Code Deviation and Fault Spread

Consider one block of $m$ tensor factors $H_1\otimes \cdots \otimes H_m$, its code subspace $Q$, and an arbitrary finite reference system $R$. For an integer $s\ge 0$, let ${\mathcal{E}}_s$ be the complex linear span of all operators on the block supported on at most $s$ physical coordinates, with support and weight as in Support and weight of an operator on a finite register.

A density operator $\rho$ on the block and $R$ is $s$-deviated from a code-supported density operator $\sigma$ if there is a CPTP map $\mathcal{N}$ on the block with a finite Kraus family $K_1,\dots ,K_t\in {\mathcal{E}}_s$ such that $\rho =(\mathcal{N}\otimes {\mathrm{id}}_R)(\sigma )={\sum }_{j=1}^t(K_j\otimes I_R)\sigma (K_j^{†}\otimes I_R)$ and ${\sum }_{j=1}^t{K}_j^{†}{K}_j=I$. Here $\sigma$ is code-supported when its joint support is contained in $Q\otimes R$. The reference factor is retained in the identity above. Equality of reduced states outside a coordinate set is not the definition of deviation here.

For a finite family of live blocks indexed by $B$, use the joint error space ${\mathcal{E}}_s^B:=\mathrm{span}\{{⨂}_{b\in B}{A}_b:A_b\text{ is supported on at most }s\text{ coordinates of block }b\}$. Joint $s$-deviation means the same Kraus formula with a single CPTP map on all live blocks, every Kraus operator in ${\mathcal{E}}_s^B$, and identity on the external reference alone. The ideal joint state is supported in ${⨂}_{b\in B}{Q}_b\otimes R$. This includes the original one-block definition when $\mathrm{\mid }B\mathrm{\mid }=1$. The joint error map need not factor across blocks: a faulty multi-block gate can correlate their errors. Other live data blocks are part of the data system, not part of the unchanged external reference. Statements about deviation per block in a multi-block computation use this joint convention.

There is also a weaker encoded instrument error-sector relation, which does not require a CPTP extension to the whole physical input space. Fix the joint encoding $V_B={⨂}_{b\in B}{V}_b$. An $s$-supported encoded instrument is a finite family of completely positive branch maps from the joint logical register to the physical register, with Kraus operators

$A_{\beta j}=E_{\beta j}{V}_B,E_{\beta j}\in {\mathcal{E}}_s^B,{\sum }_{\beta ,j}{A}_{\beta j}^{\ast }{A}_{\beta j}=I_{H_L}\mathrm{.}$

Thus its full instrument is trace preserving on the admitted encoded input domain. A branch retains its actual subnormalized probability; neither it nor its normalized conditional output is required to extend to a full-space CPTP error map. With an ideal logical unitary $g$, replace $V_B$ in this formula by $V_{B}g$. For other ideal logical instruments, state the analogous Kraus identities and output registers explicitly. Every equality is tensored with identity on an arbitrary finite external reference. With auxiliary fault memory $M$, allow coefficients on $M$, that is Kraus sums ${\sum }_a(E_a{V}_B)\otimes M_a$, and impose the full instrument's trace-preservation identity on $H_L\otimes M$. Memory output may change; after its trace, the logical/reference channel is the one required by the declared decoded-recovery identity.

A correcting gadget states the stronger input-sector/output-sector operator identity it needs: on every admitted incoming error term $E{V}_B$, its branch Kraus maps are sums of $E^{\mathrm{\prime }}{V}_{B}g$ with the prescribed output support budget. Coefficients may depend on the incoming error term and on retained branch/memory registers, but not act on the logical factor. Linearity covers coherent incoming errors and correlated blocks. This is the sector convention for noisy correcting instruments. The earlier full-space CPTP deviation relation remains a sufficient, stronger relation; it is not redefined to mean this weaker encoded relation.

For an integer $\mathrm{\ell }\ge 1$, literal support-propagation spread applies to a specified unitary circuit segment: conjugating a fault operator supported on $S$ through the remaining fault-free unitary gates produces an operator supported on at most $\mathrm{\ell }\mathrm{\mid }S\mathrm{\mid }$ output coordinates per block. For several faults, expand the finite Kraus families and take the union of their propagated supports. This gives the sum of the individual support bounds. The bound must account for all input blocks of a multi-block gate.

For a correcting, preparing or measuring gadget, an effective spread budget is instead part of an explicit input-sector/output-sector contract. The gadget declares its admitted joint error sectors, allowed fault supports, ideal logical channel or instrument, and joint residual-error sector for each allowed fault count. The required identities compare its decoded output with that ideal logical channel or instrument on every admitted input, with an arbitrary finite external reference. Measurement identities are identities of the subnormalized branch maps, not equations obtained by ignoring their probabilities. Any auxiliary fault memory is retained in the description of the faulty maps and may change there; it is distinct from the untouched external reference, and nonfaulty locations leave that memory alone. An identity that retains fault memory must specify its output memory action rather than silently identify it with the reference identity.

The numerical computation and correction budgets needed for sparse induction are explicitly imposed in the separately stated proper-code gadget contract; the word spread alone supplies none of those identities. In particular, incoming-error propagation and the full noisy correction action must be checked. Zero-fault recovery on arbitrary inputs remains a separate requirement. No global equation moving arbitrary input faults through a recovery channel on its whole input space is asserted.

The distinction between full-space CPTP deviation and encoded instruments, a concrete nonextension example on the $[\text{\hspace{0.17em}⁣}[49,1,9]\text{\hspace{0.17em}⁣}]$ code, and transfer of a fixed decoded recovery to the encoded instrument relation are proved in Encoded error instruments and full-space CPTP error maps. That comparison is not needed to define either relation.