Correcting quantum errors in higher system

I construct a quantum error correction code (QECC) in higher spin systems using the idea of multiplicative group character. Each $N$ -state quantum particle is encoded as five $N$ -state quantum registers. By doing so, this code can correct any quantum error arising from any one of the five quantum registers. This code generalizes the well-known five qubit perfect code in spin-1/2 systems and is shown to be optimal for higher spin systems. I also report a simple algorithm for encoding. The importance of multiplicative group character in constructing QECCs will be addressed.I construct a quantum error correction code (QECC) in higher spin systems using the idea of multiplicative group character. Each $N$-state quantum particle is encoded as five $N$-state quantum registers. By doing so, this code can correct any quantum error arising from any one of the five quantum registers. This code generalizes the well-known five qubit perfect code in spin-1/2 systems and is shown to be optimal for higher spin systems. I also report a simple algorithm for encoding. The importance of multiplicative group character in constructing QECCs will be addressed.

arXiv:quant-ph/9702033Physical Review AJul 1, 199742 citations

How to Correct Small Quantum Errors

Michael Keyl, Reinhard F. Werner

arXiv:quant-ph/0206086Lecture notes in physicsJan 1, 20028 citations

Error Correction in Quantum Computing

Jesús Yaljá Montiel-Pérez, Jordi Fabián González-Contreras, Erik Zamora +1 more

Studies in computational intelligenceJan 1, 20250 citations

Correcting quantum errors in higher spin systems

H. F. Chau

I consider the theory of the quantum error correcting code (QECC), where each quantum particle has more than two possible eigenstates. In this higher spin system, I report an explicit QECC that is related to the symmetry group ${\mathrm{Z}}_{2}^{\ensuremath{\bigotimes}(\mathrm{N}\mathrm{\ensuremath{-}}1)}$\ensuremath{\bigotimes} ${\mathrm{S}}_{\mathrm{N}}$ . This QECC, which generalizes Shor's simple majority vote code [Phys. Rev. A 52, 2493 (1995)], is able to correct errors arising from exactly one quantum particle. I also provide a simple encoding algorithm.I consider the theory of the quantum error correcting code (QECC), where each quantum particle has more than two possible eigenstates. In this higher spin system, I report an explicit QECC that is related to the symmetry group ${\mathrm{Z}}_{2}^{\ensuremath{\bigotimes}(\mathrm{N}\mathrm{\ensuremath{-}}1)}$\ensuremath{\bigotimes}${\mathrm{S}}_{\mathrm{N}}$. This QECC, which generalizes Shor's simple majority vote code [Phys. Rev. A 52, 2493 (1995)], is able to correct errors arising from exactly one quantum particle. I also provide a simple encoding algorithm.

arXiv:quant-ph/9610023Physical Review AFeb 1, 199739 citations

Exploiting degeneracy to construct good ternary quantum error correcting code

Ritajit Majumdar, Susmita Sur‐Kolay

Quantum error-correcting code for higher dimensional systems can, in general, be directly constructed from the codes for qubit systems. What remains unknown is whether there exist efficient code design techniques for higher dimensional systems. In this paper, we propose a 7-qutrit error-correcting code for the ternary quantum system and show that this design formulation has no equivalence in qubit systems. This code is optimum in the number of qutrits required to correct a single error while maintaining the CSS structure. This degenerate CSS code can (i) correct up to seven simultaneous phase errors and a single bit error, (ii) correct two simultaneous bit errors on pre-defined pairs of qutrits on eighteen out of twenty-one possible pairs, and (iii) in terms of the cost of implementation, the depth of the circuit of this code is only two more than that of the ternary Steane code. Our proposed code shows that it is possible to design better codes explicitly for ternary quantum systems instead of simply carrying over codes from binary quantum systems.Quantum error-correcting code for higher dimensional systems can, in general, be directly constructed from the codes for qubit systems. What remains unknown is whether there exist efficient code design techniques for higher dimensional systems. In this paper, we propose a 7-qutrit error-correcting code for the ternary quantum system and show that this design formulation has no equivalence in qubit systems. This code is optimum in the number of qutrits required to correct a single error while maintaining the CSS structure. This degenerate CSS code can (i) correct up to seven simultaneous phase errors and a single bit error, (ii) correct two simultaneous bit errors on pre-defined pairs of qutrits on eighteen out of twenty-one possible pairs, and (iii) in terms of the cost of implementation, the depth of the circuit of this code is only two more than that of the ternary Steane code. Our proposed code shows that it is possible to design better codes explicitly for ternary quantum systems instead of simply carrying over codes from binary quantum systems.

arXiv:2008.00713Aug 3, 20200 citations

Error Correction for Quantum Computers

WORLD SCIENTIFIC eBooksJul 1, 19980 citations

I construct a quantum error correction code (QECC) in higher spin systems using the idea of multiplicative group character. Each

N

-state quantum particle is encoded as five

N

-state quantum registers. By doing so, this code can correct any quantum error arising from any one of the five quantum registers. This code generalizes the well-known five qubit perfect code in spin-1/2 systems and is shown to be optimal for higher spin systems. I also report a simple algorithm for encoding. The importance of multiplicative group character in constructing QECCs will be addressed.

{\mathrm{S}}_{\mathrm{N}}

. This QECC, which generalizes Shor's simple majority vote code [Phys. Rev. A 52, 2493 (1995)], is able to correct errors arising from exactly one quantum particle. I also provide a simple encoding algorithm.