Fault-Tolerant Gadget and Proper-Code Contract

A proper fault-tolerant code contract consists of a finite-dimensional code subspace $Q$ in a fixed block of $m$ physical coordinates, a finite logical gate alphabet $G$, and a fixed finite family of encoder, initialization, error-correction, decoder/readout, gate, syndrome-measurement, reset, and idle gadgets. Throughout this page, universal means that the subgroup generated by $G$ and its adjoints is dense in the unitary group on the logical space, in the operator norm of The operator norm as the least bound and as the unit-sphere or unit-ball supremum (the supremum over the unit ball). The parameter $d$ is an unlocated-error correction capacity: for an encoding isometry $V:H_L\to H_P$ and the declared decoded recovery channel $\mathcal{R}$, every channel $\mathcal{N}$ with Kraus operators supported on at most $d$ physical coordinates satisfies $\mathcal{R}\circ \mathcal{N}\circ {\mathrm{Ad}}_V={\mathrm{id}}_{\mathcal{B}(H_L)}$, including after tensoring the identity on any finite reference. The support may differ between Kraus operators and is not supplied to the fixed recovery. This capacity is distinct from the stabilizer distance $d(S)$ of Stabilizer distance, the minimum weight of a Pauli centralizer class outside scalar phases times the stabilizer; a distance argument certifies capacity $d$ only after separately verifying $2d<d(S)$ and the relevant error-family hypotheses. When the concatenated-threshold theorem uses $m\ge 2d+1$, this inequality is an explicit hypothesis.

For several live blocks, the noisy stage requirements use the joint encoded-instrument error sectors of Code Deviation and Fault Spread. Full-space CPTP deviations remain sufficient inputs, but an output instrument is not required to have a full-space CPTP error extension. The fixed-recovery transfer proved in Encoded error instruments and full-space CPTP error maps supplies exact decoding of these sectors. Thus a fault may have arbitrary correlations between blocks while its support budget is bounded in each block; the untouched reference is external to all live data. The tensor product of the declared block recoveries must recover every trace-preserving encoded instrument in the joint $d$-error span, with that external reference retained after internal outcomes and fault memory are traced. Recorded logical measurement outcomes retain their stated ideal instrument law.

Each gadget specifies its finite input and output registers, ideal logical channel or instrument, fault locations and supports, depth, size, literal unitary support bounds and effective spread budget $\mathrm{\ell }$ in the sense of Code Deviation and Fault Spread, and a uniform upper bound $a$ on its child rectangles. Zero-fault gadgets implement their ideal channels exactly on encoded inputs, with the reference system preserved. The encoder implements its exact isometry on arbitrary logical inputs in the zero-fault case; the noisy guarantees begin at an admitted encoded input or the protected prescribed initialization. Every encoder location inside such a protected factory is counted and its faults must be handled. No guarantee corrects an arbitrary first fault on an unknown unprotected logical input. For every integer $q\ge 1$ with $2q\mathrm{\ell }\le d$, the checked computation-stage contract must say that an incoming joint encoded error instrument of budget $q\mathrm{\ell }$ and at most $q$ bad child rectangles has joint output budget $2q\mathrm{\ell }$ before correction; the checked correction-stage contract must correct every admitted joint input sector of budget at most $d$ and, with at most $q$ bad children, return joint residual budget $q\mathrm{\ell }$ with the same decoded logical channel. Residual branch maps have the encoded Kraus form specified above. Their full instrument is trace preserving on the admitted domain, and after declared decoded recovery and trace of internal records it equals the ideal logical channel. These are subnormalized branch identities for every measurement outcome, with any finite external reference retained; a branch is not replaced by a normalized physical CPTP map. They quantify over arbitrary CPTP maps supported on the union of the faulty child supports, not only product fault maps. An error-correction gadget must also map arbitrary inputs into the code subspace in the zero-fault case. Its noisy identities are required on the specified joint encoded error sectors, rather than as a full-space intertwining of every input fault through ideal recovery. Any auxiliary fault memory may be updated at a faulty location and is carried in the channel/instrument description; only the external reference is constrained to the identity, and nonfaulty locations leave that memory alone. All preparation, syndrome, readout, classical-control, reset, and scheduled idle locations are included in the size and spread bounds.

The finite physical gate alphabet is closed under recursive replacement: every operation used by each gadget is itself in the declared alphabet. The contract makes no claim that a concrete code or gadget family satisfying it has been constructed. That existence and verification obligation is a separate theorem.