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By Kazem Mahdavi, Deborah Koslover, Leonard L., III Brown

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1. 14) H1 = kU † H2 U, for some real k = 0. 15) e−itH1 / = e−ikt U † e−itH2 / U. 2. 18) ⎪ ⎪ 0 −1 1 0 ⎪ ⎪ ⎩ Eigenvalues of HSQUID : δ · 1 (triplet), δ · (−3) (singlet). Obviously, HCQED cannot be conjugate, in the usual sense of the word, to either HQD or HSQUID since their spectra differ. On the other hand, the multiplicity (degeneracy) structures of their eigenvalues are identical, and this suggests that there is a close connection between these Hamiltonians. 20) 4 HCQED E0 (I4 is the identity operator on H) HCQED ≡ E0 I4 − ⎤ ⎡ ⎧ 1 0 0 0 ⎪ ⎪ ⎪ ⎢0 1 0 0 ⎥ ⎪ ⎨ H ⎥, ⎢ CQED = E0 ⎣ 0 0 1 0⎦ ⎪ ⎪ 0 0 0 −3 ⎪ ⎪ ⎩ Eigenvalues of HCQED : E0 · 1 (triplet), E0 · (−3) (singlet).

The running time of the whole algorithm is O(n4 ). A satisfiability problem in which clauses involve k bits is known as k-SAT. Quantum analogue of k-SAT can be specified by a family of k-qubit projectors {ΠS }, where S ⊆ {1, . . , n} runs over all subsets of cardinality k. One needs to verify whether there exists an n-qubit state |Ψ such that ΠS |Ψ = 0 for all S. Since k-SAT is known to be NP-complete for k ≥ 3, one can ask whether quantum k-SAT is QMA-complete for k ≥ 3? Before addressing this question it should be noted that the definition of quantum k-SAT given above is not quite satisfactory.

Once, after allowing for the above restriction, one has found the S corresponding to a given Type I Cartan decomposition it is possible to write algebraic conditions on the matrix (or matrices) M which will render the corresponding Cartan decomposition conjugate to the standard Type I Cartan decomposition. In the case of su(4) and the pair THREE DISTINCT TYPES OF 2-QUBIT QUANTUM COMPUTING DEVICES UNIVERSALITY FOR THREE TYPES OF 2-QUBIT QUANTUM COMPUTING DEVICES 31 15 (K1 , P1 ), it turns out that S = J2 ⊗ J2 (see [3]).

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