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An analytic theory for the dynamics of wide quantum neural networks

Junyu Liu, Khadijeh Najafi, Kunal Sharma, Francesco Tacchino, Liang Jiang, Antonio Mezzacapo

Apr 01 2022 quant-ph cs.AI cs.LG stat.ML arXiv:2203.16711v1

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Parametrized quantum circuits can be used as quantum neural networks and have the potential to outperform their classical counterparts when trained for addressing learning problems. To date, much of the results on their performance on practical problems are heuristic in nature. In particular, the convergence rate for the training of quantum neural networks is not fully understood. Here, we analyze the dynamics of gradient descent for the training error of a class of variational quantum machine learning models. We define wide quantum neural networks as parameterized quantum circuits in the limit of a large number of qubits and variational parameters. We then find a simple analytic formula that captures the average behavior of their loss function and discuss the consequences of our findings. For example, for random quantum circuits, we predict and characterize an exponential decay of the residual training error as a function of the parameters of the system. We finally validate our analytic results with numerical experiments.

Classically-Boosted Quantum Optimization Algorithm

Guoming Wang

Mar 29 2022 quant-ph cs.DS arXiv:2203.13936v1

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Considerable effort has been made recently in the development of heuristic quantum algorithms for solving combinatorial optimization problems. Meanwhile, these problems have been studied extensively in classical computing for decades. In this paper, we explore a natural approach to leveraging existing classical techniques to enhance quantum optimization. Specifically, we run a classical algorithm to find an approximate solution and then use a quantum circuit to search its “neighborhood” for higher-quality solutions. We propose the Classically-Boosted Quantum Optimization Algorithm (CBQOA) that is based on this idea and can solve a wide range of combinatorial optimization problems, including all unconstrained problems and many important constrained problems such as Max Bisection, Maximum Independent Set, Minimum Vertex Cover, Portfolio Optimization, Traveling Salesperson and so on. A crucial component of this algorithm is an efficiently-implementable continuous-time quantum walk (CTQW) on a properly-constructed graph that connects the feasible solutions. CBQOA utilizes this CTQW and the output of an efficient classical procedure to create a suitable superposition of the feasible solutions which is then processed in certain way. This algorithm has the merits that it solves constrained problems without modifying their cost functions, confines the evolution of the quantum state to the feasible subspace, and does not rely on efficient indexing of the feasible solutions. We demonstrate the applications of CBQOA to Max 3SAT and Max Bisection, and provide empirical evidence that it outperforms previous approaches on these problems.

Characterization and Verification of Trotterized Digital Quantum Simulation via Hamiltonian and Liouvillian Learning

Lorenzo Pastori, Tobias Olsacher, Christian Kokail, Peter Zoller

Mar 31 2022 quant-ph cond-mat.quant-gas arXiv:2203.15846v1

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The goal of digital quantum simulation is to approximate the dynamics of a given target Hamiltonian via a sequence of quantum gates, a procedure known as Trotterization. The quality of this approximation can be controlled by the so called Trotter step, that governs the number of required quantum gates per unit simulation time, and is intimately related to the existence of a time-independent, quasilocal Hamiltonian that governs the stroboscopic dynamics, refered to as the Floquet Hamiltonian of the Trotterization. In this work, we propose a Hamiltonian learning scheme to reconstruct the implemented Floquet Hamiltonian order-by-order in the Trotter step: this procedure is efficient, i.e., it requires a number of measurements that scales polynomially in the system size, and can be readily implemented in state-of-the-art experiments. With numerical examples, we propose several applications of our method in the context of verification of quantum devices, from the characterization of the distinct sources of errors in digital quantum simulators to the design of new types of quantum gates. Furthermore, we show how our approach can be extended to the case of non-unitary dynamics and used to learn Floquet Liouvillians, thereby offering a way of characterizing the dissipative processes present in NISQ quantum devices.

Local variational quantum compilation of a large-scale Hamiltonian dynamics

Kaoru Mizuta, Yuya O. Nakagawa, Kosuke Mitarai, Keisuke Fujii

Mar 30 2022 quant-ph cond-mat.mtrl-sci cond-mat.str-el arXiv:2203.15484v1

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Implementing time evolution operators on quantum circuits is important for quantum simulation. However, the standard way, Trotterization, requires a huge numbers of gates to achieve desirable accuracy. Here, we propose a local variational quantum compilation (LVQC) algorithm, which allows to accurately and efficiently compile a time evolution operators on a large-scale quantum system by the optimization with smaller-size quantum systems. LVQC utilizes a subsystem cost function, which approximates the fidelity of the whole circuit, defined for each subsystem as large as approximate causal cones brought by the Lieb-Robinson (LR) bound. We rigorously derive its scaling property with respect to the subsystem size, and show that the optimization conducted on the subsystem size leads to the compilation of whole-system time evolution operators. As a result, LVQC runs with limited-size quantum computers or classical simulators that can handle such smaller quantum systems. For instance, finite-ranged and short-ranged interacting LL-size systems can be compiled with O(L0)O(L0)- or O(logL)O(log⁡L)-size quantum systems depending on observables of interest. Furthermore, since this formalism relies only on the LR bound, it can efficiently construct time evolution operators of various systems in generic dimension involving finite-, short-, and long-ranged interactions. We also numerically demonstrate the LVQC algorithm for one-dimensional systems. Employing classical simulation by time-evolving block decimation, we succeed in compressing the depth of a time evolution operators up to 4040 qubits by the compilation for 2020 qubits. LVQC not only provides classical protocols for designing large-scale quantum circuits, but also will shed light on applications of intermediate-scale quantum devices in implementing algorithms in larger-scale quantum devices.

Proofs of network quantum nonlocality aided by machine learning

Alejandro Pozas-Kerstjens, Nicolas Gisin, Marc-Olivier Renou

Apr 01 2022 quant-ph arXiv:2203.16543v1

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The study of nonlocality in scenarios that depart from the bipartite Einstein-Podolsky-Rosen setup is allowing to uncover many fundamental features of quantum mechanics. Recently, an approach to building network-local models based on machine learning lead to the conjecture that the family of quantum triangle distributions of [DOI:10.1103/PhysRevLett.123.140401] did not admit triangle-local models in a larger range than the original proof. We prove part of this conjecture in the affirmative. Our approach consists in reducing the family of original, four-outcome distributions to families of binary-outcome ones, which we prove not to admit triangle-local models by means of the inflation technique. In the process, we produce a large collection of network Bell inequalities for the triangle scenario with binary outcomes, which are of independent interest.

Variational Quantum Pulse Learning

Zhiding Liang, Hanrui Wang, Jinglei Cheng, Yongshan Ding, Hang Ren, Xuehai Qian, Song Han, Weiwen Jiang, Yiyu Shi

Apr 01 2022 quant-ph arXiv:2203.17267v1

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Quantum computing is among the most promising emerging techniques to solve problems that are computationally intractable on classical hardware. A large body of existing works focus on using variational quantum algorithms on the gate level for machine learning tasks, such as the variational quantum circuit (VQC). However, VQC has limited flexibility and expressibility due to limited number of parameters, e.g. only one parameter can be trained in one rotation gate. On the other hand, we observe that quantum pulses are lower than quantum gates in the stack of quantum computing and offers more control parameters. Inspired by the promising performance of VQC, in this paper we propose variational quantum pulses (VQP), a novel paradigm to directly train quantum pulses for learning tasks. The proposed method manipulates variational quantum pulses by pulling and pushing the amplitudes of pulses in an optimization framework. Similar to variational quantum algorithms, our framework to train pulses maintains the robustness to noise on Noisy Intermediate-Scale Quantum (NISQ) computers. In an example task of binary classification, VQP learning achieves 30% and 40% higher accuracy compared with VQC learning on the qiskit noise simulatosr (FakeMachines) and ibmq-jarkata, respectively, demonstrating its effectiveness and feasibility. Stability for VQP to obtain reliable results has also been verified in the presence of noise.

Monte Carlo Tree Search based Hybrid Optimization of Variational Quantum Circuits

Jiahao Yao, Haoya Li, Marin Bukov, Lin Lin, Lexing Ying

Apr 01 2022 quant-ph cond-mat.quant-gas cs.LG physics.comp-ph arXiv:2203.16707v1

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Variational quantum algorithms stand at the forefront of simulations on near-term and future fault-tolerant quantum devices. While most variational quantum algorithms involve only continuous optimization variables, the representational power of the variational ansatz can sometimes be significantly enhanced by adding certain discrete optimization variables, as is exemplified by the generalized quantum approximate optimization algorithm (QAOA). However, the hybrid discrete-continuous optimization problem in the generalized QAOA poses a challenge to the optimization. We propose a new algorithm called MCTS-QAOA, which combines a Monte Carlo tree search method with an improved natural policy gradient solver to optimize the discrete and continuous variables in the quantum circuit, respectively. We find that MCTS-QAOA has excellent noise-resilience properties and outperforms prior algorithms in challenging instances of the generalized QAOA.

Variational quantum simulation of long-range interacting systems

Xiaoyu Tang, Chufan Lyu, Junning Li, Xusheng Xu, Man-Hong Yung, Abolfazl Bayat

Mar 29 2022 quant-ph arXiv:2203.14281v1

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Current quantum simulators suffer from multiple limitations such as short coherence time, noisy operations, faulty readout and restricted qubit connectivity. Variational quantum algorithms are the most promising approach in near-term quantum simulation to achieve quantum advantage over classical computers. Here, we explore variational quantum algorithms, with different levels of qubit connectivity, for digital simulation of the ground state of long-range interacting systems. We find that as the interaction becomes more long-ranged, the variational algorithms become less efficient, achieving lower fidelity and demanding more optimization iterations. In particular, when the system is near its criticality the efficiency is even lower. Increasing the connectivity between distant qubits improves the results, even with less quantum and classical resources. Our results show that by mixing circuit layers with different levels of connectivity one can sensibly improve the results. Interestingly, the order of layers becomes very important and grouping the layers with long-distance connectivity at the beginning of the circuit outperforms other permutations. The same design of circuits can also be used to variationally produce spin squeezed states, as a resource for quantum metrology. The quantum variational method indeed outperforms the ground state and quench dynamics approach for creating spin squeezing.

Multiclass classification using quantum convolutional neural networks with hybrid quantum-classical learning

Denis Bokhan, Alena S. Mastiukova, Aleksey S. Boev, Dmitrii N. Trubnikov, Aleksey K. Fedorov

Mar 30 2022 quant-ph cs.LG arXiv:2203.15368v1

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Multiclass classification is of great interest for various machine learning applications, for example, it is a common task in computer vision, where one needs to categorize an image into three or more classes. Here we propose a quantum machine learning approach based on quantum convolutional neural networks for solving this problem. The corresponding learning procedure is implemented via TensorFlowQuantum as a hybrid quantum-classical (variational) model, where quantum output results are fed to softmax cost function with subsequent minimization of it via optimization of parameters of quantum circuit. Our conceptional improvements include a new model for quantum perceptron and optimized structure of the quantum circuit. We use the proposed approach to demonstrate the 4-class classification for the case of the MNIST dataset using eight qubits for data encoding and four acnilla qubits. Our results demonstrate comparable accuracy of our solution with classical convolutional neural networks with comparable numbers of trainable parameters. We expect that our finding provide a new step towards the use of quantum machine learning for solving practically relevant problems in the NISQ era and beyond.

Quantum-Aided Meta-Learning for Bayesian Binary Neural Networks via Born Machines

Ivana Nikoloska, Osvaldo Simeone

Apr 01 2022 quant-ph cs.LG arXiv:2203.17089v1

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Near-term noisy intermediate-scale quantum circuits can efficiently implement implicit probabilistic models in discrete spaces, supporting distributions that are practically infeasible to sample from using classical means. One of the possible applications of such models, also known as Born machines, is probabilistic inference, which is at the core of Bayesian methods. This paper studies the use of Born machines for the problem of training binary Bayesian neural networks. In the proposed approach, a Born machine is used to model the variational distribution of the binary weights of the neural network, and data from multiple tasks is used to reduce training data requirements on new tasks. The method combines gradient-based meta-learning and variational inference via Born machines, and is shown in a prototypical regression problem to outperform conventional joint learning strategies.

Direct parameter estimations from machine-learning enhanced quantum state tomography

Hsien-Yi Hsieh, Jingyu Ning, Yi-Ru Chen, Hsun-Chung Wu, Hua Li Chen, Chien-Ming Wu, Ray-Kuang Lee

Mar 31 2022 quant-ph arXiv:2203.16385v1

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With the capability to find the best fit to arbitrarily complicated data patterns, machine-learning (ML) enhanced quantum state tomography (QST) has demonstrated its advantages in extracting complete information about the quantum states. Instead of using the reconstruction model in training a truncated density matrix, we develop a high-performance, lightweight, and easy-to-install supervised characteristic model by generating the target parameters directly. Such a characteristic model-based ML-QST can avoid the problem of dealing with large Hilbert space, but keep feature extractions with high precision. With the experimentally measured data generated from the balanced homodyne detectors, we compare the degradation information about quantum noise squeezed states predicted by the reconstruction and characteristic models, both give agreement to the empirically fitting curves obtained from the covariance method. Such a ML-QST with direct parameter estimations illustrates a crucial diagnostic toolbox for applications with squeezed states, including advanced gravitational wave detectors, quantum metrology, macroscopic quantum state generation, and quantum information process.

Adaptive Quantum State Tomography with Active Learning

Hannah Lange, Matjaž Kebrič, Maximilian Buser, Ulrich Schollwöck, Fabian Grusdt, Annabelle Bohrdt

Mar 30 2022 quant-ph cond-mat.dis-nn cond-mat.quant-gas arXiv:2203.15719v1

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Recently, tremendous progress has been made in the field of quantum science and technologies: different platforms for quantum simulation as well as quantum computing, ranging from superconducting qubits to neutral atoms, are starting to reach unprecedentedly large systems. In order to benchmark these systems and gain physical insights, the need for efficient tools to characterize quantum states arises. The exponential growth of the Hilbert space with system size renders a full reconstruction of the quantum state prohibitively demanding in terms of the number of necessary measurements. Here we propose and implement an efficient scheme for quantum state tomography using active learning. Based on a few initial measurements, the active learning protocol proposes the next measurement basis, designed to yield the maximum information gain. For a fixed total number of measurements and basis configurations, our algorithm maximizes the information one can obtain about the quantum state under consideration. We apply the active learning quantum state tomography scheme to reconstruct different multi-qubit states with varying degree of entanglement as well as to ground states of a kinetically constrained spin chain. In all cases, we obtain a significantly improved reconstruction as compared to a reconstruction based on the exact same number of measurements, but with randomly chosen basis configurations. Our scheme is highly relevant to gain physical insights in quantum many-body systems as well as for the characterization of quantum devices, and paves the way for benchmarking and probing large quantum systems.

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