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

ISBN-10: 0821849751

ISBN-13: 9780821849750

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4 that there is a 4 × 4 special orthogonal matrix which renders these two subalgebras conjugate, and which, therefore, diagonalizes HSQUID . Thus, in particular, a real matrix will achieve the conjugation (since special orthogonal matrices are real). Of course, this special orthogonal matrix renders the two Cartan decompositions of su(4) conjugate as well. In this case this conjugation can be found by inspection. Indeed, if we can ﬁnd a 2 × 2 (real) special orthogonal matrix V such that V † σx V is diagonal, then V ⊗ V will diagonalize HSQUID .

S Thus the property “H has zero ground state energy” is equivalent to the quantum 4-SAT {ΠS } having a satisfying assignment. Let L = Lyes ∪ Lno be a language from QMA1 , x ∈ L be a binary string, and U (x) be a verifying circuit see Deﬁnition 3. Using the majority voting to amplify the gap in acceptance probabilities, see [1], we can assume that • If x ∈ Lyes then AP (U, ψwit ) = 1 for some input witness state |ψwit , • If x ∈ Lno then AP (U, ψwit ) ≤ , = 1/p(|x|), for all |ψwit , where U is a circuit implementing several copies of U (x) and the majority voting (obviously it can be realized using the gate set G).

All common linear algebra tasks for operators whose matrix elements are algebraic numbers can be solved eﬃciently, see books [5, 6] for the subject. The rest of the paper is organized as follows. Eﬃcient algorithm for quantum 2SAT is presented in Section 2 (for the sake of completeness we outline the standard algorithm solving classical 2-SAT in Appendix B). QMA1 -completeness of quantum k-SAT, k ≥ 4, is proved in Section 3. A technical lemma needed for this proof concerning universality of three-qubit quantum gates with matrix elements from a ﬁxed ﬁeld is placed in Appendix A.

### Cross Disciplinary Advances in Quantum Computing by Kazem Mahdavi, Deborah Koslover, Leonard L., III Brown

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