scholarly journals QUANTUM f-DIVERGENCES AND ERROR CORRECTION

2011 ◽  
Vol 23 (07) ◽  
pp. 691-747 ◽  
Author(s):  
FUMIO HIAI ◽  
MILÁN MOSONYI ◽  
DÉNES PETZ ◽  
CÉDRIC BÉNY
Keyword(s):  
Error Correction ◽  
Matrix Analysis ◽  
Special Cases ◽  
Monotonicity Inequality ◽  
Operator Convex ◽  
Quantum Error ◽  
Operator Convex Functions ◽  
Special Case ◽  
Stochastic Map ◽  
Positive Half

Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f-divergences. Special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Hölder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.

scholarly journals Operator quantum error correction

2006 ◽  
Vol 6 (4&5) ◽  
pp. 382-399
Author(s):  
D.W. Kribs ◽  
R. Laflamme ◽  
D. Poulin ◽  
M. Lesosky
Keyword(s):  
Error Correction ◽  
Standard Error ◽  
Error Correction Model ◽  
Quantum Error Correction ◽  
Mathematical Framework ◽  
Correction Model ◽  
Quantum Operations ◽  
Special Cases ◽  
Quantum Error ◽  
Arbitrary Quantum

This paper is an expanded and more detailed version of the work \cite{KLP04} in which the Operator Quantum Error Correction formalism was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques --- i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method --- as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of "unitarily noiseless subsystems''.

scholarly journals Quantum Graphs as Quantum Relations

2021 ◽  
Author(s):  
Nik Weaver
Keyword(s):  
Error Correction ◽  
Channel Coding ◽  
Error Correcting Codes ◽  
Quantum Channels ◽  
Information Encoding ◽  
Graph Theoretic ◽  
Graph Homomorphisms ◽  
Special Cases ◽  
Quantum Error ◽  
Lovász Number

AbstractThe “noncommutative graphs” which arise in quantum error correction are a special case of the quantum relations introduced in Weaver (Quantum relations. Mem Am Math Soc 215(v–vi):81–140, 2012). We use this perspective to interpret the Knill–Laflamme error-correction conditions (Knill and Laflamme in Theory of quantum error-correcting codes. Phys Rev A 55:900-911, 1997) in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke’s noncommutative graph homomorphisms (Stahlke in Quantum zero-error source-channel coding and non-commutative graph theory. IEEE Trans Inf Theory 62:554–577, 2016) and Duan, Severini, and Winter’s noncommutative bipartite graphs (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013), and to realize the noncommutative confusability graph associated to a quantum channel (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013) as the pullback of a diagonal relation. Our framework includes as special cases not only purely classical and purely quantum information theory, but also the “mixed” setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.

scholarly journals 2-designs and redundant syndrome extraction for quantum error correction

2021 ◽  
Vol 20 (3) ◽  
Author(s):  
Vickram N. Premakumar ◽  
Hele Sha ◽  
Daniel Crow ◽  
Eric Bach ◽  
Robert Joynt
Keyword(s):  
Error Correction ◽  
Quantum Error Correction ◽  
Quantum Error

scholarly journals Exponential suppression of bit or phase errors with cyclic error correction

Nature ◽  
2021 ◽  
Vol 595 (7867) ◽  
pp. 383-387
Author(s):  
◽  
Zijun Chen ◽  
Kevin J. Satzinger ◽  
Juan Atalaya ◽  
Alexander N. Korotkov ◽  
...  
Keyword(s):  
Error Correction ◽  
Error Detection ◽  
Fault Tolerant ◽  
Error Rates ◽  
Quantum Error Correction ◽  
Superconducting Qubits ◽  
Two Dimensional ◽  
Logical Error ◽  
Quantum Error ◽  
Logical Qubit

AbstractRealizing the potential of quantum computing requires sufficiently low logical error rates1. Many applications call for error rates as low as 10−15 (refs. 2–9), but state-of-the-art quantum platforms typically have physical error rates near 10−3 (refs. 10–14). Quantum error correction15–17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.

scholarly journals Impact of correlations and heavy tails on quantum error correction

2021 ◽  
Vol 103 (5) ◽  
Author(s):  
B. D. Clader ◽  
Colin J. Trout ◽  
Jeff P. Barnes ◽  
Kevin Schultz ◽  
Gregory Quiroz ◽  
...  
Keyword(s):  
Error Correction ◽  
Heavy Tails ◽  
Quantum Error Correction ◽  
Quantum Error
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