Quantum Inspired Computational Intelligence: Research and Applications

By Siddhartha Bhattacharyya, Ujjwal Maulik and Paramartha Dutta

Quantum Inspired Computational Intelligence: Research and Applications explores the latest quantum computational intelligence approaches, initiatives, and applications in computing, engineering, science, and business. The book explores this emerging field of research that applies principles of quantum mechanics to develop more efficient and robust intelligent systems. Conventional computational intelligence—or soft computing—is conjoined with quantum computing to achieve this objective. The models covered can be applied to any endeavor which handles complex and meaningful information.

• Brings together quantum computing with computational intelligence to achieve enhanced performance and robust solutions
• Includes numerous case studies, tools, and technologies to apply the concepts to real world practice
• Provides the missing link between the research and practice

• Language: English
• Publisher: Elsevier Science
• Release date: Sep 20, 2016
• ISBN: 9780128044377

Siddhartha Bhattacharyya

Siddhartha Bhattacharyya is a Senior Researcher in the Faculty of Electrical Engineering and Computer Science of VSB Technical University of Ostrava, Czech Republic. He is also serving as the Scientific Advisor of Algebra University College, Zagreb, Croatia. Prior to this, he served as the Principal of Rajnagar Mahavidyalaya, Rajnagar, Birbhum. He was a professor at CHRIST (Deemed to be University), Bangalore, India, and also served as the Principal of RCC Institute of Information Technology, Kolkata, India. He is the recipient of several coveted national and international awards. He received the Honorary Doctorate Award (D. Litt.) from the University of South America and the SEARCC International Digital Award ICT Educator of the Year in 2017. He was appointed as the ACM Distinguished Speaker for the tenure 2018-2020. He has been appointed as the IEEE Computer Society Distinguished Visitor for the tenure 2021-2023. He has co-authored six books, co-edited 75 books, and has more than 300 research publications in international journals and conference proceedings to his credit.

Book preview

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Part I

Research

Chapter 1

Quantum neural computation of entanglement is robust to noise and decoherence

E.C. Behrman; N.H. Nguyen; J.E. Steck; M. McCann    Wichita State University, Wichita, KS, United States

Abstract

Measurement and witnesses of entanglement remain an important issue in quantum computing. Most witnesses will work for only a very restricted class of states, while measurements commonly require lengthy procedures. Quantum neural entanglement indicators are both more general and easier to implement. The neural network entanglement indicator can be used for a pure or a mixed state, and the system need not be close to any particular state; moreover, as the size of the system grows, the amount of additional training necessary diminishes. Here we show that the indicator is stable to noise and decoherence.

Keywords

Quantum algorithm; Entanglement; Dynamic learning; Noise; Decoherence

Acknowledgments

This work was supported in part by the National Science Foundation under grant no. NSF PHY05-51164, through the KITP Scholars program (E.C.B.), at the Kavli Institute for Theoretical Physics, University of California, Santa Barbara, Santa Barbara, California, United States.

1 Introduction and Literature Background

The use and manipulation of entanglement is central to the exploitation of quantum computation (see, e.g., [1–8]). The quantum system obviously knows what its own entanglement is, although extraction of that information is not obvious; thus we use dynamic learning methods [9, 10] to map this information onto a single experimental measurement which is our entanglement indicator [11]. Our method does not require prior state reconstruction or lengthy optimization [4, 5] nor must the system be close to a given entangled state [7]. An entanglement witness emerges from the learning process. We use knowledge of the smaller two-qubit system as a means of bootstrapping to larger systems [12]. As the size of the system grows, the amount of additional training necessary diminishes [13, 14], unlike in other methods, for example, which require knowledge or reconstruction of the density matrix [1–3]; thus our method potentially may be of general applicability even to large-scale quantum computers, once they have been built.

In any experimental implementation, we need also to consider that no setup is perfect: there will always be some uncertainty due to extraneous effects. The problem of fault tolerance in quantum computing has received considerable attention. One approach is to design fault-tolerant algorithms [15]. Another is to implement feedback control [16, 17]. Still another is to use repeated weak measurement [18]. Dong et al. [19] use an approach in some ways similar to ours that they call sampling-based learning control. Several authors have established robustness of quantum computing to noise [20]. In quantum systems there is also the problem of decoherence. Under certain conditions [17, 21] or for certain classes of problems [20, 22] quantum computers can be robust to decoherence.

Over the past couple of decades interest has grown in the use of a machine learning or neural network approach to quantum computing [23]. Several different approaches have been used [19, 20, 24–27]. For example, Bisio et al. [26] trained a network to reproduce an unknown (unitary) transformation. We imagine, instead, that we have a particular quantity (here the entanglement) that we wish to measure, and we train the network to give an indicator of that quantity.

Machine learning [19, 24, 25] shows particular promise for dealing with general problems. Classically learning systems such as neural networks have proven fault tolerant and robust to noise; they are also famously used for noise reduction in signals [28]. A machine learning approach would seem to be an excellent one for issues such as noise, decoherence, or missing or damaged data. Here we show that this is in fact the case, using as a test bed our entanglement indicator on the simple two-qubit system.

2 Dynamic Learning of an Entanglement Indicator

In previous work we showed we could successfully train a quantum system to estimate its own degree of entanglement, by mapping a measurable output at the final time, to give an indicator of the entanglement of the prepared, initial state. Briefly, the method was as follows; for full details the reader is referred to [11, 13, 14, 29, 30]

We begin with the Schrödinger equation for the time evolution of the density matrix ρ [31]:

   (1)

where H is the Hamiltonian. We consider an N-qubit system whose Hamiltonian is

   (2)

where {σ} are the Pauli operators corresponding to each of the qubits, {K} are the tunneling amplitudes, {ε} are the biases, and {ζ} are the qubit-qubit couplings. We choose the usual charge basis, in which each qubit’s state is given as 0 or 1; for a system of N qubits there are 2N states, each labeled by a bit string, each of whose numbers corresponds to the state of each qubit, in order. The amplitude for each qubit to tunnel to its opposing state (i.e., switch between the 0 and 1 states) is its K value; each qubit has an external bias represented by its ε value; and each qubit is coupled to each of the other qubits, with a strength represented by the appropriate ζ value. Note that, for example, the operator σx1 = σx ⊗ I … ⊗ I, where there are (N − 1) outer products, acts nontrivially only on qubit 1.

The parameter functions {K(t), ε(t), ζ(t)} direct the time evolution of the system in the sense that if at least one of them is changed, the way a given initial state will evolve in time will also change because of Eqs. (1) and (2). This is the basis for the use of our quantum system as a neural network. The role of the input vector is played by the initial density matrix ρ(0), the role of the output is played by (some function of) the density matrix at the final time, ρ(tf), and the role of the weights of the network is played by the parameter functions of the Hamiltonian, {K, ε, ζ}, all of which can be adjusted experimentally [8]. By adjusting these parameters using a machine learning algorithm, we can train the system to evolve in time from an input state to a set of particular final states at the final time tf. Because the time evolution is quantum mechanical (and, we assume, coherent), a quantum mechanical function, like an entanglement witness of the initial state, can be mapped to an observable of the system’s final state, a measurement made at the final time tf. Complete details, including a derivation of the quantum dynamic learning paradigm using quantum backpropagation [9] in time [10], are given in [11]. We call this quantum system a quantum neural network