Five quantum register error correction code for higher spin systems
TL;DR: A quantum error correction code ~QECC! in higher spin systems using the idea of multiplicative group character, which generalizes the well-known five qubit perfect code in spin-1/2 systems and is shown to be optimal for higherspin systems.
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Abstract: 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.
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References
Mixed State Entanglement and Quantum Error Correction
Charles H. Bennett,Charles H. Bennett,Charles H. Bennett,David P. DiVincenzo,David P. DiVincenzo,David P. DiVincenzo,John A. Smolin,John A. Smolin,John A. Smolin,William K. Wootters,William K. Wootters,William K. Wootters +11 more
TL;DR: It is proved that an EPP involving one-way classical communication and acting on mixed state M (obtained by sharing halves of Einstein-Podolsky-Rosen pairs through a channel) yields a QECC on \ensuremath{\chi} with rate Q=D, and vice versa, and it is proved Q is not increased by adding one- way classical communication.
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Scheme for reducing decoherence in quantum computer memory
TL;DR: In the mid-1990s, theorists devised methods to preserve the integrity of quantum bits\char22{}techniques that may become the key to practical quantum computing on a large scale.
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Elementary gates for quantum computation.
Adriano Barenco,Charles H. Bennett,Richard Cleve,David P. DiVincenzo,Norman Margolus,Peter W. Shor,Tycho Sleator,John A. Smolin,Harald Weinfurter +8 more
TL;DR: U(2) gates are derived, which derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number of unitary operations on arbitrarily many bits.
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A classical introduction to modern number theory
Kenneth Ireland,Michael Rosen +1 more
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TL;DR: This second edition has been corrected and contains two new chapters which provide a complete proof of the Mordell-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curve.
Good quantum error-correcting codes exist
A. R. Calderbank,Peter W. Shor +1 more
TL;DR: The techniques investigated in this paper can be extended so as to reduce the accuracy required for factorization of numbers large enough to be difficult on conventional computers appears to be closer to one part in billions.