Qector Space Time

QECTOR Space-Time and Streaming

Source of authority: v1.0.0 reference manual, chapter 10.

Why space-time

Physical measurements are noisy, so a detector event must be attributable either to a data fault or to a measurement fault. The space-time approach builds a (2+1)-dimensional detector lattice over T rounds and solves a single matching problem on it, with anisotropic edge weights.

Detector formulation (manual 10.1)

d_{c,t} = s_{c,t} xor s_{c,t-1}

• A data error in round t flips the same checks in round t and t+1, producing two detection events in consecutive layers.
• A measurement error flips one detector in a single layer.

The lifted matrix H_ST contains data columns and measurement columns, with:

• Spatial weights derived from the data-error probability.
• Temporal weights derived from the measurement-error probability.

Theorem 13 (space-time lifting faithfulness, manual 10.2)

Let H_ST be the lifted parity-check matrix with data and measurement columns. A correction c_ST whose boundary in the detector graph equals d satisfies H_ST c_ST = d (mod 2); projecting the measurement columns away yields a spatial correction whose syndrome differs from the final raw round only by the final-round measurement error term.

The proof sketch: the lifted matrix is a chain complex whose boundary operator applies the spatial syndrome to each time layer and the difference operator across layers. Any matching whose paths pair the defects of d reproduces d by Theorem 3, with temporal degree cancellation handled by the boundary term of time-like paths. The projection statement follows from the telescope identity XOR_r d_{c,r} = s_{c,T} over raw rounds, leaving the measurement-boundary term.

Validation patterns (manual 10.2)

The SpaceTimeDecoder is validated against two failure modes:

• A lone measurement glitch that reverts on the next round must be absorbed as two time-like edges and must not corrupt the spatial correction.
• A persistent data error must not trigger the single-round fallback.

Both are locked by regression tests.

Worked example (manual 10.4)

Take a ring of 8 checks over 6 rounds. Rounds 1-2 clean, round 3 reports a single spurious flip at check 2, round 4 is clean again. XOR differencing produces two detection events (appearance and disappearance) paired by a temporal edge: the spatial correction is empty, as it must be - no data error was ever present.

This is the regression locked by the space-time tests.

The streaming layer (manual 10.3, 20.5)

• Compiled core: StreamingDecoder (OR-accumulated history) and SlidingWindowDecoder (exponentially decayed window).
• Higher-level Python layer: decodes each round independently. It explicitly does not claim full circuit-level space-time matching; its window controls commit latency, batching, and telemetry. For stateless inner decoders the per-round result is window-invariant.

Truncation bound (manual 10.3)

For the decayed window with decay factor lambda in [0, 1):

|| S_W - S_inf ||_1  <=  lambda^W / (1 - lambda) * || s ||_inf

by the geometric tail series. The excess logical-error implication is qualitative (O(lambda^W)); no numerical fidelity loss is claimed without a surviving artifact.

Scoping rules

• The Python streaming layer and the Rust streaming primitives decode per-round or windowed syndromes; they do not implement full circuit-level space-time matching unless the 3D SpaceTimeDecoder is used (manual 20.5).
• The sliding-window decoder is a simulation workflow, not hardware control (manual 4, Table 4.1).

Common pitfalls

• Using the Python streaming layer for full space-time matching -> it is per-round by design; route to SpaceTimeDecoder for (2+1)D.
• Picking lambda close to 1 -> the truncation bound explodes; a value of 0.95 over W = 100 rounds leaves a tail of order 0.95^100 / 0.05 ~ 5.9e-3 * ||s||_inf.
• Claiming "streaming beats single-shot" without an artifact -> per-machine, per-workload only (manual 22.5).

How the bench server helps

• qector-research.env_block returns the chapter 22.3 environment block for the space-time artifact.
• qector-research.artifact_metadata_check generates the metadata block (decoder class, mode, weight scheme) for a space-time run.
• qector-research.decode_faithfulness_check re-verifies H c = s on the spatial projection of the lifted correction.
• qector-research.hot_path_microbench produces a per-machine latency sample (always per-machine, never portable).