Amplitudes, tensor products, and phase-only gates

Advanced Quantum Machine Learning

Created by Pavel · 17.03.2026 at 07:05 UTC · 1 completed

Tensor products multiply single-qubit amplitudes when the state stays separable; that is how you read off coefficients such as $|11\rangle$ in a product of $R_y$ rotations. Separating population changes from phase-only $R_z$ factors avoids false intuition about measurement.

Global phase on $|0\rangle$ is unobservable, but once superposition exists, relative phases steer interference.

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Tasks

Question 1

For the vector $(0.3, 0.7, 0.5, 0.1)$, what is the squared norm before normalization?

Hint

Add the squares of all entries.

$0.64$

$0.84$

$1.16$

$1.84$

Question 2

For the product state $R_y(x_1)|0\rangle \otimes R_y(x_2)|0\rangle$, what is the coefficient of $|11\rangle$?

Hint

Each qubit must contribute its $|1\rangle$ component.

$\cos(x_1/2)\cos(x_2/2)$

$\cos(x_1/2)\sin(x_2/2)$

$\sin(x_1/2)\cos(x_2/2)$

$\sin(x_1/2)\sin(x_2/2)$

Question 3

What is the immediate effect of applying only $R_z(\theta)$ to the initial state $|0\rangle$?

Hint

Compare state change with probability change.

It changes $P(0)$ and $P(1)$ immediately

It maps $|0\rangle$ to $|1\rangle$

It adds only a global phase to $|0\rangle$

It creates entanglement with an implicit ancilla

Question 4

Suppose every qubit starts in $|0\rangle$ and the encoding applies only single-qubit $R_z(x_i)$ gates, with no Hadamards and no entangling gates. What can be said about measurement probabilities in the computational basis?

Hint

Generalize the one-qubit $R_z$ observation.

They depend strongly on all $x_i$

They are unchanged from the initial state

They become uniform over all basis states

They cannot be defined because the state is complex-valued

Question 5

Implement ry11_amplitude(x1: float, x2: float) -> float returning the $|11\rangle$ amplitude coefficient $\sin(x_1/2)\sin(x_2/2)$ for a product of single-qubit $R_y$ rotations applied to $|0\rangle$ each.

Hint

import math; use sin.

import math


def ry11_amplitude(x1: float, x2: float) -> float:
    # TODO
    pass

Starter code is prefilled; replace TODO blocks with your solution.

Runtime output (stdout/stderr)

2 test cases will be used for grading

Run checks runtime behavior only. Final correctness is evaluated when you submit.

Card Info

Topic: Quantum Machine Learning
Difficulty: Advanced
Completed: 1 users

Creator

Pavel