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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.

Richard Feynman examined the role quantum mechanics can play in the development of future computer hardware and demonstrated [

The physically allowed degree of entanglement and mixture is a timely issue given that the entangled mixed states could be advantageous for certain quantum information situation [

Starting with the theoretical basis of quantum computing in the present paper, 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. It has been shown that the degree of entanglement for a two-qubit state depends on the extent of fractionalization of its density matrix and that the entanglement is completely a quantum phenomenon without any classical analogue. A reliable measure of entanglement of two-qubit states has also been expressed in terms of concurrence [

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. It has been emphasized that the newly constructed maximally entangled two-qubit states, constituting new eigen basis, may be the most appropriate choice for utilizing entanglement in quantum neural computation. It has been shown that in quantum approach to neural networks all patterns can be stored as a superposition, where each of the patterns can be considered as existing in a separate quantum universe. It has also been shown that in neural networks the integrity of a stored pattern (bases states) is due to entanglement and the quantum associate memory (Qu AM) is the realization of the extreme condition of many Hopfield networks each storing a single pattern in parallel quantum universes.

At quantum level an electron can be in a superposition of many different energy states, which is not possible classically. Similarly, any Physical system is described by quantum state

which is a linear superposition of basis states

Electron-spin is a two state system with elements

As long as system maintains coherence, it cannot be said to be in either spin-up or spin-down. When it decoheres, it can be in either of these states. Such a simple two- state quantum system is the basic unit of quantum computation: quantum-bit (qu-bit) where we rename two states as 0-state, and 1-state. Smallest unit of information stored in a two-state quantum computer is called a qu-bit. If there is a system of m qu-bits, it can represent

Qubit is simply a two-level system with generic state as

a two-dimensional complex vector, where a and b are complex coefficients specifying the probability amplitudes of corresponding states such that

Qubit individual is defined by a string of qubits. An operator on a Hilbert space describes how one eigen state is changed into other. Thus a quantum operator is a q-gate and represented by a square matrix.

State of a qubit can be changed by the operation with a quantum gate which derives the individuals towards better solution (eventually towards a single state). A quantum gate is a reversible gate and can be represented as a unitary matrix U acting on a qubit basis state. Q-gates operating on just two bits at a time are sufficient to construct ageneral quantum circuit (based on Lie-Grouptheory). Thus quantum operator may be made [

Wave-peaks in phase interfere constructively and those out of phase destructively

Let

and an operator represented by matrix

Then we have

ÞAmplitude of

Quantum Computation (QC) can be defined as representing the problem to be solved in the language of quantum states and producing operators that derive the system to a final state such that when system is observed there is high probability of finding a solution. QC consists of state preparation; useful time evolution of quantum system; and measurement of the system to obtain information. Upon measurement system will collapse to a single basis state. Object of QC is to ensure that measured basis state is with high probability. There are three different approaches to state preparation, based on information in set T of (n + 1) two states quantum systems (

1) Inclusion, 2) Exclusion, 3) Phase Inversion.

Inclusion is most intuitive where basis states not in T have zero coefficients and those in T have non-zero coefficients in the superposition:

Exclusion is an opposite approach, where basis state in T has zero coefficients and those not in T have non-zero coefficients in the superposition:

In Phase Inversion all basis states are included with coefficients of equal amplitudes but with different phases based on membership in T:

After state preparation, the pattern classification may be performed in straight forward approach employing the method of Grover’s [

If

Let us consider the case of n = 2 and T = {(001), (111)}. Then we have

and

where

Here probability

highest conditional probability

For implementing quantum computation there are following five requirements:

1) A scalable system with well characterized qubits;

2) Ability to initialize the state of qubits to a simple feudal state

3) Long relevant decoherence time (longer than gate operation time);

4) A universal set of quantum gates;

5) A qubit-specific measurement capability;

For quantum communication there are two more requirements;

1) Ability to interconvert stationary and flying qubits;

2) Ability to faithful transmit flying qubits between specific locations.

It is the correlation that can exist between different qu-bits (very little understood). When superposition is destroyed, the proper correlation is communicated between the qu-bits. It is this correlation that is the crux of entanglement.

Mathematically, it is described using density matrix formulation.

Density matrix of state

The state for which density matrix cannot be factorized is said to be entangled while those with fully factorized density matrix are not entangled at all. For instance, let us consider a two-qubit state

which appears in matrix form as

where “1” denotes the presence of the corresponding eigen state in the superposition and ‘0’ denotes its absence, i.e. “1” for

which cannot be factorized at all and the state

Let us now consider the following quantum state as superposition of qubits

or

Its density matrix is

which is completely factorized. This state is not entangled at all.

Another quantum state with as superposition of qubits with least Hamming spread may be written as

with density matrix

which is fully factorized.

On the other hand the quantum state as superposition of qubits

Its density matrix is

which can be only partially factorized as

and hence the state is partially entangled. Thus the degree of entanglement for a two-qubit state depends on the extent of fractionalization of its density matrix and the entanglement is completely quantum phenomena without classical analogue.

It may readily be shown that the density matrix for the following two-qubit states (Bell States) cannot be factorized at all;

And hence all these states are maximally entangled states (MES). The matrices of these states satisfy the condition

or

The states given by Equation (3.8) also satisfy the condition

These Equations (3.9) and (3.10) show that the states, given by Equation (3.8), constitute the orthonormal complete set and hence form the eigen-basis (magic basis) of the space of two level qubits. These states are maximally entangled states (MES) and also the eigen states of the unitary operator

where

where

and

For pure state

with its concurrence defined as [

If the concurrence

For

the state

The concurrence of a state is as reliable measure of degree of entanglement as the extent of factorization of its density matrix while Hamming spread of a two-qubit state is not that reliable measure of the entanglement since the states

In terms of Z-components of spins of two electrons, the states

and

which consists of qubits with anti-parallel spins. On the other hand, the states

and

with one combination of parallel spins and other of anti-parallel spins. In states

Various qubits of two-qubit states may be written as follows in magic basis;

A general two-qubit state may be written as

where

Using the relations (3.19), this state may be written as

and using relation (3.15), its concurrence becomes

Thus for non-entangled state (i.e. separable state), we have

and for partially entangled states,

For MES, we have

or

which can be true either for

or for

These are the necessary conditions for the state

and

Bell states (i.e. magic bases) given by Equation (3.8) may readily be obtained from the state

For these sets of values of a and b, the state

tated by

Other maximally entangled two-qubit states which form the orthonormal complete set (i.e. eigen bases) may be obtained as follows by putting

with their density matrices respectively given by

None of which can be factorized at all. The concurrence for each of these states is unity and these states constitute the orthonormal set since

And

Other six MES obtained from

States given by Equation (4.11) also constitute the eigen basis (different from magic basis given by Equation (3.8)) of the space of two-qubit system. In this basis, various qubits of two-qubit states may be written as

Substituting these relations in Equation (3.8), Bell states may be constructed as follows in this basis;

Concurrence of each of Bell states in this basis also is unity showing the invariance of concurrence in different bases.

Condition (4.6) for partial entanglement shows that if any coefficient of qubits in the state

uation (4.1) is vanishing, then the state is necessarily partially entangled and its concurrence is

squares of moduli of non-zero coefficients is 3. For instance, let

currence given by Equation (4.4) becomes

are partially entangled with concurrence

We have following motivations for applying capabilities of quantum computation to neural networks.

1) To compensate for ever decreasing scale in hardware development;

2) To produce computational capability not available using classical neural computation;

3) Recent Demonstration of superiority of Quantum Neural Network (QNN) over Classical ones [

In Quantum neural computing, the phenomenon of entanglement can be viewed as playing arole similar to that of weighted connections in the classical neural network, producing correlations between different parts of the system. The quantum computational systems that make use of entangled states have the potential functionality of quantum neural networks (QNN). For instance, let us consider the entangled three-qubit state

which can be interpreted as computing the XOR function [

The probability of finding the input in the state

followed by the operator

is operated upon the state

Thus the correct computation of XOR function requires the proper correlation between the input and output qubits. The presence of appropriate entanglement in the system guarantees this correlation. For entangled quantum states local operations on some qubits affect the states of all qubits in the system. Maximally entangled two-qubit states, constructed in Section 4, are the most appropriate choice for utilizing entanglement in quantum neural computation.

Two main difficulties faced in the implementation of QNN are related with linearity of quantum theory (while neural-computing depends upon non-linear data processing) and unitarity of evolutionary operators in quantum mechanics (while the pattern recall problem in QNN is equivalent to a search of a random data base). In the case of storage algorithm, evolution processes are a necessity (since the system must maintain a coherent superposition that represents the stored pattern) but requiring the recall mechanism to be evolutionary will limit the efficiency with which the recall may be accomplished and hence the recall is needed through non-evolutionary (i.e. non- unitary) process. These difficulties may be removed in the many universe interpretation of quantum mechanics, where decoherence or collapse of wave-function is only an illusion and the effect of measurement is split in to a number of copies each observing just one of the possible results of the measurement, unaware of the other possible outcomes. In this approach, there exist many mutually unobservable but equally real universes, each corresponding to a single possible outcome of the measurement and correlating through maximally entangled states. This combines the field of ANN with quantum computation in a natural way.

Hopfield neural network is best suited for the extraction of the locally most plausible version of a single prototype. If we generate multiple classical Hopfield networks which store only one pattern each, we lose any parallelism in processing the information. But in quantum approach, we can store all patterns as the quantum superposition

where each of the patterns p can be considered as existing in a separate universe. Interaction of a superposition with the environment is performed in parallel. Each of the basis states in superposition will play the role of a single memory state independent of the number of them that exist in superposition.

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. Equations (3.3), (3.5), (3.6a) and (3.7a) show that the degree of entanglement for a two-qubit state depends on the extent of fractionalization of its density matrix and that the entanglement is completely a quantum phenomenon without any classical analogue. Equations (3.9) and (3.10) show that the maximally entangled Bell states, given by Equation (3.8), constitute the orthonormal complete set and hence form the eigen basis of the space of two-qubit states. These states have also been shown to be the eigen states of the unitary operator, defined by Equation (3.11), with the corresponding eigen values given by Equation (3.13). A reliable measure of entanglement of two-qubit states has also been expressed in terms of concurrence defined by Equation (3.15) and it has been shown by Equations (3.17) and (3.18) that in a free two-qubit system the states with both combinations of parallel spins (i.e. states with maximum Hamming spread) are definitely maximally entangled states (MES) while among the states with minimum Hamming spread, those with both anti-parallel combinations are MES and those with one combination of parallel spins and other with anti-parallel spins are not entangled at all. Equation (3.19) represents various qubits of two-qubit states in magic bases. Equation (4.7) gives the necessary and sufficient conditions for the general state, given by Equation (4.1), to be maximally entangled state. Equation (4.5) gives the condition for this state to be non-entangle (i.e. separable) while (4.6) gives the condition for this state to be partially entangled. Equations (4.8) and (4.9) give two different sets of maximally entangled two-qubit states, where it has been demonstrated that Bell states may be obtained from the state of Equation (4.8) by substituting

been shown to produce Bell states

rotated by

Equation (4.10) gives other MES, different from Bell states, forming the orthonormal complete set (i.e. eigen basis) and it has been shown that, besides these two sets, there is no other orthonormal complete set of MES in two-qubit systems. Thus the set of Bell states is not the only eigen basis (magic eigen basis) of the space of two- qubit system, the set of MES given by Equation (3.10) also constitutes a very powerful and reliable eigen basis of two-qubit systems. This is the new eigen basis, being introduced for the first time, and to differentiate it from the already known Bell’s bases, let us call it Singh-Rajput basis for its possible use in future in the literature. The MES constructed in form given by Equation (4.10) may be correspondingly called Singh-Rajput states which generate Bell’s States in the form given by Equation (4.12). In terms of these states, all the qubits of two-qubit system may be obtained in terms of Equation (4.11). For symmetry purpose, for establishing functional dependence of entanglement on spin operators of qubits constituting MES and for the representations of SU(2) group and three dimensional rotation group, the use of these states may be more convenient. These possibilities will be demonstrated in our forthcoming papers.

Equations (5.1) and (5.4) demonstrate that the correct computation of XOR function in QNN requires the proper correlation between the input and output qubits. The presence of appropriate entanglement in the system guarantees this correlation. For entangled quantum states, local operations on some qubits affect the states of all qubits in the system. Maximally entangled two-qubit states (Singh-Rajput States), constructed in Section 4, may be the most appropriate choice for utilizing entanglement in quantum neural computation. In quantum approach to neural networks, all patterns can be stored as superposition given by Equation (5.5), where each of the patterns p can be considered as existing in a separate quantum universe. In this quantum analogue of Hopfield neural network, the integrity of a stored pattern (basis states) is due to entanglement. It leads to all known quantum algorithms. Quantum associate memory (Qu AM) is the realization of the extreme condition of many Hopfield networks, each storing a single pattern in parallel quantum universes.

Manu P.Singh,B. S.Rajput, (2015) Role of Entanglement in Quantum Neural Networks (QNN). Journal of Modern Physics,06,1908-1920. doi: 10.4236/jmp.2015.613196