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arXiv:0705.4314 · Physical Review A

Entanglement-assisted quantum error correction with linear optics

Mark M. Wilde, Hari Krovi, Todd A. Brun

Published Nov 9, 2007• 24 citations
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Abstract

We construct a theory of continuous-variable entanglement-assisted quantum error correction. We present an example of a continuous-variable entanglement-assisted code that corrects for an arbitrary single-mode error. We also show how to implement encoding circuits using passive optical devices, homodyne measurements, feedforward classical communication, conditional displacements, and off-line squeezers.

DOI
10.1103/physreva.76.052308

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We construct a theory of continuous-variable entanglement-assisted quantum error correction. We present an example of a continuous-variable entanglement-assisted code that corrects for an arbitrary single-mode error. We also show how to implement encoding circuits using passive optical devices, homodyne measurements, feedforward classical communication, conditional displacements, and off-line squeezers.
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Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard QECCs, which can only be constructed from dual-containing codes. Operator quantum error-correcting codes allow certain errors to be corrected (or prevented) passively, reducing the complexity of the correction procedure. We combine these two extensions of standard quantum error correction into a unified entanglement-assisted quantum error-correction formalism. This new scheme, which we call entanglement-assisted operator quantum error correction (EAOQEC), is the most general and powerful quantum error-correcting technique known, retaining the advantages of both entanglement-assistance and passive correction. We present the formalism, show the considerable freedom in constructing EAOQECCs from classical codes, and demonstrate the construction with examples.
Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of pre-existing entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard QECCs which can only be constructed from dual-containing codes. Operator quantum error-correcting codes (OQECCs) allow certain errors to be corrected (or prevented) passively, reducing the complexity of the correction procedure. We combine these two extensions of standard quantum error correction into a unified entanglement- assisted quantum error correction formalism. This new scheme, which we call entanglement-assisted operator quantum error correction (EAOQEC), is the most general and powerful quantum error-correcting technique known, retaining the advantages of both entanglement-assistance and passive correction. We present the formalism, show the considerable freedom in constructing EAOQECCs from classical codes, and demonstrate the construction with examples.
Quantum information processing promises exponential advantages for certain computational and communication tasks, but practical quantum devices are inherently vulnerable to decoherence and operational imperfections. This thesis develops new algebraic and analytical tools for quantum error correction (QEC) aimed at unifying entanglement-assisted coding theory with operator-algebra frameworks, while also providing mathematically rigorous insights for continuous-variable photonic implementations. First, we introduce the framework of entanglement-assisted operator algebra quantum error correction (EA-OAQEC), extending entanglement-assisted QEC in an operator-algebra context. Built on a stabilizer-based formalism, EA-OAQEC unifies and generalizes subspace, subsystem, and hybrid classical-quantum entanglement-assisted codes. We establish a master error-correction theorem, define a generalized distance notion, and develop constructive techniques including gauge fixing, clean-qubit transformations, and entanglement-assisted gauge-fixing procedures. Second, we study general logical operators: unitaries that preserve a stabilizer codespace beyond the traditional Pauli normalizer. We derive explicit structure theorems for the centralizer and normalizer in the full unitary setting and show how normalizer transformations can be parameterized via classical bit matrix multiply complement (BMMC) permutations. Finally, motivated by photonic quantum computing, we analyze finite-rank compressions of the continuous position and momentum operators relevant to Gottesman-Kitaev-Preskill (GKP) codes. We prove that the spectra of these compressions are governed by the roots of classical Hermite polynomials and connect the compressed operators to physically realizable approximations of displacement operations and finite-energy GKP states.
We propose a linear-optical implementation of a hyperentanglement-assisted quantum error-correcting code. The code is hyperentanglement assisted because the shared entanglement resource is a photonic state hyperentangled in polarization and orbital angular momentum. It is possible to encode, decode, and diagnose channel errors using linear-optical techniques. The code corrects for polarization ``flip'' errors and is thus suitable only for a proof-of-principle experiment. The encoding and decoding circuits use a Knill-Laflamme-Milburn-like scheme for transforming polarization and orbital angular momentum photonic qubits. A numerical optimization algorithm finds a unit-fidelity encoding circuit that requires only three ancilla modes and has success probability equal to 0.0097.
We show how entanglement shared between encoder and decoder can simplify the theory of quantum error correction. The entanglement-assisted quantum codes we describe do not require the dual-containing constraint necessary for standard quantum error-correcting codes, thus allowing us to "quantize" all of classical linear coding theory. In particular, efficient modern classical codes that attain the Shannon capacity can be made into entanglement-assisted quantum codes attaining the hashing bound (closely related to the quantum capacity). For systems without large amounts of shared entanglement, these codes can also be used as catalytic codes, in which a small amount of initial entanglement enables quantum communication.