How quantum computers work

A classical computer encodes bits that are definitively 0 or 1. Logic gates flip and combine those bits with deterministic (or well-characterized probabilistic) rules. Scaling means more transistors, more memory, better algorithms.

A quantum computer encodes qubits whose state can be a superposition of 0 and 1, and whose qubits can be entangled. Computation is a carefully choreographed sequence of unitary operations (gates), followed by measurement that collapses amplitudes into classical outcomes.

The point is not that every step is “both 0 and 1 at once equals infinite parallelism.” Interference — constructive for right answers, destructive for wrong ones — is the real engine. Without a structure that creates useful interference, a quantum device is just an expensive random-bit generator.

Classically hard problems are not automatically good quantum targets. Many NP-hard tasks remain hard on quantum machines under standard complexity beliefs. The interesting cases are problems where quantum linear algebra, interference, or quantum simulation map cleanly onto the question.

Write a single-qubit pure state as |ψ⟩ = α|0⟩ + β|1⟩ with |α|² + |β|² = 1. In polar form, α = cos(θ/2), β = e^{iφ} sin(θ/2). The angles (θ, φ) are coordinates on the Bloch sphere: north pole |0⟩, south pole |1⟩, equator equal superpositions with a relative phase.

Measuring in the computational basis yields 0 with probability |α|² and 1 with |β|², and the state collapses to the observed basis vector. Phase between amplitudes is invisible to that single measurement — but it becomes crucial when gates create interference across paths.

Superposition is not “the qubit is randomly 0 or 1 before you look.” Before measurement there is a definite state vector; randomness appears in the Born-rule sampling of that vector. After measurement, phases and coherences that encoded the prior superposition are gone for that copy.

X (NOT) flips |0⟩↔|1⟩. Z leaves |0⟩ alone and maps |1⟩→−|1⟩ (phase flip). H (Hadamard) maps |0⟩→|+⟩ and |1⟩→|−⟩ — the usual way to create equal superposition from a computational basis state.

Any single-qubit unitary is a rotation of the Bloch vector. Circuits compose gates; global phases don’t affect measurement probabilities, but relative phases do.

To build a Bell state |Φ⁺⟩ = (|00⟩+|11⟩)/√2: start from |00⟩, apply H on qubit A, then CNOT with A as control and B as target. Measuring A alone looks random; measuring B afterward is perfectly correlated.

Entanglement does not allow faster-than-light signaling. Local measurement outcomes are random; only when compared (classically) do correlations appear. That is why teleportation and device-independent protocols still need classical communication channels.

Given a black-box function that marks one (or a few) items in an unstructured list of N, Grover’s algorithm rotates the state in the plane spanned by the uniform superposition and the marked subspace. Each oracle + diffusion iteration boosts the marked amplitude. Optimal iterations ≈ π/4 √N.

Estimate the ground-state energy of a Hamiltonian by preparing a parameterized ansatz on a QPU, measuring Pauli expectations, and letting a classical optimizer update the parameters. Fits NISQ-era exploration of chemistry/materials — but ansatz choice, barren plateaus, and noise mean VQE is not automatic quantum advantage.

Every platform trades coherence, gate speed, connectivity, operating temperature, and manufacturability. Roadmaps converge on needing logical qubits: encode many noisy physical qubits into fewer protected logical ones.

Hover a node to highlight its links. Click a chapter node to jump there. Each earlier chapter also has a smaller local mindmap under its lead.