Quantum states have high affinity for errors and hence error correction is of utmost importance to realise a quantum computer. Laflamme showed that 5 qubits are necessary to correct a single error on a qubit. In a Pauli error model, four different types of errors can occur on a qubit. Maximally entangled states are orthogonal to each other and hence can be uniquely distinguished by a measurement in the Bell basis. Thus a measurement in Bell basis and a unitary transformation is sufficient to correct error in Bell states. However, such a measurement is not possible for non-maximally entangled states. In this work we show that the 16 possible errors for a non-maximally entangled two qubit system map to only 8 distinct error states. Hence, it is possible to correct the error without perfect knowledge of the type of error. Furthermore, we show that the possible errors can be grouped in such a way that all 4 errors can occur on one qubit, whereas only bit flip error can occur on the second qubit. As a consequence, instead of 10, only 8 qubits are sufficient to correct a single error. We propose an 8-qubit error correcting code to correct a single error in a non-maximally entangled state. We further argue that for an $n$ -qubit non-maximally entangled state of the form $a|0>^{n} + b|1>^{n}$ , it is always possible to correct a single error with fewer than $5n$ qubits, in fact only $3n+2$ qubits suffice.Quantum states have high affinity for errors and hence error correction is of utmost importance to realise a quantum computer. Laflamme showed that 5 qubits are necessary to correct a single error on a qubit. In a Pauli error model, four different types of errors can occur on a qubit. Maximally entangled states are orthogonal to each other and hence can be uniquely distinguished by a measurement in the Bell basis. Thus a measurement in Bell basis and a unitary transformation is sufficient to correct error in Bell states. However, such a measurement is not possible for non-maximally entangled states. In this work we show that the 16 possible errors for a non-maximally entangled two qubit system map to only 8 distinct error states. Hence, it is possible to correct the error without perfect knowledge of the type of error. Furthermore, we show that the possible errors can be grouped in such a way that all 4 errors can occur on one qubit, whereas only bit flip error can occur on the second qubit. As a consequence, instead of 10, only 8 qubits are sufficient to correct a single error. We propose an 8-qubit error correcting code to correct a single error in a non-maximally entangled state. We further argue that for an $n$-qubit non-maximally entangled state of the form $a|0>^{n} + b|1>^{n}$, it is always possible to correct a single error with fewer than $5n$ qubits, in fact only $3n+2$ qubits suffice.

arXiv:1709.04301Sep 13, 20170 citations

Title: Self-Organizing Quantum Error Correction via Entanglement Swapping

Jincheng Zhang

Quantum error correction is a critical component of quantum computing, enabling the reliable transmission and manipulation of quantum information. Traditional error correction schemes, however, often struggle to scale effectively due to the inherent complexity of quantum systems. This paper presents a novel approach to quantum error correction utilizing entanglement swapping to dynamically reconstruct and correct errors across a network of qubits. We propose a self-organizing mechanism where entangled pairs of qubits are continuously swapped, amplifying the corrective signal and effectively mitigating errors. This dynamic adaptation contrasts with fixed protocols, offering a potential pathway towards more robust and scalable quantum error correction. The theoretical framework and preliminary experimental results demonstrate the feasibility of this approach.Quantum error correction is a critical component of quantum computing, enabling the reliable transmission and manipulation of quantum information. Traditional error correction schemes, however, often struggle to scale effectively due to the inherent complexity of quantum systems. This paper presents a novel approach to quantum error correction utilizing entanglement swapping to dynamically reconstruct and correct errors across a network of qubits. We propose a self-organizing mechanism where entangled pairs of qubits are continuously swapped, amplifying the corrective signal and effectively mitigating errors. This dynamic adaptation contrasts with fixed protocols, offering a potential pathway towards more robust and scalable quantum error correction. The theoretical framework and preliminary experimental results demonstrate the feasibility of this approach.

Aug 29, 20260 citations

Quantum error correction beyond qubits

Takao Aoki, Go Takahashi, Tadashi Kajiya +4 more

arXiv:0811.3734Nature PhysicsJun 28, 2009144 citations

Protecting entanglement between logical qubits via quantum error correction

Weizhou Cai, Xianghao Mu, Weiting Wang +11 more

arXiv:2302.13027Nature PhysicsMar 6, 202429 citations

Systems of quantum communication with linear optical

Daniel Barbosa de Brito

In this dissertation it is considered a quantum error correction system for coherent states of light and a complete optical system for probabilistic teleportation of single-photon polarization encoded qubit. Initially, the quantum error correction with the use of redundancy is analyzed. After that, it is presented a form of error correction using time-bin qubit without redundancies. Based on this last approach, a passive quantum error correction system, that is, without external control and synchronization, for quantum communication system employing coherent states, using only linear optical devices, is proposed. It is shown that the quantum error correction system also works for bipartite states of qubits. Following, the teleportation of quantum states is analyzed and an optical system for probabilistic teleportation of single-photon polarization encoded qubit, using only linear optical devices, is proposed. Finally, the error correction and teleportation topics are put together in the proposal of a probabilistic teleportation system using an error correction system for distribution of the entangled pair of photonsIn this dissertation it is considered a quantum error correction system for coherent states of light and a complete optical system for probabilistic teleportation of single-photon polarization encoded qubit. Initially, the quantum error correction with the use of redundancy is analyzed. After that, it is presented a form of error correction using time-bin qubit without redundancies. Based on this last approach, a passive quantum error correction system, that is, without external control and synchronization, for quantum communication system employing coherent states, using only linear optical devices, is proposed. It is shown that the quantum error correction system also works for bipartite states of qubits. Following, the teleportation of quantum states is analyzed and an optical system for probabilistic teleportation of single-photon polarization encoded qubit, using only linear optical devices, is proposed. Finally, the error correction and teleportation topics are put together in the proposal of a probabilistic teleportation system using an error correction system for distribution of the entangled pair of photons

Jan 1, 20070 citations

n

-qubit non-maximally entangled state of the form

a|0>^{n} + b|1>^{n}

, it is always possible to correct a single error with fewer than

5n

qubits, in fact only

3n+2

qubits suffice.