Entanglement-Assisted Quantum
Error-Correcting Codes
Igor Devetak, Todd A. Brun and Min-Hsiu Hsieh
Abstract We develop the theory of entanglement-assisted quantum error correcting codes (EAQECCs), a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to pre-shared entanglement. Conventional stabilizer codes are equivalent to self-orthogonal symplectic codes. In contrast, EAQECCs do not require self-orthogonality, which greatly simplifies their construction. We show how any classical quaternary block code can be made into a EAQECC. Furthermore, the error-correcting power of the quantum codes follows directly from the power of the classical codes.
1 Introduction
One of the most important discoveries in quantum information science was the existence of quantum error correcting codes (QECCs) in 1995 [16]. Up to that point, there was a widespread belief that decoherence–environmental noise–would doom any chance of building large scale quantum computers or quantum communication protocols [18], and the equally widespread belief that any analogue of classical error correction was impossible in quantum mechanics. The discovery of quantum
Igor Devetak
Ming Hsieh Electrical Engineering Department, University of Southern California, Los Angeles, CA 90089, USA, e-mail: devetak@usc.edu
Todd A. Brun
Communication Sciences Institute, Department of Electrical Engineering Systems, University of Southern California, 3740 McClintock Ave., EEB 502, Los Angeles, CA 90089-2565 USA, e-mail: tbrun@usc.edu
Min-Hsiu Hsieh
Ming Hsieh Electrical Engineering Department, University of Southern California, Los Angeles, CA 90089, USA, e-mail: minhsiuh@gmail.com
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Igor Devetak, Todd A. Brun and Min-Hsiu Hsieh |
error correcting codes defied these expectations, and were quickly developed into the theory of stabilizer codes [5, 10]. Moreover, a construction of Calderbank, Shor and Steane [4, 17] showed that it was possible to construct quantum stabilizer codes from classical linear codes–the CSS codes–thereby drawing on the well-studied theory of classical error correction.
Important as these results were, they fell short of doing everything that one might wish. The connection between classical codes and quantum codes was not universal. Rather, only classical codes which satisfied a dual-containing constraint could be used to construct quantum codes. While this constraint was not too difficult to satisfy for relatively small codes, it is a substantial barrier to the use of highly efficient modern codes, such as Turbo codes [1] and Low-Density Parity Check (LDPC) codes [8], in quantum information theory. These codes are capable of achieving the classical capacity; but the difficulty of constructing dual-containing versions of them has made progress toward finding quantum versions very slow.
The results of our entanglement-assisted stabilizer formalism generalize the existing theory of quantum error correction. If the CSS construction for quantum codes is applied to a classical code which is not dual-containing, the resulting “stabilizer” group is not commuting, and thus has no code space. We are able to avoid this problem by making use of pre-existing entanglement. This noncommuting stabilizer group can be embedded in a larger space, which makes the group commute, and allows a code space to be defined. Moreover, this construction can be applied to any classical quaternary code, not just dual-containing ones. The existing theory of quantum error correcting codes thus becomes a special case of our theory: dual-containing classical codes give rise to standard quantum codes, while non- dual-containing classical codes give rise to entanglement-assisted quantum error correction codes (EAQECCs).
2 Notations
Consider an n-qubit system corresponding to the tensor product Hilbert space H 2 n. Define an n-qubit Pauli matrix S to be of the form S = S1 S2 · · · Sn, where Sj {I, X, Y, Z} is an element of the set of Pauli matrices. Let G n be the group of all 4n n-qubit Pauli matrices with all possible phases. Define the equivalence class
[S] = {βS | β C, |β| = 1}. Then
[S][T] = [S1T1] [S2T2] · · · [SnTn] = [ST].
Thus the set [G n] = {[S] : S G n} is a commutative multiplicative group.
Now consider the group/vector space (Z2)2n of binary vectors of length 2n. Its elements may be written as u = (z|x), z = z1 . . . zn (Z2)n, x = x1 . . . xn (Z2)n. We shall think of u, z and x as row vectors. The symplectic product of u = (z|x) and v = (z |x ) is given by