Optimal direct sum results for deterministic and randomized decision tree complexity
Information Processing Letters
New developments in quantum algorithms
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Quantum interpolation of polynomilas
Quantum Information & Computation
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Faster quantum algorithm for evaluating game trees
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Reflections for quantum query algorithms
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Exact Quantum Algorithms for the Leader Election Problem
ACM Transactions on Computation Theory (TOCT)
The quantum query complexity of AC0
Quantum Information & Computation
Quantum adversary (upper) bound
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Quantum counterfeit coin problems
Theoretical Computer Science
Span programs and quantum algorithms for st-connectivity and claw detection
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
The quantum query complexity of read-many formulas
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Average-case lower bounds for formula size
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Superlinear advantage for exact quantum algorithms
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
ACM Transactions on Computation Theory (TOCT) - Special issue on innovations in theoretical computer science 2012
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The general adversary bound is a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. We turn this lower bound into an upper bound, by giving a quantum walk algorithm based on the dual SDP that has query complexity at most the general adversary bound, up to a logarithmic factor. In more detail, the proof has two steps, each based on "span programs," a certain linear-algebraic model of computation. First, we give an SDP that outputs for any boolean function a span program computing it that has optimal "witness size." The optimal witness size is shown to coincide with the general adversary lower bound. Second, we give a quantum algorithm for evaluating span programs with only a logarithmic query overhead on the witness size. The first result is motivated by a quantum algorithm for evaluating composed span programs. The algorithm is known to be optimal for evaluating a large class of formulas. The allowed gates include all constant-size functions for which there is an optimal span program. So far, good span programs have been found in an ad hoc manner, and the SDP automates this procedure. Surprisingly, the SDP's value equals the general adversary bound. A corollary is an optimal quantum algorithm for evaluating "balanced" formulas over any finite boolean gate set. The second result extends span programs' applicability beyond the formula-evaluation problem. We extend the analysis of the quantum algorithm for evaluating span programs. The previous analysis shows that a corresponding bipartite graph has a large spectral gap, but only works when applied to the composition of constant-size span programs. We show generally that properties of eigenvalue-zero eigenvectors in fact imply an "effective" spectral gap around zero. A strong universality result for span programs follows. A good quantum query algorithm for a problem implies a good span program, and vice versa. Although nearly tight, this equivalence is nontrivial. Span programs are a promising model for developing more quantum algorithms.