PhD position in Robust Tensor Networks Algorithms for Quantum Computing at Lille University

Abstract: Quantum computing is at the cutting edge of technological innovation, offering the potential to solve complex problems that classical "binary" computers cannot address. Ten sor algebra, with its comprehensive mathematical framework, offers crucial tools for modeling and approximating large multidimensional datasets. This thesis seeks to investigate the interplay between tensor networks and quantum computing by proposing original, robust (to decoherence of qubits) quantum algorithms that utilize tensor structures to improve computational efficiency and capabilities. This research requires a multidisciplinary understanding of quantum physics and linear algebra. This thesis topic will benefit from the complementary expertises of Remy Boyer (CRISTAL/SIGMA) for the multilinear algebra aspect and Giuseppe Patera (PhLAM, Quantum Information team) for the quantum physics aspect. Key-words: quantum processor, low-rank decomposition, tensor network, qubits, coherence, curse of dimensionality Contact: For application, please contact •Remy BOYER, University of Lille, CRISTAl Lab. , remy.boyer@univ-lille.fr •Giuseppe PATERA, University of Lille, PhLAM Lab. , giuseppe.patera@univ-lille.fr Why quantum processors are attractive solutions ? 1. Quantum processors are based on the superposition principle [1]. In brief, unlike classical bit-based processor where the information is encoded in two states "0 excluding 1" or "1 1 excluding 0", quantum bits (qubits) |0⟩and |1⟩can exist in multiple states simultaneously according to a linear combination of the qubits alphabet α|0⟩+ β|1⟩where |α|2 + |β|2 = 1. 2. Quantum processors are based on the entanglement principle. Qubits can be entangled or correlated, meaning the state of one qubit is directly related to the state of another, regardlessofaphaseparameter. Consequently, α andβ cannotbereducedtoaprobabilistic point of view as the qubit probabilities but include the relative interdependence in the form of a phase-relation between the two states |0⟩and |1⟩. This means that knowing the state of one qubit allows to instantly deduce the state of the other. The two above principles (superposition and entanglement) allow quantum algorithms to perform many calculations in parallel. This leads to potential speed-ups for many important problems. Tensor-based processing Tensor algebra is a powerful mathematical framework [6] that extends the concepts of scalars, vectors, and matrices to higher dimensions, known as tensors. Tensor algebra allows the compact (i.e. low-rank) representation of massive data in multidimensional arrays. The applications are for instance Physics, Machine Learning, Data Science, Computer Graphics, Robotics and Control Systems, etc. Multi-Linear algebra and quantum systems 1. Entanglement and Singular Value Decomposition (SVD) are strongly linked [3]. SVD gives the degree of communication between two subsystems and the entanglement is measured by the number of nonzero singular values of a particular matrix associated to the reshaping of the quantum state. 2. TNs and quantum computing are highly interconnected concepts [2]. They provide an efficient way to graphically represent complex quantum states into connected core tensors (3-order tensors). A quantum state of multiple qubits can be expressed as a graph of core tensors, capturing entanglements between qubits in a more compact form. Some quantum algorithms can benefit from the structure of tensor networks. For example, Matrix Product States (MPS) [7] and Projected Entangled Pair States (PEPS) utilize tensor networks to efficiently represent and manipulate quantum. Quantum architecture and decoherence A typical Quantum architecture is composed by three main steps: 1. Data encoding via Tensor Networks (quantum state preparation), 2. data processing (multi-qubit quantum gates), 3. measurement (quantum state tomography). The proposed work will be mainly focused on step 1 with respect to the constraints of the two other steps. A major drawback of the quantum framework is the decoherence of qubits. Qubits are highly susceptible to environmental interference, which can cause them to lose their quantum state or also their coherence. This phenomenon is known as decoherence [10]. Briefly, decoherent quantum computing is classical "bit"-based computing. Research Objectives 1. Investigate TN in the context of the curse of dimensionality: One of the objectifs of this work is to explore the interest of the different TN topologies focusing on their capability to mitigate the "curse of dimensionality" [4]. 2. Develop novel on-line/streaming algorithms: Batch-mode processing is quite inefficient for streaming data. So, there is a need to propose adaptive (over time) implementation of TN [8]. 3. Propose new TN-based algorithm robust to qubits decoherence. Robustness allows to increase the number of qubits in a quantum system while maintaining performance ("scal- ability"). 4. The SVD is the basic building block of TN algorithms. Recently, randomized methods [9] also known under the name of "compressed sensing" [5] allow to speed-up the SVD at the price of a bounded error. References [1] A. Steane, Quantum computing. Reports on Progress in Physics, 61(2), 117, 1998. [2] R. Orus, Tensor networks for complex quantum systems. Nature Reviews Physics, 1(9), 2019. [3] R. Orus, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of Physics, Vol. 349, 2014. [4] A. Cichocki; N. Lee; I. Oseledets; A.-H. Phan; Q. Zhao; D. P. Mandic, Tensor Networks for Dimensionality Reduction and Large-scale Optimization, Foundations and Trends in Machine Learning, Vol. 9, No. 4-5, 2016. [5] D.L. Donoho, Compressed sensing, IEEE Transactions on IT. 52 (4), 2006. [6] T. G. Kolda and B. W. Bader, Tensor Decompositions and Applications, SIAM REVIEW, Vol. 51, No. 3, 2009. [7] Y. Zniyed, R. Boyer, A. De Almeida, and G. Favier. A TT-based hierarchical framework for decomposing high-order tensors. SIAM Journal on Scientific Computing, vol. 42, 2020. [8] L. T. Thanh, K. Abed-Meraim, N. L. Trung and R. Boyer, "Adaptive Algorithms for Track- ing Tensor-Train Decomposition of Streaming Tensors," 28th European Signal Processing Conference (EUSIPCO), 2021. [9] N.Halko, P.G.Martinsson, andJ.A.Tropp, Findingstructurewithrandomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM review, 53(2), 2011. [10] M. L., Hu, and H. Fan, Robustness of quantum correlations against decoherence. 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