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Quantum generalisation of feedforward neural networks
Kwok Ho Wan, Oscar Dahlsten, Hlér Kristjánsson, Robert Gardner, M. S. Kim
TL;DR
The paper addresses how classical feedforward neural networks can be generalised to process quantum inputs despite the requirement for reversible quantum dynamics. It makes neurons reversible and then unitary, trains the resulting network by global gradient descent, and demonstrates quantum autoencoding and teleportation rediscovery. The authors conclude that these networks can serve as versatile tools for quantum information processing, with photonic implementation as a natural platform.
Problem
Quantum neural-network designs face a clash between nonlinear, dissipative neural computation and linear, reversible quantum computation, while the connection between quantum machine learning and neural networks remains limited.
Method
The paper renders classical feedforward neurons reversible with ancillary bits, generalises them to unitary quantum operations, and trains the network using global gradient descent on a quantum cost function.
Results
The network compressed quantum states, including entangled inputs, and learned the teleportation protocol from the task alone; teleportation training converged toward zero cost for randomly initialised weights.
Takeaways & Limitations
The proposed networks provide a framework for processing quantum inputs coherently while remaining strict generalisations of classical neural networks, and may become versatile tools for quantum information scientists.
Takeaways & Limitations
Training efficiency is established only in the sense that weight counts and cost-function evaluations per iteration grow polynomially; polynomial iteration counts and classical simulation efficiency are not established.
Abstract
from arXiv · showhide
We propose a quantum generalisation of a classical neural network. The classical neurons are firstly rendered reversible by adding ancillary bits. Then they are generalised to being quantum reversible, i.e.\ unitary. (The classical networks we generalise are called feedforward, and have step-function activation functions.) The quantum network can be trained efficiently using gradient descent on a cost function to perform quantum generalisations of classical tasks. We demonstrate numerically that it can: (i) compress quantum states onto a minimal number of qubits, creating a quantum autoencoder, and (ii) discover quantum communication protocols such as teleportation. Our general recipe is theoretical and implementation-independent. The quantum neuron module can naturally be implemented photonically.
INTRODUCTION
The paper develops a quantum generalisation of feedforward neural networks by converting classical step-function neurons into reversible and then unitary operations. It introduces an efficient gradient-descent training approach and demonstrates quantum autoencoding and teleportation applications.
- Quantum machine learning has focused less on neural-network connections, despite using quantum information processing based on superposition and phase.
- The proposed networks process quantum inputs coherently and strictly generalise classical neural networks, enabling direct comparison between quantum and classical models.
- The network is trained by global gradient descent on a quantum cost function and applied to quantum autoencoding and rediscovery of teleportation protocols.
- Classical neurons use weighted inputs and a Heaviside activation function, producing 1 above a threshold and 0 otherwise.
- Each irreversible neuron is made reversible by adding a dummy input bit, yielding a permutation-matrix operation that preserves the original inputs and records the output.
- The reversible construction is then generalised to a unitary operation on quantum states, with classical computation recovered when inputs are computational-basis mixtures and the unitary is a permutation matrix.
The network
The quantum network preserves feedforward structure while replacing irreversible copying with reversible operations. Because quantum states cannot generally be cloned, the design uses trainable fan-out unitaries that distribute information without ordinary copying.
- Feedforward networks arrange neurons in layers, connecting each neuron to neurons in adjacent layers without within-layer connections.
- Classical copying is rendered reversible by mapping (b, 0) to (b, b) with a permutation-matrix gate.
- The quantum no-cloning theorem prevents naive copying of arbitrary quantum states, although a CNOT copies computational-basis states.
- The quantum network replaces CNOT copying with a trainable two-qubit fan-out unitary that distributes input-state information without generally copying non-orthogonal states.
Efficient training with gradient descent
The quantum network uses continuously parameterized unitaries and global gradient descent to minimize a quantum cost function. The training is efficient in the number of cost-function evaluations, although convergence is not proved polynomial and classical simulation is generally inefficient.
- Gradient descent: Global gradient descent updates each parameter in the direction that decreases the cost function, using numerical partial derivatives and an adjustable learning parameter η.Each derivative is approximated from cost-function evaluations at shifted parameter values.
- Quantum generalisation: Classical neural-network gradient descent is extended by parameterizing quantum unitaries with continuous variables.General unitaries are expressed through Hermitian generators and Pauli-matrix tensor products, allowing the classical weight-update rule to act on unitary parameters.
- Quantum cost function: The quantum cost function compares actual and desired local Pauli expectation values on individual output qubits.In classical operation, only the Z-Pauli expectation is nonzero, reducing the quantum cost function to the classical form up to a simple transformation.
- Efficiency and scope: The training procedure requires polynomially many cost-function evaluations per iteration because both the number of weights and parameter shifts grow polynomially with network size.The paper does not prove that convergence to zero cost always occurs in polynomially many iterations.
Classical autoencoder
A classical autoencoder compresses inputs through a smaller bottleneck layer and trains later layers to reconstruct the original signal. Removing the post-bottleneck layers exposes the compressed representation.
- Architecture: A classical autoencoder compresses a signal by passing it through a bottleneck layer with fewer neurons than the input layer.The minimum useful bottleneck size depends on the input data set.
- Training objective: The network is trained to recreate the input at a later layer, requiring reversible compression to the bottleneck size.The bottleneck can be varied during training to search for the smallest compression size possible.
- Training objective: For a two-bit input compressed to one bit, training uses a cost function and the stated learning rule, with complete training when outputs match inputs to numerical precision.Removing the post-bottleneck layers then reveals the compressed data in the second-last layer.
Quantum autoencoder
The quantum autoencoder replaces classical neurons with unitaries and adds a fan-out gate to compress quantum inputs while reconstructing them at later output qubits. Gradient-descent training reached near-zero cost across tested orthogonal input sets, including superpositions.
- Quantum autoencoder: The quantum autoencoder generalizes classical neurons into unitary matrices and adds a fan-out gate to the circuit.The input state is processed through U1, U2, U3, and UF, with final neurons reconstructing the input on designated output qubits.
- Quantum autoencoder: FIG. 4 depicts a quantum autoencoder accommodating two entangled input qubits, with the blue box marking the trained quantum compression device.The figure identifies the compression device after training rather than reporting a numerical performance value.
- Training: Global gradient descent trains the unitary parameters using independently sampled states from the specified input set.The training uses standard speed-up techniques such as a momentum term.
- Results: The quantum autoencoder’s cost function converged toward zero for tested sets of orthogonal input states, including superposition states.An additional cost term can force the compressed output to be diagonal in a chosen basis by driving selected Pauli-X and Pauli-Y expectations to zero.
- Quantum protocol discovery: The teleportation network is designed to discover a standard protocol using quantum entanglement and restricted classical communication.The circuit includes only two dephased connections, matching the protocol’s two classical communication bits.
- Quantum protocol discovery: A fully trained teleportation network transfers Alice’s arbitrary input state |ψ⟩ to Bob’s output port on qubit 6.Training on randomly selected Bloch-sphere axis-intersection states prevented the network from simply learning to copy the input, and the cost function converged toward zero.
Quantum vs. classical
The quantum networks process quantum inputs that classical networks cannot distinguish, while ancillary qubits preserve strict classical generalisation at an experimental and numerical cost.
- Quantum networks can process |+⟩ and |−⟩ differently, enabling quantum tasks unavailable to classical networks, including state teleportation.
- The comparison is well-defined because classical networks correspond to a particular parameter regime of the quantum networks.
- The teleportation cost converges to zero, with 1000 Haar-random states yielding mean 5.0371 × 10−4 and standard deviation 1.7802 × 10−4.
- Allowing unitary parameters to occupy superpositions is proposed as a possible route to further advantages but would be experimentally more challenging.
- Omitting ancillary qubits simplifies implementation and numerics but sacrifices classical-mode performance and may underperform an equally sized classical network.
Visualising the cost function landscape
A reduced two-parameter view makes gradient descent visually tractable by plotting a one-input neuron's cost surface and optimization path.
- The visualization uses a one-input neuron represented by a 4×4 unitary matrix and a dummy-then-output second qubit.
- The optimized mapping sends |+⟩⊗|0⟩ to |+⟩⊗|0⟩ and |−⟩⊗|0⟩ to |−⟩⊗|1⟩.
- The unitary is optimized by gradient descent over the variables θ and φ.
- The cost function of a 2-qubit unitary is plotted against θ and φ, with gradient descent shown as a red path from one starting point.
Scaling to bigger networks
The quantum-neuron construction extends to multiple inputs and outputs through a unitary followed by fan-out, with circuit restrictions needed for scalable parameter counts and cost evaluation.
- The generalized network supports an M-qubit input quantum neuron followed by an N-qubit fan-out gate.
- In the circuit diagram, U [M+1] represents the quantum neuron and U [N] fans out its final-port output in a chosen basis.
- Restricting the unitary to a polynomial-length circuit keeps the number of free parameters polynomial in the number of inputs.
- Restricting the cost to local Pauli expectation values requires only a vector of 3n expectation values for n qubits.
QUANTUM PHOTONICS NEURON MODULE
The proposed photonic neuron is a connectable module using a single photon’s polarization and spatial modes to encode qubits, with tunable optical components implementing updates.
- Quantum photonics is selected for room-temperature operation, miniaturization through integrated circuits, and established optical-fibre technology.
- The neuron is designed as a module that connects to other neurons, making its experimental construction, operation, and training concrete.
- The C-A-K protocol encodes three qubits in one photon using polarization and four spatial modes.
- The design initially uses one input qubit and one dummy input so the designated output can feed the next neuron.
- The module routes the input through the polarization qubit, while swaps ensure subsequent neuron modules receive input through polarization.
- Network outputs are updated by tuning polarization rotators and spatial phase shifters in Mach-Zehnder interferometers.
- Silicon or silica waveguides can improve robustness and miniaturization, with phase shifting implemented by heaters or electro-optic effects.
SUMMARY AND OUTLOOK
The paper presents quantum feedforward networks that process quantum inputs and demonstrates them on quantum autoencoding and protocol discovery tasks.
- The protocol generalises classical feedforward step-function neural networks into networks that take and process quantum inputs.
- The networks perform quantum autoencoding by compressing entangled inputs.
- The networks can discover quantum information-processing protocols such as teleportation from the task alone.