arXiv subjects

Eric Sabo

Publications and source records attributed to Eric Sabo.

5 recordsLinked to original sources

arXiv2026

Spectral Theory of Semisimple Bivariate Bicycle Codes

Extending the classical theory of two-dimensional cyclic codes, we develop an algebraic approach to bivariate bicycle codes. Using Frobenius-orbit idempotents, formulas for logical dimensions are derived and lower bounds on minimum distances are established. A systematic theory of code symmetries is formulated to construct a structured block-monomial subgroup of coordinate permutations. Several explicit examples show how to generate these codes from first principles without relying on numerical searches. An appendix extends the analysis to BCH-based product constructions.

quant-ph↗

arXiv2024

Weight Reduced Stabilizer Codes with Lower Overhead

Stabilizer codes are the most widely studied class of quantum error-correcting codes and form the basis of most proposals for a fault-tolerant quantum computer. A stabilizer code is defined by a set of parity-check operators, which are measured in order to infer information about errors that may have occurred. In typical settings, measuring these operators is itself a noisy process and the noise strength scales with the number of qubits involved in a given parity check, or its weight. Hastings proposed a method for reducing the weights of the parity checks of a stabilizer code, though it has previously only been studied in the asymptotic regime. Here, we instead focus on the regime of small-to-medium size codes suitable for quantum computing hardware. We provide both a fully explicit description of Hastings's method and propose a substantially simplified weight reduction method that is applicable to the class of quantum product codes. Our simplified method allows us to reduce the check weights of hypergraph and lifted product codes to at most six, while preserving the number of logical qubits and at least retaining (in fact often increasing) the code distance. The price we pay is an increase in the number of physical qubits by a constant factor, but we find that our method is much more efficient than Hastings's method in this regard. We benchmark the performance of our codes in a photonic quantum computing architecture based on GKP qubits and passive linear optics, finding that our weight reduction method substantially improves code performance.

quant-ph↗

arXiv2022

Trellis Decoding For Qudit Stabilizer Codes And Its Application To Qubit Topological Codes

Trellis decoders are a general decoding technique first applied to qubit-based quantum error correction codes by Ollivier and Tillich in 2006. Here we improve the scalability and practicality of their theory, show that it has strong structure, extend the results using classical coding theory as a guide, and demonstrate a canonical form from which the structural properties of the decoding graph may be computed. The resulting formalism is valid for any prime-dimensional quantum system. The modified decoder works for any stabilizer code $S$ and separates into two parts: a one-time, offline computation which builds a compact, graphical representation of the normalizer of the code, $S^\perp$, and a quick, parallel, online query of the resulting vertices using the Viterbi algorithm. We show the utility of trellis decoding by applying it to four high-density, length 20 stabilizer codes for depolarizing noise and the well-studied Steane, rotated surface, and 4.8.8/6.6.6 color codes for $Z$-only noise. Numerical simulations demonstrate a 20\% improvement in the code-capacity threshold for color codes with boundaries by avoiding the mapping from color codes to surface codes. We identify trellis edge number as a key metric of difficulty of decoding, allowing us to quantify the advantage of single-axis decoding for Calderbank-Steane-Shor codes and block-decoding for concatenated codes.

quant-ph↗

arXiv2015

Exact computation of the n-loop invariants of knots

The loop invariants of Dimofte-Garoufalidis is a formal power series with arithmetically interesting coefficients that conjecturally appears in the asymptotics of the Kashaev invariant of a knot to all orders in $1/N$. We develop methods implemented in SnapPy that compute the first 6 coefficients of the formal power series of a knot. We give examples that illustrate our method and its results.

math.GT↗

arXiv2013

$R$-Symmetry Breaking in Supersymmetric Hybrid Inflation

We consider a supersymmetric hybrid inflation scenario in which the U(1) $R$-symmetry is explicitly broken by Planck scale suppressed operators in the superpotential. We provide an example with minimal Kähler potential, with the $R$-symmetry breaking term relevant during inflation being $αS^4$, where $S$ denotes the well-known gauge singlet inflaton superfield. The inflationary potential takes into account the radiative and supergravity corrections, as well as the soft supersymmetry breaking terms. For successful inflation, with the scalar spectral index in the currently preferred range, $n_s \approx 0.97 \pm 0.010$, $| α|$ $\lesssim 10^{-7}$. The tensor to scalar ratio $r \lesssim 10^{-4}$, while $| \text{d} n_s / \text{d} \ln k | \sim \mathcal{O}(10^{-3})-\mathcal{O}(10^{-4})$

hep-ph↗