65 research outputs found
Entanglement-assisted codeword stabilized quantum codes with imperfect ebits
Quantum error correcting codes (QECCs) in quantum communi- cation systems has
been known to exhibit improved performance with the use of error-free
entanglement bits (ebits). In practical situations, ebits inevitably suffer
from errors, and as a result, the error-correcting capability of the code is
diminished. Prior studies have proposed two different schemes as a solu- tion.
One uses only one QECC to correct errors on the receiver's side (i.e., Bob) and
on the sender's side (i.e., Alice). The other uses different QECCs on each
side. In this paper, we present a method to correct errors on both sides by
using single nonadditive Entanglement-assisted codeword stabilized quantum
error correcting code(EACWS QECC). We use the property that the number of
effective error patterns decreases as much as the number of ebits. This
property results in a greater number of logical codewords using the same number
of physical qubits
The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure
Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing
and protecting fragile qubits against the undesirable effects of quantum
decoherence. Similar to classical codes, hashing bound approaching QECCs may be
designed by exploiting a concatenated code structure, which invokes iterative
decoding. Therefore, in this paper we provide an extensive step-by-step
tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided
concatenated quantum codes based on the underlying quantum-to-classical
isomorphism. These design lessons are then exemplified in the context of our
proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the
outer component of a concatenated quantum code. The proposed QIRCC can be
dynamically adapted to match any given inner code using EXIT charts, hence
achieving a performance close to the hashing bound. It is demonstrated that our
QIRCC-based optimized design is capable of operating within 0.4 dB of the noise
limit
Entanglement-assisted codeword stabilized quantum codes
Entangled qubit can increase the capacity of quantum error correcting codes
based on stabilizer codes. In addition, by using entanglement quantum
stabilizer codes can be construct from classical linear codes that do not
satisfy the dual-containing constraint. We show that it is possible to
construct both additive and non-additive quantum codes using the codeword
stabilized quantum code framework. Nonadditive codes may offer improved
performance over the more common sta- bilizer codes. Like other
entanglement-assisted codes, the encoding procedure acts only the qubits on
Alice's side, and only these qubits are assumed to pass through the channel.
However, errors the codeword stabilized quantum code framework gives rise to
effective Z errors on Bob side. We use this scheme to construct new
entanglement-assisted non-additive quantum codes, in particular, ((5,16,2;1))
and ((7,4,5;4)) codes
Semidefinite programming bounds on the size of entanglement-assisted codeword stabilized quantum codes
In this paper, we explore the application of semidefinite programming to the
realm of quantum codes, specifically focusing on codeword stabilized (CWS)
codes with entanglement assistance. Notably, we utilize the isotropic subgroup
of the CWS group and the set of word operators of a CWS-type quantum code to
derive an upper bound on the minimum distance. Furthermore, this
characterization can be incorporated into the associated distance enumerators,
enabling us to construct semidefinite constraints that lead to SDP bounds on
the minimum distance or size of CWS-type quantum codes. We illustrate several
instances where SDP bounds outperform LP bounds, and there are even cases where
LP fails to yield meaningful results, while SDP consistently provides tight and
relevant bounds. Finally, we also provide interpretations of the Shor-Laflamme
weight enumerators and shadow enumerators for codeword stabilized codes,
enhancing our understanding of quantum codes.Comment: 20 pages, 1 tabl
The Encoding and Decoding Complexities of Entanglement-Assisted Quantum Stabilizer Codes
Quantum error-correcting codes are used to protect quantum information from
decoherence. A raw state is mapped, by an encoding circuit, to a codeword so
that the most likely quantum errors from a noisy quantum channel can be removed
after a decoding process.
A good encoding circuit should have some desired features, such as low depth,
few gates, and so on. In this paper, we show how to practically implement an
encoding circuit of gate complexity for an
quantum stabilizer code with the help of pairs of maximally-entangled
states. For the special case of an stabilizer code with , the
encoding complexity is , which is previously known to be
. For this suggests that the benefits from shared
entanglement come at an additional cost of encoding complexity.
Finally we discuss decoding of entanglement-assisted quantum stabilizer codes
and extend previously known computational hardness results on decoding quantum
stabilizer codes.Comment: accepted by the 2019 IEEE International Symposium on Information
Theory (ISIT2019
Quantum Error Correcting Codes and Fault-Tolerant Quantum Computation over Nice Rings
Quantum error correcting codes play an essential role in protecting quantum information from the noise and the decoherence. Most quantum codes have been constructed based on the Pauli basis indexed by a finite field. With a newly introduced algebraic class called a nice ring, it is possible to construct the quantum codes such that their alphabet sizes are not restricted to powers of a prime.
Subsystem codes are quantum error correcting schemes unifying stabilizer codes, decoherence free subspaces and noiseless subsystems. We show a generalization of subsystem codes over nice rings. Furthermore, we prove that free subsystem codes over a finite chain ring cannot outperform those over a finite field. We also generalize entanglement-assisted quantum error correcting codes to nice rings. With the help of the entanglement, any classical code can be used to derive the corresponding quantum codes, even if such codes are not self-orthogonal. We prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs using symplectic geometry over rings. So, we can find hyperbolic pairs and commuting generators generating the check matrix of the entanglement-assisted quantum code.
Fault-tolerant quantum computation has been also studied over a finite field. Transversal operations are the simplest way to implement fault-tolerant quantum gates. We derive transversal Clifford operations for CSS codes over nice rings, including Fourier transforms, SUM gates, and phase gates. Since transversal operations alone cannot provide a computationally universal set of gates, we add fault-tolerant implementations of doubly-controlled Z gates for triorthogonal stabilizer codes over nice rings.
Finally, we investigate optimal key exchange protocols for unconditionally secure key distribution schemes. We prove how many rounds are needed for the key exchange between any pair of the group on star networks, linear-chain networks, and general networks
High-Rate Quantum Low-Density Parity-Check Codes Assisted by Reliable Qubits
Quantum error correction is an important building block for reliable quantum information processing. A challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error-correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation. A typical obstacle to constructing a variety of strong quantum error-correcting codes is the complicated restrictions imposed on the structure of a code. Recently, promising solutions to this problem have been proposed in quantum information science, where in principle any binary linear code can be turned into a quantum error-correcting code by assuming a small number of reliable quantum bits. This paper studies how best to take advantage of these latest ideas to construct desirable quantum error-correcting codes of very high information rate. Our methods exploit structured high-rate low-density parity-check codes available in the classical domain and provide quantum analogues that inherit their characteristic low decoding complexity and high error correction performance even at moderate code lengths. Our approach to designing high-rate quantum error-correcting codes also allows for making direct use of other major syndrome decoding methods for linear codes, making it possible to deal with a situation where promising quantum analogues of low-density parity-check codes are difficult to find
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