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Apply Hadamard gates to $|0^m⟩$ to prepareReversibly compute the efficiently decidable predicate $[x<Q]$ into an ancillary qubit and measure it. The success probability is $Q/2^m>1/2$, and conditioned on success the first register is exactlyRestart after a failed measurement. After $L=\lceil {\log }_2(1/\delta )\rceil$ attempts, the probability that all attempts fail is below $\delta$, so the state is prepared with probability at least $1-\delta$ using a number of gates polynomial in $m$ and $\log (1/\delta )$.

Apply a Hadamard gate to Alice's qubit, followed by a controlled-NOT gate from Alice to Bob and another from Alice to Charlie:Thus this circuit gives the required quantum circuit preparation of a three-qubit GHZ state.

A unitary operation on a fixed number of qubits used as a component of a quantum circuit. Examples include the Hadamard gate and Controlled-Z gate. Measurements are described separately by quantum measurement; a gate is a reversible operation on its full input state.

In an $N$-dimensional computational basis, the uniform superposition state isMeasurement in that basis gives the uniform distribution on a finite set. For $N=2^n$, Hadamard gates prepare it from $|0^n⟩$. Its reflection $2|s_N⟩⟨s_N|-I$ is a Grover diffusion operator.