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IEEE Transactions on Information Theory

Beyond stabilizer codes .I. Nice error bases

Andreas Klappenecker, Martin Rötteler

Published Aug 1, 2002• 81 citations
Publisher

Abstract

Nice error bases have been introduced by Knill (1996) as a generalization of the Pauli basis. These bases are shown to be projective representations of finite groups. We classify all nice error bases of small degree, and all nice error bases with Abelian index groups. We show that, in general, an index group of a nice error basis is necessarily solvable.

DOI
10.1109/tit.2002.800471

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Nice error bases are a fundamental primitive of quantum information processing. For example, they govern the discretization of errors in quantum error-correcting codes. It is show that the generalized Pauli basis, the most widely used example of nice error bases, has some remarkable structural properties. However, the generalized Pauli basis is limited to dimensions that are a power of a prime, since it is constructed with the help of a finite field. A wider class of nice error bases is introduced that shares many features of the generalized Pauli basis, yet allows one to remove the restriction to prime power dimensions. The nice error bases are indexed by nearrings. Nearrings that support the construction of nice error bases are called nice. It is shown that all finite nearfields are nice. It is shown that a finite ring is nice if and only if it finite Frobenius ring. Several fundamental properties of nice nearrings are established.

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Nice error bases have been introduced by Knill (1996) as a generalization of the Pauli basis. These bases are shown to be projective representations of finite groups. We classify all nice error bases of small degree, and all nice error bases with Abelian index groups. We show that, in general, an index group of a nice error basis is necessarily solvable.
Unitary error bases generalize the Pauli matrices to higher dimensional systems. Two basic constructions of unitary error bases are known: An algebraic construction by Knill, which yields nice error bases, and a combinatorial construction by Werner, which yields shift-and-multiply bases. An open problem posed by Schlingemann and Werner (see http://www.imaph.tu-bs.de/qi/problems/6.html) relates these two constructions and asks whether each nice error basis is equivalent to a shift-and-multiply basis. We solve this problem and show that the answer is negative. However, we also show that it is always possible to find a fairly sparse representation of a nice error basis.
Nice error bases have various applications in quantum computing such as in the teleportation of quantum states and in the theory of noiseless systems. Associated in a natural way to any nice error basis there is a so-called index group and also a group of central type. In this paper, we will extend the number of known index groups for nice error bases by exploiting the connections between groups of central type and their Sylow subgroups. Our methods will allow us find all index groups of degree up to 44 with two exceptions.
Nice error bases are a fundamental primitive of quantum information processing. For example, they govern the discretization of errors in quantum error-correcting codes. It is show that the generalized Pauli basis, the most widely used example of nice error bases, has some remarkable structural properties. However, the generalized Pauli basis is limited to dimensions that are a power of a prime, since it is constructed with the help of a finite field. A wider class of nice error bases is introduced that shares many features of the generalized Pauli basis, yet allows one to remove the restriction to prime power dimensions. The nice error bases are indexed by nearrings. Nearrings that support the construction of nice error bases are called nice. It is shown that all finite nearfields are nice. It is shown that a finite ring is nice if and only if it finite Frobenius ring. Several fundamental properties of nice nearrings are established.