Structured error recovery for code-word-stabilized quantum codes

Abstract

Codeword stabilized (CWS) codes are, in general, non-additive quantum codes that can correct errors by an exhaustive search of different error patterns, similar to the way that we decode classical non-linear codes. For an n-qubit quantum code correcting errors on up to t qubits, this brute-force approach consecutively tests different errors of weight t or less, and employs a separate n-qubit measurement in each test. In this paper, we suggest an error grouping technique that allows to simultaneously test large groups of errors in a single measurement. This structured error recovery technique exponentially reduces the number of measurements by about 3 t times. While it still leaves exponentially many measurements for a generic CWS code, the technique is equivalent to syndrome-based recovery for the special case of additive CWS codes.

Clustered bounded-distance decoding of codeword-stabilized quantum codes

“…It is readily proven [10] that the corresponding measurement circuit is defined by 2(n−k) controlled n-qubit Pauli operators plus (n − k − 1) Toffoli gates [12]. Overall, this gives Lemma 3: An error-detecting measurement M Q0 of a stabilizer code Q 0 requires up to 2(n−k)(n+3) two-qubit gates.…”

“…This relation (or associativity) can be used to define the symmetric difference for several vector spaces, e.g., A B C ≡ (A B) C. The same relation implies that the measurement of M 1 ⊕ M 0 can be implemented as a concatenation of measurements M 1 and M 0 , see Fig. 2 [10]. The following two lemmas will be used to decompose the multi-qubit measurements.…”

“…In this case, for a given index set A, our scheme employs a bigger group W g ≡ E A , W g and a smaller translation set T of size m < K. The corresponding complexity of a single measurement would then be reduced to 2mn 2 , compared to 2Kn 2 for the general case in Theorem 1. Note also that for stabilizer codes, our error-grouping technique is equivalent to the syndrome-based recovery [10]. Indeed, for a stabilizer code Q =[[n, k, d]], the equivalence classes for all correctable errors form an Abelian group E = e 1 , .…”

“…This circuit uses 2K[n 2 + O(n)] two-qubit gates; another circuit with up to 2n 2 +KO(n) two-qubit gates is constructed in Ref. [10]. The operators EM Q E † stabilize orthogonal spaces for mutually non-degenerate errors E. Therefore, error correction can be done by measuring such operators for different representatives of degeneracy classes of all correctable errors, which requires up to B(n, t) measurements for a non-degenerate terror correcting code.…”

Non-Pauli observables for CWS codes

“…A generic decoding procedure for binary CWS codes was proposed in Ref. [18] and extended for nonbinary CWS codes in Ref. [19].…”