Quantum Error Correction with Only Two Extra Qubits

Noise rates in quantum computing experiments have dropped dramatically, but reliable qubits remain precious. Fault-tolerance schemes with minimal qubit overhead are therefore essential. We introduce fault-tolerant error-correction procedures that use only two extra qubits. The procedures are based on adding “flags” to catch the faults that can lead to correlated errors on the data. They work for various distance-three codes. In particular, our scheme allows one to test the $⟦5,1,3⟧$ code, the smallest error-correcting code, using only seven qubits total. Our techniques also apply to the $⟦7,1,3⟧$ and $⟦15,7,3⟧$ Hamming codes, thus allowing us to protect seven encoded qubits on a device with only 17 physical qubits.

Figure 1

Flagged syndrome extraction for the $⟦5,1,3⟧$ code. (a) Stabilizers and logical operators. (b) Circuit to extract the syndrome of the $XZZXI$ stabilizer into an extra qubit. This is not fault tolerant because a fault on the extra qubit can spread to a weight-two error on the data. (c) This circuit also extracts the $XZZXI$ syndrome, and if a single fault spreads to a data error of weight $\ge 2$, then the $X$ measurement will return $|-⟩$, flagging the failure. (d) The nontrivial data errors that can result from a single gate failure in (c) that triggers the flag; these errors are distinguishable by their syndromes and so can be corrected.

Figure 2

Flagged syndrome extraction for Hamming codes. (a) $⟦15,7,3⟧$ code parity checks. (b) Flagged circuit to extract the $Z_{\{8,\dots ,15\}}$ syndrome fault tolerantly.

Figure 3

(a) Stabilizers for an $⟦8,3,3⟧$ code. (b) Stabilizers for a $⟦10,4,3⟧$ code, based on the $⟦5,1,3⟧$ code [27]. (c) Stabilizers for an $⟦11,5,3⟧$ code.

Figure 4

(a) Circuit to extract the $Z^{\otimes n}$ syndrome. (b) Adding a flag makes it fault tolerant.

Figure 5

Comparison to previous error-correction schemes. 2-qubit error correction (orange circle) slightly outperforms Shor-style correction (green triangle) for the $⟦5,1,3⟧$ code and both Shor- and Steane-style (gray square) error correction for the $⟦7,1,3⟧$ code. For the $⟦15,7,3⟧$ code, Steane’s method performs best. Logical error rates are plotted divided by $p^2$ to reveal leading-order coefficients, and $p$, $3p$, $4p$, and $7p$, i.e., error rates without encoding, are plotted to help judge pseudothresholds [31]. 2-qubit error correction for the $⟦8,3,3⟧$ (brown circle) and $⟦10,4,3⟧$ (blue circle) codes can achieve higher pseudothresholds.

Figure 6

(a) Overlapping flags can closely localize $Z$ faults. (b), (c) Extracting multiple syndromes with a shared flag.