We study a reduced quantum circuit computation paradigm in which the only allowable gates either permute the computational basis states or else apply a “global Hadamard operation”, i.e. apply a Hadamard operation to every qubit simultaneously. In this model, we discuss complexity bounds (lower-bounding the number of global Hadamard operations) for common quantum algorithms: we illustrate upper bounds for Shor’s Algorithm, and prove lower bounds for Grover’s Algorithm. We also use our formalism to display a gate that is neither quantum-universal nor classically simulable, on the assumption that Integer Factoring is not in BPP.
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Shepherd, D.J. On the Role of Hadamard Gates in Quantum Circuits. Quantum Inf Process 5, 161–177 (2006). https://doi.org/10.1007/s11128-006-0023-4
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DOI: https://doi.org/10.1007/s11128-006-0023-4
Keywords
- Quantum depth in quantum circuits
- Fourier Hierarchy
- Shor’s algorithm with Toffoli gates only
- lower-bounding Grover’s algorithm