ar X iv : q ua nt - p h / 05 01 18 4 v 1 31 J an 2 00 5 Loss Tolerant Optical Qubits

Quantum logic gates can be built using linear optics, photon detection and ancillary resources in a scalable manner, as shown by Knill, Laflamme and Milburn (KLM) [1]. A number of experimental efforts are currently focused on testing the building blocks of linear optical quantum computing (LOQC) [2, 3, 4]. However, optimism for large scale quantum computation based on LOQC has been tempered by the major overheads inherent in the KLM scheme and the high detector and source efficiencies apparently required [1]. An alternative approach to implementing LOQC was proposed by Nielsen [5] and further developed by Browne and Rudolph [6] (see also [7] for related work). This approach combines the model of cluster-state quantum computation [8] with the non-deterministic gates presented by KLM, and achieves a very significant reduction in the overheads. The fault tolerance of the scheme has also been studied [9]. In this paper we present a new approach to LOQC based on an incremental parity encoding [10]. Our method combines ideas from both the KLM and the cluster-state approaches. Parity encoding was used in the original KLM proposal to protect against both teleporter failures (i.e. the non-determinism of the gates) and photon loss. By using parity encoding but re-encoding incrementally (instead of by concatenation) we can obtain the reduction in overheads characteristic of the cluster state approach whilst retaining the photon loss tolerance of KLM. In particular we will describe a quantum memory which is fault tolerant [11] to photon loss. Though our techniques for detecting and correcting loss are themselves themselves also subject to loss, above a particular threshold efficiency the effect of loss can be negated to arbitrary accuracy. A previous description of an optical quantum memory based on error correction did not consider fault tolerance [12]. Although we specifically only consider memory, our construction is compatible with gate operations and thus can form a template for fault tolerant quantum computation with respect to photon loss. We will deal with qubits in three different tiers of encoding: physical encoding, parity encoding and redundant encoding. The application we describe in this letter operates at the redundant-encoding level to protect information from photon loss. Physical encoding: At the first tier are the basic physical states that we will use to construct qubits, these will be the polarisation states of a photon so that |0〉 ≡ |H〉 and |1〉 ≡ |V 〉. The advantage of this choice in optics, is that we can perform any single physicalqubit unitary deterministically with passive linear optical elements. Of course gates between different physical qubits become difficult and in LOQC these are nondeterministic. Parity encoding: at the second tier of encoding are parity qubits encoded across many physical qubits. We shall use the notation |ψ〉(n) to mean the logical state |ψ〉 of a qubit, which is parity encoded across n physical qubits. In this notation the physical qubits are the first level, and we will often drop the superscript for this level as was done above. Specifically, the parity encoding is given by