Quantum Mechanics 101 · Unit 6: Quantum Technology & Frontiers

Quantum Academy · Chapter 23: Qubits & Quantum Computing

Computing with amplitudes instead of bits

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Instructions. Work through these questions with the lesson closed first, then check yourself against it. For calculations, show every step. Useful facts: H sends |0⟩ to (1/√2)(|0⟩ + |1⟩) and |1⟩ to (1/√2)(|0⟩ − |1⟩); on the Bloch sphere P(0) = cos²(θ/2); a classical simulator stores 16 bytes per amplitude; Grover needs about (π/4)√N rounds. One point for each vocabulary match and each question; the challenge is a bonus.

Part A Vocabulary matching

Write the letter of the matching definition in the blank beside each term.

1.Qubit
2.Probability amplitude
3.Bloch sphere
4.Unitary gate
5.Hadamard gate
6.Phase flip
7.CNOT gate
8.Bell state
9.Oracle
10.Quantum Fourier transform
11.Coherence time
12.Logical qubit

A.The quantum version of the discrete Fourier transform, used in Shor's algorithm to make amplitudes interfere so that measurement reveals a period.
B.A two-level quantum system used to store information, whose state is a normalized superposition α|0⟩ + β|1⟩ of two basis states.
C.A two-qubit gate that flips the target qubit if and only if the control qubit is 1; with a superposed control it creates entanglement.
D.How long a qubit keeps a well-defined relative phase before its environment scrambles it, often written T₂.
E.A unit sphere on which every pure single-qubit state is a point: |0⟩ at the north pole, |1⟩ at the south pole, equal superpositions on the equator.
F.A qubit encoded across many physical qubits by an error-correcting code, so that errors can be detected and fixed without reading the encoded information.
G.A complex number attached to each possible outcome; its squared magnitude is the outcome's probability, and amplitudes can cancel through interference.
H.A black-box subroutine that evaluates a function inside a quantum algorithm, often by marking chosen inputs with a minus sign.
I.A reversible operation that preserves total probability; on a single qubit it is a rotation of the Bloch sphere.
J.One of four maximally entangled two-qubit states, such as (1/√2)(|00⟩ + |11⟩), made from |00⟩ by a Hadamard followed by a CNOT.
K.The action of the Z gate: it leaves |0⟩ unchanged and multiplies |1⟩ by −1, changing interference without changing 0-or-1 probabilities.
L.The gate that maps |0⟩ to |+⟩ and |1⟩ to |−⟩ and back again, creating and undoing equal superpositions.

Part B Questions

1.
A qubit is in the state α|0⟩ + β|1⟩. Which statement about measuring it in the 0-or-1 basis is correct?Circle one
1. AYou read out the values of α and β directly
2. BYou get 0 with probability |α|² or 1 with probability |β|², and the superposition is lost
3. CYou get the weighted average |α|² × 0 + |β|² × 1
4. DYou always get 0 if |α| is larger than |β|, and 1 otherwise
2.
Applying the Z gate to a qubit in the state |+⟩ changes the probabilities of measuring 0 and 1.Circle one
TrueFalse
3.
Quantum gates must be operations: they preserve total probability and can always be undone.Fill in the blank
4.
Two qubits start in |00⟩. Which circuit produces the Bell state (1/√2)(|00⟩ + |11⟩)?Circle one
1. ACNOT first, then H on the first qubit
2. BH on both qubits
3. CX on the first qubit, then CNOT with the first qubit as control
4. DH on the first qubit, then CNOT with the first qubit as control
5.
A qubit is in the state (√3/2)|0⟩ + (1/2)|1⟩. Find the probabilities of measuring 0 and 1, and the polar angle θ of its Bloch vector.Show your working
Working
Answer:
6.
A classical simulator stores each amplitude in 16 bytes. How much memory does it need to hold the full state of 40 qubits?Show your working
Working
Answer:
7.
Grover's algorithm searches N = 10⁸ unsorted items for one marked item. About how many rounds does it need, how many checks does a classical search need on average, and what is the ratio?Show your working
Working
Answer:
8.
Shor's algorithm reduces factoring to finding a period. For N = 15 and a = 7, list the remainders of 7, 7², 7³ and 7⁴ when divided by 15, find the period r, and then find the factors from the greatest common divisors of 15 with the numbers one below and one above 7 raised to the power r/2.Show your working
Working
Answer:
9.
In two or three sentences, explain why 'a quantum computer tries every answer at once' is misleading.Short answer
10.
Which hardware platform holds atoms in grids of focused laser beams and entangles them by exciting them to Rydberg states?Circle one
1. ASuperconducting circuits
2. BTrapped ions
3. CNeutral atoms
4. DSpin qubits in silicon
11.
Because quantum states cannot be copied, quantum error correction is impossible.Circle one
TrueFalse
12.
Grover's algorithm gives a speed-up over classical unstructured search.Fill in the blank
13.
What is a logical qubit, and what does it mean for hardware to operate 'below threshold'?Short answer

Part C Mad scientist challenge bonus

Scale Willow. In December 2024 Google's distance-7 surface code failed in 0.143 percent of cycles, and each increase of the distance by 2 divided the logical error rate by Λ ≈ 2.14. Assume Λ stays at 2.14 as the code grows. (a) What error rates does this imply for distances 5 and 3? (b) What is the smallest odd distance whose logical error rate is at most 10⁻⁶ per cycle? (c) A distance-d surface-code patch uses 2d² − 1 physical qubits. How many is that per logical qubit at your answer to (b), and how many for 1,000 logical qubits? (d) Give two reasons real requirements might differ.

Working