Quantum Computation and Nonlinearity

In practice, the most useful model for quantum computation is the Quantum Computational Network also called Deutsch's model [5,9]. The basic information unit in this model is a qubit, which can be considered a superposition of two independent states $\mid0\rangle$ and $\mid 1\rangle$ , denoted by $\mid \psi \rangle =\alpha_{0}\mid 0\rangle +\alpha _{1}\mid 1\rangle$, where $\alpha _{0},\alpha _{1}$ are complex numbers such that $\left\vert \alpha _{0}\right\vert ^{2}+\left\vert\alpha _{1}\right\vert ^{2}=1.$ A composed system with $n$ qubits is described using $N=2^{n}$ independent states obtained through the tensor product of the Hilbert Space associated with each qubit. Thus, the resulting space has a natural basis that can be denoted by: $\left\{ \midi_{0}i_{1}...i_{n-1}\rangle ;\quad i_{j}\in \left\{ 0,1\right\} \right\} .$ This set can be indexed by $\mid i\rangle ;\quad i=0,1,...,N-1.$ Following the Quantum Mechanics Postulates, the state of the system $\mid \psi\rangle ,$ in any time $t,$ can be expanded as: $\mid \psi \rangle=\sum_{i=0}^{N-1}$$\alpha _{i}\mid i\rangle$ and $\sum_{i=0}^{N-1}$$%%\left\vert \alpha _{i}\right\vert ^{2}=1.$ The computation unit in Deutsch's model consists of quantum gates which are unitary operators that evolves an initial state performing the necessary computation to get the desired result (final state). A quantum computing algorithm can be summarized in three steps: (1) Prepare the initial state; (2) A sequence of quantum gates to evolve the system; (3) Quantum measurements. From quantum mechanics theory, the last stage performs a collapse and only what we know in advance is the probability distribution associated to the measurement operation. So, it is possible that the result obtained by measuring the system should be post-processed to achieve the target (the Deutschïs Algorithm (Chapter 6 of [9]) is a nice example). Let us return to the perceptron model of section 2. Would it be possible to implement a system analogous with it but based on quantum mechanics? Just as a matter of setting ideas, let's take the quantum inspired perceptron model proposed in [1]. In this model a quantum system with $n$ input qubits $\mid x_{0}\rangle$, $\mid x_{1}\rangle$,..., $\mid x_{n-1}\rangle$ is considered and an output is derived by the rule:

$$\mid y\rangle =\sum\limits_{j=0}^{n-1}\breve{U}_{j}\mid x_{j}\rangle$$ (3)

where $\breve{U}_{j}$ are $2\times 2$ matrices acting on the basis $\left\{\mid 0\rangle ,\mid 1\rangle \right\}$. In analogy with the classical perceptron, the following learning rule is proposed:

$$\breve{U}_{j}(t+1)=\breve{U}_{j}\left( t\right) +\eta \left......e-\mid y\left( t\right) \rangle \right) \langle x_{j}\vert$$ (4)

where $\mid d\rangle$ is the desired output. It can be shown [1] that the above rule drives the system into desired state $\mid d\rangle$. From the quantum mechanics point of view, the first problem of the above system is that the learning rule in expression (4) is not an unitary operation in general (the same is true for expression (3)). That is way we call this model quantum inspired. Besides, ANNs need activation functions, which are scalar and nonlinear functions, to be implemented. Nonlinear effects in quantum computation are discussed by Gupta at al. [5] when they proposed a new gate - a dissipative one- called D-Gate. The behavior of the D-Gate is the following: given the state system $\mid \psi \rangle=\sum_{i=0}^{N-1}$$\alpha _{i}\mid i\rangle ,$ let $\mathit{A}%%\left( \mid i\rangle \right)$ and $\mathit{A}^{\prime }\left( \mid i\rangle\right)$ respectively denote the probability amplitudes before and after the application of the D operator. Then, if $\mathit{A}\left( \mid 0\rangle\right) >\delta \Rightarrow \mathit{A}^{\prime }\left( \mid 0\rangle \right)=c$; otherwise $\mathit{A}^{\prime }\left( \mid 0\rangle \right) =0$. The value $c$ for probability amplitude denotes some constant used for encoding $%%1$ and $\delta$ is a pre-set threshold. From the point of view of Gupta at al. [5] we could postulate a quantum neural network constructed from Unitary operators and the D-Gate. In the network representation, the quantum gates are interconnected by wires indicating the information flow during the computation (see Figure 2). By convention, the computation proceeds from left to right.

Figure 2: Outputs are connected to Gates inputs in the network.

However, a quantum mechanics feasible learning rule should be designed to complete the QNN. This point is not addressed by Gupta at al.

To answer this question we need a more deeply consideration about the D-Gate and its hardware implementation. This is the starting point of our work. The D-Gate nonlinearity is due to dissipations. Such irreversible operation can be realized if full interaction with the environment is taken into account. The behavior of a system can also be nonlinear because of the interactions between its degrees of freedom (see sections 5, 6 of [5]). But, what kind of physical system in quantum mechanics can perform nonlinear operations?

What about learning rules? The expression (4) gives a rule that adapts operators which evolves the state of the system. However, in the classical perceptron, the quantities affected by the learning rule (2) are system parameters! Quantum mechanical systems have in general a set of pre-defined parameters. Could be a learning rule that adapts system parameters more feasible in practice? Is there such a rule? We believe that a possible (may be partial) solution for these questions is the model stated next.