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Conventional fault-tolerant quantum error-correction schemes require a number of extra qubits that grows linearly with the code's maximum stabilizer generator weight. For some common distance-three codes, the recent "flag paradigm" uses…
The problem of finding code distance has been long studied for the generic ensembles of linear codes and led to several algorithms that substantially reduce exponential complexity of this task. However, no asymptotic complexity bounds are…
Estimating the difference between quantum data is crucial in quantum computing. However, as typical characterizations of quantum data similarity, the trace distance and quantum fidelity are believed to be exponentially-hard to evaluate in…
In this paper, we provide a comprehensive overview of a recent debate over the quantum versus classical solvability of bounded distance decoding (BDD). Specifically, we review the work of Eldar and Hallgren [EH22], [Hal21] demonstrating a…
Quantum error correction is capable of digitizing quantum noise and increasing the robustness of qubits. Typically, error correction is designed with the target of eliminating all errors - making an error so unlikely it can be assumed that…
The stabilizer code is the most general algebraic construction of quantum error-correcting codes proposed so far. A stabilizer code can be constructed from a self-orthogonal subspace of a symplectic space over a finite field. We propose a…
We describe a quantum error correction scheme aimed at protecting a flow of quantum information over long distance communication. It is largely inspired by the theory of classical convolutional codes which are used in similar circumstances…
This paper introduces a new quantization scheme for real and complex Grassmannian sources. The proposed approach relies on a structured codebook based on a geometric construction of a collection of bent grids defined from an initial mesh on…
Linear programming approaches have been applied to derive upper bounds on the size of classical codes and quantum codes. In this paper, we derive similar results for general quantum codes with entanglement assistance, including nonadditive…
The development of accurate and fast numerical schemes for the five fold Boltzmann collision integral represents a challenging problem in scientific computing. For a particular class of interactions, including the so-called hard spheres…
The highest information rate at which quantum error-correction schemes work reliably on a channel, which is called the quantum capacity, is proven to be lower bounded by the limit of the quantity termed coherent information maximized over…
Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best…
For a number of quantum channels of interest, phase-flip errors occur far more frequently than bit-flip errors. When transmitting across these asymmetric channels, the decoding error rate can be reduced by tailoring the code used to the…
Random classical linear codes are widely believed to be hard to decode. While slightly sub-exponential time algorithms exist when the coding rate vanishes sufficiently rapidly, all known algorithms at constant rate require exponential time.…
We present a new family of quantum low-density parity-check codes, which we call radial codes, obtained from the lifted product of a specific subset of classical quasi-cyclic codes. The codes are defined using a pair of integers $(r,s)$ and…
Low check weight is practically crucial code property for fault-tolerant quantum computing, which underlies the strong interest in quantum low-density parity-check (qLDPC) codes. Here, we explore the theory of weight-constrained stabilizer…
The theory of error-correcting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous…
The quantum Wasserstein distance (W-distance) is a fundamental metric for quantifying the distinguishability of quantum operations, with critical applications in quantum error correction. However, computing the W-distance remains…
Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences…
We present an algorithm for manipulating quantum information via a sequence of projective measurements. We frame this manipulation in the language of stabilizer codes: a quantum computation approach in which errors are prevented and…