What determines the ultimate precision of a quantum computer

A quantum error correction (QEC) code uses $N_{\mathrm{c}}$ quantum bits to construct one “logical” quantum bit of better quality than the original “physical” ones. QEC theory predicts that the failure probability $p_L$ of logical qubits decreases exponentially with $N_{\mathrm{c}}$ provided the failure probability $p$ of the physical qubit is below a certain threshold $p<p_{\mathrm{th}}$. In particular, QEC theorems imply that the logical qubits can be made arbitrarily precise by simply increasing $N_c$. In this article, we search for physical mechanisms that lie outside of the hypothesis of QEC theorems and set a limit ${\eta }_{\mathrm{L}}$ to the precision of the logical qubits (irrespectively of $N_c$). ${\eta }_{\mathrm{L}}$ directly controls the maximum number of operations $\propto 1/{\eta }_{\mathrm{L}}^2$ that can be performed before the logical quantum state gets randomized, hence the depth of the quantum circuits that can be considered. We identify a type of error—silent stabilizer failure—as a mechanism responsible for finite ${\eta }_{\mathrm{L}}$ and discuss its possible causes. Using the example of the topological surface code, we show that a single local event can provoke the failure of the logical qubit, irrespectively of $N_c$.

Figure 1

Schematic of a $N_c=9$ repetition code (a) and a $N_c=72$ surface code (b); the small yellow circles indicate the data qubits. The larger circles with an X or a Z embedded indicate ancilla qubits that are used to measure the stabilizers. The X (Z) ancillas are used to measure the stabilizer $X_N{X}_E{X}_S{X}_W$ ($Z_N{Z}_E{Z}_S{Z}_W$) where $N,E,S,W$ indicate the data qubit situated north, east, south, and west with respect to the position of the ancilla. (c, d) Schematic of the circuit used to measure the X and Z type of stabilizers. The different pictograms indicate $|0\rangle$: initialization in state $|0\rangle$. H: Hadamard gate $H=(X+Z)/\sqrt{2}$. $M_Z$: measurement of the qubit along the $z$ axis. Small black circle linked to an X: Control-Not 2 qubits gate $C(1,2)=(1+Z_1+X_2-Z_1{X}_2)/2$, which flips the second qubit (X) if the first one is in state $|1\rangle$.

Figure 2

Silent stabilizer failure. In this surface code, the logical qubits A and B are created by making two large holes (per qubit) in the code. A hole simply corresponds to not measuring the corresponding stabilizers. A Control-Not gate between A and B is performed by braiding the lower hole of A between the two holes of B as indicated by the blue arrow. The failure of a single stabilizer (small hole on the left of the upper hole of qubit B) creates a silent logical qubit. The dashed lines enclose the data qubits that enter in the definition of different logical operators; see text.