In a recent study [Rohde et al., quant-ph/0603130 (2006)] of several quantum error correcting protocols designed for tolerance against qubit loss, it was shown that these protocols have the undesirable effect of magnifying the effects of depolarization noise. This raises the question of which general properties of quantum error-correcting codes might explain such an apparent trade-off between tolerance to located and unlocated error types. We extend the counting argument behind the well-known quantum Hamming bound to derive a bound on the weights of combinations of located and unlocated errors which are correctable by nondegenerate quantum codes. Numerical results show that the bound gives an excellent prediction to which combinations of unlocated and located errors can be corrected {\em with high probability} by certain large degenerate codes. The numerical results are explained partly by showing that the generalized bound, like the original, is closely connected to the information-theoretic quantity the {\em quantum coherent information}. However, we also show that as a measure of the exact performance of quantum codes, our generalized Hamming bound is provably far from tight.
In a recent study [Rohde et al., quant-ph/0603130 (2006)] of several quantum error correcting protocols designed for tolerance against qubit loss, it was shown that these protocols have the undesirable effect of magnifying the effects of depolarization noise. This raises the question of which general properties of quantum error-correcting codes might explain such an apparent trade-off between tolerance to located and unlocated error types. We extend the counting argument behind the well-known quantum Hamming bound to derive a bound on the weights of combinations of located and unlocated errors which are correctable by nondegenerate quantum codes. Numerical results show that the bound gives an excellent prediction to which combinations of unlocated and located errors can be corrected with high probability by certain large degenerate codes. The numerical results are explained partly by showing that the generalized bound, like the original, is closely connected to the information-theoretic quantity the quantum coherent information. However, we also show that as a measure of the exact performance of quantum codes, our generalized Hamming bound is provably far from tight.
The quantum Hamming bound arises from a direct dimension estimate for exact quantum error-correcting codes. For nondegenerate codes, distinct correctable errors map the code space into orthogonal subspaces, so dimension counting directly gives the bound; degeneracy allows these subspaces to overlap and invalidates the argument. Although linear-programming, asymptotic, and structural approaches have advanced the problem from several directions, the general local-dimension case has remained open. Here we prove the quantum Hamming bound for every local dimension
q≥3. Building on the exact two-center Fourier reduction, we develop new overlap estimates for nonbinary Hamming spaces. For
q≥4, Jacobi estimates and monotonicity give a uniform gap; qutrits form the boundary case where the gap vanishes and are handled by a filtered correction. This completes the finite-length nonbinary case; together with the existing qubit result, it establishes the quantum Hamming bound for arbitrary local dimension.
The quantum Hamming bound was originally put forward as an upper bound on the parameters of nondegenerate quantum codes, but over the past few decades much work has been done to show that many degenerate quantum codes must also obey this bound. In this paper, we show that there is a Hamming-like bound stricter than the quantum Hamming bound that applies to degenerate
t-error-correcting stabilizer codes of length greater than some positive integer
N(t). We show that this bound holds for all single-error-correcting degenerate stabilizer codes, forcing all but a handful of optimal distance-3 stabilizer codes to be nondegenerate.
It is well-known that non-degenerate quantum error correcting codes (QECCs) are constrained by a quantum version of the Hamming bound. Whether degenerate codes also obey such a bound, however, remains a long-standing question with practical implications for the efficacy of QECCs. We employ a combination of previously derived bounds on QECCs to demonstrate that a subset of all codes must obey the quantum Hamming bound. Specifically, we combine an analytical bound due to Rains with a numerical bound due to Li and Xing to show that no
((n,K,d))code with
d<127can violate the quantum Hamming bound.
The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group
G and two subsets of
G whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find
[[60,4,10]],[[36,6,6]],[[48,8,6]], and
[[85,8,9]] codes, all of which are not CSS; we also find several weight-7 codes with
kd>n.
Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of
0.2%, approximately matching that of the
[[144,12,12]] BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.
This thesis provides bounds on the performance of quantum error correcting codes when used for quantum communication and quantum key distribution. The first two chapters provide a bare-bones introduction to classical and quantum error correcting codes, respectively. The next four chapters present achievable rates for quantum codes in various scenarios. The final chapter is dedicated to an upper bound on the quantum channel capacity.
Chapter 3 studies coding for adversarial noise using quantum list codes, showing there exist quantum codes with high rates and short lists. These can be used, together with a very short secret key, to communicate with high fidelity at noise levels for which perfect fidelity is, impossible.
Chapter 4 explores the performance of a family of degenerate codes when used to communicate over Pauli channels, showing they can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code and that exceed those of previously known degenerate codes. By studying the scaling of the optimal block length as a function of the channel's parameters, we develop a heuristic for designing even better codes.
Chapter 5 describes an equivalence between a family of noisy preprocessing protocols for quantum key distribution and entanglement distillation protocols whose target state belongs to a class of private states called twisted states.
In Chapter 6, the codes of Chapter 4 are combined with the protocols of Chapter 5 to provide higher key rates for one-way quantum key distribution than were previously thought possible.
Finally, Chapter 7 presents a new upper bound on the quantum channel that is both additive and convex, and which can be interpreted as the of the channel for communication given access to side channels from a class of zero cloning channels. This clone assisted capacity is equal to the unassisted for channels that are degradable, which we use to find new upper bounds on the of a depolarizing channel.