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arXiv:1101.0413

Unitary Quantum Error Correction without Error Detection

Hiroyuki Tomita, Mikio Nakahara

Published Jan 2, 2011• 2 citations
arXivHTMLPDF

Abstract

We propose a quantum error correction without error detection. A quantum state $ρ_0$ combined with an ancilla state $σ$ is encoded unitarily and an error operator is applied on the encoded state. The recovery operation then produces a tensor product state $ρ_0 \otimes σ'$. The decoding operation is combined with the recovery operation and the state $ρ_0$ 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.

DOI
10.48550/arxiv.1101.0413

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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.
1. IntroductionQuantum system is vulnerable to disturbance from the external environment, which leads todecoherence in the system. We have to overcome this difficulty to realize a working quanutumcomputer and a dependable quantum information processing. Quantum error correction (QEC)[1, 2, 3] is one of the most promising candidates for overcoming decoherence.QEC proposals to date are separated roughly into two classes, one employs extra ancilla qubitsfor error syndrome readout while the other, called operator quantum error correction (OQEC)employs a higher rank projection operator. However, the schemes have not been built in actualquantum computing or quantum information processing so far. This is dueto the following obstaclesinherent in these QECs; the syndrome must be read out by introducing ancilla qubits duringcomputing/information processing in the former case, while realization of a high-rank projectionoperator is physically challenging in the latter case.It was shown in [4] that for some quantum channels there exist different QECs in which nosyndrome measurements, no ancillas and no projection operators were required. Recovery operationand decoding operation are combined into a single unitary operation and the output state is a directproductof a decoded qubitstate and an ancilla state. Aqubit state isreproducedwithout recoveringthe codeword and, moreover, the projection operation is automatically built in our output state.The purpose of this paper is to extend the results in [4] to general quantum channels. We showthat for any quantum channel there is a unitary recovery operation, with which the output stateis a tensor product of decoded qubit state and an ancilla state. As a result, a decoding scheme canbe done by a unitary gate followed by a partial trace operation.The rest of the paper is organized as follows. We introduce the basic notions of OQEC andthen prove the main theorem in Section 2. We also give simple examples demonstrating our resultand a simplified proof of a theorem given in [2] illustrating that our recovery channel can be usedto do correction for many other channels related to ours. Section 3 is devoted to summary anddiscussions.
Most quantum error-correcting codes are predicated on the assumption that there exists a reservoir of qubits in the state $|0\ensuremath{\rangle}$ which can be used as ancilla qubits to prepare multiqubit logical states. In this Brief Report, we examine the consequences of relaxing this assumption and propose a method to increase the fidelity produced by a given code when the ancilla qubits are initialized in mixed states, using the same number of qubits, at most doubling the number of gates when the recovery operation would already be implemented. The procedure implemented consists of altering the encoding operator to include the inverse of the unitary operation used to correct detected errors after decoding. This augmentation will be especially useful in quantum computing architectures that do not possess projective measurement, such as solid-state NMR quantum information processing.
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.
It is known that one can do quantum error correction without syndrome measurement, which is often done in operator quantum error correction (OQEC). However, the physical realization could be challenging, especially when the recovery process involves high-rank projection operators and a superoperator. We use operator theory to improve OQEC so that the implementation can always be done by unitary gates followed by a partial trace operation. Examples are given to show that our error correction scheme outperforms the existing ones in various scenarios.
It is known that one can do quantum error correction without syndrome measurement, which is often done in operator quantum error correction (OQEC). However, the physical realization could be challenging, especially when the recovery process involves high-rank projection operators and a superoperator. We use operator theory to improve OQEC so that the implementation can always be done by unitary gates followed by a partial trace operation. Examples are given to show that our error correction scheme outperforms the existing ones in various scenarios.
Abstract: 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 errors and a formal proof that the classical bounds on the probability of errorcorrecting codes applies to error correcting quantum codes, provided that the interaction is dominated by an identity component.