## Quantum Model

The quantum system can be put into the Hopfield form to obtain the quantum Hopfield net (QHN) as follows. Consider an array of N qubits. We write the Hamiltonian, or energy function, for each qubit j as

where ax = ^ and az = ^ are the Pauli matrices, and the full

Hamiltonian H = ^ Hj. The first term represents the flipping of the qubit from one state to the other, with amplitude K. This term allows a quantum system to switch from one state to another without having the energy to climb the associated energy barrier and is therefore called "tunneling". The effect of ax can be understood by looking at its operation on a state This state has probability amplitude S of being found in the +1 state, and y of being found in the -1 state, J ; ax operating on this state gives us

1 CO = CO, in which the amplitudes are flipped. In particular, the

+ 1 state is flipped to the -1 state, and vice versa. The second term in (10.4) represents the energy difference 2A between the two states. This difference can be the result of external fields or interaction with other qubits. The

Pauli matrix az measures the state of the system; for example,, _ 1 , ,

(-1) 01 , telling us that the system is in the -1 state. The difference between the operation of az on the +1 state and on the -1 state is (+1 — ( — 1)) = 2, so Aaz gives us an energy difference of 2A between the two states.

For the whole system we need to include all the qubits and their interactions. For a system at finite temperature we write the partition function Q = tre-pH, where tr is the trace of the matrix and ¡3 the inverse temperature in units of Boltzmann's constant. The partition function contains information about all possible states of the quantum system, and how they depend functionally on each variable and each parameter. This will lead us to our definition of the quantum Hopfield net. Following  and , we write the trace in the basis set of state variables {<Sj = ±1} , and use the Feynman path integral  formulation: