What determines the ultimate precision of a quantum computer?

Abstract

A quantum error correction (QEC) code uses $N_{\mathrm{c}}$ quantum bits to construct one "logical" quantum bits 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 $N_{\mathrm{c}}$. In this letter, 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 $N_{\mathrm{c}}$). ${\eta }_{\mathrm{L}}$ 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 $N_c$.

Published version. New discussion added at the end

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