Computing with Noisy Information
SIAM Journal on Computing
A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
SIAM Journal on Computing
On the Power of Quantum Computation
SIAM Journal on Computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Quantum lower bounds by polynomials
Journal of the ACM (JACM)
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
Quantum lower bounds by quantum arguments
Journal of Computer and System Sciences - Special issue on STOC 2000
Lower Bounds for Randomized and Quantum Query Complexity Using Kolmogorov Arguments
CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
A lower bound on the quantum query complexity of read-once functions
Journal of Computer and System Sciences
Quantum Speed-Up of Markov Chain Based Algorithms
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Polynomial degree vs. quantum query complexity
Journal of Computer and System Sciences - Special issue on FOCS 2003
THE QUANTUM ADVERSARY METHOD AND CLASSICAL FORMULA SIZE LOWER BOUNDS
Computational Complexity
On the power of Ambainis lower bounds
Theoretical Computer Science
Negative weights make adversaries stronger
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Span-program-based quantum algorithm for evaluating formulas
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Probabilistic Boolean decision trees and the complexity of evaluating game trees
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
Computational Complexity
Efficient discrete-time simulations of continuous-time quantum query algorithms
Proceedings of the forty-first annual ACM symposium on Theory of computing
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Quantum search on bounded-error inputs
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
BQP and the polynomial hierarchy
Proceedings of the forty-second ACM symposium on Theory of computing
Any AND-OR Formula of Size $N$ Can Be Evaluated in Time $N^{1/2+o(1)}$ on a Quantum Computer
SIAM Journal on Computing
On the Monte carlo boolean decision tree complexity of read‐once formulae
Random Structures & Algorithms
Quantum Information & Computation
All quantum adversary methods are equivalent
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Faster quantum algorithm for evaluating game trees
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Robust polynomials and quantum algorithms
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
New developments in quantum algorithms
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
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
A stronger LP bound for formula size lower bounds via clique constraints
Theoretical Computer Science
Span programs for functions with constant-sized 1-certificates: extended abstract
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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
Adversary lower bound for the k-sum problem
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
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
Time-Efficient quantum walks for 3-distinctness
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We show that any boolean function can be evaluated optimally by a quantum query algorithm that alternates a certain fixed, input-independent reflection with a second reflection that coherently queries the input string. Originally introduced for solving the unstructured search problem, this two-reflections structure is therefore a universal feature of quantum algorithms. Our proof goes via the general adversary bound, a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. By a quantum algorithm for evaluating span programs, this lower bound is known to be tight up to a sub-logarithmic factor. The extra factor comes from converting a continuous-time query algorithm into a discrete-query algorithm. We give a direct and simplified quantum algorithm based on the dual SDP, with a bounded-error query complexity that matches the general adversary bound. Therefore, the general adversary lower bound is tight; it is in fact an SDP for quantum query complexity. This implies that the quantum query complexity of the composition f o (g,..., g) of two boolean functions f and g matches the product of the query complexities of f and g, without a logarithmic factor for error reduction. It efficiently characterizes the quantum query complexity of a read-once formula over any finite gate set. It further shows that span programs are equivalent to quantum query algorithms.