What determines the ultimate precision of a quantum computer
Xavier Waintal · Physical Review A · 2019
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 ${\ensuremath{\eta}}_{\mathrm{L}}$ to the precision of the logical qubits (irrespectively of ${N}_{c}$). ${\ensuremath{\eta}}_{\mathrm{L}}$ directly controls the maximum number of operations $\ensuremath{\propto}1/{\ensuremath{\eta}}_{\mathrm{L}}^{2}$ 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 ${\ensuremath{\eta}}_{\mathrm{L}}$ 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 ${N}_{c}$.