Scalable quantum computing in the presence of large detected-error rates

The theoretically tolerable erasure error rate for scalable quantum computing is shown to be well above 0.1, given standard scalability assumptions. This bound is obtained by implementing computations with generic stabilizer code teleportation steps that combine the necessary operations with error correction. An interesting consequence of the technique is that the only errors that affect the maximum tolerable error rate are storage and Bell measurement errors. If storage errors are negligible, then any detected Bell measurement error below is permissible. For practical computation with high detected error rates, the implementation overheads need to be improved.

Figure 1

Teleporting with an encoded entangled state is equivalent to a syndrome measurement. The gray lines are the time lines of blocks of $n$ qubits. The boxes denote various operations. The Bell-state preparation on corresponding pairs of qubits in two blocks is depicted with a box angled to the right and labeled “Bell.” The state used for teleportation in the top diagram is obtained after Bell-state preparation by projecting one of the blocks with $\Pi (Q,0)$. (The actual preparation procedure is different but has the same output.) Projection operators are shown with boxes angled both ways with the operator written in the box. Bell measurement of corresponding pairs of qubits in two blocks is depicted with a box angled to the left and labeled “Bell.” A Bell measurement on qubits 1 and 2 may be implemented by applying a CNOT from qubit 1 to 2, performing an ${\sigma }_x$ measurement on qubit 1 and a ${\sigma }_z$ measurement on qubit 2. The top diagram is the actual network implemented. The other two are logically equivalent. The Bell measurement outcome $\mathbf{g}$ is correlated with the effective projection in the bottom diagram. If the input state has a particular syndrome, then only $\mathbf{g}$ for which the projection is onto the subspace with this syndrome have nonzero probabilities.

Figure 2

Sequence of computational steps. Each step consists of a teleportation with integrated operations and error correction. The steps required for correction determine the prepared state used in the next step. That is, a state that has the appropriate operations preapplied to the destination qubit is used. Any such state is assumed to be available at the “state preparation factory” at the next step. Zero communication and classical computation delays are assumed. In particular the classical feed forward following the Bell measurement and selection of the desired prepared state take no additional time. The state preparations are implemented so that the states are ready at selection time. The prepared states have been checked for errors, with no errors detected just before they are used. Errors in the computation are therefore due only to the Bell measurement and the storage time required for the second block of each prepared state. The storage time is the duration of the Bell measurement.

Figure 3

Region for which scalable computing with the detected-error model is possible. Tolerable error rates are in the gray region, strictly below the upper boundary. If memory and Bell measurement error are equal, then any error rate below 0.292 is tolerable. The point on the boundary corresponding to this value is shown. If memory errors are negligible, then Bell-measurement error rates close to $1∕2$ are tolerable in principle.