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arXiv:1103.0649 · Physical Review A

Approximate simulation of quantum channels

Cédric Bény, Ognyan Oreshkov

Published Aug 25, 2011• 27 citations
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Abstract

Earlier, we proved a duality between two optimizations problems [Phys. Rev. Lett. 104, 120501 (2010)]. The primary one is, given two quantum channels $\mathcal{M}$ and $\mathcal{N}$, to find a quantum channel $\mathcal{R}$ such that $\mathcal{R}\ensuremath{\circ}\mathcal{N}$ is optimally close to $\mathcal{M}$ as measured by the worst-case entanglement fidelity. The dual problem involves the information obtained by the environment through the so-called complementary channels $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{M}}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{N}}$, and consists in finding a quantum channel ${\mathcal{R}}^{\ensuremath{'}}$ such that ${\mathcal{R}}^{\ensuremath{'}}\ensuremath{\circ}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{M}}$ is optimally close to $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{N}}$. It turns out to be easier to find an approximate solution to the dual problem in certain important situations, notably when $\mathcal{M}$ is the identity channel---the problem of quantum error correction---yielding a good near-optimal worst-case entanglement fidelity as well as the corresponding near-optimal correcting channel. Here we provide more detailed proofs of these results. In addition, we generalize the main theorem to the case where there are certain constraints on the implementation of $\mathcal{R}$, namely, on the number of Kraus operators. We also offer a simple algebraic form for the near-optimal correction channel in the case $\mathcal{M}=\mathrm{id}$. For approximate error correction, we show that any $\ensuremath{\varepsilon}$-correctable channel is, up to appending an ancilla, $\ensuremath{\varepsilon}$-close to an exactly correctable one. We also demonstrate an application of our theorem to the problem of minimax state discrimination.

DOI
10.1103/physreva.84.022333

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We derive necessary and sufficient conditions for the approximate correctability of a quantum code, generalizing the Knill-Laflamme conditions for exact error correction. Our measure of success of the recovery operation is the worst-case entanglement fidelity. We show that the optimal recovery fidelity can be predicted exactly from a dual optimization problem on the environment causing the noise. We use this result to obtain an estimate of the optimal recovery fidelity as well as a way of constructing a class of near-optimal recovery channels that work within twice the minimal error. In addition to standard subspace codes, our results hold for subsystem codes and hybrid quantum-classical codes.
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We develop a theory of approximate quantum error correction (QEC) based on the error-set model, complemented by general methods for code construction. Exact QEC has a powerful error-set structure: by the Knill-Laflamme conditions, a code correcting a given error set automatically protects against every channel whose Kraus operators lie in their linear span. This linearity gives rise to code distance, the equivalence between erasures and general errors, and a theory of asymptotically good codes. A longstanding view has been that these features do not extend to AQEC, leaving the theory essentially channel-by-channel. We show instead that, although full Knill--Laflamme linearity fails, a restricted form survives and suffices to extend all three structural features to the approximate setting. Specifically, a common error-set criterion governs families of channels whose Kraus operators are linear combinations of a given error set and whose coefficient matrices satisfy a spectral constraint. Using the Bény-Oreshkov worst-case and Petz average-case frameworks, we derive uniform fidelity guarantees for these families in terms of two new code parameters--the \emph{environment-leakage distance}, controlling worst-case performance, and the \emph{Knill-Laflamme Hellinger distance}, characterizing the average-case performance of Petz recovery. To demonstrate the scope of this model, we develop partition-based constructions across diverse quantum systems and geometries, placing exact and approximate correction on equal footing. These constructions lead to a metric--error alignment hierarchy for Hilbert spaces, metrics, and error families, which in turn characterizes the resulting recovery guarantees. They yield the first known asymptotically good code families for fermionic systems, one-dimensional Rydberg-blockaded systems, and deletion errors, and extend to other physical platforms.