Quantum Information Processing, Vol. 5, No. 3, June 2006 (© 2006)
On the Role of Hadamard Gates in Quantum
D. J . Shepherd
Received October 25, 2005; accepted March 23, 2006; Published online May 27, 2006
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
KEY WORDS: Quantum depth in quantum circuits; Fourier Hierarchy; Shor’s
algorithm with Toffoli gates only; lower-bounding Grover’s algorithm.
PACS: 03.67. Lx; 03.67. Ta.
A Quantum Circuit (or Quantum Logic Network) is usually presented as
being composed both of wires that carry qubits and gates that tap those wires
to modify the qubits they carry.
In Sec. 2, we specify the notation used
to describe computation with quantum circuits, and specify exactly which
features we shall be allowing within the kinds of circuits we wish to consider.
The main focus is to enquire about the difference it makes if we
limit to using ‘classical’ gates (ones which preserve the set of computa-
tional basis states) and ‘global Hadamard transforms’ (where a Hadam-
ard gate is applied once to each qubit). We will show in Sec. 3 that our
imposed limitations do not limit computational power in any real sense,
Department of Computer Science, University of Bristol, Bristol, B58 ITH, UK. E-mail:
1570-0755/06/0600-0161/0 © 2006 Springer Science+Business Media, Inc.