Fault-tolerant thresholds for encoded ancillae with homogeneous errors

I describe a procedure for calculating thresholds for quantum computation as a function of error model given the availability of ancillae prepared in logical states with independent, identically distributed errors. The thresholds are determined via a simple counting argument performed on a single qubit of an infinitely large Calderbank-Shor-Steane code. I give concrete examples of thresholds thus achievable for both Steane and Knill style fault-tolerant implementations and investigate their relation to threshold estimates in the literature.

Figure 1

A circuit showing the homogeneous application of several encoded gates for the seven qubit code. The inset on the right shows a single strand which has been extracted from the coding blocks. Note that, modulo a label indicating the ancilla’s starting state, the strand is identical to the encoded circuit.

Figure 2

Examples of Pauli propagation. Equalities are up to an overall phase.

Figure 3

Encoded circuits for the single-coupling Steane procedure. Handling of the data is minimized at the cost of syndrome verification. (a) and (b) display the circuits for finding $X$ and $Z$ errors, respectively. (c) lists the circuits analyzed to determine the encoded error rates for this example.

Figure 4

Encoded circuits for the double-coupling Steane procedure. Syndrome information is extracted twice, and qubits implicated both times are presumed to be in error. (a) and (b) display the circuits for finding $X$ and $Z$ errors, respectively. (c) lists the circuits analyzed to determine the encoded error rates for this example.

Figure 5

Encoded circuits for the Knill procedure. Error correction is performed by teleporting the data, minus the errors, using an entangled two-logical-qubit ancilla; the precise location of errors is unimportant so long as the result of the logical-qubit measurement is correctly decoded. (a) and (b) display the circuits for correcting errors when the output of the previous step was dominated by $Z$ and $X$ errors, respectively. (c) lists the circuits analyzed to determine the encoded error rates for this example. Ancilla error distributions are used as the input to these circuits since that is the only remaining source of error after a successful teleportation correction.