Error Correction for Continuous Quantum Variables

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 spin2 systems (in terms of qubits or quantum bits) [2–7]. Some of these results have been generalized to higher-spin systems [8–11]. 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 the error syndromes and discuss potential implementation of the scheme. This paper is restricted to one-dimensional wave packets which might represent the wave function of a nonrelativistic onedimensional particle or the state of a single polarization of a transverse mode of electromagnetic radiation. We shall henceforth refer to such descriptions by the generic term wave packets [14]. Rather than starting from scratch we shall use some of the theory that has already been given for error correction on qubits. In particular, Steane has noted that the Hadamard transform,