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This work focuses on investigating an end-to-end learning approach for quantum neural networks (QNN) on noisy intermediate-scale quantum devices. The proposed model combines a quantum tensor network (QTN) with a variational quantu...
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This work focuses on investigating an end-to-end learning approach for quantum neural networks (QNN) on noisy intermediate-scale quantum devices. The proposed model combines a quantum tensor network (QTN) with a variational quantum circuit (VQC), resulting in a QTN-VQC architecture. This architecture integrates a QTN with a horizontal or vertical structure related to the implementation of quantum circuits for a tensor-train network. The study provides theoretical insights into the quantum advantages of the end-to-end learning pipeline based on QTN-VQC from two perspectives. The first perspective refers to the theoretical understanding of QTN-VQC with upper bounds on the empirical error, examining its learnability and generalization powers; The second perspective focuses on using the QTN-VQC architecture to alleviate the Barren Plateau problem in the training stage. Our experimental simulation on CPU/GPUs is performed on a handwritten digit classification dataset to corroborate our proposed methods in this work.
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In recent times, Variational Quantum Circuits (VQC) have been widely adopted to different tasks in machine learning such as Combinatorial Optimization and Supervised Learning. With the growing interest, it is pertinent to study th...
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In recent times, Variational Quantum Circuits (VQC) have been widely adopted to different tasks in machine learning such as Combinatorial Optimization and Supervised Learning. With the growing interest, it is pertinent to study the boundaries of the classical simulation of VQCs to effectively benchmark the algorithms. Classically simulating VQCs can also provide the quantum algorithms with a better initialization reducing the amount of quantum resources needed to train the algorithm. Even though Matrix Product State representations have been extensively used for quantum state approximation, their capacity is limited in simulating quantum circuits due to the exponential complexity in circuit depth. This manuscript proposes an algorithm that compresses the quantum state within a circuit using a noisy tensor ring representation which allows for the implementation of VQC based algorithms on a classical simulator at a fraction of the usual storage and computational complexity. Using the tensor ring approximation of the input quantum state, we propose a method that applies the parametrized unitary operations while retaining the low-rank structure of the tensor ring corresponding to the transformed quantum state, providing an exponential improvement of storage and computational time in the number of qubits and layers. This approximation is used to implement the tensor ring VQC (TRVQC) for the task of supervised learning on Iris and MNIST datasets to demonstrate the performance of the proposed method compared with the implementations from classical simulator using Matrix Product States (MPS). TRVQC has a test accuracy of 82.63% compared to the benchmark of 83.68% on Iris dataset whereas the former outperforms the latter on a reduced MNIST dataset with TRVQC having an accuracy of 83.73% compared to the benchmark 81.02%, showcasing the comparable performance of the proposed algorithm with the MPS framework.
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We discuss the calculus of variations in tensor representations with a special focus on tensor networks and apply it to functionals of practical interest. The survey provides all necessary ingredients for applying minimization met...
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We discuss the calculus of variations in tensor representations with a special focus on tensor networks and apply it to functionals of practical interest. The survey provides all necessary ingredients for applying minimization methods in a general setting. The important cases of target functionals which are linear and quadratic with respect to the tensor product are discussed, and combinations of these functionals are presented in detail. As an example, we consider the representation rank compression in tensor networks. For the numerical treatment, we use the nonlinear block Gauss-Seidel method. We demonstrate the rate of convergence in numerical tests.
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Tensor networks (TNs) are a family of computational methods built on graph-structured factorizations of large tensors, which have long represented state-of-the-art methods for the approximate simulation of complex quantum systems ...
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Tensor networks (TNs) are a family of computational methods built on graph-structured factorizations of large tensors, which have long represented state-of-the-art methods for the approximate simulation of complex quantum systems on classical computers. The rapid pace of recent advancements in numerical computation, notably the rise of GPU and TPU hardware accelerators, have allowed TN algorithms to scale to even larger quantum simulation problems, and to be employed more broadly for solving machine learning tasks. The 'quantum-inspired' nature of TNs permits them to be mapped to parametrized quantum circuits (PQCs), a fact which has inspired recent proposals for enhancing the performance of TN algorithms using near-term quantum devices, as well as enabling joint quantum-classical training frameworks that benefit from the distinct strengths of TN and PQC models. However, the success of any such methods depends on efficient and accurate methods for approximating TN states using realistic quantum circuits, which remains an unresolved question. This work compares a range of novel and previously-developed algorithmic protocols for decomposing matrix product states (MPS) of arbitrary bond dimension into low-depth quantum circuits consisting of stacked linear layers of two-qubit unitaries. These protocols are formed from different combinations of a preexisting analytical decomposition method together with constrained optimization of circuit unitaries, with initialization by the former method helping to avoid poor-quality local minima in the latter optimization process. While all of these protocols have efficient classical runtimes, our experimental results reveal one particular protocol employing sequential growth and optimization of the quantum circuit to outperform all others, with even greater benefits in the setting of limited computational resources. Given these promising results, we expect our proposed decomposition protocol to form a useful ingredient within any joint application of TNs and PQCs, further unlocking the rich and complementary benefits of classical and quantum computation.
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Variational quantum Monte Carlo (VMC) method~(1-6)) applied to interacting electron systems has played a pivotal role for the studies of electronic states of atoms, molecules and condensed matters. First, the many-body wave functi...
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Variational quantum Monte Carlo (VMC) method~(1-6)) applied to interacting electron systems has played a pivotal role for the studies of electronic states of atoms, molecules and condensed matters. First, the many-body wave functions optimized by the VMC method provide good guiding functions in more accurate, diffusion quantum Monte Carlo (DMC) calculations~1) for identical systems. Second, the VMC calculations themselves can give a good upper bound for the exact ground-state energy of many-electron systems when an appropriate form for the variational wave function is employed. In, addition to a conventional form for the variational wave function that is expressed by the product of the Jastrow factor and the single Slater determinant, recent investigations~(1,7,8)) have found that the expression in terms of many Slater determinants can provide a very accurate description of electronic states, especially for intrinsically multi-reference systems, which corresponds to the configuration interaction (CI) expansion method~9) in conventional quantum chemistry. Although the CI method needs an extremely large number of expansion coefficients as variational parameters, the number of parameters can be reduced efficiently by using an idea of (molecular) orbital pair correlations.~(10)) The purpose of the present paper is to examine the feasibility and Usefulness of this idea within the framework of VMC method with the Jastrow-Slater wave function, where the ground state of a water molecule is investigated as a test case.
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In this paper, we eliminate the classical outer learning loop of the quantum approximate optimization algorithm (QAOA) and present a strategy to find good parameters for QAOA based on topological arguments of the problem graph and...
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In this paper, we eliminate the classical outer learning loop of the quantum approximate optimization algorithm (QAOA) and present a strategy to find good parameters for QAOA based on topological arguments of the problem graph and tensor network techniques. Starting from the observation of the concentration of control parameters of QAOA, we find a way to classically infer parameters which scales polynomially in the number of qubits and exponentially with the depth of the circuit. Using this strategy, the quantum processing unit (QPU) is only needed to sample from the final state of QAOA. This method paves the way for a variation-free version of QAOA and makes QAOA more practical for applications on NISQ devices. We investigate the performance of the proposed approach for the initial assumptions and its resilience with respect to situations where they are not fulfilled. Moreover, we investigate the applicability of our method beyond the scope of QAOA, in improving schedules for quantum annealing.
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We study the bipartite entanglement per bond to determine characteristic features of the phase diagram of various quantum spin models in different spatial dimensions. The bipartite entanglement is obtained from a tensor network re...
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We study the bipartite entanglement per bond to determine characteristic features of the phase diagram of various quantum spin models in different spatial dimensions. The bipartite entanglement is obtained from a tensor network representation of the ground state wave-function. Three spin-1/2 models (Ising, XY, XXZ, all in a transverse field) are investigated. Infinite imaginary-time evolution (iTEBD in 1D, 'simple update' in 2D and 3D) is used to determine the ground states of these models. The phase structure of the models is discussed for all three dimensions.
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In this paper we provide a novel way to explore the relation between quantum teleportation and quantum phase transition. We construct a quantum channel with a mixed state which is made from one dimensional quantum Ising chain with...
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In this paper we provide a novel way to explore the relation between quantum teleportation and quantum phase transition. We construct a quantum channel with a mixed state which is made from one dimensional quantum Ising chain with infinite length, and then consider the teleportation with the use of entangled Werner states as input qubits. The fidelity as a figure of merit to measure how well the quantum state is transferred is studied numerically. Remarkably we find the first-order derivative of the fidelity with respect to the parameter in quantum Ising chain exhibits a logarithmic divergence at the quantum critical point. The implications of this phenomenon and possible applications are also briefly discussed. (C) 2018 Elsevier B.V. All rights reserved.
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The transverse-field Ising model on the Sierpiński fractal, which is characterized by the fractal dimension log_2 3 ≈ 1.585, is studied by a tensor-network method known as the higher-order tensor renormalization group. We analyz...
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The transverse-field Ising model on the Sierpiński fractal, which is characterized by the fractal dimension log_2 3 ≈ 1.585, is studied by a tensor-network method known as the higher-order tensor renormalization group. We analyze the ground-state energy and the spontaneous magnetization in the thermodynamic limit. The system exhibits the second-order phase transition at the critical transverse field h_c= 1.865. The critical exponents β ≈0.198 and δ ≈ 8.7 are obtained. Complementary to the tensor-network method, we make use of the real-space renormalization group and improved mean-field approximations for comparison.
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