arXiv:2011.06113 · Quantum Science and Technology

Storage capacity and learning capability of quantum neural networks

Maciej Lewenstein, Aikaterini Gratsea, Andreu Riera-Campeny, Albert Aloy, Valentin Kasper, Anna Sanpera

Published Jun 1, 2021• 26 citations

Abstract

Abstract We study the storage capacity of quantum neural networks (QNNs), described by completely positive trace preserving (CPTP) maps acting on an N-dimensional Hilbert space. We demonstrate that attractor QNNs can store in a non-trivial manner up to N linearly independent pure states. For n qubits, QNNs can reach an exponential storage capacity, O ( 2 n ) , clearly outperforming standard classical neural networks whose storage capacity scales linearly with the number of neurons n. We estimate, employing the Gardner program, the relative volume of CPTP maps with M ⩽ N stationary states and show that this volume decreases exponentially with M and shrinks to zero for M ⩾ N + 1. We generalize our results to QNNs storing mixed states as well as input–output relations for feed-forward QNNs. Our approach opens the path to relate storage properties of QNNs to the quantum features of the input–output states. This paper is dedicated to the memory of Peter Wittek.

DOI: 10.1088/2058-9565/ac070f

Related papers

Optimal storage capacity of quantum Hopfield neural networks

Lukas Bödeker, Eliana Fiorelli, Markus Müller

Quantum neural networks form one pillar of the emergent field of quantum machine learning. Here quantum generalizations of classical networks realizing associative memories---capable of retrieving patterns, or memories, from corrupted initial states---have been proposed. It is a challenging open problem to analyze quantum associative memories with an extensive number of patterns and to determine the maximal number of patterns the quantum networks can reliably store, i.e., their storage capacity. In this work, we propose and explore a general method for evaluating the maximal storage capacity of quantum neural network models. By generalizing what is known as Gardner's approach in the classical realm, we exploit the theory of classical spin glasses for deriving the optimal storage capacity of quantum networks with quenched pattern variables. As an example, we apply our method to an open-system quantum associative memory formed of interacting spin-1/2 particles realizing coupled artificial neurons. The system undergoes a Markovian time evolution resulting from a dissipative retrieval dynamics that competes with a coherent quantum dynamics. We map out the nonequilibrium phase diagram and study the effect of temperature and Hamiltonian dynamics on the storage capacity. Our method opens an avenue for a systematic characterization of the storage capacity of quantum associative memories.

arXiv:2210.07894Physical Review ResearchMay 1, 202310 citations

Quantum Storage Network

Ri‐Gui Zhou, Yongzhong Cao, Shuqun Yang +1 more

Quantum Neural Network (QNN) is a young and outlying science built upon the combination of classical neural network and quantum computing. Making the best of some characteristics of the quantum operators, this paper presents both the network structure of Quantum Storage Network (QSN) and its corresponding pattern storage algorithm. QSN can store quantum patterns by a series of quantum unitary operators and its storage capacity or memory capacity is exponentially increased. Besides, the case analysis validates the feasibility and validity of the QSN proposed in this paper.

International Conference on Natural Computation (ICNC)Jan 1, 20076 citations

High-Capacity Quantum Associative Memories

M. C. Diamantini, Carlo A. Trugenberger

We review our models of quantum associative memories that represent the “quantization” of fully coupled neural networks like the Hopfield model. The idea is to replace the classical irreversible attractor dynamics driven by an Ising model with pattern-dependent weights by the reversible rotation of an input quantum state onto an output quantum state consisting of a linear superposition with probability amplitudes peaked on the stored pattern closest to the input in Hamming distance, resulting in a high probability of measuring a memory pattern very similar to the input. The unitary operator implementing this transformation can be formulated as a sequence of one-qubit and two-qubit elementary quantum gates and is thus the exponential of an ordered quantum Ising model with sequential operations and with pattern-dependent interactions, exactly as in the classical case. Probabilistic quantum memories, that make use of postselection of the measurement result of control qubits, overcome the famed linear storage limitation of their classical counterparts because they permit to completely eliminate crosstalk and spurious memories. The number of control qubits plays the role of an inverse fictitious temperature. The accuracy of pattern retrieval can be tuned by lowering the fictitious temperature under a critical value for quantum content association while the complexity of the retrieval algorithm remains polynomial for any number of patterns polynomial in the number of qubits. These models thus solve the capacity shortage problem of classical associative memories, providing a polynomial improvement in capacity. The price to pay is the probabilistic nature of information retrieval.

arXiv:1506.01231Journal of Applied Mathematics and PhysicsJan 1, 20168 citations

Quantum Hopfield Neural Network and Image Recognition

Ding Qiu-lin

The storage capacity of the conventional Hopfield network is the 0.14 times of the number of neurons(P=0.14N). Because of the huge difficulty in recognizing a large number of images or patterns, researchers are looking for new methods at all times. Quantum neural network (QNN) is a young and outlying science built upon the combination of classical neural network and quantum computing. A Quantum Hopfield neural network (QHNN) whose elements of the storage matrix are distributed in a probability way on the base of quantum linear superposition is presented for speeding up the images recognition and increasing the number of the images recognition. Contrasting to the conventional Hopfield neural network, the storage capacity of the QHNN is increased by a factor of 2N, where N is the number of neurons. Besides, the case analysis and simulation tests have been carried out for the recognition of images in this paper. The result indicates that QHNN can recognize the images or patterns effectively and its working process accords with quantum evolvement process.

Jan 1, 20080 citations

Quantum Neuron: an elementary building block for machine learning on quantum computers

Yudong Cao, Gian Giacomo Guerreschi, Alán Aspuru‐Guzik

Even the most sophisticated artificial neural networks are built by aggregating substantially identical units called neurons. A neuron receives multiple signals, internally combines them, and applies a non-linear function to the resulting weighted sum. Several attempts to generalize neurons to the quantum regime have been proposed, but all proposals collided with the difficulty of implementing non-linear activation functions, which is essential for classical neurons, due to the linear nature of quantum mechanics. Here we propose a solution to this roadblock in the form of a small quantum circuit that naturally simulates neurons with threshold activation. Our quantum circuit defines a building block, the "quantum neuron", that can reproduce a variety of classical neural network constructions while maintaining the ability to process superpositions of inputs and preserve quantum coherence and entanglement. In the construction of feedforward networks of quantum neurons, we provide numerical evidence that the network not only can learn a function when trained with superposition of inputs and the corresponding output, but that this training suffices to learn the function on all individual inputs separately. When arranged to mimic Hopfield networks, quantum neural networks exhibit properties of associative memory. Patterns are encoded using the simple Hebbian rule for the weights and we demonstrate attractor dynamics from corrupted inputs. Finally, the fact that our quantum model closely captures (traditional) neural network dynamics implies that the vast body of literature and results on neural networks becomes directly relevant in the context of quantum machine learning.

arXiv:1711.11240Nov 30, 2017113 citations

Critically excited states with enhanced memory and pattern recognition capacities in quantum brain networks: Lesson from black holes

Gia Dvali

We implement a mechanism - originally proposed as a model for the large memory storage capacity of black holes - in quantum neural networks and show that an exponentially increased capacity of pattern storage and recognition is achieved in certain critically excited states, without involvement of synaptic plasticity. We consider a simple network of N interconnected quantum neurons with weak excitatory synaptic connections. We show that for frozen synaptic weights there exist the critical states of enhanced memory storage capacity. These states are achieved thanks to the high excitation levels of some of the neurons, which - despite of feeble synaptic connections - dramatically lower the response threshold of the remaining weaker-excited neurons. As a results, the latter neurons acquire a capacity to store an exponentially large number of patterns within a narrow energy gap. The stored patterns can be recognized and retrieved with perfect response under the influence of arbitrarily soft input stimuli. In sharp contrast, under the same stimuli the recall is absent in the ground-state of the system. The lesson is that the state with the highest micro-state entropy and memory storage capacity is not necessarily a local minimum of energy, but rather an excited critical state. The considered phenomenon has a smooth classical limit and can serve for achieving an enhanced memory storage capacity in classical brain networks.

arXiv:1711.09079Nov 24, 20178 citations