The path-integral formulation is one way of writing the trace: as a summation over all possible paths az (t) for each of the j qubits. Here, a is the Coulombic interaction between all the pairs of qubits of the array. The first term in (10.5) is the tunneling term that represents the probability amplitude that the qubit can make transition from one state to another. The second term represents the coulombic interaction between the qubits. (10.5) can be thought of as propagation in imaginary time, because the Boltzmann factor exp{—[H} is the same as the expression for time evolution exp{-itH/h} for t = —i[h, where h is Planck's constant divided by 2^.

To make progress in evaluating the path integral, we discretize the qubits of the QHN. This transforms the path integral over the continuous states of the N qubits, into a set of sums over states of the N qubits at n discrete times. Thus we can, for finite n, compute the state of the net by summing over all possible states of each of the discretized points of each of the qubits. If we let n ^ <x, the discretization is exact. The approximation of finite n will be good as long as [H/n is small. Because the partition function is a trace, the state of each qubit at "time" t = 0 is constrained to be the same as its state at "time" t = —i(3H. Thus we can picture each qubit as a loop, having n discretization points, propagating in imaginary time from 0 to [. A discretization point at any instant is the instantaneous state of the qubit at that corresponding value of inverse temperature. Figure 10.2 shows a qubit with 6 discretization points. While the loop is here drawn as a circle, it should not be imagined as being constrained so: all possible paths from the starting point to the ending point should be imagined as being included in the sum. The lines between discretization points represent the tunneling term in the Hamiltonian. (The analogous picture for a continuum position variable x is perhaps easier to understand; see, e. g., [6].) Qualitatively, we can think of the size of K, the tunneling amplitude, as indicating the floppiness of the bonds between adjacent discretization points: the larger it is, the less constrained is one point by its immediate neighbors, and thus, the more likely is the instantaneous state of the system at an intermediate imaginary time to flip from one state to the next.

Physically, the quantum-mechanical nature of the network increases as the discretization number n increases. In the picture, we can think of the amount of phase space (different states) the system is free to explore, as increasing when we increase the number of discretization points. That is, the loop can become floppier and floppier, for a given value of K, if the number of intermediate intervals is increased. On the other hand, if n is reduced, the loop will have less and less freedom to explore different states, and if we set n =1 the loop must shrink to a single point, with no freedom to flip its state at all. This is the classical situation: no quantum-mechanical tunneling.

Fig. 10.2. A qubit with 6 discretization points

We can also think of this same situation in terms of the inverse temperature. The inverse temperature 3 is the total "length" of the path the qubit has to travel, from its initial point out into the world and then back. At low temperature, which is the quantum limit, 3 is large, which means the system can explore a large amount of the phase space (and we need to set n large in order to have a good approximation); at high temperature, 3 ^ 0 and the loop necessarily shrinks to a point (the classical limit.)

Rewriting the path integral in terms of discretized sums, (10.5) becomes:

Here, Z is the tunneling term that is given by the expression Z = — 2 ln[tanh(3K/n)]. Again, we use the variable t to label the inverse temperature ranging from 0 to 3, but it is now a discrete variable, ranging from 0 to n — 1. The functions {Sj} refer to the instantaneous values assumed by each of the qubits j; i.e. the eigenvalues of the operators az(t) for each of the j qubits, frequently called "spins" in the literature. Thus, {Sj(t)} is a microstate of the system; and the summation over all the microstates

gives the partition function. The contribution of microstates to the partition function is large as their associated energy is low. For an infinite discretization value (n ^ to), the discretization representation is exact, as indicated by the equality sign in (10.6) and as noted above, but since we do not have an infinite (or quantum) computer on which to do the simulation, we will make do with finite n. As long as 3H/n is much less than one, the discretization error should be small. Only infinite n and n =1 have recoverable physical meaning, as corresponding to the quantum and classical cases, respectively; however, as we increase n we can investigate the role of increasing quantum-mechanical character of the network.

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