Related papers: Hyperbolic quantum color codes
We describe a quantum scheme to ``color-code'' a set of objects in order to record which one is which. In the classical case, N distinct colors are required to color-code N objects. We show that in the quantum case, only N/e distinct…
Quantum error correction is necessary for achieving exponential speedups on important applications. The planar surface code has remained the most studied error-correcting code for the last two decades because of its relative simplicity.…
We construct several explicit instances of quantum Tanner codes, a class of asymptotically good quantum low-density parity check (qLDPC) codes. The codes are constructed using dihedral groups and random pairs of classical codes and exhibit…
Surface and color codes are two forms of topological quantum error correction in two spatial dimensions with complementary properties. Surface codes have lower-depth error detection circuits and well-developed decoders to interpret and…
Holographic codes grown with perfect tensors on regular hyperbolic tessellations using an inflation rule protect quantum information stored in the bulk from errors on the boundary provided the code rate is less than one. Hyperbolic geometry…
We develop a heuristic graph coloring approximation algorithm that uses the D-Wave 2X as an independent set sampler and evaluate its performance against a fully classical implementation. A randomly generated set of small but hard graph…
Within the realm of advanced code retrieval, existing methods have primarily relied on intricate matching and attention-based mechanisms. However, these methods often lead to computational and memory inefficiencies, posing a significant…
We study the error threshold of color codes, a class of topological quantum codes that allow a direct implementation of quantum Clifford gates suitable for entanglement distillation, teleportation and fault-tolerant quantum computation. We…
Color codes present distinct advantages for fault-tolerant quantum computing, such as high encoding rates and the transversal implementation of Clifford gates. However, existing matching-based decoders for the color codes such as the…
Let $G$ be a graph embedded on a surface $S_\varepsilon$ with Euler genus $\varepsilon > 0$, and let $P\subseteq V(G)$ be a set of vertices mutually at distance at least 4 apart. Suppose all vertices of $G$ have $H(\varepsilon)$-lists and…
Two-level quantum systems, qubits, are not the only basis for quantum computation. Advantages exist in using qudits, d-level quantum systems, as the basic carrier of quantum information. We show that color codes, a class of topological…
Given a graph $G$, an identifying code $C \subseteq V(G)$ is a vertex set such that for any two distinct vertices $v_1,v_2\in V(G)$, the sets $N[v_1]\cap C$ and $N[v_2]\cap C$ are distinct and nonempty (here $N[v]$ denotes a vertex $v$ and…
If pairs of new colored particles are produced at the Large Hadron Collider, determining their quantum numbers, and even discovering them, can be non-trivial. We suggest that valuable information can be obtained by measuring the resonant…
Surface codes are a popular error-correction route to fault-tolerant quantum computation. The so-called exponential backlog problem that can arise when one has to do logical $T$-gates within the surface code demands real-time decoding of…
Improving a result of Dyer, Frieze and Greenhill [Journal of Combinatorial Theory, Series B, 2015], we determine the $q$-colorability threshold in random $k$-uniform hypergraphs up to an additive error of $\ln 2+\varepsilon_q$, where…
Quantum error correction is rapidly seeing first experimental implementations, but there is a significant gap between asymptotically optimal error-correcting codes and codes that are experimentally feasible. Quantum LDPC codes range from…
In the field of algebraic geometric codes (AG codes), the characterization of dual codes has long been a challenging problem which relies on differentials. In this paper, we provide some descriptions for certain differentials utilizing…
Given a metric space and a set of distances, one constructs the associated distance graph by taking as vertices the points of the space and as edges the pairs whose distance is in the given set. It is a longstanding open question to…
A $t$-tone $k$-coloring of a graph $G$ assigns a set of $t$ distinct colors from $\{1, \dots, k\}$ to each vertex so that vertices at distance $d$ share fewer than $d$ common colors. The $t$-tone chromatic number of $G$ is the minimum $k$…
We introduce a "hyperbicycle" ansatz for quantum codes which gives the hypergraph-product (generalized toric) codes by Tillich and Z\'emor and generalized bicycle codes by MacKay et al. as limiting cases. The construction allows for both…