Combining qubits multiplies their state spaces via the tensor product, so n qubits live in a 2^n-dimensional space — and the states that do not factor apart are exactly the entangled ones.
Why it matters
This is where exponential scaling actually enters the theory, and where entanglement stops being a metaphor and becomes a precise mathematical statement.
After this you will be able to
One qubit needs 2 amplitudes, two qubits need 4, three need 8. At 300 qubits you would need more amplitudes than there are atoms in the observable universe, which is why classical simulation of quantum systems fails so quickly.
A combination lock. Each extra dial multiplies the number of combinations rather than adding to it. Quantum states are similar, except the system carries a number for every combination at once.
Because the state space is exponentially large, quantum computers have exponential memory.
You cannot store or retrieve those amplitudes. 300 qubits give you 300 bits of readout. The exponential space is a computational medium, not storage.
This exponential space is why quantum computers exist at all — Feynman proposed them precisely because simulating quantum physics classically is intractable.
Hand-picked and ordered. If you only have time for one, take the first.
A rigorous but genuinely well-taught course by John Watrous covering single systems, multiple systems, quantum circuits and the core protocols. The cleanest bridge from intuition to real notation.
Free, self-contained and rigorous, with exercises. Widely used as the standard course text in Europe and the best single free replacement for a paid textbook.
Learn by writing code in the browser, exercise by exercise, from single qubits up to variational algorithms. The best option if you retain things by typing rather than reading.
Topics that list Multi-qubit states & tensor products as a prerequisite.
A small finite set of gates — for example Clifford+T, or CNOT plus arbitrary single-qubit rotations — suffices to approximate any unitary to arbitrary precision.
IntermediateA protocol that moves an unknown qubit state from A to B using one shared entangled pair and two classical bits — destroying the original in the process.
IntermediateThe mirror image of teleportation: using one pre-shared entangled pair, you can send two classical bits by transmitting only one qubit.
IntermediateThe teaching algorithms: contrived problems where a quantum computer needs one oracle query and a classical computer needs many, built entirely out of Hadamards and phase kickback.
IntermediateThe discrete Fourier transform applied to amplitudes, implementable in O(n^2) gates on n qubits — exponentially fewer operations than the classical FFT needs on 2^n numbers.
IntermediateAn experimentally testable inequality that any local hidden-variable theory must satisfy — and that quantum mechanics, and reality, provably violate.
IntermediateA protocol for two parties to agree on a shared secret key such that any eavesdropper necessarily disturbs the transmission and is detected — security resting on physics rather than on computational hardness.
IntermediateEncoding one logical qubit across many physical qubits so that errors can be detected and corrected by measuring stabilisers — without ever measuring, and thus destroying, the logical state.
AdvancedThe rigorous framework beyond pure states and unitaries: density matrices, quantum channels, POVMs, entropy, distance measures and channel capacities.
Advanced