Reported for amplitude-damping errors in ReimpellW:05. P < 0.5 and diverges thereafter, a phenomenon similar to what was This curve follows the weight-3 optimal recovery for When this recovery is applied to the weight-3Įrrors we get the dashed curve with star markers labeled “optimal
#OPTIMAL RECOVERY QUANTUM ERROR CORRECTION CODE#
Understood from the fact that the code also perfectly corrects Solving either ( 9) for the optimal γ or using theĪpproximation ( 10) and then finding R from “optimal & avg-case, weight-2 errors”) which are found by Optimal for p < 0.5 and then is identical with the optimal for p ≥ 0.5.įor weight-2 errors, perfect fidelity over the entire p-range isĪchieved by the optimal and average-case recovery (labeled Indicated by the dotted curve with triangle markers falls below the We also see that the optimal performance decreases until Solid lines is entirely due to the optimized recovery. Optimal γ using the standard encoding and then finding R from The curve with circle markers labeled “optimal weight-3Įrrors” is for an optimal recovery by solving ( 9) for the Recovery for arbitrary weight-2 or weight-3 errors, which as expected, The solid lines show the code performance with standard In this figure the code isĪlways the standard code and we compare different recoveries. Optimized recovery for weight-2,3 bit-flip errors.-įig. This is equivalent to replacing theĮrror system matrix elements by E e ↦ E β, e / √ ℓ. In the average-case, the objectives can be equivalentlyĮxpressed in the same form but with f a v g and d i n d replaced by The worst-case, these objectives can be replaced by optimizing overĪll E β. Possibilities are the worst-case and average-case. The standard errorĬorrection procedure involves CP encoding ( C), error ( E),Īnd recovery ( R) maps (or channels): ρ S C → ρ C E → σ C R → ^ ρ S, i.e., using the OSR: ^ ρ S = ∑ r, e, c ( R r E e C c ) ρ S ( R r E e C c ) † (see Ref. System density matrix and the A i are called operation elements, and In terms of a completely-positive (CP) map: ρ S → ∑ i A i ρ S A † i, a result known as the Kraus Operator Sum Noise and error correction model.- Subject to standardĪssumptions, the dynamics of any open quantum system can be described Which provides a considerable reduction in computational cost with Show, has the added advantage of incorporating an approximation method More efficient computationaly then the direct approach and, as we will However, the indirect approach is in general For a given recovery, the problem is convex in For a given encoding the problem isĬonvex in the recovery. Semidefinite programs (SDPs) BoydV:04 which can be iteratedīetween recovery and encoding. Naturally to bi-convex optimization problems, specifically, two Both the direct and indirect approaches lead Maximization based on minimizing the error between the actual channelĪnd the desired channel. Here we present an indirect approach to fidelity The process matrices associated with the encoding and/or recoveryĬhannels. To directly maximize fidelity, with the design variables being ĭesign was posed as an optimization problem ReimpellW:05 YamamotoHT:05FletcherSW:06ReimpellWA:06 KosutL:06. This assessment, which is critical toĪny specific implementation, is not knowable without performing theĪn optimization approach to QEC was reported in a number of recent papers No further increases in codespace dimension and/or levels ofĬoncatenation are necessary. Regarding cost, if the resulting robust fidelity levels are sufficiently high, then Shor:95Gott:96Steane:96 Laflamme:96 KnillL:97 NielsenC:00, canīe tailored with optimized encoding and/or recovery. Which do not satisfy the standard assumptions for perfect correction Noise channel uncertainty, resulting in highly robust errorĬorrection. Regarding robustness, we develop an approach that incorporates specific models of Present an optimization approach to QEC that addresses both these (2) Cost: The encoding and recovery effort in perfect QEC typically growsĮxponentially with the number of errors in the noise channel. Specific errors, are often not robust to even small changes in the This theory allows one to find perfect-fidelity encoding and recovery procedures for aĪpproach has two important disadvantages: (1) Robustness: Perfect QEC schemes, as well those produced by optimization tuned to A theory of quantum errorĬorrecting codes has been developed, in analogy to classical coding The scale-up of quantum information devices. Introduction.- Quantum error correction (QEC) is essential for
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