We study the approximate correctability of general algebras of observables, which represent hybrid quantum-classical information. This includes approximate quantum error correcting codes and subsystems codes. We show that the main result of arXiv:quant-ph/0605009 yields a natural generalization of the Knill-Laflamme conditions in the form of a dimension independent estimate of the optimal reconstruction error for a given encoding, measured using the trace-norm distance to a noiseless channel.
The errors that arise in a quantum channel can be corrected perfectly if and only if the channel does not decrease the coherent information of the input state. We show that, if the loss of coherent information is small, then approximate error correction is possible.
We present a comparative analysis of exact and approximate quantum error correction by means of simple unabridged analytical computations. For the sake of clarity, using primitive quantum codes, we study the exact and approximate error correction of the two simplest unital (Pauli errors) and nonunital (non-Pauli errors) noise models, respectively. The similarities and differences between the two scenarios are stressed. In addition, the performances of quantum codes quantified by means of the entanglement fidelity for different recovery schemes are taken into consideration in the approximate ca…
The concept of entropy and the correct application of the Second Law of thermodynamics are essential in order to understand the reason why quantum error correction is thermodynamically possible and no violation of the Second Law occurs during its execution. We report in this work our first steps towards an entropic analysis extended to approximate quantum error correction (QEC). Special emphasis is devoted to the link among quantum state discrimination (QSD), quantum information gain, and quantum error correction in both the exact and approximate QEC scenarios.
Based on the group structure of a unitary Lie algebra, a scheme is provided to systematically and exhaustively generate quantum error correction codes, including the additive and nonadditive codes. The syndromes in the process of error-correction distinguished by different orthogonal vector subspaces, the coset subspaces. Moreover, the generated codes can be classified into four types with respect to the spinors in the unitary Lie algebra and a chosen initial quantum state.
The breakthrough of quantum error correction brought with it the picture of quantum information as a sort of combination of two complementary types of classical information, "amplitude" and "phase". Here I show how this intuition can be used to construct two new conditions for approximate quantum error correction. The first states that entanglement is locally recoverable from a bipartite state when one system can be used to approximately predict the outcomes of two complementary observables on the other. The second, more in the spirit of the recent decoupling approach, states that entanglement…
Recent work on approximate quantum error correction (QEC) has opened up the possibility of constructing subspace codes that protect information with high fidelity in scenarios where perfect error correction is impossible. Motivated by this, we investigate the problem of approximate subsystem codes. Subsystem codes extend the standard formalism of subspace QEC to codes in which only a subsystem within a subspace of states is used to store information in a noise-resilient fashion. Here, we demonstrate easily checkable sufficient conditions for the existence of approximate subsystem codes. Furthe…
Operator quantum error-correction is a technique for robustly storing quantum information in the presence of noise. It generalizes the standard theory of quantum error-correction, and provides a unified framework for topics such as quantum error-correction, decoherence-free subspaces, and noiseless subsystems. This paper develops (a) easily applied algebraic and information-theoretic conditions which characterize when operator quantum error-correction is feasible; (b) a representation theorem for a class of noise processes which can be corrected using operator quantum error-correction; and (c)…
Most of the research done on quantum error correction studies an error model in which each qubit is affected by noise, independently of the other qubits. In this paper we study a different noise model -- one in which the noise may be correlated with the qubits it acts upon. We show both positive and negative results. On the one hand, we show controlled-X errors cannot be perfectly corrected, yet can be approximately corrected with sub-constant approximation error. On the other hand, we show that no non-trivial quantum error correcting code can approximately correct controlled phase error with …
We present relaxed criteria for quantum error correction which are useful when the specific dominant noise process is known. These criteria have no classical analogue. As an example, we provide a four-bit code which corrects for a single amplitude damping error. This code violates the usual Hamming bound calculated for a Pauli description of the error process, and does not fit into the GF(4) classification.
Approximate quantum error correction theory of non-isometric codes
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Joint Center for Quantum Information and Computer Science, University of Maryland（马里兰大学量子信息与计算机科学联合中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
It is well known that no quantum error correcting code of rate $R$ can correct adversarial errors on more than a $(1-R)/4$ fraction of symbols. But what if we only require our codes to *approximately* recover the message? We construct efficiently-decodable approximate quantum codes against adversarial error rates approaching the quantum Singleton bound of $(1-R)/2$, for any constant rate $R$. Moreover, the size of the alphabet is a constant independent of the message length and the recovery error is exponentially small in the message length. Central to our construction is a notion of quantum l…
We demonstrate that there exists a universal, near-optimal recovery map---the transpose channel---for approximate quantum error-correcting codes, where optimality is defined using the worst-case fidelity. Using the transpose channel, we provide an alternative interpretation of the standard quantum error correction (QEC) conditions, and generalize them to a set of conditions for approximate QEC (AQEC) codes. This forms the basis of a simple algorithm for finding AQEC codes. Our analytical approach is a departure from earlier work relying on exhaustive numerical search for the optimal recovery m…
We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. The generalization yields new classes of quantum error correcting codes that have no finite-dimensional counterparts. The error correction theory we develop begins with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of algebras of observables given by finite-dimensional von Neumann factors of type I. Our generalization allows for the correction of codes characterized by any von Neumann algebra and we give examples, in…
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 of the overall process. 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 easy-to-calculate estimate of the optimal recovery fidelity as well as a way of constructing a class of near-optimal recovery channels that work …
We present analytic estimates of the performances of various approximate quantum error correction schemes for the generalized amplitude damping (GAD) qubit channel. Specifically, we consider both stabilizer and nonadditive quantum codes. The performance of such error-correcting schemes is quantified by means of the entanglement fidelity as a function of the damping probability and the non-zero environmental temperature. The recovery scheme employed throughout our work applies, in principle, to arbitrary quantum codes and is the analogue of the perfect Knill-Laflamme recovery scheme adapted to …
Classical simulations of noisy stabilizer circuits are often used to estimate the threshold of a quantum error-correcting code. Physical noise sources are efficiently approximated by random insertions of Pauli operators. For a single qubit, more accurate approximations that still allow for efficient simulation can be obtained by including Clifford operators and Pauli operators conditional on measurement in the noise model. We examine the feasibility of employing these expanded error approximations to obtain better threshold estimates. We calculate the level-1 pseudothreshold for the Steane [[7…
It is shown that, if the loss of entanglement along a quantum channel is sufficiently small, then approximate quantum error correction is possible, thereby generalizing what happens for coherent information. Explicit bounds are obtained for the entanglement of formation and the distillable entanglement, and their validity naturally extends to other bipartite entanglement measures in between. Robustness of derived criteria is analyzed and their tightness compared. Finally, as a byproduct, we prove a bound quantifying how large the gap between entanglement of formation and distillable entangleme…
Approximate Quantum Error Correction with 1D Log-Depth Circuits
Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院量子信息中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
We present a simple proof of the approximate Eastin-Knill theorem, which connects the quality of a quantum error-correcting code (QECC) with its ability to achieve a universal set of transversal logical gates. Our derivation employs powerful bounds on the quantum Fisher information in generic quantum metrological protocols to characterize the QECC performance measured in terms of the worst-case entanglement fidelity. The theorem is applicable to a large class of decoherence models, including independent erasure and depolarizing noise. Our approach is unorthodox, as instead of following the est…
For a generic set of Markovian noise models, the estimation precision of a parameter associated with the Hamiltonian is limited by the $1/\sqrt{t}$ scaling where $t$ is the total probing time, in which case the maximal possible quantum improvement in the asymptotic limit of large $t$ is restricted to a constant factor. However, situations arise where the constant factor improvement could be significant, yet no effective quantum strategies are known. Here we propose an optimal approximate quantum error correction (AQEC) strategy asymptotically saturating the precision lower bound in the most ge…
The uncertainty principle can be understood as constraining the probability of winning a game in which Alice measures one of two conjugate observables, such as position or momentum, on a system provided by Bob, and he is to guess the outcome. Two variants are possible: either Alice tells Bob which observable she measured, or he has to furnish guesses for both cases. Here I derive new uncertainty relations for both, formulated directly in terms of Bob's guessing probabilities. For the former these relate to the entanglement that can be recovered by action on Bob's system alone. This gives a con…
The Eastin-Knill theorem is a central result of quantum error correction theory and states that a quantum code cannot correct errors exactly, possess continuous symmetries, and implement a universal set of gates transversely. As a way to circumvent this result, there are several approaches in which one gives up on either exact error correction or continuous symmetries. In this context, it is common to employ a complementary measure of fidelity as a way to quantify quantum state distinguishability and benchmark approximations in error correction. Despite having useful properties, evaluating fid…
We compare exact and SU(2)-cluster approximate calculation schemes to determine dynamics of the second-harmonic generation model using its reformulation in terms of a polynomial Lie algebra $su_{pd}(2)$ and related spectral representations of the model evolution operator realized in algorithmic forms. It enabled us to implement computer experiments exhibiting a satisfactory accuracy of the cluster approximations in a large range of characteristic model parameters.
We show that the theory of operator quantum error correction can be naturally generalized by allowing constraints not only on states but also on observables. The resulting theory describes the correction of algebras of observables (and may therefore suitably be called ``operator algebra quantum error correction''). In particular, the approach provides a framework for the correction of hybrid quantum-classical information and it does not require the state to be entirely in one of the corresponding subspaces or subsystems. We discuss applications to quantum teleportation and to the study of info…
We work out a theory of approximate quantum error correction that allows us to derive a general lower bound for the entanglement fidelity of a quantum code. The lower bound is given in terms of Kraus operators of the quantum noise. This result is then used to analyze the average error correcting performance of codes that are randomly drawn from unitarily invariant code ensembles. Our results confirm that random codes of sufficiently large block size are highly suitable for quantum error correction. Moreover, employing a lemma of Bennett, Shor, Smolin, and Thapliyal, we prove that random coding…
In renormalization group (RG) flow, the low energy states form a code subspace that is approximately protected against the local short-distance errors. We motivate this connection with an example of spin-blocking RG in classical spin models. We consider the continuous multi-scale renormalization ansatz (cMERA) for massive free fields as a concrete example of real-space RG in quantum field theory (QFT) and show that the low-energy coherent states are approximately protected from the errors caused by the high-energy localized coherent operators. In holographic RG flows, we study the phase transi…
We consider a class of holographic tensor networks that are efficiently contractible variational ansatze, manifestly (approximate) quantum error correction codes, and can support power-law correlation functions. In the case when the network consists of a single type of tensor that also acts as an erasure correction code, we show that it cannot be both locally contractible and sustain power-law correlation functions. Motivated by this no-go theorem, and the desirability of local contractibility for an efficient variational ansatz, we provide guidelines for constructing networks consisting of mu…
The ideal realization of quantum teleportation relies on having access to a maximally entangled state; however, in practice, such an ideal state is typically not available and one can instead only realize an approximate teleportation. With this in mind, we present a method to quantify the performance of approximate teleportation when using an arbitrary resource state. More specifically, after framing the task of approximate teleportation as an optimization of a simulation error over one-way local operations and classical communication (LOCC) channels, we establish a semi-definite relaxation of…
We show that the approximate quasiorthogonality of two operator algebras is equivalent to the algebras being approximately private relative to their conditional expectation quantum channels. Our analysis is based on a characterization of the measure of orthogonality in terms of Choi matrices and Kraus operators for completely positive maps. We present examples drawn from different areas of quantum information.
We introduce a fully constructive characterisation of holographic quantum error-correcting codes. That is, given a code and an erasure error we give a recipe to explicitly compute the terms in the RT formula. Using this formalism, we employ quantum circuits to construct a number of examples of holographic codes. Our codes have nontrivial holographic properties and are simpler than existing approaches built on tensor networks. Finally, leveraging a connection between correctable and private systems we prove the uniqueness of the algebra satisfying complementary recovery. The material is present…
We generalize our results in paper I in this series to quantum channels between general v. Neumann algebras, proving the approximate recoverability of states which undergo a small change in relative entropy through the channel. To this end, we derive a strengthened form of the quantum data processing inequality for the change in relative entropy of two states under a channel between two v. Neumann algebras. Compared to the usual inequality, there is an explicit lower bound involving the fidelity between the original state and a recovery channel.
In the framework of vector model of spin, we discuss the problem of a covariant formalism \cite{Pomeranskii1998} concerning the discrepancy between relativistic and Pauli Hamiltonians. We show how the spin-induced non commutativity of a position accounts the discrepancy on the classical level, without appeal to the Dirac equation and Foldy-Wouthuysen transformation.
The eigenstate thermalization hypothesis (ETH) is a powerful conjecture for understanding how statistical mechanics emerges in a large class of many-body quantum systems. It has also been interpreted in a CFT context, and, in particular, holographic CFTs are expected to satisfy ETH. Recently, it was observed that the ETH condition corresponds to a necessary and sufficient condition for an approximate quantum error correcting code (AQECC), implying the presence of AQECCs in systems satisfying ETH. In this paper, we explore the properties of ETH as an error correcting code and show that there ex…
The algebraic approach to black hole quantization requires the horizon area eigenvalues to be equally spaced. As shown previously, for a neutral non-rotating black hole, such eigenvalues must be $2^{n}$-fold degenerate if one constructs the black hole stationary states by means of a pair of creation operators subject to a specific algebra. We show that the algebra of these two building blocks exhibits $U(2)\equiv U(1)\times SU(2)$ symmetry, where the area operator generates the U(1) symmetry. The three generators of the SU(2) symmetry represent a {\it global} quantum number (hyperspin) of the …
Approximate QCAs in one dimension using approximate algebras
California Institute of Technology（加州理工学院）；Ludwig-Maximiliations-Universität München（慕尼黑大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；CWI（荷兰数学与计算机科学研究中心）
There are two parts to this work: First, we study the error correction properties of the real-space renormalization group (RG). The long-distance operators are the (approximately) correctable operators encoded in the physical algebra of short-distance operators. This is closely related to modeling the holographic map as a quantum error correction code. As opposed to holography, the real-space RG of a many-body quantum system does not have the complementary recovery property. We discuss the role of large $N$ and a large gap in the spectrum of operators in the emergence of complementary recovery…
We improve on the Thomas-Fermi approximation for the single-particle density of fermions by introducing inhomogeneity corrections. Rather than invoking a gradient expansion, we relate the density to the unitary evolution operator for the given effective potential energy and approximate this operator by a Suzuki-Trotter factorization. This yields a hierarchy of approximations, one for each approximate factorization. For the purpose of a first benchmarking, we examine the approximate densities for a few cases with known exact densities and observe a very satisfactory, and encouraging, performanc…
We establish that, in an appropriate limit, qubits of communication should be regarded as composite resources, decomposing cleanly into independent correlation and transmission components. Because qubits of communication can establish ebits of entanglement, qubits are more powerful resources than ebits. We identify a new communications resource, the zero-bit, which is precisely half the gap between them, replacing classical bits by zero-bits makes teleportation asymptotically reversible. The decomposition of a qubit into an ebit and two zero-bits has wide-ranging consequences including applica…
