Quantum Error Correction of Time-correlated Errors

The complexity of the error correction circuitry forces us to design quantum error correction codes capable of correcting a single error per error correction cycle. Yet, time-correlated error are common for physical implementations of quantum systems; an error corrected during the previous cycle may reoccur in the next cycle due to physical processes specific for each physical implementation of the qubits. In this paper we study quantum error correction for a restricted class of time-correlated errors in a spin-boson model. The algorithm we propose allows the correction of two errors per error correction cycle, provided that one of them time-correlated. The algorithm can be applied to any quantum error correcting code when the two logical qubits | 0L〉 and | 1L〉 are entangled states of a 2 n basis states in H2n . 1 Quantum Error Correction Since the early days of computing, reliability has been a major concern. Knowing that quantum states are subject to decoherence, the question whether a reliable quantum computer could be built was asked early on. A “pure state” | φ〉 = α0 | 0〉 + α1 | 1〉 may be transformed as a result of the interaction with the environment into a “mixed state” with density matrix: ρ =| α0 | | 0〉〈0 | + | α1 | | 1〉〈1 | . Other forms of decoherence, e.g. leakage may affect the state probability amplitude as well. The initial thought was that a quantum computation could only be carried out successfully if its duration is shorter than the decoherence time of the quantum computer. As we shall see in Section 2, the decoherence time ranges from about 10 seconds for the nuclear spin embodiment of a qubit, to 10 seconds for quantum dots based upon charge. Thus, it seemed very problematic that a quantum computer could be built unless we have a mechanism to deal periodically with errors. Now we know [19] that quantum error correcting codes can be used to ensure fault-tolerant quantum computing; quantum error correction allows us to deal algorithmically with decoherence. There is a significant price to pay to achieve fault-tolerance through error correction: the number of qubits required to correct errors could be several orders of magnitude larger than the number of “useful” qubits [7]. In 1996, Shor [19] showed how to perform reliable quantum computations when the probability of a qubit or quantum gate error decays polylogarithmically with the size of the computation, a rather unrealistic assumption. The “quantum threshold theorem” ensures that that arbitrary long computations can be carried out with high reliability provided that error rate is below an accuracy threshold according to Knill, Laflamme, and Zurek [12]. In 1999, Aharonow and Ben-Or [1] proved that reliable computing is possible when the error rate is smaller than a constant threshold, but the cost is polylogarithmic in time and space. In practice, error correction is successful for a quantum