Hadamard gate ($H$)

The Hadamard gate is the unitary operatorwhere $I$ is the identity operator. It maps the computational basis to the Hadamard basis:

These outputs have equal computational-basis measurement probabilities, but their relative quantum phase differs. Applying another Hadamard gate sends them to different definite outputs. Calling this gate a classical coin toss would discard precisely the information that makes this possible. ^[ref].

Starting from $|0⟩$, compare two quantum circuits, reading matrix products from right to left:Here $Z$ is the Pauli Z gate. After the first Hadamard gate, the probability amplitudes are $(1/\sqrt{2},1/\sqrt{2})$. Inserting $Z$ changes them to $(1/\sqrt{2},-1/\sqrt{2})$ without changing their squared magnitudes. At the final gate, the contributions to output zero cancel and those to output one add. Without $Z$, cancellation occurs at output one instead. This is controlled interference, with the quantum phase determining which answer emerges.

The same calculation works for any phase gate $P_{\varphi }=\mathrm{diag}(1,e^{i\varphi })$:For example, $\varphi =\pi /3$ gives $P(1)=1/4$, while $\varphi =\pi /2$ gives $P(1)=1/2$. The output-one amplitude at $\varphi =\pi /2$ is $(1-i)/2$: its two complex contributions add before the Born rule is applied. A phase gate alone leaves the populations unchanged; recombination converts its relative quantum phase into observable probabilities.

If quantum decoherence completely removes the computational basis coherence between the gates, the intermediate density matrix becomes $I/2$. Both the phase gate and final Hadamard gate leave $I/2$ unchanged, so $P(1)=1/2$ for every $\varphi$. The disappearing fringe explains why preserving quantum coherence matters: it enables the gate sequence to carry phase information into the final measurement.

For $|\pm i⟩=(|0⟩\pm i|1⟩)/\sqrt{2}$, direct matrix multiplication by the Hadamard gate gives coefficients $(1\pm i,1\mp i)/2$, proving the displayed identities. This exchanges the two Pauli eigenstates of the Pauli Y gate and agrees with $HYH=-Y$. The phases in the formula are global phases for each separate state; the relative phase inside each state is essential.

For $P_{\alpha }=\mathrm{diag}(1,e^{i\alpha })$, conjugation by the Hadamard gate turns a relative quantum phase into an $x$-axis Bloch vector rotation. In the convention $\rho =(I+\mathbf{P}\cdot \mathbf{\sigma })/2$, the map is $(P_x,P_y,P_z)\mapsto (P_x,P_y\cos \alpha -P_z\sin \alpha ,P_y\sin \alpha +P_z\cos \alpha )$. An $x$-polarized input is unchanged, whereas a $z$-polarized input gives a sinusoidal population difference. A maximally mixed input remains maximally mixed under every unitary time evolution.

A flat reference over $k$ adjacent Fock states has overlap $r_k$ with a one-photon shift and $s_k$ with a two-photon shift. These overlaps, obtained by counting shared number states, determine the reduced qubit channel of a number-conserving Hadamard interaction. For $k\ge 2$, its squared fidelity with the desired pure output is $1-(1+xz)/(2k)$ for input Bloch vector $(x,y,z)$. Its worst-case infidelity is $3/(4k)$ because $|xz|\le 1/2$. At $k=1$ the two-shift overlap is zero, not $-1$, and the fidelity is $1/2+(y^2-xz)/4$. Broad coherent number support, rather than merely a large fixed photon number, suppresses distinguishability of the outgoing field states.

The uniform superposition of all $n$-bit strings is