Gates · Rotation gate

RZ rotation gate

A parametrized single-qubit rotation about the Z axis that adds a relative phase between |0⟩ and |1⟩ without changing measurement probabilities in the computational basis.

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A small worked example, computed in your browser. It is not a result from this record's papers.

RZ(θ), a rotation that only moves phase

One qubit, put into |+⟩ = (|0⟩ + |1⟩)/√2 first, because on |0⟩ or |1⟩ alone this gate changes nothing a measurement or another gate can ever detect.

Step 1 of 2 · H

Probability of each outcome after this step

050%
150%

Puts the qubit into an equal superposition. Without this there is no second amplitude for the gate's phase to be measured against, and a phase on the whole state is not observable.
Probability spreads from 1 outcome to 2. All 2 are equally likely.
Multiplies the |1⟩ half of the state by a phase of π/2 at this angle, and leaves |0⟩ alone. RZ splits the angle between the two halves rather than putting all of it on |1⟩, so it differs from P by a phase on the whole state — which is to say, by nothing any measurement can find. The two outcomes stay equally likely — the difference is carried entirely in the phase between them, which is what the panel beside the bars shows.
The outcome probabilities do not move at all. What this step changes is phase — 2 distinct relative phases across the state now — which no measurement can see until a later step turns it back into probability.

Readout Measuring gives 0 or 1, each half the time — exactly as it did before the gate. A single measurement of this qubit cannot detect this gate at all. Its effect becomes visible only when the phase is turned back into probability by a later interference step, which is what every algorithm that uses it does.

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Steps1
Wires1
Qubits1
Depth1 gate (often 0-duration on hardware)
Parameterθ (radians)
Papers2

Definition

R_Z(\theta) = e^{-i\theta Z/2} = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}

$R_Z$ is diagonal in the computational basis, so it commutes with any other $R_Z(\phi)$ and with measurement in the $Z$ basis.

Action on basis states

R_Z(\theta)|0\rangle = e^{-i\theta/2}|0\rangle, \qquad R_Z(\theta)|1\rangle = e^{i\theta/2}|1\rangle

Each basis state only picks up a phase, so $|\langle 0|R_Z(\theta)|0\rangle|^2 = |\langle 1|R_Z(\theta)|1\rangle|^2 = 1$ for every $\theta$ — a $Z$ rotation is invisible to a $Z$ -basis measurement on its own. Its effect only shows up on a qubit already in superposition, e.g. $R_Z(\theta)\tfrac{1}{\sqrt2}(|0\rangle+|1\rangle) = \tfrac{1}{\sqrt2}(e^{-i\theta/2}|0\rangle + e^{i\theta/2}|1\rangle)$ , which differs from the input by a relative phase of $e^{i\theta}$ .

Key identities

$P(\lambda) = e^{i\lambda/2}R_Z(\lambda)$ : the OpenQASM phase gate $P$ is $R_Z$ up to an overall phase — $P$ fixes the $|0\rangle$ amplitude at exactly $1$ instead of splitting the phase symmetrically.
$S = R_Z(\pi/2)$ and $T = R_Z(\pi/4)$ up to global phase, so $R_Z$ generalizes the whole Clifford/non-Clifford phase-gate ladder to a continuous parameter.
$R_X(\theta) = H R_Z(\theta) H$ and $R_Y(\theta) = R_Z(-\pi/2) R_X(\theta) R_Z(\pi/2)$ : composing conjugations by $H$ and by $R_Z(\pm\pi/2)$ lets a compiler reach any rotation axis from $R_Z$ alone.
On superconducting hardware, $R_Z(\theta)$ is frequently realized as a virtual zero-duration frame update rather than a physical microwave pulse, making it effectively free and noiseless compared to $R_X$ / $R_Y$ .

Quantum vs classical

Classical baseline

Use a classical state-vector or matrix simulation at the same width, precision, and measurement objective.

Quantum claim

The quantum record demonstrates a state or operator behavior; it does not make classical simulation or communication costs disappear.

How to compare

Compare fidelity, samples, gate depth, noise, memory, and the cost of preparing and reading the state.

References

Quantum Computation and Quantum Information: 10th Anniversary Edition
Michael A. Nielsen and Isaac L. Chuang · 2010
Standard reference for single- and multi-qubit gate matrices, Euler-angle decomposition, and universal gate sets.
Efficient Z-Gates for Quantum Computing
David C. McKay, Christopher J. Wood, Sarah Sheldon, Jerry M. Chow, Jay M. Gambetta · 2016
Shows how a fixed physical X_{π/2} pulse (the SX gate) combined with zero-duration virtual Z rotations synthesizes any single-qubit unitary on superconducting hardware.

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