Related papers: Structure Theorem for Quantum Replacer Codes
Error-correcting codes were invented to correct errors on noisy communication channels. Quantum error correction (QEC), however, may have a wider range of uses, including information transmission, quantum simulation/computation, and…
Decoding algorithms are essential to fault-tolerant quantum-computing architectures. In this perspective we explore decoding algorithms for the surface code; a prototypical quantum low-density parity-check code that underlies many of the…
In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful…
We show how entanglement-assisted codes can be constructed from arbitrary quantum codes by associating them with quantum codes for erasure channels. If a subset of physical qubits is correctable for an erasure error, then it naturally forms…
We introduce a new graphical framework for designing quantum error correction codes based on classical principles. A key feature of this graphical language, over previous approaches, is that it is closely related to that of factor graphs or…
A quantum code is a subspace of a Hilbert space of a physical system chosen to be correctable against a given class of errors, where information can be encoded. Ideally, the quantum code lies within the ground space of the physical system.…
Quantum algorithms have the potential to provide exponential speedups over some of the best known classical algorithms. These speedups may enable quantum devices to solve currently intractable problems such as those in the fields of…
We show that the multiplicative domain of a completely positive map yields a new class of quantum error correcting codes. In the case of a unital quantum channel, these are precisely the codes that do not require a measurement as part of…
We make an explicit connection between fundamental notions in quantum cryptography and quantum error correction. Error-correcting subsystems (and subspaces) for quantum channels are the key vehicles for contending with noise in physical…
We investigate the usage of highly efficient error correcting codes of multilevel systems to protect encoded quantum information from erasure errors and implementation to repetitively correct these errors. Our scheme makes use of quantum…
A quantum algorithm for general combinatorial search that uses the underlying structure of the search space to increase the probability of finding a solution is presented. This algorithm shows how coherent quantum systems can be matched to…
We introduce a model of quantum insertion-deletion (insdel) channels. Insdel channels are meant to represent, for example, synchronization errors arising in data transmission. In the classical setting, they represent a strict generalization…
Quantum error correction and the use of quantum error correction codes is likely to be essential for the realisation of practical quantum computing. Because the error models of quantum devices vary widely, quantum codes which are tailored…
Typical stabilizer codes aim to solve the general problem of fault-tolerance without regard for the structure of a specific system. By incorporating a broader representation-theoretic perspective, we provide a generalized framework that…
Quantum computers theoretically are able to solve certain problems more quickly than any deterministic or probabilistic computers. A quantum computer exploits the rules of quantum mechanics to speed up computations. However, one has to…
In this work, we introduce a new concatenation scheme which aims at protecting information against the occurrence of both computational errors and quantum erasures. According to our scheme, the internal code must be a quantum…
Galois rings are regarded as "building blocks" of a finite commutative ring with identity. There have been many papers on classical error correction codes over Galois rings published. As an important warm-up before exploring quantum…
These notes introduce quantum computation and quantum error correction, emphasising the importance of stabilisers and the mathematical foundations in basic Lie theory. We begin by using the double cover map $\mathrm{SU}_2 \rightarrow…
Methods of quantum mechanics promise information-theoretic security for various protocols in cryptography. However, impossibility of some cryptographic applications such as standard bit commitment, oblivious transfer, multiparty secure…
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…