Quantum Neural Networks
Quantum neural networks are parameterized quantum circuits trained like neural networks, borrowing the layered structure and gradient-based optimization of deep learning.
The analogy and its limits
A quantum neural network (QNN) is a parameterized quantum circuit whose gate angles play the role of weights. Layers of single-qubit rotations act like activations applied per neuron, and entangling gates act like connections between neurons that create correlations no classical product state can hold. Training minimizes a loss by gradient descent, exactly as in classical deep learning.
The analogy breaks in important ways. A quantum state evolves by unitary, hence linear and reversible, operations; nonlinearity enters only through measurement, data re-uploading, or mid-circuit resets. There is no direct quantum equivalent of an arbitrary nonlinear activation applied to a hidden vector, so QNNs realize a restricted, structured function class rather than the universal approximators of classical nets.
Sources of nonlinearity
- Measurement: expectation values are nonlinear in the state amplitudes.
- Data re-uploading: interleaving encoding and trainable layers builds up richer functions of the input.
- Non-unitary operations: mid-circuit measurement and reset introduce controlled nonlinearity at the cost of extra hardware demands.
Expressivity through Fourier series
A clean result describes a QNN with repeated data encoding as a truncated Fourier series in the input: the encoding frequencies set which harmonics appear, and the trainable gates set their coefficients. This makes the model's reachable functions concrete and shows why deeper re-uploading enlarges the fittable function space. See encoding and Fourier expressivity.
Training obstacles
QNNs inherit every difficulty of variational training: barren plateaus that flatten the landscape, shot noise in every gradient estimate, and hardware noise that biases outputs. Cost-function locality and clever initialization help, but there is no backpropagation on hardware; gradients come from the parameter-shift rule and cost circuit evaluations that grow with the number of parameters.
Where they may earn their keep
The most promising setting is quantum data: states produced by a quantum sensor, chemistry simulation, or another circuit. There the encoding step is free and the network processes information that has no compact classical description. For classical data, QNNs remain a research subject without a demonstrated general advantage over classical networks.