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Using this quantum feed-forward network, deep neural networks can be executed and trained efficiently. A deep neural network is essentially a network with many hidden-layers, as seen in the sample model neural network above. Since the Quantum neural network being discussed utilizes fan-out Unitary operators, and each operator only acts on its respective input, only two layers are used at any given time.<ref name=":0" /> In other words, no Unitary operator is acting on the entire network at any given time, meaning the number of qubits required for a given step depends on the number of inputs in a given layer. Since Quantum Computers are notorious for their ability to run multiple iterations in a short period of time, the efficiency of a quantum neural network is solely dependent on the number of qubits in any given layer, and not on the depth of the network.<ref name=":1" />
=== Cost
To determine the effectiveness of a neural network, a cost function is used, which essentially measures the proximity of the network’s output to the expected or desired output. In a Classical Neural Network, the weights (<math>w </math>) and biases (<math>b </math>) at each step determine the outcome of the cost function <math>C(w, b)</math>.<ref name=":0" /> When training a Classical Neural network, the weights and biases are adjusted after each iteration, and given equation 1 below, where <math>y(x)</math> is the desired output and <math>a^\text{out}(x)</math> is the actual output, the cost function is optimized when <math>C(w, b)</math>= 0. For a quantum neural network, the cost function is determined by measuring the fidelity of the outcome state (<math>\rho^\text{out}</math>) with the desired outcome state (<math>\phi^\text{out}</math>), seen in Equation 2 below. In this case, the Unitary operators are adjusted after each iteration, and the cost function is optimized when C = 1.<ref name=":0" />
Equation 1 <math>C(w,b)={1 \over N}\sum_{x}{||y(x)-a^\text{out}(x)|| \over 2}</math>
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