Towards practical classical processing for the surface code

The surface code is unarguably the leading quantum error correction code for 2-D nearest neighbor architectures, featuring a high threshold error rate of approximately 1%, low overhead implementations of the entire Clifford group, and flexible, arbitrarily long-range logical gates. These highly desirable features come at the cost of significant classical processing complexity. We show how to perform the processing associated with an nxn lattice of qubits, each being manipulated in a realistic, fault-tolerant manner, in O(n^2) average time per round of error correction. We also describe how to parallelize the algorithm to achieve O(1) average processing per round, using only constant computing resources per unit area and local communication. Both of these complexities are optimal.Comment: 5 pages, 6 figures, published version with some additional tex

The Extragalactic Distance Scale Key Project XXVII. A Derivation of the Hubble Constant Using the Fundamental Plane and Dn-Sigma Relations in Leo I, Virgo, and Fornax

Using published photometry and spectroscopy, we construct the fundamental plane and D_n-Sigma relations in Leo I, Virgo and Fornax. The published Cepheid P-L relations to spirals in these clusters fixes the relation between angular size and metric distance for both the fundamental plane and D_n-Sigma relations. Using the locally calibrated fundamental plane, we infer distances to a sample of clusters with a mean redshift of cz \approx 6000 \kms, and derive a value of H_0=78+- 5+- 9 km/s/Mpc (random, systematic) for the local expansion rate. This value includes a correction for depth effects in the Cepheid distances to the nearby clusters, which decreased the deduced value of the expansion rate by 5% +- 5%. If one further adopts the metallicity correction to the Cepheid PL relation, as derived by the Key Project, the value of the Hubble constant would decrease by a further 6%+- 4%. These two sources of systematic error, when combined with a +- 6% error due to the uncertainty in the distance to the Large Magellanic Cloud, a +- 4% error due to uncertainties in the WFPC2 calibration, and several small sources of uncertainty in the fundamental plane analysis, combine to yield a total systematic uncertainty of +- 11%. We find that the values obtained using either the CMB, or a flow-field model, for the reference frame of the distant clusters, agree to within 1%. The Dn-Sigma relation also produces similar results, as expected from the correlated nature of the two scaling relations. A complete discussion of the sources of random and systematic error in this determination of the Hubble constant is also given, in order to facilitate comparison with the other secondary indicators being used by the Key Project.Comment: 21 pages, 3 figures, Accepted for publication in Ap

Relaxed Locally Correctable Codes

Locally decodable codes (LDCs) and locally correctable codes (LCCs) are error-correcting codes in which individual bits of the message and codeword, respectively, can be recovered by querying only few bits from a noisy codeword. These codes have found numerous applications both in theory and in practice. A natural relaxation of LDCs, introduced by Ben-Sasson et al. (SICOMP, 2006), allows the decoder to reject (i.e., refuse to answer) in case it detects that the codeword is corrupt. They call such a decoder a relaxed decoder and construct a constant-query relaxed LDC with almost-linear blocklength, which is sub-exponentially better than what is known for (full-fledged) LDCs in the constant-query regime. We consider an analogous relaxation for local correction. Thus, a relaxed local corrector reads only few bits from a (possibly) corrupt codeword and either recovers the desired bit of the codeword, or rejects in case it detects a corruption. We give two constructions of relaxed LCCs in two regimes, where the first optimizes the query complexity and the second optimizes the rate: 1. Constant Query Complexity: A relaxed LCC with polynomial blocklength whose corrector only reads a constant number of bits of the codeword. This is a sub-exponential improvement over the best constant query (full-fledged) LCCs that are known. 2. Constant Rate: A relaxed LCC with constant rate (i.e., linear blocklength) with quasi-polylogarithmic query complexity. This is a nearly sub-exponential improvement over the query complexity of a recent (full-fledged) constant-rate LCC of Kopparty et al. (STOC, 2016)

Locally Decodable/Correctable Codes for Insertions and Deletions

Recent efforts in coding theory have focused on building codes for insertions and deletions, called insdel codes, with optimal trade-offs between their redundancy and their error-correction capabilities, as well as efficient encoding and decoding algorithms. In many applications, polynomial running time may still be prohibitively expensive, which has motivated the study of codes with super-efficient decoding algorithms. These have led to the well-studied notions of Locally Decodable Codes (LDCs) and Locally Correctable Codes (LCCs). Inspired by these notions, Ostrovsky and Paskin-Cherniavsky (Information Theoretic Security, 2015) generalized Hamming LDCs to insertions and deletions. To the best of our knowledge, these are the only known results that study the analogues of Hamming LDCs in channels performing insertions and deletions. Here we continue the study of insdel codes that admit local algorithms. Specifically, we reprove the results of Ostrovsky and Paskin-Cherniavsky for insdel LDCs using a different set of techniques. We also observe that the techniques extend to constructions of LCCs. Specifically, we obtain insdel LDCs and LCCs from their Hamming LDCs and LCCs analogues, respectively. The rate and error-correction capability blow up only by a constant factor, while the query complexity blows up by a poly log factor in the block length. Since insdel locally decodable/correctble codes are scarcely studied in the literature, we believe our results and techniques may lead to further research. In particular, we conjecture that constant-query insdel LDCs/LCCs do not exist