scholarly journals Fault-tolerant gates via homological product codes

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 120 ◽  
Author(s):  
Tomas Jochym-O'Connor
Keyword(s):  
Fault Tolerant ◽  
General Scheme ◽  
Quantum Codes ◽  
Product Codes ◽  
Logical State

A method for the implementation of a universal set of fault-tolerant logical gates is presented using homological product codes. In particular, it is shown that one can fault-tolerantly map between different encoded representations of a given logical state, enabling the application of different classes of transversal gates belonging to the underlying quantum codes. This allows for the circumvention of no-go results pertaining to universal sets of transversal gates and provides a general scheme for fault-tolerant computation while keeping the stabilizer generators of the code sparse.

scholarly journals Fault-Tolerant Gates on Hypergraph Product Codes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Anirudh Krishna ◽  
David Poulin
Keyword(s):  
Fault Tolerant ◽  
Product Codes

scholarly journals Study of a Computer-Based Control System with Fault-Tolerant and Fail-Safe Behavior

2015 ◽  
Vol 13 (3-4) ◽  
pp. 34-45
Author(s):  
Mariya Hristova
Keyword(s):  
Probabilistic Models ◽  
Fault Tolerant ◽  
General Scheme ◽  
Base System ◽  
Critical Systems ◽  
Critical Technology ◽  
Previous Edition ◽  
System 2 ◽  
Software And Hardware ◽  
Fail Safe

Abstract The software and hardware of Safety Critical Systems - SCS, which control special critical technology process or operation, are subject of enhanced requirements for reliability and inadmissibility of any incorrect controlling influences (safety) after failures. This paper suggests and investigates a hybrid computerbased fail-safe/fault-tolerance FST-structure with single reservation, which has the qualities to meet these requirements. The base system 2 ∨ 2, based on which it is built, is studied in the previous edition of Information Technjlogies and Control. For derivation of probabilistic models through which to establish the efficiency of the strutucral redundancy in a fault-tolerant structure, in this paper are used the published results. Formulas are derived for the probability of failure-free operation (availability coefficient), for safe failure and for unidentified (dangerous) failure. Models are found for calculation of the enhancement of reliability and the variation of safety relative to the base system 2 ∨ 2. Subject of analysis are type software and hardware modifications of the proposed general scheme used in the practice of different companies, which manufacture and operate SCSs. It is proven that at the expense of acceptable hardware redundancy and insignificant increase of dangerous failures the probability of interruption of the operation of the FST- system and its down-time due to failure may decrease by tenths of thousands of times.

scholarly journals New quantum codes from evaluation and matrix-product codes

2015 ◽  
Vol 36 ◽  
pp. 98-120 ◽  
Author(s):  
Carlos Galindo ◽  
Fernando Hernando ◽  
Diego Ruano
Keyword(s):  
Quantum Codes ◽  
Matrix Product ◽  
Product Codes

scholarly journals Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes

2021 ◽  
Author(s):  
Dongsheng Wang ◽  
Yunjiang Wang ◽  
Ningping Cao ◽  
Bei Zeng ◽  
Raymond Lafflamme
Keyword(s):  
Error Correction ◽  
Quantum Computation ◽  
Fault Tolerant ◽  
Valence Bond ◽  
Quantum Error Correction ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Exact Quantum ◽  
Valence Bond Solid ◽  
Quantum Error

Abstract In this work, we develop the theory of quasi-exact fault-tolerant quantum (QEQ) computation, which uses qubits encoded into quasi-exact quantum error-correction codes (``quasi codes''). By definition, a quasi code is a parametric approximate code that can become exact by tuning its parameters. The model of QEQ computation lies in between the two well-known ones: the usual noisy quantum computation without error correction and the usual fault-tolerant quantum computation, but closer to the later. Many notions of exact quantum codes need to be adjusted for the quasi setting. Here we develop quasi error-correction theory using quantum instrument, the notions of quasi universality, quasi code distances, and quasi thresholds, etc. We find a wide class of quasi codes which are called valence-bond-solid codes, and we use them as concrete examples to demonstrate QEQ computation.

scholarly journals Degenerate Quantum LDPC Codes With Good Finite Length Performance

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 585
Author(s):  
Pavel Panteleev ◽  
Gleb Kalachev
Keyword(s):  
Minimum Distance ◽  
Belief Propagation ◽  
Fault Tolerant ◽  
Large Family ◽  
Parity Check ◽  
Product Codes ◽  
Medium Length ◽  
Surface Code ◽  
Depolarizing Channel ◽  
Better Than

We study the performance of medium-length quantum LDPC (QLDPC) codes in the depolarizing channel. Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. We also construct several interesting examples of short generalized bicycle codes. Some of them have an additional property that their syndromes are protected by small BCH codes, which may be useful for the fault-tolerant syndrome measurement. We also propose a new large family of QLDPC codes that contains the class of hypergraph product codes, where one of the used parity-check matrices is square. It is shown that in some cases such codes have better performance than hypergraph product codes. Finally, we demonstrate that the performance of the proposed BP-OSD decoder for some of the constructed codes is better than for a relatively large surface code decoded by a near-optimal decoder.

New quantum codes from matrix-product codes over small fields

2020 ◽  
Vol 19 (8) ◽  
Author(s):  
Hao Song ◽  
Ruihu Li ◽  
Yang Liu ◽  
Guanmin Guo
Keyword(s):  
Quantum Codes ◽  
Matrix Product ◽  
Product Codes ◽  
Small Fields

scholarly journals C_3, semi-Clifford and genralized semi-Clifford operations

2010 ◽  
Vol 10 (1&2) ◽  
pp. 41-59
Author(s):  
S. Beigi ◽  
P.W. Shor
Keyword(s):  
Quantum Computation ◽  
Basic Problem ◽  
Fault Tolerant ◽  
Quantum Codes ◽  
Stabilizer Codes

Fault-tolerant quantum computation is a basic problem in quantum computation, and teleportation is one of the main techniques in this theory. Using teleportation on stabilizer codes, the most well-known quantum codes, Pauli gates and Clifford operators can be applied fault-tolerantly. Indeed, this technique can be generalized for an extended set of gates, the so called ${\mathcal{C}}_k$ hierarchy gates, introduced by Gottesman and Chuang (Nature, 402, 390-392). ${\mathcal{C}}_k$ gates are a generalization of Clifford operators, but our knowledge of these sets is not as rich as our knowledge of Clifford gates. Zeng et al. in (Phys. Rev. A 77, 042313) raise the question of the relation between ${\mathcal{C}}_k$ hierarchy and the set of semi-Clifford and generalized semi-Clifford operators. They conjecture that any ${\mathcal{C}}_k$ gate is a generalized semi-Clifford operator. In this paper, we prove this conjecture for $k=3$. Using the techniques that we develop, we obtain more insight on how to characterize ${\mathcal{C}}_3$ gates. Indeed, the more we understand ${\mathcal{C}}_3$, the more intuition we have on ${\mathcal{C}}_k$, $k\geq 4$, and then we have a way of attacking the conjecture for larger $k$.

scholarly journals Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 180 ◽  
Author(s):  
Ali Lavasani ◽  
Guanyu Zhu ◽  
Maissam Barkeshli
Keyword(s):  
Quantum Computation ◽  
Fault Tolerant ◽  
Hyperbolic Surface ◽  
Constant Factor ◽  
Basic Question ◽  
Quantum Codes ◽  
Constant Depth ◽  
Dehn Twists ◽  
Local Operators ◽  
Time Overhead

A basic question in the theory of fault-tolerant quantum computation is to understand the fundamental resource costs for performing a universal logical set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distance d through the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci Turaev-Viro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggest the possibility of universal fault tolerant quantum computation with constant space overhead and time overhead of O(d/log⁡d). For quantum circuits that allow parallel gate operations, this yields the optimal scaling of space-time overhead known to date.

scholarly journals Flag fault-tolerant error correction with arbitrary distance codes

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 53 ◽  
Author(s):  
Christopher Chamberland ◽  
Michael E. Beverland
Keyword(s):  
Error Correction ◽  
Fault Tolerant ◽  
Quantum Error Correction ◽  
Quantum Codes ◽  
Parity Check ◽  
Code Length ◽  
Stabilizer Codes ◽  
Low Density Parity Check ◽  
Quantum Error ◽  
First Time

In this paper we introduce a general fault-tolerant quantum error correction protocol using flag circuits for measuring stabilizers of arbitrary distance codes. In addition to extending flag error correction beyond distance-three codes for the first time, our protocol also applies to a broader class of distance-three codes than was previously known. Flag circuits use extra ancilla qubits to signal when errors resulting fromvfaults in the circuit have weight greater thanv. The flag error correction protocol is applicable to stabilizer codes of arbitrary distance which satisfy a set of conditions and uses fewer qubits than other schemes such as Shor, Steane and Knill error correction. We give examples of infinite code families which satisfy these conditions and analyze the behaviour of distance-three and -five examples numerically. Requiring fewer resources than Shor error correction, flag error correction could potentially be used in low-overhead fault-tolerant error correction protocols using low density parity check quantum codes of large code length.

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