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Rx Gate

The Rx gate performs a rotation by angle theta about the x-axis of the Bloch sphere, enabling continuous single-qubit rotations.

What it means

The Rx(theta) gate is a parameterized single-qubit gate that rotates the qubit state by angle theta about the x-axis of the Bloch sphere.Its matrix is [[cos(theta/2), -i*sin(theta/2)], [-i*sin(theta/2), cos(theta/2)]].When theta = pi, Rx reduces to the Pauli-X gate (up to global phase).The Rx gate is one of the three rotation gates (Rx, Ry, Rz) that together can implement any single-qubit unitary operation.These parameterized gates are central to variational quantum algorithms like VQE and QAOA, where the rotation angles are optimized classically.

Everyday analogy

The Rx gate is like a dial that can rotate a compass needle by any chosen angle around the east-west axis.
Think of it as tilting a globe around its equatorial axis by a precise amount -- small angles give gentle tilts, pi gives a full flip.

Common misconceptions

  • Rx(pi) is NOT exactly the same as the Pauli-X gate -- they differ by a global phase of -i, though this is physically unobservable for single qubits.
  • The angle theta in Rx(theta) is the rotation angle on the Bloch sphere, NOT half the rotation angle -- the factor of 1/2 is already built into the matrix.

Key takeaways

  • Rx(theta) rotates the qubit state by angle theta about the x-axis of the Bloch sphere.
  • Rx(pi) = -iX (Pauli-X up to global phase).
  • Together with Ry and Rz, Rx can decompose any single-qubit unitary.

Check your understanding

What gate does Rx(pi) correspond to (up to global phase)?

  1. A.Hadamard
  2. B.Pauli-X
  3. C.Pauli-Z
  4. D.Identity
Show the answer

Answer: B. Pauli-X

Why: Rx(pi) = -iX. Up to the global phase factor -i, this is the Pauli-X gate.

Builds on

QubitBloch SpherePauli-X Gate

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

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