By Robabeh Rahimi, Akira SaiToh, Mikio Nakahara

ISBN-10: 9814295833

ISBN-13: 9789814295833

This quantity offers an summary at the decoherence suppression equipment in quantum computing, open quantum structures, quantum errors correction and fault-tolerant quantum computing. it is usually innovations in geometric quantum computing through composite pulses. Quantum wipe impression is defined as an strategy for suppressing decoherence of the process. a brief contribution at the implementation of holonomic quantum gates with NMR (Nuclear Magnetic Resonance) is gifted. The lecture notes contributed to this quantity are ready in a self-contained demeanour therefore readers with constrained wisdom at the issues may well comprehend the discussions by means of following the series of chapters which commence with mathematical frameworks and development to the main up to date results of the fields. the amount may be worthwhile for a large viewers from graduate scholars to researchers drawn to the sphere.

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**Sample text**

Let HE be a Hilbert space with dimension d and let d {ek }k=1 be an orthonormal basis of HE . Then, there exists a Unitary operator U on HS ⊗ HE such that d′ U (ψ ⊗ e1 ) := (Vk ψ) ⊗ ek , k=1 October 19, 2009 11:29 WSPC - Proceedings Trim Size: 9in x 6in proceedings 23 for any ψ ∈ HS . [Hint: First check an operator U defined on the subspace {ψ ⊗ e1 |ψ ∈ HS } ⊂ H ⊗ HE preserves an inner product. ] ′ Proof of Theorem 3 Let {Ek }dk=1 satisfy (28), and let HE be a Hilbert ′ space with dimension d′ and let {ek }dk=1 be an orthonormal basis of HE .

Let A be an Hermitian operator on H, and let Φ be a unital d d TPCP map on L(H). Let A = i=1 ai |ψi ψi | and Φ(A) = i=1 a′i |ψi′ ψi′ | be eigenvalue decompositions of A and Φ(A), respectively. Then, there exists d a d × d doubly stochastic matrix B such that a′i = i=1 Bij aj . A d×d matrix B is called a stochastic matrix if Bij ≥ 0 for all i, j = 1, . . , d and i Bij = 1 for all j = 1, . . , d. It is called a doubly stochastic matrix if B is a stochastic matrix and j=1 Bij = 1 for all i = 1, .

Then, by noting i |φi φi | *26 Show i = |ψki φi | so that Vk φi † I, we have k Vk AVk (36) = = that, indeed, any state ψ which has the maximal Schmidt number d can also be used for such a state. d *27 Show that i,j=1 Aij ⊗ Eij = 0 ⇔ Aij = 0 (∀i, j = 1, . . , d). October 19, 2009 11:29 WSPC - Proceedings Trim Size: 9in x 6in proceedings 30 Vk ( i |φi φi |)A( j |φj φj |)Vk† = i,j,k φi |Aφj (|Vk φi Vk φj |) = φ |Aφ ( |ψ ψ |) = φ |Aφ i j ki kj i j (Φ(Eij )) i,j k i,j = Φ( i |φi φi |A j |φj φj |) = Φ(A) from (36) for any A ∈ L(H).

### Decoherence Suppression in Quantum Systems 2008 by Robabeh Rahimi, Akira SaiToh, Mikio Nakahara

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