Error Correction for Continuous Quantum Variables

Abstract

We propose an error correction coding algorithm for continuous quantum variables. We use this algorithm to construct a highly efficient 5-wave-packet code which can correct arbitrary single wave-packet errors. We show that this class of continuous variable codes is robust against imprecision in the error syndromes. A potential implementation of the scheme is presented. Quantum computers hold the promise for efficiently factoring large integers [1]. However, to do this beyond a most modest scale they will require quantum error correction [2]. The theory of quantum error correction is already well studied in two-level or spin-! systems (in terms of qubits or quantum bits) [2, 3, 4, 5, 6, 7]. Some of these results have been generalized to higher-spin systems [8, 9, 10, 11]. This work applies to discrete systems like the hyperfine levels in ions but is not suitable for systems with continuous spectra, such as unbound wave packets. Simultaneously with this paper, Lloyd and Slotine present the first treatment of a quantum error correction code for continuous quantum variables [12], demonstrating a 9-wave-packet code in analogy with Shor's 9-qubit coding scheme [2]. Such codes hold exciting prospects for the complete manipulation of quantum systems, including both discrete and continuous degrees of freedom, in the presence of inevitable noise [13]. In this Letter we consider a highly efficient and compact error correction coding algorithm for continuous quantum variables. As an example, we construct a 5-wave-packet code which can correct arbitrary single-wave-packet errors. We show that such continuous variable codes are robust against imprecision in * s.