Quantum Machine IntelligenceSource publication:
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Dissipative quantum neural networks (DQNNs) previously used unitary perceptrons, so each building block could only realize a restricted class of quantum transformations. A new architecture rewrites the perceptrons as isometries and adds ancilla neurons, making each building block quantum-channel universal, able to implement any completely positive trace-preserving map. On a minimal three-qubit network trained with Choi training, the Hilbert-Schmidt distance reaches a mean diamond distance of 4.55×10⁻⁴ over 100 random qubit-qubit channels, while the quantum Chernoff bound and quantum relative entropy reach only 0.102 and 0.386.
The smallest building block of a dissipative quantum neural network can now exactly implement any qubit-to-qubit quantum channel, rather than only the maps expressible by unitary transformations. Earlier DQNNs wrote weights and biases as unitary gates, so a network with one input, one hidden, and one output neuron produced its output from a product of two isometries; a general channel's Stinespring isometry cannot be written as such a product, so that network was not quantum-channel universal. On a minimal extended network of three qubits and a single perceptron, the training target was a random qubit channel, and the architecture guarantees the target channel can be represented exactly, removing the possibility that the target itself is unlearnable.
Rewriting perceptrons from unitaries to isometries noticeably reduces the number of parameters that must be optimized. A unitary perceptron acting on d-dimensional input and hidden qudits has d⁴ free parameters, while an isometry perceptron over the same qudits has only d²(2d−1); at d=4 this more than halves the parameter count. The count follows from adapting the composite parametrization of the unitary group, and a one-to-one parametrization of isometries makes the reduction usable in gradient optimization.
Under Choi training, the Hilbert-Schmidt distance is among the best-performing and most readily measurable cost functions. After 1000 training iterations on 100 random qubit channels, D₁ and D_HS reached mean diamond distances of 3.43×10⁻⁴ and 4.55×10⁻⁴, better than any fidelity; the quantum Chernoff bound and quantum relative entropy, both tied to asymptotic hypothesis testing, performed worst at 0.102 and 0.386. The comparison used the same 100 random target channels and the same 1000 ADAM iterations, with diamond distance as a channel-distinguishability measure independent of the cost function.
Random state training converges faster but ends about two orders of magnitude less accurate than Choi training. Random state training converged to roughly 5×10⁻² diamond distance for almost all cost functions, whereas Hilbert-Schmidt distance under Choi training reached 4.55×10⁻⁴; F₂ and D₂ performed significantly worse under random state training. Random state training sampled 32 input states per iteration from the Hilbert-Schmidt distribution in eight batches of four, run on the same minimal network as Choi training.
The results concern shallow DQNNs with small output Hilbert spaces, and the numerical experiments use a minimal extended network of three qubits and one perceptron over 1000 training iterations. That setting can represent any qubit-qubit channel exactly, so the architecture and cost-function comparison applies at this scale.
The extended architecture guarantees only that each perceptron is quantum-channel universal, not that a whole multilayer network is; in a five-qubit example the parameter count already falls short of a general isometry. Choi training requires the target channel to be fully known and applies the cost function on a larger Hilbert space, and the authors connect its slower convergence to barren plateaus. Convergence for the Werner channel correlates with its parameter α, and the completely depolarizing channel at α=0 only begins to drop rapidly after about 350 steps.