Quantum Gate Decomposer | mysimulator.uk

Target unitary U

Axis polar θ_n 90°

Axis azimuth φ_n 0°

Rotation angle θ 90°

Compiled sequence Rz(α)·Ry(β)·Rz(γ)

γ (1st, Rz)

0.00°

β (2nd, Ry)

0.00°

α (3rd, Rz)

0.00°

Verification

Max |ΔU| element error

0.0e+0

Reconstruction

–

Target U–

Rz(α)·Ry(β)·Rz(γ) reconstructed–

How it works

A quantum programming language's compiler never gets to assume the hardware natively supports every rotation an algorithm asks for — it must rewrite an arbitrary single-qubit unitary U into gates the device actually implements. Every U ∈ SU(2) (2×2 complex, unitary, determinant 1) can be written exactly as U = R_z(α)·R_y(β)·R_z(γ) for real Euler angles α, β, γ — this is the same ZYZ decomposition real compilers (Qiskit's transpiler, Cirq's optimizers) use to target Rz/Ry-only or Rz/√X-only hardware.

beta  = 2*atan2(|U10|, |U00|)
alpha = arg(U10) - arg(U00)
gamma = -arg(U00) - arg(U10)
U =?= Rz(alpha) . Ry(beta) . Rz(gamma)   (verified below)

The sliders build a target U = exp(−iθ/2·(n_xX + n_yY + n_zZ)), the general axis–angle rotation on the Bloch sphere — any point you can dial in is a valid SU(2) target.
α, β, γ are computed directly from U's matrix elements by the formulas above, not looked up or faked.
The reconstructed matrix multiplies the three decomposed gates back together; the max element-wise error against the true target is the actual proof the decomposition is correct, not just plausible.

Single-Qubit Gate Decomposition: Rz·Ry·Rz Compiler