Lesson S4 · Special topics

Quantum computing basics

A qubit is not a faster bit. Draw its state as an arrow on a sphere, turn it with a Hadamard gate into a superposition, and measure it to get a plain 0 or 1 at random. Then watch two H gates interfere so the answer is always 0, link two qubits by entanglement, and see Grover search boost the right answer. Finish with why noise (decoherence) is the big problem, how error correction fights it, and why a quantum chip is a special helper, not a faster CPU.

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Fig S4 · Interactive

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01A qubit is in the state (|0⟩ + |1⟩)/√2. You measure it once. What do you get?

Each amplitude is 1/√2, and squaring it gives 1/2. A measurement always returns one plain bit, here 0 or 1 with a 50% chance each.

02What does the Hadamard gate (H) do to a qubit in |0⟩?

H rotates the arrow from the top of the sphere to the equator: (|0⟩ + |1⟩)/√2, an equal superposition.

03You apply H, then H again, to |0⟩ and measure. Why do you always get 0?

Amplitudes are signed numbers. After the second H the two contributions to |1⟩ are +0.5 and -0.5 and cancel; the contributions to |0⟩ add to 1.

04Two qubits are entangled in the state (|00⟩ + |11⟩)/√2. You measure both. Which results can you see?

Each qubit alone is a 50/50 coin, but the two results always agree. That perfect agreement, with neither qubit holding a set value beforehand, is one sign of entanglement.

05Which statement about today's quantum computers is true?

Current machines range from tens up to about a thousand noisy qubits, with no full error correction (Preskill's "NISQ" era). They are special-purpose helpers driven by classical computers, not faster general CPUs.

Key terms

Glossary

Qubit: A quantum bit. Its state is a|0⟩ + b|1⟩; measuring it gives 0 with probability |a|² and 1 with probability |b|².
Superposition: A qubit state that is a weighted blend of |0⟩ and |1⟩. The weights (amplitudes) set the odds of each measurement result.
Amplitude: The signed number in front of |0⟩ or |1⟩. Square it to get a probability; amplitudes can cancel or add up.
Measurement: Reading a qubit. It gives one plain 0 or 1 at random and leaves the qubit in that state.
Hadamard gate (H): A quantum gate that turns |0⟩ into (|0⟩ + |1⟩)/√2 and |1⟩ into (|0⟩ - |1⟩)/√2. Applying it twice undoes it.
Interference: Amplitudes adding up or cancelling, which makes some results more likely and others impossible. Quantum algorithms are built on it.
Entanglement: A link between qubits: measured, their results are correlated (here they always agree) even though neither qubit had a set value beforehand. Measured in other ways, the correlations are stronger than any pair of separate bits can produce.
Decoherence: The loss of a qubit's quantum state as it interacts with its surroundings (heat, stray fields). Error correction fights it.

Fine print

Notes and sources

The figure follows the standard textbook model (Nielsen and Chuang, Quantum Computation and Quantum Information): a qubit is a|0> + b|1>, a measurement gives 0 with probability |a|^2. Only real amplitudes are used, so every arrow stays on one circle of the sphere.
The measurement results shown (1 0 0 1 1 0 and 00 11 11 00) are one example run. Real results are random; only the probabilities are fixed.
Grover search needs about (pi/4)·sqrt(N) steps for N items (Grover, 1996); for N = 4 one step finds the marked item with certainty. Shor (1994; arXiv 1995) showed a quantum computer could factor large numbers efficiently.
Google Quantum AI, Willow (December 2024): 105 qubits, T1 approaching 100 µs; a 101-qubit distance-7 surface code lowered the logical error rate by 2.14x per step in code size (Nature, 2025).
"NISQ", noisy intermediate-scale quantum, is John Preskill's name (2018) for today's machines: useful for experiments, too noisy for long algorithms.

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