Quantum DEQ

This compact example runs a four-wire exact statevector transition inside a SILVA Broyden equilibrium and differentiates the task output with JFB. The circuit-equilibrium construction follows Quantum Deep Equilibrium Models [90].

"""Run a compact QDEQ with an exact four-wire statevector circuit."""

from __future__ import annotations

import torch

from silva_networks import SILVAQuantumDEQ, SILVAStatevectorQuantumCircuit, SolverConfig


def main() -> None:
    torch.manual_seed(23)
    model = SILVAQuantumDEQ(
        input_dim=6,
        output_dim=4,
        n_qubits=4,
        circuit=SILVAStatevectorQuantumCircuit(n_qubits=4, fixed_depth=2),
        config=SolverConfig(
            solver="broyden",
            max_iter=5,
            tol=1e-4,
            history=4,
            backward_mode="jfb",
        ),
    )
    result = model(torch.randn(2, 6), return_result=True)
    result.output.square().mean().backward()
    print(
        "QDEQ",
        result.output.shape,
        "iterations",
        result.solver_result.iterations,
        "residual",
        result.solver_result.residual,
    )


if __name__ == "__main__":
    main()

Run it from the repository root:

python examples/quantum_deq.py

The gate derivation, source architecture, circuit adapter, and complete experiment route are in Quantum Equilibria.

Complete Worked Study

The short construction above identifies the main API. A complete study must also distinguish the state equation, task objective, numerical residual, gradient path, and scale transfer. In this example, the equilibrium state is the measured circuit feature vector z, the condition is encoded classical features and optional circuit condition, and the repeated map is a feature-injected quantum circuit followed by real-valued measurements.

Derivation From Transition to Reported Result

The forward solve is defined by

\[ z^\star = T_\theta(z^\star,x). \]

The task output and task objective are separate from convergence:

\[ \widehat y = R_\phi(z^\star), \qquad \mathcal L_{\mathrm{task}}=\ell(\widehat y,y). \]

For a computed state \(z_K\), the normalized fixed-point residual is

\[ r_K = \frac{\lVert T_\theta(z_K,x)-z_K\rVert_2} {\lVert z_K\rVert_2+\varepsilon}. \]

A small task loss does not imply a small \(r_K\), and a small \(r_K\) does not establish task quality. Both belong in the result. For implicit training, the parameter sensitivity follows

\[ \frac{\mathrm d z^\star}{\mathrm d\theta} = \left(I-\partial_z T_\theta(z^\star,x)\right)^{-1} \partial_\theta T_\theta(z^\star,x). \]

This is why the example checks gradients in addition to forward convergence. The reader-facing evidence for this route is circuit measurements, equilibrium residual, task output, and circuit gradients. The invariants that must remain true are wire count, encoding width, measurement shape, normalization, and differentiability.

Run the Complete Example

python examples/quantum_deq.py

Measured Compact Output

The following output was produced by the executable program in the current repository. Floating-point values may vary slightly across devices and library builds, while shapes, finite values, invariants, and declared tolerances must remain stable.

QDEQ torch.Size([2, 4]) iterations 5 residual 0.01891479454934597

Interpret the Output

Evidence	What it answers	What would require investigation
Tensor shapes	Did every source, state, branch, and readout preserve its declared contract?	A changed entity, channel, token, or spatial dimension
Task metric	Did the compact task execute and produce finite evidence?	Non-finite loss, a missing mask, or a metric computed on the wrong split
Fixed-point residual	Did the returned state satisfy the repeated transition to the requested tolerance?	A residual plateau, rising trajectory, or convergence flag inconsistent with the value
Iteration or trajectory data	How much numerical work was required?	Solver effort that grows sharply under a small input or resolution change
Gradient evidence	Can the loss reach every trainable component through the selected backward mode?	Missing, non-finite, or implausibly large gradients
Domain invariant	Did the method retain positivity, feasibility, boundary values, permutation behavior, or another structural requirement?	A task metric that looks acceptable while the structural contract fails

The compact output is a mechanism check, not a paper-scale benchmark claim. It shows that data enter the intended construction, the transition executes, the solver returns diagnostics, and differentiation reaches trainable parameters.

Add a Solver and Scale Sweep

The next run should hold model parameters and data fixed while changing one numerical control at a time. A complete experiment record can use this schema:

experiment:
  example: quantum-deq
  state: the measured circuit feature vector z
  condition: encoded classical features and optional circuit condition
  repeated_transition: a feature-injected quantum circuit followed by real-valued measurements
  invariant_checks: wire count, encoding width, measurement shape, normalization, and differentiability
  compact_evidence: circuit measurements, equilibrium residual, task output, and circuit gradients
  scale_axes: wire count, statevector or shot budget, circuit depth, and solver evaluations
solver_sweep:
  methods: [picard, anderson, broyden]
  tolerances: [1.0e-4, 1.0e-6, 1.0e-8]
  maximum_iterations: [25, 50, 100]
report:
  - task_metric
  - fixed_point_residual
  - backward_linear_residual
  - iterations
  - wall_time
  - peak_memory
  - gradient_norm

At full scale, move toward MNIST-4, MNIST, Fashion-MNIST, or CIFAR-10 with the source circuit protocol. Increase only one of wire count, statevector or shot budget, circuit depth, and solver evaluations at a time. Retain this compact run as a regression test, preserve the source split and preprocessing receipt, archive the resolved configuration and checkpoint, and report convergence failures rather than discarding them.

Where to Go Next

Question	Page
How is the circuit transition derived?	Quantum Equilibria
Which circuit and model classes are public?	Quantum Equilibria API
Where is the executed image and training lab?	Quantum DEQ Notebook
How does QDEQ compare with other equilibrium placements?	Equilibrium Expansion Atlas