Level 7 · Capstone Topics
Practice Problem Set
This is the capstone exercise for the QuantumAtlas Learning Center — a set of problems spanning everything from Level 0 through Level 7. Try working through each one yourself before revealing the worked solution. There's no grading or submission — this is purely for your own understanding.
Problem 1: Basic probability
A qubit is in state |ψ⟩ = 0.6|0⟩ + β|1⟩, where β is a positive real number. What is β, and what is the probability of measuring 1?
Show worked solution
Since |α|² + |β|² = 1 (see our Qubit entry), and α = 0.6, we have 0.36 + β² = 1, so β² = 0.64, giving β = 0.8. The probability of measuring 1 is |β|² = 0.64, or 64%. You can verify this using our Probability Amplitude Calculator tool.
Problem 2: Identifying entanglement
Is the state (1/2)(|00⟩ + |01⟩ + |10⟩ + |11⟩) entangled? Why or why not?
Show worked solution
This state is not entangled — it's a product state. You can verify this by checking whether it factors into a tensor product of two single-qubit states: (1/√2)(|0⟩ + |1⟩) ⊗ (1/√2)(|0⟩ + |1⟩) expands to exactly this state. Compare this to the genuinely entangled Bell state (1/√2)(|00⟩ + |11⟩), which cannot be factored this way — see our Multi-Qubit Systems & Tensor Products article for the full explanation.
Problem 3: Gate sequences
Using our Mini Quantum Circuit Builder tool (or by hand), determine what applying H, then X, then H does to a qubit starting in state |0⟩.
Show worked solution
H applied to |0⟩ gives (1/√2)(|0⟩ + |1⟩). Applying X (which swaps the |0⟩ and |1⟩ components) to this gives (1/√2)(|1⟩ + |0⟩), which is the same state — equal superposition is unchanged by a bit flip. Applying H again returns the qubit to |0⟩. So H-X-H, applied to |0⟩, returns |0⟩ unchanged. Try this in the circuit builder tool to confirm.
Problem 4: Algorithm speedup classification
Without looking it up, guess whether each of these offers an exponential, quadratic, or no proven speedup: (a) Shor's Algorithm, (b) Grover's Algorithm, (c) QAOA.
Show worked solution
(a) Shor's Algorithm offers an exponential speedup for factoring. (b) Grover's Algorithm offers a quadratic speedup for unstructured search. (c) QAOA is a heuristic algorithm with no proven speedup over the best classical methods — see our Algorithms Database to check your answers against all 50 entries.
Problem 5: Hardware tradeoffs
A trapped-ion processor has fewer qubits than a superconducting processor, but much higher gate fidelity. Using the concept of algorithmic qubits, explain why the trapped-ion system might still be more useful for running a specific algorithm.
Show worked solution
Algorithmic qubits accounts for both qubit count and gate fidelity together. A processor with fewer, higher-fidelity qubits can run deeper, more complex circuits reliably before errors dominate, while a larger but noisier processor may only support very shallow circuits before results become unreliable. Use our Compare Processors tool to see this tradeoff with real hardware specs.
Problem 6: Teleportation conceptual check
After Alice performs her measurement in the quantum teleportation protocol, has Bob's qubit already become the teleported state, or does he need to wait for something else first?
Show worked solution
Bob's qubit has already collapsed into a state correlated with Alice's original qubit — but Bob doesn't yet know which of the four possible corrections to apply without Alice's classical measurement result. He must wait for her classical message before he can complete the protocol and recover the exact original state. See our Quantum Teleportation lesson for the full step-by-step walkthrough.
Problem 7: Complexity theory reasoning
A friend claims: "Since Shor's Algorithm proves quantum computers are exponentially faster at one NP problem, they must be exponentially faster at all NP-hard problems too." What's wrong with this reasoning?
Show worked solution
Factoring (the problem Shor's Algorithm solves) is believed to be in NP but is not believed to be NP-complete — meaning it isn't representative of the hardest problems in NP. Quantum computers having a speedup for factoring doesn't imply a speedup for NP-hard problems like the traveling salesman problem. See our Quantum Complexity Theory article for the full explanation of this frequently confused distinction.
Problem 8: Design pattern identification
You're designing a new quantum algorithm to estimate an unknown probability more efficiently than classical Monte Carlo sampling. Which design pattern from our Quantum Algorithm Design Patterns article is most relevant?
Show worked solution
This points toward amplitude amplification — and in fact, this is exactly the design pattern behind Quantum Amplitude Estimation, which provides a proven quadratic speedup over classical Monte Carlo methods for precisely this kind of probability estimation task.
How did you do?
If most of these felt approachable, you've built a genuinely solid working understanding of quantum computing — spanning the math foundations, core principles, major algorithms, hardware realities, and the field's complexity-theoretic boundaries. If some problems were tricky, revisit the linked articles for those specific topics; that's exactly what they're there for.