Download E-books Quantum Information and Quantum Computing (Kinki University Series on Quantum Computing, Volume 6) PDF

The open examine heart venture "Interdisciplinary primary study towards awareness of a quantum computer" has been supported through the Ministry of schooling, Japan for 5 years. this can be a number of the study results by means of the contributors engaged within the undertaking. To make the presentation self-contained, it starts off with an outline by way of Mikio Nakahara, which serves as a concise creation to quantum details and quantum computing. next contributions comprise topics from physics, chemistry, arithmetic, and knowledge technological know-how, reflecting upon the big variety of scientists operating lower than this venture. those contributions introduce NMR quantum computing and comparable innovations, quantity concept and coding conception, quantum mistakes correction, photosynthesis, non-classical correlations and entanglement, impartial atom quantum desktop, between others. every one of contributions will function a brief advent to those innovative study fields.

Readership: Graduate scholars and researchers in physics, chemistry, arithmetic, informatics and machine technology. additionally obtainable to complex undergraduate scholars.

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Read or Download Quantum Information and Quantum Computing (Kinki University Series on Quantum Computing, Volume 6) PDF

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2 2 hence they're very with regards to the “type II codes” (see pp. 189-205 of ConwaySloane3 or p. 139 of Pless12 ), yet are individual from them by way of the second one August three, 2012 13:45 8568: Quantum Information.. Vol-6 main˙orc-symp11 177 situation. In Chinen,1 the extremal estate of the formal weight enumerators are defined and an analogue of the Mallows-Sloane certain is received. additionally, the Riemann speculation is formulated and a few examples of extremal formal weight enumerators which fulfill the Riemann speculation are given (Section three of Chinen1 ). It additionally considers the formal weight enumerators over F3 (Section four of Chinen1 ). Chinen2 considers one other generalization within the similar line. It bargains with √ homogeneous polynomials within the ring C[x+( q−1)y, y(x−y)] (see Chap. 19 of MacWilliams-Sloane8 for normal concept on invariant polynomial rings). The polynomials during this ring have the valuables that they're invariant lower than taking twin over Fq . as a result their zeta features have the practical equation within the kind of Theorem three. 1 and we will be able to formulate the Riemann √ speculation. Chinen2 finds such a lot of homogeneous polynomials in C[x+( q− 1)y, y(x − y)] which fulfill the Riemann speculation. First it proposes a manner of constructing an invariant polynomial (i. e. a member √ of C[x + ( q − 1)y, y(x − y)]) from the burden enumerator WC (x, y) of arbitrary linear code C (see part 2 of Chinen2 ): 1 {WC (x, y) + q k−n/2 WC ⊥ (x, y)}. 1 + q k−n/2 ˜ C (x, y) ∈ C[x + (√q − 1)y, y(x − y)] from Theorem we will be able to simply see W 2. 1, and will build invariant polynomials from codes which aren't selfdual. In Chinen,2 the subsequent periods of codes are mentioned: ˜ C (x, y) := W (i) the MDS codes, (ii) the overall Hamming [(q r − 1)/(q − 1) = n, n − r, three] codes over Fq , (iii) the binary [23, 12, 7] and the ternary [11, 6, five] Golay codes. We quote the most result of Chinen:2 Theorem four. 1. (i) If C is an MDS code which isn't self-dual, then the ˜ C (x, y) satisfies the Riemann speculation. zeta polynomial P˜C (T ) of W (ii) enable C = Ham(r, q) be the Hamming [(q r − 1)/(q − 1) = n, n − r, three] code over Fq . If r ≥ three and q ≥ four, then P˜C (T ) satisfies the Riemann speculation. (iii) enable C be the binary [23, 12, 7] or the ternary [11, 6, five] Golay code, then the P˜C (T ) satisfies the Riemann speculation. those statements are proved utilizing different tools. between them, the subsequent theorem (Theorem 6 in Chinen2 ) is used to end up (ii) above and will be of self sufficient curiosity: Theorem four. 2. If f (T ) = a0 +a1 T +· · ·+ak T okay +ak T m−k +ak−1 T m−k+1 + August three, 2012 13:45 8568: Quantum Information.. Vol-6 main˙orc-symp11 178 · · · + a0 T m (m > 2k) satisfies a0 > a1 > · · · > ak > zero, then all of the roots of f (T ) lie at the unit circle. There also are a few new effects at the line of the Duursma conception. For the Riemann speculation for self-dual codes, Okuda10 proved the next in 2008: Theorem four. three. All extremal self-dual codes over F4 of code size 6k − 2 (k ∈ N) fulfill the Riemann speculation. this can be the first outcome after Duursma proving the Riemann speculation for a chain of extremal self-dual codes.

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