Rotation gates and parameterised circuits

Meet the angle-based rx, ry and rz gates, build circuits whose angles are supplied later with Parameter and see the idea behind variational algorithms.

Turning by an angle: rx, ry, rz

Gates such as H and X always do the same thing. Rotation gates take an angle and turn the qubit by that much. For a |0⟩ qubit turned with ry(θ), the probability of measuring 1 is sin²(θ/2): θ = 0 does nothing, θ = π/2 gives 50–50 like H and θ = π gives a certain 1 like X.

rz(θ) and p(θ) only change the phase. Gates you already know are special angles of these: Z = p(π), S = p(π/2), T = p(π/4).

import math
from qiskit import QuantumCircuit

qc = QuantumCircuit(1)
qc.ry(math.pi / 3, 0)   # probability of 1 is sin²(π/6) = 0.25
qc.measure_all()

Parameter: supply the angle later

When you want to try the same circuit with different angles you do not need to build a new circuit each time. Parameter leaves the angle as a name and assign_parameters gives it a value before running. The loop below gives always 0, 50–50 and always 1 for the angles 0, π/2 and π.

import math
from qiskit import QuantumCircuit
from qiskit.circuit import Parameter
from qiskit.primitives import StatevectorSampler

theta = Parameter('theta')
qc = QuantumCircuit(1)
qc.ry(theta, 0)
qc.measure_all()

sampler = StatevectorSampler()
for angle in [0, math.pi / 2, math.pi]:
    circuit = qc.assign_parameters({theta: angle})
    counts = sampler.run([circuit], shots=1000).result()[0].data.meas.get_counts()
    print(round(angle, 2), counts)

Why it matters: variational algorithms

The methods most tried on today’s noisy devices rely on parameterised circuits. In variational algorithms such as VQE and QAOA the quantum computer runs a parameterised circuit and computes an energy or cost with the Estimator, while a classical optimiser adjusts the angles to make that value smaller. The loop repeats until the best angles are found.

“Quantum neural networks” in quantum machine learning follow the same idea: the weights being trained are the angles of rotation gates.

Try it yourself

The playground has no rotation gates, but you can see two special angles: ry(π/2) gives the same 50–50 result as H, and ry(π) the same certain 1 as X. Compare the two examples.

Open the circuit in your browser

The circuit playground is in Turkish. Button guide: “1000 kez ölç” = measure 1000 times, “Durum vektörünü göster” = show the state vector, “Bu devrenin Qiskit kodu” = this circuit’s Qiskit code, “Geri al” = undo, “Temizle” = clear. Click a gate in the circuit to remove it.

Key points

For a |0⟩ qubit turned with ry(θ), the probability of measuring 1 is sin²(θ/2).
rz and p only change the phase: Z = p(π), S = p(π/2), T = p(π/4).
Parameter leaves an angle as a name; assign_parameters gives it a value before running.
Variational algorithms tune the angles with a classical optimiser.

Test yourself

For a |0⟩ qubit, which θ in ry(θ) gives a 50% chance of measuring 1?

π/4
π/2
π

Show the answer

π/2. The probability of 1 is sin²(θ/2). For θ = π/2 that is sin²(π/4) = 0.5. π/4 gives about 15% and π gives 100%.

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