Quantum error correction and generalized numerical ranges

Physical Review A, 1997

Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-errorcorrecting quantum codes, provided that the interaction is dominated by an identity component.

Journal of Mathematical Physics, 2009

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 particular, of codes defined by infinite-dimensional algebras.

arXiv (Cornell University), 2023

We establish rigorous connections between quantum circuit complexity and approximate quantum error correction (AQEC) capability, two properties of fundamental importance to the physics and practical use of quantum many-body systems, covering both all-to-all and geometric scenarios like lattice systems in finite spatial dimensions. To this end, we introduce a type of code parameter that we call subsystem variance, which is closely related to the optimal AQEC precision. Our key finding is that if the subsystem variance is below an O(k/n) threshold then any state in the code subspace must obey certain circuit complexity lower bounds, which identify nontrivial "phases" of codes. Based on our results, we propose O(k/n) as a boundary between subspaces that should and should not count as AQEC codes. This theory of AQEC provides a versatile framework for understanding the quantum complexity and order of many-body quantum systems, offering new insights for wide-ranging physical scenarios, in particular topological order and critical quantum systems which are of outstanding importance in many-body and high energy physics. We observe from various different perspectives that roughly O(1/n) represents a common, physically significant "scaling threshold" of subsystem variance for features associated with nontrivial quantum order.

Physical Review A, 2010

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 map, with optimality defined based on entanglement fidelity. For the practically useful case of codes encoding a single qubit of information, our algorithm is particularly easy to implement.

Quantum information & computation

In a recent study [Rohde et al., quant-ph/0603130 (2006)] of several quantum error correcting protocols designed for tolerance against qubit loss, it was shown that these protocols have the undesirable effect of magnifying the effects of depolarization noise. This raises the question of which general properties of quantum error-correcting codes might explain such an apparent trade-off between tolerance to located and unlocated error types. We extend the counting argument behind the well-known quantum Hamming bound to derive a bound on the weights of combinations of located and unlocated errors which are correctable by nondegenerate quantum codes. Numerical results show that the bound gives an excellent prediction to which combinations of unlocated and located errors can be corrected with high probability by certain large degenerate codes. The numerical results are explained partly by showing that the generalized bound, like the original, is closely connected to the information-theoretic quantity the quantum coherent information. However, we also show that as a measure of the exact performance of quantum codes, our generalized Hamming bound is provably far from tight.

IEEE Transactions on Information Theory, 2015

We construct explicitly two infinite families of genuine nonadditive 1-error correcting quantum codes and prove that their coding subspaces are 50% larger than those of the optimal stabilizer codes of the same parameters via the linear programming bound. All these nonadditive codes can be characterized by a stabilizer-like structure and thus their encoding circuits can be designed in a straightforward manner.

2012

In the present paper we initiate the study of the product higher rank numerical range. The latter, being a variant of the higher rank numerical range [M.-D. Choi et al., Rep. Math. Phys. 58, 77 (2006); Lin. Alg. Appl. 418, 828 (2006)], is a natural tool for studying a construction of quantum error correction codes for multiple access channels. We review properties of this set and relate it to other numerical ranges, which were recently introduced in the literature. Further, the concept is applied to the construction of codes for bi-unitary two-access channels with a hermitian noise model. Analytical techniques for both outerbounding the product higher rank numerical range and determining its exact shape are developed for this case. Finally, the reverse problem of constructing a noise model for a given product range is considered.

Research Square (Research Square), 2024

Quantum error correction (QEC) codes are vital for scalable and robust computations of quantum operations. The main challenge lies in introducing ancilla qubits with minimal side effect in the resultant computation, particularly considering state-of-the-art near-term devices. With the aim of identifying error value ranges that make encoding advantageous, the Shor code stands out as a viable first approach for a physical implementation of quantum error correction, due to its simpler structure. The noise models utilized, through their Operator Sum Representation (OSR), can be extrapolated, as these are typical of the majority of current technologies. In this article, error widths are analytically determined such that it becomes worthwhile to transmit information through encoding.

Physical Review A, 2013

Errors are inevitable during all kinds quantum informational tasks and quantum error-correcting codes (QECCs) are powerful tools to fight various quantum noises. For standard QECCs physical systems have the same number of energy levels. Here we shall propose QECCs over mixed alphabets, i.e., physical systems of different dimensions, and investigate their constructions as well as their quantum Singleton bound. We propose two kinds of constructions: a graphical construction based a graph-theoretical object composite coding clique and a projection-based construction. We illustrate our ideas using two alphabets by finding out some 1-error correcting or detecting codes over mixed alphabets, e.g., optimal ((6, 8, 3)) 4 5 2 1 , ((6, 4, 3)) 4 4 2 2 and ((5, 16, 2)) 4 3 2 2 code and suboptimal ((5, 9, 2)) 3 4 2 1 code. Our methods also shed light to the constructions of standard QECCs, e.g., the construction of the optimal ((6, 16, 3))4 code as well as the optimal ((2n + 3, p 2n+1 , 2))p codes with p = 4k.

submitted to Springer- …, 2004