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Pattern Association Using New Maximally Entangled States in a Two-qubit System

Manu Pratap Singh, B. S. Rajput, Shireesh Reddy Pyreddy, Mudasir Manzoor Kirman, Syed Mohsin Saif, Ahmad Talha Siddiqui

Published Apr 27, 2016• 0 citations
OpenAlex

Abstract

New set of maximally entangled states (Singh-Rajput MES), constituting orthonormal eigen bases, has been revisited and its superiority and suitability in the processes of quantum associative memory (QuAM) have been demonstrated. Using these MES as memory states in the evolutionary process of pattern storage, the suitability and superiority of these MES over Bell’s MES have been demonstrated and it has been shown that, under the operations of all the possible memorization operators for a two-qubit system, the first two states of Singh-Rajput MES are useful for storing the pattern ǀ11 > and the last two of these MES are useful in storing the pattern ǀ10 > while Bell’s MES are not much suitable as memory states in a valid memorization process. Recall operations of quantum associate memory (QuAM) have been conducted through evolutionary process in terms of unitary operators by separately choosing Singh-Rajput MES and Bell’s MES as memory states for various queries and it has been shown that in each case the choices of Singh-Rajput MES as valid memory states are much more suitable than those of Bell’s MES.

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New set of maximally entangled states (Singh-Rajput MES), constituting orthonormal eigen bases, has been revisited and its superiority and suitability in the processes of quantum associative memory (QuAM) have been demonstrated. Using these MES as memory states in the evolutionary process of pattern storage, the suitability and superiority of these MES over Bell’s MES have been demonstrated and it has been shown that, under the operations of all the possible memorization operators for a two-qubit system, the first two states of Singh-Rajput MES are useful for storing the pattern ǀ11 > and the last two of these MES are useful in storing the pattern ǀ10 > while Bell’s MES are not much suitable as memory states in a valid memorization process. Recall operations of quantum associate memory (QuAM) have been conducted through evolutionary process in terms of unitary operators by separately choosing Singh-Rajput MES and Bell’s MES as memory states for various queries and it has been shown that in each case the choices of Singh-Rajput MES as valid memory states are much more suitable than those of Bell’s MES.
Abstract This paper discusses an approach to constructing an artificial quantum associative memory (QuAM). The QuAM makes use of two quantum computational algorithms, one for pattern storage and the other for pattern recall. The result is an exponential increase in the capacity of the memory when compared to traditional associative memories such as the Hopfield network. Further, the paper argues for considering pattern recall as a non-unitary process and demonstrates the utility of non-unitary operators for improving the pattern recall performance of the QuAM. 1.
Starting with the theoretical basis of quantum computing, entanglement has been explored as one of the key resources required for quantum computation, the functional dependence of the entanglement measures on spin correlation functions has been established and the role of entanglement in implementation of QNN has been emphasized. Necessary and sufficient conditions for the general two-qubit state to be maximally entangled state (MES) have been obtained and a new set of MES constituting a very powerful and reliable eigen basis (different from magic bases) of two-qubit systems has been constructed. In terms of the MES constituting this basis, Bell’s States have been generated and all the qubits of two-qubit system have been obtained. Carrying out the correct computation of XOR function in neural network, it has been shown that QNN requires the proper correlation between the input and output qubits and the presence of appropriate entanglement in the system guarantees this correlation.
There is increasing interest in quantum computing, because of its enormous computing potential. A small number of powerful quantum algorithms have been proposed to date; however, the development of new quantum algorithms for practical use remains essential. Parallel computing with a neural network has successfully realized certain unique functions such as learning and recognition; therefore, the introduction of certain neural computing techniques into quantum computing to enlarge the quantum computing application field is worthwhile. In this paper, a novel quantum associative memory (QuAM) is proposed, which is achieved with a quantum neural network by employing adiabatic Hamiltonian evolution. The memorization and retrieval procedures are inspired by the concept of associative memory realized with an artificial neural network. To study the detailed dynamics of our QuAM, we examine two types of Hamiltonians for pattern memorization. The first is a Hamiltonian having diagonal elements, which is known as an Ising Hamiltonian and which is similar to the cost function of a Hopfield network. The second is a Hamiltonian having non-diagonal elements, which is known as a neuro-inspired Hamiltonian and which is based on interactions between qubits. Numerical simulations indicate that the proposed methods for pattern memorization and retrieval work well with both types of Hamiltonians. Further, both Hamiltonians yield almost identical performance, although their retrieval properties differ. The QuAM exhibits new and unique features, such as a large memory capacity, which differs from a conventional neural associative memory.
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.
We generalize the canonical Hopfield model of associative memory, which stores classical states of individual spins, to store entangled quantum states. The construction is based on truncated projector-sum Hamiltonian, and we explicitly consider a special case that stores dimer singlet coverings of a spin-1/2 system. We analyze the stability of the stored memories and find that the capacity -- the number of stable patterns -- scales faster with the system size than one would expect from a naive analogy with the standard Hopfield model, suggesting a possible quantum advantage.
Every Maximally Entangled State (MES) of two d-dimensional particles is shown to be a product state of suitably chosen collective coordinates. The state may be viewed as defining a "point" in a "phase space" like d^2 array representing d^2 orthonormal Maximally Entangled States basis for the Hilbert space. A finite geometry view of MES is presented and its relation with the afore mentioned "phase space" is outlined: "straight lines" in the space depict product of single particle mutually unbiased basis (MUB) states, inverting thereby Schmidt's diagonalization scheme in giving a product single particle states as a d-terms sum of maximally entangled states. To assure self sufficiency the essential mathematical results are summarized in the appendices.