Rotates the qubit around the Y axis of the Bloch sphere by a chosen angle .
RY rotates a qubit around the Y axis of the Bloch sphere. Its distinguishing property is that it produces superpositions with purely real amplitudes. Starting from , an RY rotation creates states like , with no imaginary parts. RX and RZ both introduce imaginary phase factors, but RY does not. This makes RY the cleanest gate for preparing states with a specific probability distribution.
RYI − i Y. Because the Pauli-Y matrix is purely imaginary in the computational basis, the resulting rotation matrix has only real entries: [[, −], []]. This means RY maps real-valued states to real-valued states. No complex phases are introduced. At , RY equals the Y gate up to a global phase (a prefactor that multiplies the entire state and does not affect measurements).
Creates superpositions with tunable measurement probabilities, controlled directly by the angle .
At , this becomes the Y gate up to global phase.
Rotates the Bloch vector by around the Y axis. The motion stays in the XZ plane. From , an RY rotation reaches the equator with a real-valued superposition.
It is tempting to think RX and RY are interchangeable since both create superpositions. That sounds plausible because both tilt the Bloch vector toward the equator. What actually happens is that RY uniquely produces real amplitudes from basis states, while RX introduces imaginary phase factors. This distinction matters for circuit analysis and appears in decomposition theorems.
RY is the workhorse of state preparation. When you need to encode specific probability distributions into a qubit, RY gives you direct, clean control over the amplitudes without introducing unwanted imaginary phases.
Why does RY uniquely produce real amplitudes from |0⟩?