Probabilities of Failure for Quantum Error Correction

Xavier Waintal

A quantum error correction (QEC) code uses $N_{\rm c}$ quantum bits to construct one quantum bits of better quality than the original ones. QEC theory predicts that the failure probability $p_L$ of logical qubits decreases exponentially with $N_{\rm c}$ provided the failure probability $p$ of the physical qubit is below a certain threshold $p<p_{\rm th}$. In particular QEC theorems imply that the logical qubits can be made arbitrarily precise by simply increasing $N_{\rm c}$. In this letter, we search for physical mechanisms that lie outside of the hypothesis of QEC theorems and set a limit $\eta_{\rm L}$ to the precision of the logical qubits (irrespectively of $N_{\rm c}$). $\eta_{\rm L}$ directly controls the maximum number of operations $\propto 1/\eta_{\rm L}^2$ 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 $\eta_{\rm L}$ 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 $N_c$.

Wen-Ching Li

Lorenzo Valentini, Diego Forlivesi, Marco Chiani

Quantum error correction (QEC) is essential for realizing scalable quantum computation. However, when evaluating its benefits, most analyses assume idealized components, overlooking the imperfections inherent in realistic fault-tolerant (FT) implementations. In this paper, we investigate the performance of QEC schemes taking into account that quantum gates and measurements are themselves error-prone. We derive bounds for the failure probability of Shor-style FT-QEC schemes using limited structural information, such as the number of flag qubits and quantum gates. Our analysis separates and quantifies two key contributors to the failure rate: decoding errors and residual errors arising from circuit-level faults. The derived bounds highlight fundamental limitations in Shor-style FT-QEC performance and quantify how circuit imperfections degrade error correction capabilities, under the assumption of depolarizing noise.

John Chiaverini, Dietrich G. Leibfried, Tobias Schaetz +9 more

Scott, A J

We investigate the integrity of a quantum error-correcting code when pushed beyond its intended capacity, presenting formulae for the probability of failure when the errors affect more qudits than that specified by the code's minimum distance. We consider both error detection and error correction. Explicit examples are treated in the case of stabilizer codes constructed from qubits and encoding a single qubit.