Matrix analysis
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
SIAM Journal on Computing
Parallel Quantum Computation and Quantum Codes
SIAM Journal on Computing
Fast parallel circuits for the quantum Fourier transform
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Algorithms for quantum computation: discrete logarithms and factoring
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Counting, fanout and the complexity of quantum ACC
Quantum Information & Computation
Quantum lower bounds for fanout
Quantum Information & Computation
Bounds on the power of constant-depth quantum circuits
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
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We demonstrate that the unbounded fan-out gate is very powerful. Constant-depth polynomial-size quantum circuits with bounded fan-in and unbounded fan-out over a fixed basis (denoted by QNCf0) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, threshold[t], exact[q], and counting. Classically, we need logarithmic depth even if we can use unbounded fan-in gates. If we allow arbitrary one-qubit gates instead of a fixed basis, then these circuits can also be made exact in log-star depth. Sorting, arithmetical operations, phase estimation, and the quantum Fourier transform can also be approximated in constant depth.