Every potential benefit of quantum computers would be lost without methods to protect the computations from error. Quantum channels provide a framework for understanding error in quantum systems. In general, for a system of size d, a quantum channel may require as many as ~d^4 parameters to characterize it. We introduce the leading Kraus approximation, a simplification which reduces the number of parameters to ~d^2, while accurately approximating two figures of merit important to the experimentalist: the unitarity and average process fidelity. Additionally, applying the leading Kraus approximation declutters investigations into the set of quantum channels. When applied, a natural decomposition of channels arises, separating behaviour into coherent and decoherent contributions. We find that eliminating the coherent contribution provides the greatest increase to the fidelity, while decoherent processes provide a generalization of depolarizing channels.
Every potential benefit of quantum computers would be lost without methods to protect the computations from error. Quantum channels provide a framework for understanding error in quantum systems. In general, for a system of size d, a quantum channel may require as many as ~d^4 parameters to characterize it. We introduce the leading Kraus approximation, a simplification which reduces the number of parameters to ~d^2, while accurately approximating two figures of merit important to the experimentalist: the unitarity and average process fidelity. Additionally, applying the leading Kraus approximation declutters investigations into the set of quantum channels. When applied, a natural decomposition of channels arises, separating behaviour into coherent and decoherent contributions. We find that eliminating the coherent contribution provides the greatest increase to the fidelity, while decoherent processes provide a generalization of depolarizing channels.
Inevitably, assessing the overall performance of a quantum computer must rely on characterizing some of its elementary constituents and, from this information, formulate a broader statement concerning more complex constructions thereof. However, given the vastitude of possible quantum errors as well as their coherent nature, accurately inferring the quality of composite operations is generally difficult. To navigate through this jumble, we introduce a non-physical simplification of quantum maps that we refer to as the leading Kraus (LK) approximation. The uncluttered parameterization of LK approximated maps naturally suggests the introduction of a unitary-decoherent polar factorization for quantum channels in any dimension. We then leverage this structural dichotomy to bound the evolution -- as circuits grow in depth -- of two of the most experimentally relevant figures of merit, namely the average process fidelity and the unitarity. We demonstrate that the leeway in the behavior of the process fidelity is essentially taken into account by physical unitary operations.
Quantum information is prone to suffer from errors caused by the so-called decoherence, which describes the loss in coherence of quantum states associated to their interactions with the surrounding environment. This decoherence phenomenon is present in every quantum information task, be it transmission, processing or even storage of quantum information. Consequently, the protection of quantum information via quantum error correction codes (QECC) is of paramount importance to construct fully operational quantum computers. Understanding environmental decoherence processes and the way they are modeled is fundamental in order to construct effective error correction methods capable of protecting quantum information. Moreover, quantum channel models that are efficiently implementable and manageable on classical computers are required in order to design and simulate such error correction schemes. In this article, we present a survey of decoherence models, reviewing the manner in which these models can be approximated into quantum Pauli channel models, which can be efficiently implemented on classical computers. We also explain how certain families of quantum error correction codes can be entirely simulated in the classical domain, without the explicit need of a quantum computer. A quantum error correction code for the approximated channel is also a correctable code for the original channel, and its performance can be obtained by Monte Carlo simulations on a classical computer.
Earlier, we proved a duality between two optimizations problems [Phys. Rev. Lett. 104, 120501 (2010)]. The primary one is, given two quantum channels M and N, to find a quantum channel R such that $\mathcal{R}\ensuremath{\circ}\mathcal{N}$ is optimally close to 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 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 R, namely, on the number of Kraus operators. We also offer a simple algebraic form for the near-optimal correction channel in the case M=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.
The quantum capacity of a memoryless channel determines the maximal rate at which we can communicate reliably over asymptotically many uses of the channel. Here we illustrate that this asymptotic characterization is insufficient in practical scenarios where decoherence severely limits our ability to manipulate large quantum systems in the encoder and decoder. In practical settings, we should instead focus on the optimal trade-off between three parameters: the rate of the code, the size of the quantum devices at the encoder and decoder, and the fidelity of the transmission. We find approximate and exact characterizations of this trade-off for various channels of interest, including dephasing, depolarizing and erasure channels. In each case, the trade-off is parameterized by the capacity and a second channel parameter, the quantum channel dispersion. In the process, we develop several bounds that are valid for general quantum channels and can be computed for small instances.
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.
Quantum error correction is the art of protecting quantum states from the detrimental influence from the environment. To master this art, one must understand how the system interacts with the environment and gives rise to a full set of quantum phenomena, many of which have no correspondence in classical information theory. Such phenomena include decoherence, an effect that in general destroys superpositions of pure states as a consequence of entanglement with the environment. But decoherence can also be understood as “information leakage”, i.e., when knowledge of an encoded code block is transferred to the environment. In this event, the block’s information or entanglement content is typically lost.In a typical scenario, however, not all types of destructive events are likely to occur, but only those allowed by the information carrier, the type of interaction with the environment, and how the environment “picks up” information of the error events. These characteristics can be incorporated into a code, i.e., a channel-adapted quantum error-correcting code.Often, it is assumed that the environment’s ability to distinguish between error events is small, and I will denote such environments “memory-less”. But this assumption is not always valid, since the ability to distinguish error events is related to the temperature of the environment, and in the particular case of information coded onto photons, kBTR «ℏω typically holds, and one must then assume that the environment has a “memory”. In the thesis I describe a short quantum error-correction code adapted for photons interacting with a “cold” reservoir, i.e., a reservoir which continuously probes what error occurred in the coded state.I also study other types of environments, and show how to distill meaningful figures of merit from codes adapted for these channels, as it turns out that resource-based figures reflecting both information and entanglement can be calculated exactly for a well-studied class of channels: the Pauli channels. Starting from these resource-based figures, I establish the notion of efficiency and quality and show that there will be a trade-off between efficiency and quality for short codes. Finally I show how to incorporate, into these calculations, the choices one has to make when handling quantum states that have been detected as incorrect, but where no prospect of correcting them exists, i.e., so-called detection errors.