Single-Qubit Gates and Circuit Composition

Beginner Quantum Gates

Created by Pavel · 11.03.2026 at 14:32 UTC

Single-qubit gates are $2\times 2$ unitaries composed in time order. $X$ swaps populations, $Z$ adds phase to the $|1\rangle$ component, $H$ bridges computational and Hadamard bases; identities such as $HZH=X$ show how basis changes move effects between “phase” and “bit-flip” stories.

Order matters because matrices rarely commute—swap two gates and you may change the entire measurement distribution. [1].

Sources

[1]https://en.wikipedia.org/wiki/Quantum_logic_gate Return to text

University approvals: 0

Tasks

Question 1

Applying $X$ to $|0\rangle$ gives:

Hint

X is a bit flip.

$|0\rangle$

$|1\rangle$

$\frac{|0\rangle+|1\rangle}{\sqrt{2}}$

-$|0\rangle$

Question 2

Applying $H$ to $|0\rangle$ gives:

Hint

The equal positive superposition.

$|1\rangle$

$\frac{|0\rangle-|1\rangle}{\sqrt{2}}$

$\frac{|0\rangle+|1\rangle}{\sqrt{2}}$

$|0\rangle$

Question 3

What does Z primarily change on a qubit state?

Hint

Not population swap.

phase

amplitude only

wire label

backend latency

Question 4

A short circuit with H then Z differs from H then X mainly because:

Hint

Compare action type.

Z and X are physically identical

Z acts on phase while X acts on computational basis populations

X cannot be represented as a matrix

H cannot compose with Z

Question 5

In one sentence, why do we call these operations unitary?

Hint

Norm preservation.

they preserve total probability

none of the above

all of the above

insufficient information

Question 6

Starting from $|0\rangle$, what is $P(\text{measure }1)$ after the sequence $H \to Z \to H$?

Hint

H Z H equals X.

Question 7

What does the identity H Z H = X most clearly illustrate?

Hint

Think basis transformation.

Gate order never matters

Basis change can convert phase effects into bit-flip effects

Z and X are numerically identical in every basis

Hadamard is not unitary

Question 8

Why is non-commutativity important when designing circuits?

Hint

Matrix multiplication order matters in general.

Because any gate sequence yields the same output

Because changing order can change the final state and measured distribution

Because only classical circuits are order-sensitive

Because unitary matrices always commute

Question 9

Implement hadamard_plus_amplitudes() -> tuple[complex, complex] returning complex amplitudes $(a_0, a_1)$ of $H|0\rangle$ in the computational basis, i.e. $(1/\sqrt2, 1/\sqrt2)$.

Hint

Use math.sqrt for denominators; return two complex numbers.

import math


def hadamard_plus_amplitudes() -> tuple[complex, complex]:
    # TODO: return (1/sqrt(2), 1/sqrt(2)) as complex values
    pass

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

Runtime output (stdout/stderr)

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

Card Info

Topic: Quantum Gates
Difficulty: Beginner
Completed: 0 users

Creator

Pavel