2026-09-25
cvQNN inserts a trainable imaginary-time layer, cutting regression MSE from 9.55e-2 to 3.69e-4; a 4-qubit single-layer net hits ~96% on MNIST 0/1.
Variational quantum neural networks encode classical data into quantum states, train a parameterized circuit, and read out an observable. Encoding angles and trainable weights stay real, and every layer stays unitary. Prior Fourier analysis shows the model output is a truncated Fourier series whose frequencies come from the encoding Hamiltonian's eigenvalues. The reachable coefficient range is small. When the target function has large amplitudes, the fit collapses.
Complex numbers are native to quantum mechanics: the Schrödinger equation carries an imaginary unit, and a wave function stores both amplitude and phase. Classical complex-valued nets already gain capacity by moving weights onto the complex plane. Doing the same on a quantum circuit requires non-unitary evolution. Existing non-unitary learners (trace out ancillas, engineered dissipation, linear combination of unitaries) still train real parameters. Imaginarity is not treated as a trainable resource.
cvQNN splits each complex operator into a unitary real part \(Rl(\alpha^R)=\exp(-i \alpha^R Hl)\) and a non-unitary imaginary part \(Il(\alpha^I)=\exp(\alpha^I Hl)\). That is real-time evolution interleaved with imaginary-time evolution. The imaginary-time piece pushes the state toward the ground subspace, so training is no longer gradient-only.
The imaginary part is implemented with one ancilla and post-selection: diagonalize the Pauli tensor product, apply \(O(n)\) CNOTs plus a controlled-Rx with angle \(\varphi=2\arccos(\exp(-2|\alpha^I|))\), then keep the desired ancilla outcome. Extra gates scale linearly in the number of complex operators \(q\), but post-selection success decays exponentially in \(q\). The intended use is sparse insertion.
Experiments tie real and imaginary parts to the same parameter, so the parameter count does not grow.
On the encoding block, the output remains a Fourier series, now with an extra \(e^{(wj+wk)x}\) factor. Coefficient ratios can sit in \((0,1)\), and components that share a frequency difference but not a frequency sum become distinguishable. Gradients with respect to the complex parameter \(\alphaj\) are linear combinations of two observables, \(Vj O\) and \(O Vj\). Parameter-shift still applies.
Two mixed models combine the encoding and variational tricks:
A covering-number bound says that, at the same parameter count, cvQNN's hypothesis class is larger than a purely unitary net. The unitary case is the special value \(B=2^{-1/4}\approx 0.8409\). A trainable \(B\) is the knob between expressivity and generalization.
Regression target: a two-term Fourier function \(y1(x)\). Pennylane, Adam at learning rate 0.3, at most 200 steps.
| setting | method | MSE |
| A=0.5 | standard one-layer QNN | \(1.50\times 10^{-4}\) (60 steps) |
| A=1.0 | standard QNN | \(9.55\times 10^{-2}\) |
| A=1.0 | cvQNN-MM1 | \(3.69\times 10^{-4}\) (75 steps) |
| A=1.0 | cvQNN-MM2 | \(1.20\times 10^{-4}\) (110 steps) |
Inserting an extra unitary \(U(\alpha)\) into the baseline still fails to fit. The gain is the non-unitary imaginary piece, not an extra parameter. As amplitude \(A\) grows, the baseline error climbs; cvQNN keeps converging.
With two data-reuploading layers on \(y2(x)\) (frequencies 0, 1, 2), both families fit well at \(A1=A2=0.6\). At 0.9 the baseline degrades and both cvQNN variants hold. KL divergence on 2000 random samples (lower means more expressive): 0.03297 (baseline), 0.02575 (MM1), 0.02674 (MM2).
MNIST digits 0 vs 1: 12,665 train / 2,115 test. PCA then angle encoding, 4-qubit hardware-efficient ansatz, cross-entropy plus L2 (\(\lambda=0.01\)), Adam for 100 steps, five random inits. One imaginary operator: +8.6% train / +9.7% test vs the unitary baseline. Two operators: +9.1% / +10.1%. Extra insertions saturate quickly. A single-layer cvQNN with one imaginary operator reaches about 96% accuracy, above a many-layer unitary net.
Hardware: WuYue-QPU-002, five transmons, 4-qubit cvQNN-q(0) plus one ancilla. Models are trained classically; the chip only infers 50 images at 5,000 shots. Test accuracy 82%. Relative performance loss vs noiseless theory: 4.93% on loss, 6.82% on accuracy.
For NISQ variational work, this is a way to buy expressivity without deeper circuits or extra data qubits: insert a few non-unitary imaginary-time layers. A shallow cvQNN beating a deeper unitary net is the part that matters on hardware with limited depth. Simulations use Pennylane; code is public.
The gain is incremental, not a new learning paradigm. Tasks stop at synthetic regression and MNIST 0/1. There is no head-to-head against dissipative QNNs or LCU non-unitary models. Covering numbers compare upper bounds, not proven generalization.
Post-selection success drops exponentially with the number of complex operators, so measurement cost blows up. Sparse insertion is mandatory. The barren-plateau calculation still shows exponential decay of gradient variance in qubit number \(n\). The extra non-unitary term \(g(\theta)\) might slightly lift gradients; that claim is not tested.
The chip experiment is inference only, on 50 samples. MNIST is 0/1 after PCA. Data are withheld because of unpublished follow-up work. A sinh mixed model zeros coefficients when \(wj=-wk\) and fits poorly. Tying real and imaginary parts to one parameter leaves unused degrees of freedom on the complex plane.