Generalized Performance of Concatenated Quantum Codes—A Dynamical Systems Approach

Abstract

We apply a dynamical systems approach to concatenation of quantum error correcting codes, extending and generalizing the results of Rahn et al.[1] to both diagonal and non-diagonal channels. Our point of view is global: instead of focusing on particular types of noise channels, we study the geometry of the coding map as a discrete-time dynamical system on the entire space of noise channels.In the case of diagonal channels, we show that any code with distance at least three corrects (in the infinite concatenation limit) an open set of errors. For Calderbank-Shor-Steane (CSS) codes, we give a more precise characterization of that set. We show how to incorporate noise in the gates, thus completing the framework. We derive some general bounds for noise channels, which allows us to analyze several codes in detail. Index Terms-Quantum error correction, quantum channels, quantum fault toleranceJ. Fern is with the

Modeling coherent errors in quantum error correction

“…We work with the repetition code, which, though not a full quantum code in that it cannot correct both X and Z errors, has the advantage of being analytically tractable and yet nontrivial. Indeed, we find that it reproduces the key features of generic codes, saturating the bounds on error channel parameters under concatenation given in [12].…”

“…Using a formalism developed by Rahn et al [13] for general noise, these authors found that coherent errors in the physical error channel can lead to coherent errors in the logical channel, as manifested by off-diagonal elements in the superoperators for these channels. For the specific example of the d=3 Steane code, [12] found that an off-diagonal element of order ò in the unencoded error superoperator leads to an encoded (logical) error superoperator with offdiagonals of order  3 and diagonals of order  4 . This leads to a diamond-distance logical error rate of order  1 greater than would be obtained by replacing the physical error by its Pauli twirl.…”

“…Such information can be useful for determining parameter regimes where a Pauli model is valid, and for providing independent validation of numerical results. Indeed, [12] considered general channels, deriving upper bounds on superoperator coefficients for the logical error, but not their actual value except for the d=3 Steane code. The results of [13] were limited to diagonal channels, while [14,15] evaluated the logical noise maps numerically, a technique which does not make explicit the scaling of the logical error parameters with d. The latter references also considered coherent and incoherent errors individually but not simultaneously.…”

Quantum Error Correction Decoheres Noise

“…Determining the performance of an errorcorrecting code at the logical level under general noise is complicated because such noise is harder to simulate. Previous approaches have expanded the class of errors to some larger class that can still be efficiently simulated [2], performed full density-matrix simulations [3], used tensor network descriptions of specific codes [4,5] or effective logical process matrices [6][7][8]. These meth- * sbeale@uwaterloo.ca † jwallman@uwaterloo.ca ods are suboptimal because they either require a huge amount of resources to simulate or are indirect approximations.…”

Untitled

“…Before deriving an expression for the coding map, let us first recall some properties of the Stokes representation from [14,15]. First of all the matrix entries of a valid quantum channel are real-valued and lie within the interval [−1, 1].…”