive advances both q2Bn in quantum computing hardware and algorithms have been where q is a set of n binary variables, and P is a symmetric demonstrated over the last thirty years [40, 30, 61, 58, 42, matrix of weights between the variables. The operational 19, 25, 49, 65, 48]. Quantum computers are not universally principle of AQA is grounded on the adiabatic theorem of faster than conventional machines, but they can natively ex- quantum mechanics [17] which states that ecute algorithms relying on quantum parallelism, i.e., the ability to perform operations on exponentially many super- if a quantum-mechanical system is in the ground state imposed memory states simultaneously [59]. of a time-dependent Hamiltonian and parameters of this Hamiltonian are changing gradually enough, the (2) To harness the advantages, carefully designed algorithms system will continue to remain in the ground state dur- are required. Nowadays, the motivation to take advantage of ing the evolution (see Table1 for quantum notions ). quantum effects in computing is also facilitated by the clas- sical computing paradigm approaching its limits, since the In their seminal paper, Farhi et al. [26] have shown quantum effects are becoming non-neglectable while man- that the adiabatic principle (2) can be used for solving ufacturing and using conventional CPUs. As a result, al- NP-complete optimisation problems and laid the founda- ternative paradigms such as massively parallel computing tion for adiabatic quantum computing. Several years later, devices have been brought into being. Aharonov et al. [2] theoretically showed the equivalence be- While universal gate quantum computer technology has tween classical quantum computing and quantum anneal- not yet reached the maturity, modern adiabatic quantum an- ing models. As of 2019-2020, general-purpose quantum nealers (AQA) are already capable of solving difficult real- computers accessible for research purposes and applications world combinatorial optimisation problems [15, 14, 23, 49]. contain up to 20 qubits [19]. In contrast, the latest quantum 10 1 The primary difference of universal gate quantum comput- annealers support up to 2 qubits [21] . Nevertheless, due ing and AQA is that the latter can address objectives formu- 1the amount of logical qubits which are available on this system is an lated as quadratic unconstrained binary optimisation prob- order of magnitude lower since most qubits are used for the error correction 1 to design and practical restrictions, quantum algorithms for quantum notion classical counterpart qubit (states j0i and j1i) bit (states 0 and 1) the gates model such as Shor’s prime number factorisation (time-dependent) Hamiltonian energy functional [61] or Grover’s search algorithms [30] cannot be imple- eigenstate some energy state mented on current quantum annealers. ground state globally optimal energy state Motivation and Contributions. Considering recent suc- quantum system evolution optimisation process quantum annealing [26] simulated annealing [39] cessful applications of AQA in several fields of computa- tional science [42, 49, 65], we are motivated to investigate Table 1: Quantum notions and their counterparts in computer vision. how useful AQA can be for computer vision and which problems can be potentially solved on the new hardware. 2. Preliminaries, Definitions and Notations The vast majority of available materials about quantum an- nealers are either oriented to physicists or lack technical de- In this section, we introduce the reader into the basics of tails and clarity. Our goal is to fill this gap, introduce the quantum computing. See Table1 for a lookup of notions reader into the modern AQA and provide all notions and specific to AQA which have counterparts and interpretation the background to understand, analyse, simulate and design in the classical optimisation theory for computer vision. quantum algorithms for computer vision which can poten- Qubit. Quantum computing encompasses tasks which can tially run on modern AQA, as well as interpret the results. be performed on quantum-mechanical systems [53]. Quan- We consider correspondence problems on point sets tum superposition and entanglement are two forms of par- which have various applications in computer vision. They allelism evidenced in quantum computers. A qubit is a consist in finding an optimal rigid transformation between quantum-mechanical equivalent of a classical bit. A qubit inputs [34, 13, 47, 66]. While transformation estimation as- jφi — written in the Dirac notation — can be in the state sumes known matches, point set alignment is more general j0i, j1i or an arbitrary superposition of both states denoted and targets, in addition, the recovery of correspondences. by jφi = αj0i + βj1i, where α and β are the (generally, We consider two inputs, i.e., a fixed reference point set and a complex) probability amplitudes satisfying jαj2 +jβj2 = 1. j0i+j1i template undergoing a rigid transformation. Thus, our goal In quantum computing, the state p denoted by j+i is 2 is to design a quantum approach for point set alignment often used for initialisation of a qubit register. The state of which can potentially run on AQA and show that it offers a qubit remains hidden during the entire computation and advantages compared to the classical counterparts. reveals when measured. If qubits are entangled, measur- Therefore, we adapt the recent progress in rigid point ing one of them influences the measurement outcome of the set alignment and formulate a globally multiply-linked en- other one [59]. During the measurement, the qubit’s state ergy functional which does not require any intermediate irreversibly collapses to one of the basis states j0i or j1i. correspondence updates [29]. In the gravitational approach Efficient physical realisation of a qubit demand very low (GA) [29], the optimal alignment is achieved when the grav- temperatures. Otherwise, thermal fluctuations will destroy itational potential energy (GPE) of the system with two in- it and lead to arbitrary changes of the measured qubit state. teracting particle swarms is locally minimal. Proceeding One possible physical implementation of a qubit is an from GA, we build the weight matrix P for the associated electron which possesses a spin, i.e., its intrinsic magnetic QUBOP (1) which is unalterably valid in the course of the moment [53, 63]. The spin of an electron can be manip- optimisation. Along with that, we are targeting at a method ulated and brought to the state spin down, spin up, or a which is implementable on classical hardware and can solve superposition of both. A concrete experimentally realised real-world problems, cf. Fig.1. To summarise, the main scheme that uses this property is represented by an atom contributions of this paper are: of phosphorus 31P embedded into a 28Si silicon lattice at- • A self-contained and detailed introduction into modern tached to a transistor [37, 46, 67]. The nucleus of 31P has quantum annealers for computer vision problems, in- a positive charge compensated by electrons. The bundle of cluding notions from quantum physics and computing electrons in the transistor is filled up to the energetic level (Sec.2), modern adiabatic quantum annealers (Sec.3) between the energy of spin-down and spin-up state of 31P. including D-WAVE (Sec. 3.2), and previous and related To change a state of a 31P–28Si qubit, a microwave pulse of works from quantum computing (Sec.4). the frequency — which is equal to the resonance frequency of the atom — is applied to it. The new state jφi depends on • The first quantum approach (QA) to transformation esti- the duration of the exposure. A transistor is used to measure mation (Sec.5) and point set alignment (Sec.6) which a state of the 31P–28Si qubit. If the extra electron of 31P tun- can run on the upcoming quantum annealers (Sec. 6.2). nels into the electron bundle, a positive charge is measured • Experimental analysis of the proposed method in a simu- in the transistor indicating the spin-up state (e.g., j1i). lated environment on several datasets (Sec.7). Fig.2-(a) visualises a qubit with a so-called Bloch sphere. Every qubit can be both in a superposition and en- 2 ... 0 1 0 −i 1 0 (a) z (b) s = 0: + + + + + σx = ; σy = ; σz = : (4) ] 1 0 i 0 0 −1 1 ; 0 [ ... s ... The Pauli matrices are 2×2 Hermitian and unitary. Together g 0 2×2 n + i σ = I l with the identity 2×2, they form a basis for C . a e . x . n y . σ flips the probabilities to measure j0i and j1i, whereas n a z z x σ j0i = j0i, and σ j1i = −|1i.
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