Scale-TimeDynamics
A phase error is a lost branch
Quantum error correction protects information by spreading it across many physical qubits and reading parity checks that never touch the logical state itself. This paper does not propose a replacement. It proposes a second way of reading the same problem, borrowed from signal processing, and asks whether that reading suggests anything the standard picture does not.
The reframing is simple. If you observe a rotating phase by taking snapshots, you only ever see the phase modulo one full turn. The whole number of turns is missing, and it is missing in a strong sense: no estimator can recover it from the wrapped data alone, at any record length and any noise level. It has to be supplied from outside. In that language, a phase error is not a random flip. It is a lost branch, and correction is the work of tracking which branch you are on.
Two results that shape everything
Resolution is a budget, not a magic number. Across a wide scan of sampling depths, phase estimation error falls smoothly at the theoretical limit. There is no privileged depth, integer, half-integer or dyadic, at which stability spontaneously appears. The depth needed to resolve a given structure is computed from the noise scale and the size of the structure, and it scales with the square of the noise. Any scheme that pins its guarantees to a favourite oversampling number is unsupported.
Estimation is not protection. Deep sampling answers where the phase is. It says nothing about whether the phase will stay there. Without coupling, exact recurrence happens only at isolated rational values with no basin around them and no robustness to noise. Add a weak coupling between the tracked phase and its reference frame and those points inflate into finite windows: rigid against noise on the inside, freely diffusing on the outside. In simulation the contrast is stark. A locked ensemble holds its spread bounded over hundreds of steps while an unlocked one under identical noise diffuses past the point where the branch is lost.
The compact form: oversampling estimates, coupling protects.
Five layers
The architecture is called HCOPL, for Harmonic Consistency and Oversampled Phase-Lock, and it is a pipeline rather than a code.
- Encode. Distribute the logical phase relation across integer-harmonic readout channels, at a sampling depth computed from the resolvability budget rather than chosen.
- Sense. Extract syndromes as harmonic congruence checks. For a consistent register the harmonic content cancels exactly, so a residual is a fault.
- Decide. Recover the correct branch with a weighted congruence solver seeded by a continuity prior, with a computable probability of choosing wrong.
- Hold. Keep the accepted branch inside a lock window between syndrome decisions, using a servo that acts only on the classical reference frame.
- Anchor. Bound long-term integer drift with fiducial reference channels, since a slip of exactly one full cycle is invisible to congruence by construction.
What it never touches
The logical amplitudes are never a sampling target, and the substrate gives that rule an independent reason rather than leaving it as good manners. When a superposition is sampled deeply and directly, the outcome statistics drift away from amplitude-squared weighting toward a deterministic majority readout of the dominant component. Deep direct sampling of a logical mode behaves like a strong measurement. Every deep channel in the architecture therefore has to carry the phase relation through a harmonic, ancilla or reference degree of freedom.
The servo obeys the same discipline. It rotates the controller's reference frame, changing what the controller calls zero, not what the register is.
What would settle it
This is a hypothesis, and it is written to be killed rather than admired. The scope is stated narrowly: dephasing, phase drift, coherent over-rotation and branch ambiguity. Amplitude damping, leakage and correlated crosstalk have no mapping yet, and the paper says so. There is also an internal limitation it makes explicit rather than implicit, that common-mode whole-cycle events cannot be caught by congruence at all.
The immediate next step is one closed-loop classical simulation of the whole pipeline with seven named failure points, including a prediction that the slip rate falls smoothly with depth, with no plateaus or steps at any special sampling value. A reproducible bump anywhere in that curve would contradict the substrate and falsify the budget principle the architecture rests on.