2026-10-23
CohAlign predicts, from the circuit and noise model, the first-order decay of one QNN gradient's second moment. At eight qubits the alignment RMSE across eight channels is 0.009.
Variational quantum circuits learn by measuring a cost and moving parameters. Barren plateaus are one failure mode: on the noiseless landscape the gradient shrinks exponentially with the number of qubits. Hardware noise is a different failure. A landscape that is perfectly usable without noise still has its measured gradients squeezed as noise accumulates layer by layer. Pushed far enough, that becomes a noise-induced plateau that no initialization can repair. Theory already says this happens. The missing number is how fast one chosen parameter's gradient decays, computed before anyone sweeps a noisy circuit.
A single worst-case coherence rate does not answer that. A charge-conserving brickwork network keeps the evolution inside one charge sector, and a local readout sees only part of the coherence there. Two channels with the same worst-case rate can damage gradients at very different speeds. A channel can cut almost every coherence direction and leave the measured gradient untouched, provided the directions it cuts are the ones the readout cannot see.
CohAlign is an open-source toolkit under the MIT licence, released as ugail/CohAlign. The input is a classical description of the circuit, the noise model, the prepared state, the readout and the initialization prior. It does not predict whether training converges, and it does not predict final accuracy.
The circuit is a ring of n qubits, with an open-chain variant that drops the closing edge. Each layer applies a trainable Rz on every site, then a frozen hopping gate exp(−iβ(XX+YY)) on alternating edges. Every gate commutes with the total charge, so sectors do not mix. The readout is Z on site 0 after projection onto the sector with r excitations. In the main experiment the hopping angles are drawn once from [−0.5, 0.5] with seed 42 and frozen. Trainable phases sit in a box of width 0.05 around zero, the usual small-angle initialization for avoiding noiseless plateaus. A noise channel acts once after every layer.
Three quantities ask sharper and sharper versions of the same question. λcoh is the worst-case contraction rate on the sector-coherence block. λmode is how fast the channel contracts the operator mode that carries the derivative of one parameter. Once the channel, architecture, parameter and prior are fixed, λmode does not depend on the prepared state. Λresp is the first-order susceptibility of the ensemble second moment of the measured gradient. The channel, architecture, parameter, readout, state and prior all enter that cross moment. The predicted relative decay is 2γΛresp. The alignment ratio ωresp = Λresp / (L λcoh) is a convenient scale for contractive channels, and the paper does not claim it as a theorem. When the generator has a coherent Hamiltonian piece, the susceptibility is signed, and the toolkit reports the signed value.
The generator is extracted at a probe strength of 10⁻³, restricted to the off-diagonal block of the sector. The estimator uses 40 draws. The measurement uses a separate set of 30 draws, shared between the noiseless and noisy passes. Gradients use the parameter-shift rule with shift π/2. The main validation is an exact density-matrix simulation at 8 qubits, three layers and one excitation. Paper mode takes about two hours on an ordinary cloud CPU. Every "measurement" in the paper is that simulation, not a run on hardware.
The sharpest control is correlated dephasing. Coherence inside each paired edge is preserved, while coherence across edges decays at nearly the worst-case rate. At layer 1, site 0, the mode diagnostic returns ωmode = 0.969, so the gradient-carrying operator is almost fully contracted. The response prediction is 0. Paired measurements put the degradation below 10⁻¹⁴. A finite-difference extrapolation to zero noise leaves a residual of 1.1×10⁻¹¹. An operator-only analysis would call this parameter fully exposed. The gradient that training actually measures does not move.
At 8 qubits, three layers and one excitation, Λresp for the dissipative channels lands on the identity line against the finite-difference limit. Relative error runs from 3×10⁻⁸ for amplitude damping to 3.4×10⁻³ for the fastest probes, the same order as the probe bias. For the isotropic channels the mode rate recovers the worst-case rate with relative error below 3×10⁻⁴. The four reference values are 1.000, 3.996, 6.648 and 7.973.
The signed, site-dependent Z over-rotation needs 400 matched draws before the sign is stable. Λresp = −0.584, with a bootstrap interval from −1.06 to −0.16 that excludes zero. The finite-difference limit agrees to 3.5×10⁻⁵. At first order this coherent perturbation increases the gradient second moment. No diagnostic that only reports a contraction rate can express that sign.
At depth 4 the light cone breaks the shallow geometric degeneracy. An activity preflight keeps 9 parameter locations, 7 of them measurable, across 5 channels, 35 pairs in all. Susceptibilities run from 2.73 to 27.4, a factor of ten, and every point still sits on the identity line. Bias runs from 1.9×10⁻⁸ for amplitude damping to 3.0×10⁻³ for the covariance-breaking probe. The location dephasing hits hardest, 9.36, is the location the correlated control protects most, 2.73.
The primary parameter is layer 2, site 1. The table gives predicted alignment and the zero-noise intercept inside the accumulated-defect window γL λcoh ≤ 0.4.
| Channel | Predicted ωresp | Measured intercept |
| Amplitude damping | 1.0000 | 0.9962 |
| Dephasing | 0.6667 | 0.6635 |
| Depolarising | 0.9335 | 0.9235 |
| Inhomogeneous dephasing | 0.2669 | 0.2669 |
| Site-dependent damping | 0.4000 | 0.3997 |
| Correlated dephasing | 0 | 0 |
| X error | 1.0000 | 0.9820 |
| Biased Pauli | 0.9631 | 0.9501 |
Root-mean-square error across the eight distinct channels is 0.009. Using the raw ratio, with no intercept, raises it to 0.075, and every residual is negative: that is second-order contamination. The six phase-covariant channels agree at 0.005, the two covariance-breaking probes at 0.016, and the largest single residual is the X error at 0.018. A coherent-dissipative row duplicates amplitude damping exactly. It is a null control and is excluded from those eight.
On the same grid the measured degradation stays within about 4% of the prediction when 2γΛresp is below 0.1, and within about 10% when it is below 0.2. At the strongest point on the grid the median ratio of measurement to prediction falls from 0.98 to 0.52.
Dephasing has a closed form by layer. The last noise slot does not contribute to the derivative, so the alignment is 2/3 at three layers and 1/2 at four. The measurements are 0.652 and 0.492. From depth 3 to 6 the median absolute difference between predicted and measured alignment is 0.009. The correlated control stays protected while the light cone is shallow, then turns on once the cone crosses a paired edge: 0.503 predicted against 0.488 measured at four layers, and 0.400 against 0.395 at five.
The same identity holds in other geometries. On the open chain, dephasing sits at the probe bias of 1.0×10⁻³, amplitude damping is reproduced to 9×10⁻⁹, and the correlated control has a residual of 4×10⁻¹¹. The ten-qubit check is a reduced setting, 20 draws and a two-point strength grid. Dephasing again sits at the probe bias, and amplitude damping at 1.7×10⁻⁸. Moving the readout site moves the dephasing susceptibility from 7.99 to 7.38.
Two worked examples turn the susceptibility into a placement decision. At depth 4 and strength 0.01 per layer, location (1,0) under dephasing has predicted retention 0.813 and measured retention 0.829. The same location under the correlated control has predicted 0.945 and measured 0.949. Change the noise model, and the preferred placement reverses. A composite of dephasing and amplitude damping in the ratio 0.7 to 0.3 has a first-order predicted degradation of 0.155 and a measured value of 0.144. Halving the dephasing strength cuts the measured degradation by 0.058. Halving the damping cuts it by 0.010.
In the two-excitation sector the coherence block at 8 qubits has 756 dimensions, and the estimator uses 20 draws. Alignment RMSE across the eight channels is 0.037. The correlated control, protected in the shallow one-excitation geometry, is no longer protected for the chosen delocalised input: 0.1531 predicted, 0.1517 measured. Across three input seeds the predictions move by about 0.04, with dephasing falling between 0.148 and 0.192.
This is a design-time diagnostic. A symmetry audit says whether the architecture and the channel preserve charge. CohAlign says how fast the coherence that the readout can still see is destroyed, resolved by parameter and by noise slot. Someone choosing an architecture and a noise model at 8 to 10 qubits can mark which parameter is fragile, and which noise component is worth cutting, before running a noisy sweep.
It does not replace a hardware experiment, and it does not return a trained accuracy. The calculation still needs a classical simulation of the noiseless circuit. The results live at sizes where an exact density matrix fits.
The six-layer correlated control is unresolved. Predicted alignment is 0.301, with an interval from 0.204 to 0.359. The measurement is 0.453, outside that interval. About 11 of 40 estimator draws carry the weight. The authors file it as a deep-circuit stress point, outside the main evidence. The cause is not established.
The predicted object is the ensemble second moment of one gradient under a stated prior. It is not the gradient at a partly trained parameter vector, and it is not optimizer convergence, task accuracy or shot count. The nine channels are a designed benchmark, not noise reconstructed from a device. The implementation puts one channel, uniformly, after every layer. Device noise varies by gate and by layer. Feeding in gate-set tomography would still require an error generator relative to the ideal gate, a branch choice for the channel logarithm, and a Markovianity check. None of that is done here.
The validation is two topologies of the brickwork family at 8 qubits, plus a reduced check at 10. Propagation still runs in the full space of dimension 2ⁿ, so time and memory grow exponentially with qubit number. The one- and two-excitation sectors are only polynomially large. The half-filling sector is exponentially large. The construction is written so that it applies there, and it was not tested there. The accumulated-defect window used for the fits was fixed during revision, not in a pre-registered plan. The worst-case rate is a quadratic form and the response is a bilinear cross moment, so no inequality follows. The tables, where the mode ratio never falls below the response ratio, are not a law.
Replacing the worst-case rate with the response rate lifts the pooled R² at the primary window from 0.828 to 0.998. That regression is a consistency check. A channel-cluster bootstrap puts the interval on the R² gap from just below 0 to 0.28, because only seven clusters enter it. The finite-difference identity is the harder result.