A stabilizer code for uncorrelated errors can correct spatially correlated ones

Abstract

It is shown that errors on qubits caused by spatially correlated noise of quantum Brownian motion can be corrected with a stabilizer code (a quantum error correction code) and recovery operations prepared for uncorrelated noise without modifications. The analysis is made by means of the quantum stochastic Liouville equation approach which has been developed within the canonical operator formalism for dissipative systems called non-equilibrium thermo field dynamics. This approach yields a transparent procedure to derive completely positive maps describing errors on qubits under given statistics of noise.

Non-Equilibrium Thermo Field Dynamics and Its Application to Error-Correction for Spatially Correlated Quantum Errors

“…Similar results hold forÅ Å ' ð1 ' < mÞ. Thus, we obtain [26] Non-Equilibrium Thermo Field Dynamics and Its Application to Error-Correction R R mÊ EðtÞP 0P P 0 ¼ P 0P P 0 ð106Þ up to O À ðtÅ ÅÞ m Á for a positive integer m. Within the present model of spatially correlated noise, we have confirmed through the result (106) that the spatially correlated errors happening on quantum codeword are something which can be dealt with, at each order inÅ Å, by the existing stabilizer code for independent errors, or something which does not contribute to the final state after the correction procedure. This is the reason why spatially correlated errors can be corrected by the same stabilizer code and recovery operation as those for independent errors.…”

“…with the spatially correlated error operator [26] 1 In precise, the equation is the one in the interaction representation.…”

Bath‐Induced Collective Phenomena on Superconducting Qubits: Synchronization, Subradiance, and Entanglement Generation

“…[27] The interest in correlated noise is motivated, for example, by its detrimental effects on the performance of quantum error correction protocols, [28] which must be correspondingly modified. [11,[29][30][31][32][33][34][35][36][37][38][39][40][41][42] A key factor that for this purpose needs to be taken into account is the generation of qubit-qubit entanglement due to the action of the common bath during the evolution. [35,43,44] The use of synchronization and/or subradiance as local signatures of entanglement generation may represent a useful tool for the analysis of these scenarios.…”

Quantum Error-Correction for Spatially Correlated Errors

“…Similar results hold forΠ (1 ≤ < m). Thus, we obtain [15] R mÊ (t)P 0P0 = P 0P0 (25) up to O (tΠ ) m for a positive integer m. Within the present model of spatially correlated noise, we have confirmed through the result (25) that the spatially correlated errors happening on quantum codeword are something which can be dealt with, at each order inΠ , by the existing stabilizer code for independent errors, or something which does not contribute to the final state after the correction procedure. This is the reason why spatially correlated errors can be corrected by the same stabilizer code and recovery operation as those for independent errors.…”