Related papers: Hypermap-Homology Quantum Codes (Ph.D. thesis)
Error-correcting codes over the real field are studied which can locate outlying computational errors when performing approximate computing of real vector--matrix multiplication on resistive crossbars. Prior work has concentrated on…
Quantum error correction is a critical component for scaling up quantum computing. Given a quantum code, an optimal decoder maps the measured code violations to the most likely error that occurred, but its cost scales exponentially with the…
Deciding if a given family of quantum states is topologically ordered is an important but nontrivial problem in condensed matter physics and quantum information theory. We derive necessary and sufficient conditions for a family of graph…
In this article, we establish a mathematical framework that utilizes concepts from graph theory to formalize the parity transformation, an encoding strategy for compiling optimization problems on quantum devices. We introduce the…
Long quantum codes using projective Reed-Muller codes are constructed. Projective Reed-Muller codes are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum…
Linear error-correcting codes form the mathematical backbone of modern digital communication and storage systems, but identifying champion linear codes (linear codes achieving or exceeding the best known minimum Hamming distance) remains…
It is conjectured that quantum computers are able to solve certain problems more quickly than any deterministic or probabilistic computer. A quantum computer exploits the rules of quantum mechanics to speed up computations. However, it is a…
There is an advantage in simultaneously transmitting both classical and quantum information over a quantum channel compared to sending independent transmissions. The successful implementation of simultaneous transmissions of quantum and…
We study ensembles of codes on graphs (generalized low-density parity-check, or LDPC codes) constructed from random graphs and fixed local constrained codes, and their extension to codes on hypergraphs. It is known that the average minimum…
Quantum error correction (QEC) is often implemented on hardware that experiences biased noise, where dephasing errors occur more frequently than other errors. This has motivated many recent efforts to develop bias-tailored QEC codes, such…
We introduce graphcodes, a novel multi-scale summary of the topological properties of a dataset that is based on the well-established theory of persistent homology. Graphcodes handle datasets that are filtered along two real-valued scale…
We provide a detailed study of the general structure of two-dimensional topological stabilizer quantum error correcting codes, including subsystem codes. Under the sole assumption of translational invariance, we show that all such codes can…
We introduce a high-level graphical framework for designing and analysing quantum error correcting codes, centred on what we term the coherent parity check (CPC). The graphical formulation is based on the diagrammatic tools of the…
A distributed quantum storage code maps a quantum message to N storage nodes, of arbitrary specified sizes, such that the stored message is robust to an arbitrary specified set of erasure patterns. The sizes of the storage nodes, and…
The surface code scheme for quantum computation features a 2d array of nearest-neighbor coupled qubits yet claims a threshold error rate approaching 1% (NJoP 9:199, 2007). This result was obtained for the toric code, from which the surface…
We present a general framework of quantum error-correcting codes (QECCs) as a subspace of a complex Hilbert space and the corresponding error models. Then we illustrate how QECCs can be constructed using techniques from algebraic coding…
If the cyclic sequences of {face types} {at} all vertices in a map are the same, then the map is said to be a semi-equivelar map. In particular, a semi-equivelar map is equivelar if the faces are the same type. Homological quantum codes…
We show how entanglement shared between encoder and decoder can simplify the theory of quantum error correction. The entanglement-assisted quantum codes we describe do not require the dual-containing constraint necessary for standard…
Local decoders provide a promising approach to real-time quantum error-correction by replacing centralized classical decoding, with significant hardware constraints, by a fully distributed architecture based on a simple, local update rule.…
Some combinatorial designs, such as Hadamard matrices, have been extensively researched and are familiar to readers across the spectrum of Science and Engineering. They arise in diverse fields such as cryptography, communication theory, and…