Perfect Quantum Error Correcting Code

Quantum error correction is a set of methods to protect quantum information—that is, quantum states—from unwanted environmental interactions (decoherence) and other forms of noise. The information is stored in a quantum error-correcting code, which is a subspace in a larger Hilbert space. This code is designed so that the most common errors move the state into an error space orthogonal to the original code space while preserving the information in the state. It is possible to determine whether an error has occurred by a suitable measurement and to apply a unitary correction that returns the state to the code space without measuring (and hence disturbing) the protected state itself. In general, codewords of a quantum code are entangled states. No code that stores information can protect against all possible errors; instead, codes are designed to correct a specific error set, which should be chosen to match the most likely types of noise. An error set is represented by a set of operators that can multiply the codeword state. Most work on quantum error correction has focused on systems of quantum bits, or qubits, which are two-level quantum systems. These can be physically realized by the states of a spin-1/2 particle, the polarization of a single photon, two distinguished levels of a trapped atom or ion, the current states of a microscopic superconducting loop, or many other physical systems. The most widely used codes are the stabilizer codes, which are closely related to classical linear codes. The code space is the joint +1 eigenspace of a set of commuting Pauli operators on n qubits, called stabilizer generators; the error syndrome is determined by measuring these operators, which allows errors to be diagnosed and corrected. A stabilizer code is characterized by three parameters [[n,k,d]], where n is the number of physical qubits, k is the number of encoded logical qubits, and d is the minimum distance of the code (the smallest number of simultaneous qubit errors that can transform one valid codeword into another). Every useful code has n>k; this physical redundancy is necessary to detect and correct errors without disturbing the logical state. Quantum error correction is used to protect information in quantum communication (where quantum states pass through noisy channels) and quantum computation (where quantum states are transformed through a sequence of imperfect computational steps in the presence of environmental decoherence to solve a computational problem). In quantum computation, error correction is just one component of fault-tolerant design. Other approaches to error mitigation in quantum systems include decoherence-free subspaces, noiseless subsystems, and dynamical decoupling.

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Improvement of quantum correlations by repetitive quantum error correction

International Journal of Quantum Information ◽

10.1142/s0219749919500448 ◽

2019 ◽

Vol 17 (05) ◽

pp. 1950044

Author(s):

A. El Allati ◽

H. Amellal ◽

A. Meslouhi

Keyword(s):

Error Correction ◽

Quantum Correlation ◽

Coherent States ◽

Quantum Discord ◽

Error Correcting Code ◽

Quantum Correlations ◽

Quantum Error Correction ◽

Optical Field ◽

Quantum Error ◽

Entangled Coherent States

A quantum error-correcting code is established in entangled coherent states (CSs) with Markovian and non-Markovian environments. However, the dynamic behavior of these optical states is discussed in terms of quantum correlation measurements, entanglement and discord. By using the correcting codes, these correlations can be as robust as possible against environmental effects. As the number of redundant CSs increases due to the repetitive error correction, the probabilities of success also increase significantly. Based on different optical field parameters, the discord can withstand more than an entanglement. Furthermore, the behavior of quantum discord under decoherence may exhibit sudden death and sudden birth phenomena as functions of dimensionless parameters.

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Five-wave-packet linear optics quantum-error-correcting code

Physical Review A ◽

10.1103/physreva.81.062305 ◽

2010 ◽

Vol 81 (6) ◽

Cited By ~ 2

Author(s):

Thomas A. Walker ◽

Samuel L. Braunstein

Keyword(s):

Wave Packet ◽

Error Correcting Code ◽

Linear Optics ◽

Quantum Error

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ON THE AUTOMORPHISM GROUPS OF A FAMILY OF BINARY QUANTUM ERROR-CORRECTING CODES

International Journal of Quantum Information ◽

10.1142/s0219749906002377 ◽

2006 ◽

Vol 04 (06) ◽

pp. 1013-1022

Author(s):

TAILIN LIU ◽

FENGTONG WEN ◽

QIAOYAN WEN

Keyword(s):

Semidirect Product ◽

Automorphism Groups ◽

Error Correcting Code ◽

Error Correcting Codes ◽

Index Formula ◽

Product Group ◽

Simplex Code ◽

Quantum Error ◽

The Relationship ◽

Free Element

Based on the classical binary simplex code [Formula: see text] and any fixed-point-free element f of [Formula: see text], Calderbank et al. constructed a binary quantum error-correcting code [Formula: see text]. They proved that [Formula: see text] has a normal subgroup H, which is a semidirect product group of the centralizer Z(f) of f in GLm(2) with [Formula: see text], and the index [Formula: see text] is the number of elements of Ff = {f, 1 - f, 1/f, 1 - 1/f, 1/(1 - f), f/(1 - f)} that are conjugate to f. In this paper, a theorem to describe the relationship between the quotient group [Formula: see text] and the set Ff is presented, and a way to find the elements of Ff that are conjugate to f is proposed. Then we prove that [Formula: see text] is isomorphic to S3 and H is a semidirect product group of [Formula: see text] with [Formula: see text] in the linear case. Finally, we generalize a result due to Calderbank et al.

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Limitations on transversal computation through quantum homomorphic encryption

Quantum Information and Computation ◽

10.26421/qic18.11-12-3 ◽

2018 ◽

Vol 18 (11&12) ◽

pp. 927-948

Author(s):

Michael Newman ◽

Yaoyun Shi

Keyword(s):

Quantum Computation ◽

Homomorphic Encryption ◽

Error Correcting Code ◽

Toffoli Gate ◽

Additive Codes ◽

Important Open Problem ◽

Small Set ◽

Quantum Error ◽

Universal Gate ◽

Rule Out

Transversality is a simple and effective method for implementing quantum computation fault-tolerantly. However, no quantum error-correcting code (QECC) can transversally implement a quantum universal gate set (Eastin and Knill, {\em Phys. Rev. Lett.}, 102, 110502). Since reversible classical computation is often a dominating part of useful quantum computation, whether or not it can be implemented transversally is an important open problem. We show that, other than a small set of non-additive codes that we cannot rule out, no binary QECC can transversally implement a classical reversible universal gate set. In particular, no such QECC can implement the Toffoli gate transversally.}{We prove our result by constructing an information theoretically secure (but inefficient) quantum homomorphic encryption (ITS-QHE) scheme inspired by Ouyang {\em et al.} (arXiv:1508.00938). Homomorphic encryption allows the implementation of certain functions directly on encrypted data, i.e. homomorphically. Our scheme builds on almost any QECC, and implements that code's transversal gate set homomorphically. We observe a restriction imposed by Nayak's bound ({\em FOCS} 1999) on ITS-QHE, implying that any ITS quantum {\em fully} homomorphic scheme (ITS-QFHE) implementing the full set of classical reversible functions must be highly inefficient. While our scheme incurs exponential overhead, any such QECC implementing Toffoli transversally would still violate this lower bound through our scheme.

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Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code

Quantum Information and Computation ◽

10.26421/qic12.11-12-10 ◽

2012 ◽

Vol 12 (11&12) ◽

pp. 1034-1080

Author(s):

Adam Paetznick ◽

Ben W. Reichardt

Keyword(s):

Error Propagation ◽

Fault Tolerant ◽

Error Correcting Code ◽

Block Size ◽

Golay Code ◽

Important Target ◽

Noise Threshold ◽

The Cost ◽

Quantum Error ◽

Logical Qubit

In fault-tolerant quantum computing schemes, the overhead is often dominated by the cost of preparing codewords reliably. This cost generally increases quadratically with the block size of the underlying quantum error-correcting code. In consequence, large codes that are otherwise very efficient have found limited fault-tolerance applications. Fault-tolerant preparation circuits therefore are an important target for optimization. We study the Golay code, a $23$-qubit quantum error-correcting code that protects the logical qubit to a distance of seven. In simulations, even using a na{\"i}ve ancilla preparation procedure, the Golay code is competitive with other codes both in terms of overhead and the tolerable noise threshold. We provide two simplified circuits for fault-tolerant preparation of Golay code-encoded ancillas. The new circuits minimize error propagation, reducing the overhead by roughly a factor of four compared to standard encoding circuits. By adapting the malignant set counting technique to depolarizing noise, we further prove a threshold above $\threshOverlap$ noise per gate.

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