Concatenated quantum codes

A concatenated quantum code encodes each outer-code carrier into an inner code, creating recursively scalable protection. It is not one code family: the result depends on inner and outer codes, concatenation order, logical gadgets, and the decoder at every level.

For outer [[n1,k1,d1]] and inner [[n2,k2,d2]] blocks, a standard concatenation has length n1n2, logical dimension k1k2, and distance at least d1d2, subject to the carrier-to-inner-logical mapping. The recursion tree and qubit order are construction data.

Each level treats lower-level logical failures as higher-level physical faults, producing an error-rate recursion under independent or local-stochastic assumptions. Different levels may target bias or leakage, while correlated faults invalidate simple multiplicative estimates.

Recursing the same [[n,k,d]] block for r levels gives ideal length n^r, distance at least d^r, and rate (k/n)^r. Actual resources also include level-specific ancilla factories, verification, waiting, and classical control, often far exceeding data-block counts.

Each logical gate is replaced recursively by a lower-level gadget, with transversality limiting spread inside a subblock. Missing gates are supplied by magic states, teleportation, or code switching; the gate set and concatenation structure must be recorded together.

Decoding typically proceeds from the innermost level outward, passing each subblock's syndrome and confidence to its parent. Hard decisions discard soft information; joint or message-passing decoders exploit cross-level correlations at greater computational cost.

Concatenated-code threshold theorems depend on fault-tolerant gadgets, extended rectangles, malignant-fault counting, and an explicit noise model. Additional levels suppress logical error only when the physical rate is below the threshold of those gadgets.

Small experiments often implement one or two concatenation levels or repeated error-detecting blocks. Reports should distinguish genuine nested correction, preparation of a concatenated codeword, and experiments that replace recovery with postselection.

The Shor code is itself a heterogeneous concatenation of repetition codes, while blocks such as Steane are often recursively self-concatenated. Heterogeneous A-after-B and B-after-A constructions generally differ in distance, gates, and noise adaptation even if parameter products match.

A two-level Steane [[49,1,9]] record includes a definite recursion tree. An arbitrary [[49,1,9]] stabilizer with equal parameters is not the same instance; exact mapping needs base code, level count, ordering, generators, and logical conventions.