Quantum Error Correction and Fault-Tolerance

I give an overview of the basic concepts behind quantum error correction and quantum fault tolerance. This includes the quantum error correction conditions, stabilizer codes, CSS codes, transversal gates, fault-tolerant error correction, and the threshold theorem. 1 Quantum Error Correction Building a quantum computer or a quantum communications device in the real world means having to deal with errors. Any qubit stored unprotected or one transmitted through a communications channel will inevitably come out at least slightly changed. The theory of quantum error-correcting codes has been developed to counteract noise introduced in this way. By adding extra qubits and carefully encoding the quantum state we wish to protect, a quantum system can be insulated to great extent against errors. To build a quantum computer, we face an even more daunting task: If our quantum gates are imperfect, everything we do will add to the error. The theory of fault-tolerant quantum computation tells us how to perform operations on states encoded in a quantum error-correcting code without compromising the code’s ability to protect against errors. In general, a quantum error-correcting code is a subspace of a Hilbert space designed so that any of a set of possible errors can be corrected by an appropriate quantum operation. Specifically: Definition 1 Let Hn be a 2 -dimensional Hilbert space (n qubits), and let C be a K-dimensional subspace of Hn. Then C is an ((n,K)) (binary) quantum error-correcting code (QECC) correcting the set of errors E = {Ea} iff ∃R s.t. R is a quantum operation and (R ◦ Ea)(|ψ〉) = |ψ〉 for all Ea ∈ E, |ψ〉 ∈ C. R is called the recovery or decoding operation and serves to actually perform the correction of the state. The decoder is sometimes also taken to map Hn into an unencoded Hilbert space HlogK isomorphic to C. This should be distinguished from the encoding operation which maps Hlog K into Hn, determining the imbedding of C. The computational complexity of the encoder is frequently a great deal lower than that of the decoder. In particular, the task of determining what error has occurred can be computationally difficult (NP-hard, in fact), and designing codes with efficient decoding algorithms is an important task in quantum error correction, as in classical error correction. This article will cover only binary quantum codes, built with qubits as registers, but all of the techniques discussed here can be generalized to higherdimensional registers, or qudits. To determine whether a given subspace is able to correct a given set of errors, we can apply the quantum error-correction conditions [2, 7]: Theorem 1 A QECC C corrects the set of errors E iff 〈ψi|E † aEb|ψj〉 = Cabδij , (1) where Ea, Eb ∈ E and {|ψi〉} form an orthonormal basis for C. The salient point in these error-correction conditions is that the matrix element Cab does not depend on the encoded basis states i and j, which roughly speaking indicates that neither the environment nor the decoding operation learns any information about the encoded state. We can imagine the various possible errors taking the subspace C into other subspaces