Quantum artificial neural networks with applications

doi:10.1016/j.ins.2014.08.033

Information Sciences

Volume 290, 1 January 2015, Pages 1-6

https://doi.org/10.1016/j.ins.2014.08.033 Get rights and content

Highlights

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A new type of neural networks has been proposed, named a quantum artificial neural network (QANN).
•
A QANN is presented as a system of interconnected ”quantum neurons” simulated on quantum computers.
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The ability of approximation of a QANN has been proved by a universal approximation theorem (UAT).
•
The UAT implies that a QANN may be a potential computing tool for dealing with quantum information.

Abstract

Since simulations of classical artificial neural networks (CANNs) run on classical computers, the massive parallel processing speed advantage of a neural network is lost. A quantum computer is a computation device that makes direct use of quantum–mechanical phenomena while large-scale quantum computers will be able to solve certain problems much quicker than any classical computer using the best currently known algorithms. Combining the advantages of quantum computers and the idea of CANNs, we propose in this paper a new type of neural networks, named a quantum artificial neural network (QANN), which is presented as a system of interconnected “quantum neurons” which can compute quantum states from input-quantum states by feeding information through the network and can be simulated on quantum computers. To show the ability of approximation of a QANN, we prove a universal approximation theorem (UAT) which reads every continuous mapping that transforms n quantum states as a non-normalized quantum state can be uniformly approximated by a QANN. The UAT implies that QANNs would suggest a potential computing tool for dealing with quantum information. For instance, we prove that the state of a quantum system driven by a time-dependent Hamiltonian can be approximated uniformly by a QANN. This provides a possible way for finding approximate solution to a Schrödinger equation with a time-dependent Hamiltonian.

Introduction

In computer science and related fields, classical artificial neural networks (CANNs) are computational models and capable of machine learning and pattern recognition. A CANN is usually presented as a system of interconnected “neurons” which can compute values from inputs by feeding information through the network. Since simulations of artificial neural networks run on classical computers, the massive parallel processing speed advantage of a neural network is lost [22]. Clearly, it would be better to utilize the intrinsic physics of a physical system to perform the computation. Many efforts have been expended in this direction, using systems ranging from nonlinear optical materials to proteins [15]. At the same time, many other researchers have been exploring the possibility of building quantum computers [2], [1], [6]. By using arrays of coupled quantum dot molecules, a quantum cellular automata has been posed in [11], which provides a valuable concrete example of quantum computation in which a number of fundamental issues come to light. An architecture for a quantum neural computer has been proposed in [14] in light of the real time evolution of quantum dot molecules, and simulations have proved that such an architecture can perform any classical logic gate, which can be used to calculate a purely quantum gate (a unitary matrix).

In this paper, we propose an analog of a CANN, named a quantum artificial neural network (QANN), and prove that every continuous mapping that maps n quantum states as a non-normalized quantum state can be uniformly approximated by a QANN. As an application, we show that the state of a quantum system driven by a time-dependent Hamiltonian can be approximated uniformly by a QANN.

Section snippets

Construction of a quantum artificial neural network

Let $C^{d} = {{(z_{1}, z_{2}, \dots, z_{d})}^{T} : z_{k} \in C (k = 1, 2, \dots, d)}$ be the d-dimensional complex Hilbert space with the inner product $〈 x | y 〉 = \sum_{k = 1}^{d} \overline{x_{k}} y_{k}$ for all elements $| x 〉 = {(x_{1}, x_{2}, \dots, x_{d})}^{T}$ and $| y 〉 = {(y_{1}, y_{2}, \dots, y_{d})}^{T}$ , where $\overline{x_{k}}$ denotes the conjugate of the complex number $x_{k}$ and $〈 x | = | x 〉^{†} = (\overline{x_{1}}, \overline{x_{2}}, \dots, \overline{x_{d}}) .$ The norm induced by the inner product above reads $‖ | x 〉 ‖ = 〈 x | x 〉^{1 / 2} = {(\sum_{k = 1}^{d} | x_{k} |^{2})}^{1 / 2} .$

In quantum mechanics, a d-dimensional quantum system is described by the Hilbert space $C^{d}$ and quantum states of the system are described by unit vectors in $C^{d}$

Universal approximation theorem and applications

Recall that a function $σ : R \to R$ is called a sigmoidal function if it satisfies $\lim_{t \to - \infty} σ (t) = 0, \lim_{t \to + \infty} σ (t) = 1 .$

Let $C (S_{d}^{n} (C), C^{M})$ be the set of all continuous mappings from $S_{d}^{n} (C)$ into $C^{M}$ .

Theorem 3.1

Let $σ_{k} : R \to R (k = 1, 2, \dots, M)$ be continuous sigmoidal functions. Then for every $f \in C (S_{d}^{n} (C), C^{M})$ and every positive number ε, there exists a QANN $Q (| x 〉)$ such that $‖ f (| x 〉) - Q (| x 〉) ‖ < ε, \forall | x 〉 \in S_{d}^{n} (C) .$

Proof

Let $f \in C (S_{d}^{n} (C), C^{M})$ and $ε > 0$ . Then f can be written as $f (| x 〉) = {(f_{1} (| x 〉), f_{2} (| x 〉), \dots, f_{M} (| x 〉))}^{T},$ where $f_{k} : S_{d}^{n} (C) \to C (k = 1, 2, \dots, M)$ are all continuous.

Conclusions

As an analog of the classical artificial neural network (CANN), a new type of neural networks, named quantum artificial neural network (QANN), is introduced in this paper. Universal approximation theorem (Theorem 3.1) shows that every continuous mapping that maps continuously n quantum states (d-dimensional complex unit vectors) as a non-normalized quantum state (an M-dimensional complex vector) can be uniformly approximated by the proposed quantum artificial neural networks. This implies that

Acknowledgments

The authors would like to thank the editor and the referees of this paper for their kind comments and valuable suggestions.

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^☆: This subject was supported by the NNSF Grants of China (Nos. 11171197, 11371012, 61272023) and the FRF for the Central Universities (No. GK201301007)

View full text

Quantum artificial neural networks with applications☆

Highlights

Abstract

Introduction

Section snippets

Construction of a quantum artificial neural network

Universal approximation theorem and applications

Conclusions

Acknowledgments

Conditional quantum dynamics and logic gates

Phys. Rev. Lett.

Protein-based computers

Sci. Am.

Beweis des adiabatensatzes

Z. Phys.

Quantitative sufficient conditions for adiabatic approximation

Sci. China-Phys. Mech. Astron.

Long-range adiabatic quantum state transfer through a linear array of quantum dots

Sci. China-Phys. Mech. Astron.

Quantum computations with cold trapped ions

Phys. Rev. Lett.

Trajectory tracking control of quantum systems

Chin. Sci. Bull.

Approximation by superpositions of a sigmoidal function

Math. Control Signals Syst.

Adiabatic approximation in PT-sysmetric quantum mechanics

Sci. China-Phys. Mech. Astron.

General Topology

Quantum cellular automata: the physics of computing with arrays of quantum dot molecules

Why the quantitative condition fails to reveal quantum adiabaticity

New J. Phys.