arXiv subjects

Pradeep Kiran Sarvepalli

Publications and source records attributed to Pradeep Kiran Sarvepalli.

33 records · Page 2Linked to original sources

arXiv2008

Quantum stabilizer codes and beyond

The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. Firstly, it extends the framework of an important class of quantum codes -- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work establishes a close link between subsystem codes and classical codes showing that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels.

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arXiv2008

Encoding Subsystem Codes

In this paper we investigate the encoding of operator quantum error correcting codes i.e. subsystem codes. We show that encoding of subsystem codes can be reduced to encoding of a related stabilizer code making it possible to use all the known results on encoding of stabilizer codes. Along the way we also show how Clifford codes can be encoded. We also show that gauge qubits can be exploited to reduce the encoding complexity.

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arXiv2008

Asymmetric Quantum LDPC Codes

Recently, quantum error-correcting codes were proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit flip and phase flip errors. An example for a channel which exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit flips and phase flips can be related to relaxation and dephasing time, respectively. We give systematic constructions of asymmetric quantum stabilizer codes that exploit this asymmetry. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.

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arXiv2007

Asymptotics of the quantum Hamming bound for subsystem codes

Ashikhmin and Litsyn showed that all binary stabilizer codes - pure or impure - of sufficiently large length obey the quantum Hamming bound, ruling out the possibility that impure codes of large length can outperform pure codes with respect to sphere packing. In contrast we show that impure subsystem codes do not obey the quantum Hamming bound for pure subsystem codes, not even asymptotically. We show that there exist arbitrarily long Bacon-Shor codes that violate the quantum Hamming bound.

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arXiv2007

Quantum Convolutional BCH Codes

Quantum convolutional codes can be used to protect a sequence of qubits of arbitrary length against decoherence. We introduce two new families of quantum convolutional codes. Our construction is based on an algebraic method which allows to construct classical convolutional codes from block codes, in particular BCH codes. These codes have the property that they contain their Euclidean, respectively Hermitian, dual codes. Hence, they can be used to define quantum convolutional codes by the stabilizer code construction. We compute BCH-like bounds on the free distances which can be controlled as in the case of block codes, and establish that the codes have non-catastrophic encoders.

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arXiv2007

On Subsystem Codes Beating the Hamming or Singleton Bound

Subsystem codes are a generalization of noiseless subsystems, decoherence free subspaces, and quantum error-correcting codes. We prove a Singleton bound for GF(q)-linear subsystem codes. It follows that no subsystem code over a prime field can beat the Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the Hamming bound. A number of open problems concern the comparison in performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal MDS stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.

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arXiv2007

Quantum Convolutional Codes Derived From Reed-Solomon and Reed-Muller Codes

Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. Two families of quantum convolutional codes are derived from generalized Reed-Solomon codes and from Reed- Muller codes. A Singleton bound for pure convolutional stabilizer codes is given.

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arXiv2007

Duadic Group Algebra Codes

Duadic group algebra codes are a generalization of quadratic residue codes. This paper settles an open problem raised by Zhu concerning the existence of duadic group algebra codes. These codes can be used to construct degenerate quantum stabilizer codes that have the nice feature that many errors of small weight do not need error correction; this fact is illustrated by an example.

cs.IT↗

arXiv2006

Clifford Code Constructions of Operator Quantum Error Correcting Codes

Recently, operator quantum error-correcting codes have been proposed to unify and generalize decoherence free subspaces, noiseless subsystems, and quantum error-correcting codes. This note introduces a natural construction of such codes in terms of Clifford codes, an elegant generalization of stabilizer codes due to Knill. Character-theoretic methods are used to derive a simple method to construct operator quantum error-correcting codes from any classical additive code over a finite field.

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arXiv2006

Subsystem Codes

We investigate various aspects of operator quantum error-correcting codes or, as we prefer to call them, subsystem codes. We give various methods to derive subsystem codes from classical codes. We give a proof for the existence of subsystem codes using a counting argument similar to the quantum Gilbert-Varshamov bound. We derive linear programming bounds and other upper bounds. We answer the question whether or not there exist [[n,n-2d+2,r>0,d]] q subsystem codes. Finally, we compare stabilizer and subsystem codes with respect to the required number of syndrome qudits.

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arXiv2006

On Quantum and Classical BCH Codes

Classical BCH codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of length n can contain its dual code only if its designed distance d=O(sqrt(n)), and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and - consequently - the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters.

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arXiv2006

Primitive Quantum BCH Codes over Finite Fields

An attractive feature of BCH codes is that one can infer valuable information from their design parameters (length, size of the finite field, and designed distance), such as bounds on the minimum distance and dimension of the code. In this paper, it is shown that one can also deduce from the design parameters whether or not a primitive, narrow-sense BCH contains its Euclidean or Hermitian dual code. This information is invaluable in the construction of quantum BCH codes. A new proof is provided for the dimension of BCH codes with small designed distance, and simple bounds on the minimum distance of such codes and their duals are derived as a consequence. These results allow us to derive the parameters of two families of primitive quantum BCH codes as a function of their design parameters.

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arXiv2006

Remarkable Degenerate Quantum Stabilizer Codes Derived from Duadic Codes

Good quantum codes, such as quantum MDS codes, are typically nondegenerate, meaning that errors of small weight require active error-correction, which is--paradoxically--itself prone to errors. Decoherence free subspaces, on the other hand, do not require active error correction, but perform poorly in terms of minimum distance. In this paper, examples of degenerate quantum codes are constructed that have better minimum distance than decoherence free subspaces and allow some errors of small weight that do not require active error correction. In particular, two new families of [[n,1,>= sqrt(n)]]_q degenerate quantum codes are derived from classical duadic codes.

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arXiv2005

Nonbinary stabilizer codes over finite fields

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over GF(q) in terms of classical codes over GF(q^2) is provided that generalizes the well-known notion of additive codes over GF(4) of the binary case. This paper derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum BCH codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.

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arXiv2005

Nonbinary Quantum Reed-Muller Codes

We construct nonbinary quantum codes from classical generalized Reed-Muller codes and derive the conditions under which these quantum codes can be punctured. We provide a partial answer to a question raised by Grassl, Beth and Roetteler on the existence of q-ary quantum MDS codes of length n with q\le n\le q^2-1.

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