Source-verifiedLevel 1

Pauli-Y Gate

The Pauli-Y gate performs a 180-degree rotation about the y-axis of the Bloch sphere, mapping |0⟩ to i|1⟩ and |1⟩ to -i|0⟩.

What it means

The Pauli-Y gate combines both a bit-flip and a phase-flip operation.Its matrix representation is sigma_y = [[0,-i],[i,0]].On the Bloch sphere, it corresponds to a pi rotation about the y-axis.Like all Pauli matrices, it is Hermitian and unitary (Y = Y-dagger = Y-inverse), self-inverse (Y-squared = I), and has eigenvalues plus/minus 1.The Pauli-Y gate relates to X and Z via Y = iXZ.It appears in quantum error correction for combined bit-phase errors and in the decomposition of arbitrary single-qubit rotations.

Everyday analogy

The Pauli-Y gate is like simultaneously flipping a coin and adding a twist -- it not only swaps heads and tails (bit-flip) but also adds a specific rotation (phase-flip) in one combined operation.
Think of it as a gymnast doing a combined flip and twist -- two distinct transformations executed as one smooth motion.

Common misconceptions

  • The Pauli-Y gate is NOT simply X followed by Z -- it is Y = iXZ, with the crucial factor of i that affects the global phase.
  • The imaginary factors (i and -i) in the Y gate are NOT negligible -- they are essential for maintaining unitarity and affect interference in multi-qubit circuits.

Key takeaways

  • The Pauli-Y gate combines bit-flip and phase-flip: Y|0⟩ = i|1⟩, Y|1⟩ = -i|0⟩.
  • On the Bloch sphere, it is a 180-degree rotation about the y-axis.
  • Y = iXZ, connecting all three Pauli matrices algebraically.

Check your understanding

What are the eigenvalues of the Pauli-Y gate?

  1. A.0 and 1
  2. B.+1 and -1
  3. C.+i and -i
  4. D.1 and i
Show the answer

Answer: B. +1 and -1

Why: Like all Pauli matrices, the Pauli-Y gate has eigenvalues +1 and -1.

Builds on

QubitDirac (Bra-Ket) NotationBloch Sphere

Primary source: Barenco et al., Phys. Rev. A 52, 3457 (1995), doi:10.1103/PhysRevA.52.3457

Learn it hands-on

This concept is part of a 46-level curriculum with an interactive simulator and Lumen, a tutor whose answers are verified before you see them. Levels 1–5 are free.

Start freeSee the curriculum