A quantum Hopfield network (QHN) has an enormously expanded range of states as the qubits can exist in any superposition of the two states. This is not the same as a continuum classical net, since the state of the system is not between the two states, but both states at the same time: the system carries information on as many channels as there are states contributing to the superposition. Under some circumstances that information is accessible, as in the operation of some quantum algorithms [16, 9].

Recall that the contribution of microstates to the partition function is large if their associated energy is low. In (10.6), the states that are more probable could be obtained by minimizing the argument of the exponential. We rearrange that argument to correspond with the L function for a Hopfield network. That is, we define the L function for the QHN as the integral of the Hamiltonian over the imaginary time loop, divided by 3, i.e. the quantity in curly brackets in (10.6):

(t + 1)+ £ £S(t)ajj'MS"(f) j=1 t=0 j=j' = 1 t=0

This is the QHN. Classically, the high-gain Hopfield net replaces a continuum of possible values of each neuron with a finite number; in the QHN it becomes again a continuum, as described below. Information in the Hopfield net is stored as stable states. Classically the discrete net converges to the corners of an N-dimensional box [11], where the neuron outputs necessarily equal plus or minus 1. In a network of N classical neurons there are 2N possible states (corners of the box.) Once we discretize the quantum Hop-field net to discretization number n, there are 2Nn possible states: that is, each corner of the N-dimensional classical Hopfield box has 2n states. The greater the discretization n, the greater the number of possible states. When the temperature is lower we need a larger discretization number; thus, the lower the temperature the greater the storage capacity of the QHN. For infinite n we have the fully quantum-mechanical system, and have a fully faithful representation of the quantum dispersion - all possible superposition states.

To see how our quantum loop picture works with the Hopfield formulation, consider now the discretized path integral picture for more than one qubit, a QHN at finite n. Figure 10.3 shows a picture of the net for two qubits (N = 2) and two discretization points (n = 2). Each loop represents a qubit or two-level system (TLS), propagating in imaginary time (inverse temperature), from t = 0 to t = [. Each dot on a loop represents the instantaneous state of the qubit at a particular value of t, and these dots interact with their nearest neighbors along the loop (the first term in (10.6) or (10.7)). This is the tunneling term. The arrows show the direction of integration, from t = 0 to t = [. Each dot also interacts with other dots on other loops according to the second term in (10.6). In the simplest possible model, which we consider here, the two qubits interact only at equal imaginary times t, and the Lyapunov function from (10.7) becomes:

c where the (primed, unprimed) S functions refer to the two interacting loops.

Fig. 10.3. Representation of two interacting qubits, with discretization n = 2. Each loop represents a qubit, propagating in imaginary time (inverse temperature) from 0 to ¡3. The interactions between discretization points on the same loop have a strength of the (bidirectional) interactions between points on different loops, of a

Fig. 10.3. Representation of two interacting qubits, with discretization n = 2. Each loop represents a qubit, propagating in imaginary time (inverse temperature) from 0 to ¡3. The interactions between discretization points on the same loop have a strength of the (bidirectional) interactions between points on different loops, of a

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