New PDF release: Cross Disciplinary Advances in Quantum Computing

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

ISBN-10: 0821849751

ISBN-13: 9780821849750

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Let F ⊂ C be a field that we use for exact representation of complex numbers (see the remark in Introduction). We shall consider a gate set G consisting of all three-qubit unitary operators whose matrix elements in the standard basis belong to F (we include three-qubit gates into G just to simplify the proofs). Definition 3. A language L = Lyes ∪ Lno ⊂ Σ∗ belongs to QMA1 iff there exists a polynomial p and uniform family of quantum circuits {U (x)}, x ∈ L, such that U (x) has a length at most p(|x|), U (x) uses only gates from G, |Rin | + |Rwit | + |Rout | ≤ p(|x|), and • If x ∈ Lyes then there exists a witness state |ψwit such that AP (U (x), ψwit ) = 1; • If x ∈ Lno then for any witness state |ψwit one has AP (U (x), ψwit ) < 1 − 1/p(|x|).

Thus we can apply ideas of the previous subsection to merge a and c into a single logical qubit and obtain a new instance of quantum 2-SAT with n − 1 qubits which is equivalent to the original one. Let us keep trying to generate new constraints by probing different pairs of edges until we either encounter the case 4 reducing the number of qubits from n to n − 1 (the algorithm terminates with Output 3), or after O(n3 ) steps we arrive to a homogeneous instance with a complete set of constraints which is defined below.

Let ω (a,c) be a tensor defined in Eq. (7). One has to consider the following possibilities: (1) ω (a,c) = 0. We get no additional constraints on ψ. (2) ω (a,c) = 0, but (a, c) ∈ E and ω (a,c) is proportional to φ(a,c) . Again, we get no additional constraints on ψ. (3) ω (a,c) = 0 and (a, c) ∈ / E. In this case Eq. (6) provides a new constraint on ψ. Let us add (a, c) into the list of edges E and set φ(a,c) = ω (a,c) . (4) ω (a,c) = 0, (a, c) ∈ E, and ω (a,c) is not proportional to φ(a,c) . In this case we get two independent constraints for the pair of qubits a and c.

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

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