The problem of finding quantum-error-correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.

It has recently been realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties in realizing quantum computation is that decoherence tends to destroy the information in a superposition of states in a quantum computer, making long computations impossible. A further difficulty is that inaccuracies in quantum state transformations throughout the computation accumulate, rendering long computations unreliable. However, these obstacles may not be as formidable as originally believed. For any quantum computation with t gates, we show how to build a polynomial size quantum circuit that tolerates O(1 / log c t) amounts of inaccuracy and decoherence per gate, for some constant c; the previous bound was O(1 /t). We do this by showing that operations can be performed on quantum data encoded by quantum error-correcting codes without decoding this data. 1.

The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. Hence, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per quantum gate is less than a certain critical value, the accuracy threshold. A quantum computer storing about 106 qubits, with a probability of error per quantum gate of order 10-6, would be a formidable factoring engine. Even a smaller less-accurate quantum computer would be able to perform many useful tasks. This paper is based on a talk presented at the ITP Conference on Quantum Coherence

Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate w...

Since Shor’s discovery of an algorithm to factor numbers on a quantum computer in polynomial time, quantum computation has become a subject of immense interest. Unfortunately, one of the key features of quantum computers — the difficulty of describing them on classical computers — also makes it difficult to describe and understand precisely what can be done with them. A formalism describing the evolution of operators rather than states has proven extremely fruitful in understanding an important class of quantum operations. States used in error correction and certain communication protocols can be described by their stabilizer, a group of tensor products of Pauli matrices. Even this simple group structure is sufficient to allow a rich range of quantum effects, although it falls short of the full power of quantum computation. 1

We define and show how to construct nonbinary quantum stabilizer codes. Our approach is based on nonbinary error bases. It generalizes the relationship between selforthogonal codes over F4 and binary quantum codes to one between selforthogonal codes over Fq2 and q-ary quantum codes for any prime power q. Index Terms — quantum stabilizer codes, nonbinary quantum codes, selforthogonal codes. 1

Error operator bases for systems of any dimension are defined and natural generalizations of the bit/sign flip error basis for qubits are given. These bases allow generalizing the construction of quantum codes based on eigenspaces of Abelian groups. As a consequence, quantum codes can be constructed from linear codes over Zn for any n. The generalization of the punctured code construction leads to many codes which permit transversal (i.e. fault tolerant) implementations of certain operations compatible with the error basis. Note: This report is preliminary. Please contact the author if you wish to be notified of updates. A continuation can be found in "Group representations, error bases and quantum codes" [5]. 1 Overview Quantum error correction on products of qubits is increasingly well understood [10, 12, 3, 1, 6, 2]. See the references and the many papers on quantum computation and error correction on http://xxx.lanl.gov/ in the quant-ph section for background information on the s...

This report continues the discussion of unitary error bases and quantum codes begun in [8]. Nice error bases are characterized in terms of the existence of certain characters in a group. A general construction for error bases which are non-abelian over the center is given. The method for obtaining codes due to Calderbank et al. [2] is generalized and expressed purely in representation theoretic terms. The significance of the inertia subgroup both for constructing codes and obtaining the set of transversally implementable operations is demonstrated. Note: This report is preliminary. Please contact the author if you wish to be notified of updates. 1 Overview This report discusses the construction of quantum codes based on nice error bases [8]. The main conclusion is that much of the relevant theory can be cast in terms of representations of finite groups. It is shown that nice error bases are equivalent to the existence of an irreducible character with non-zero values only on the cent...

The Gottesman-Knill theorem says that a stabilizer circuit—that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates—can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. • By removing the need for Gaussian elimination, we make the simulation algorithm much faster at the cost of a factor-2 increase in the number of bits needed to represent a state. We have implemented the improved algorithm in a freely-available program called CHP (CNOT-Hadamard-Phase), which can handle thousands of qubits easily. • We show that the problem of simulating stabilizer circuits is complete for the classical complexity class ⊕L, which means that stabilizer circuits are probably not even universal for classical computation. • We give efficient algorithms for computing the inner product between two stabilizer states, putting any n-qubit stabilizer circuit into a “canonical form ” that requires at most O ( n 2 /log n) gates, and other useful tasks. • We extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of non-stabilizer gates, and circuits acting on general tensor-product initial states but containing only a limited number of measurements. 1

Abstract — We show the equivalence of two different notions of quantum channel capacity: that which uses the entanglement fidelity as its criterion for success in transmission, and that which uses the minimum fidelity of pure states in a subspace of the input Hilbert space as its criterion. As a corollary, any source with entropy less than the capacity may be transmitted with high entanglement fidelity. We also show that a restricted class of encodings is sufficient to transmit any quantum source which may be transmitted on a given channel. This enables us to simplify a known upper bound for the channel capacity. It also enables us to show that the availability of an auxiliary classical channel from encoder to decoder does not increase the quantum capacity.