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arXiv:0909.1584 · Physical Review A

Optical implementation of a unitarily correctable code

K. M. Schreiter, Aron Pasieka, Rainer Kaltenbaek, Kevin J. Resch, David W. Kribs

Published Aug 7, 2009• 6 citations
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Abstract

Noise poses a challenge for any real-world implementation in quantum information science. The theory of quantum error correction deals with this problem via methods to encode and recover quantum information in a way that is resilient against that noise. Unitarily correctable codes are an error correction technique wherein a single unitary recovery operation is applied without the need for an ancilla Hilbert space. Here, we present an optical implementation of a nontrivial unitarily correctable code for a noisy quantum channel with no decoherence-free subspaces or noiseless subsystems. We show that recovery of our initial states is achieved with high fidelity $(\ensuremath{\ge}0.97)$, quantitatively proving the efficacy of this unitarily correctable code.

DOI
10.1103/physreva.80.022311

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We propose a quantum error correction without error detection. A quantum state ρ0ρ_0ρ0​ combined with an ancilla state σσσ is encoded unitarily and an error operator is applied on the encoded state. The recovery operation then produces a tensor product state ρ0⊗σ′ρ_0 \otimes σ'ρ0​⊗σ′. The decoding operation is combined with the recovery operation and the state ρ0ρ_0ρ0​ is directly reproduced without referring to the code word. A higher rank projection operator required for a conventional operator quantum error correction is not necessary to implement. Encoding and the recovery operations are implemented with unitary operators only, which makes quantum error correction much easier than any other proposals.
This paper is an expanded and more detailed version of the work \cite{KLP04} in which the Operator Quantum Error Correction formalism was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques --- i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method --- as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of "unitarily noiseless subsystems''.
This paper is an expanded and more detailed version of our recent work in which the Operator Quantum Error Correction formalism was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques - i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method - as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of ``unitarily noiseless subsystems''.
Quantum error correction is a set of methods to protect quantum information--that is, quantum states--from unwanted environmental interactions (decoherence) and other forms of noise. The information is stored in a quantum error-correcting code, which is a subspace in a larger Hilbert space. This code is designed so that the most common errors move the state into an error space orthogonal to the original code space while preserving the information in the state. It is possible to determine whether an error has occurred by a suitable measurement and to apply a unitary correction that returns the state to the code space, without measuring (and hence disturbing) the protected state itself. In general, codewords of a quantum code are entangled states. No code that stores information can protect against all possible errors; instead, codes are designed to correct a specific error set, which should be chosen to match the most likely types of noise. An error set is represented by a set of operators that can multiply the codeword state. Quantum error correction is used to protect information in quantum communication (where quantum states pass through noisy channels) and quantum computation (where quantum states are transformed through a sequence of imperfect computational steps in the presence of environmental decoherence to solve a computational problem). In quantum computation, error correction is just one component of fault-tolerant design.
Quantum error correction is a set of methods to protect quantum information--that is, quantum states--from unwanted environmental interactions (decoherence) and other forms of noise. The information is stored in a quantum error-correcting code, which is a subspace in a larger Hilbert space. This code is designed so that the most common errors move the state into an error space orthogonal to the original code space while preserving the information in the state. It is possible to determine whether an error has occurred by a suitable measurement and to apply a unitary correction that returns the state to the code space, without measuring (and hence disturbing) the protected state itself. In general, codewords of a quantum code are entangled states. No code that stores information can protect against all possible errors; instead, codes are designed to correct a specific error set, which should be chosen to match the most likely types of noise. An error set is represented by a set of operators that can multiply the codeword state. Quantum error correction is used to protect information in quantum communication (where quantum states pass through noisy channels) and quantum computation (where quantum states are transformed through a sequence of imperfect computational steps in the presence of environmental decoherence to solve a computational problem). In quantum computation, error correction is just one component of fault-tolerant design.
This paper is an expanded and more detailed version of the work [1] in which the Operator Quantum Error Correction protocol was introduced.This is a new scheme for the error correction of quantum operations that incorporates the known techniques -i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem methodas special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations.We also discuss a number of examples and introduce the notion of "unitarily noiseless subsystems".