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What is quantum computing?
A computer that stores and manipulates information using quantum physics, letting it explore many computational paths at once and interfere them so the wrong answers cancel out.
8 resources · BeginnerQubits
The quantum version of a bit. Instead of being 0 or 1, a qubit carries two numbers (amplitudes) that say how much of "0" and how much of "1" it holds — including negative and complex values.
10 resources · BeginnerSuperposition
A qubit holding a nonzero amount of both 0 and 1 at the same time — not "secretly one of them", and not "both in separate universes", but a genuine third kind of state with no classical counterpart.
7 resources · BeginnerMeasurement & collapse
Reading a qubit forces it to pick 0 or 1, with probability equal to the square of the corresponding amplitude — and destroys the superposition in the process.
6 resources · BeginnerQuantum interference
Because amplitudes can be negative (or complex), contributions to a wrong answer can cancel each other out while contributions to the right answer add up.
3 resources · BeginnerEntanglement
Two or more qubits sharing a single joint state, so that their measurement results are correlated in ways no classical shared-secret arrangement can reproduce — even though neither qubit has a definite state of its own.
5 resources · BeginnerQuantum gates
Reversible operations that rotate qubit states. X flips, H creates superposition, Z flips a sign, and CNOT lets one qubit control another — that small set already gets you a long way.
10 resources · BeginnerQuantum circuits
The standard notation for a quantum program: horizontal wires are qubits, time flows left to right, boxes are gates, and the meter symbols at the end are measurements.
5 resources · BeginnerWhat quantum computers can’t do
Quantum computers give large speedups on a narrow, structured set of problems. They are not faster at general computing, they do not "solve NP-complete problems instantly", and they will not run your spreadsheet.
5 resources · BeginnerNoise & decoherence
Qubits leak their quantum information into the environment within microseconds to milliseconds, and every gate applied to them is slightly wrong. That is why building a useful quantum computer is hard.
7 resources · BeginnerMath prerequisites
To go past intuition you need complex numbers, vectors, matrices, inner products and eigenvectors. That is roughly one solid linear algebra course — not a physics degree.
6 resources · BeginnerProgramming prerequisites
Basic Python plus NumPy is enough to start. Every mainstream quantum SDK — Qiskit, Cirq, PennyLane, Braket — is a Python library.
3 resources · BeginnerPick up the math. Write and run real circuits.
Dirac (bra-ket) notation
The compact notation physicists use for quantum states: |psi> is a column vector, <psi| its conjugate transpose, and <a|b> their inner product.
3 resources · IntermediateThe Bloch sphere
A geometric picture where every pure single-qubit state is a point on a sphere, and every single-qubit gate is a rotation of that sphere.
4 resources · IntermediateMulti-qubit states & tensor products
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.
3 resources · IntermediateUniversal gate sets
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.
2 resources · IntermediateQuantum teleportation
A 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.
7 resources · IntermediateSuperdense coding
The mirror image of teleportation: using one pre-shared entangled pair, you can send two classical bits by transmitting only one qubit.
3 resources · IntermediateDeutsch–Jozsa & Bernstein–Vazirani
The 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.
5 resources · IntermediateGrover's algorithm
Finds a marked item among N possibilities in about sqrt(N) queries instead of N, by repeatedly reflecting the state to pump amplitude onto the answer.
9 resources · IntermediateQuantum Fourier transform
The 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.
5 resources · IntermediateQuantum phase estimation
Given a unitary and one of its eigenvectors, extract the eigenvalue phase to n bits of precision using controlled applications of the unitary plus an inverse QFT.
4 resources · IntermediateShor's algorithm
Factors large integers in polynomial time by reducing factoring to finding the period of a modular exponentiation function, which the QFT does efficiently.
7 resources · IntermediateProgramming with Qiskit
IBM's open-source Python SDK for building, simulating, transpiling and running quantum circuits — including on IBM's real hardware.
8 resources · IntermediateOther SDKs: Cirq, PennyLane, Q#, Braket
Qiskit is not the only option: Cirq targets Google hardware, PennyLane specialises in differentiable quantum programming, Q# is a dedicated quantum language, and Braket gives one API across several vendors.
13 resources · IntermediateBell inequalities & nonlocality
An experimentally testable inequality that any local hidden-variable theory must satisfy — and that quantum mechanics, and reality, provably violate.
3 resources · IntermediateQuantum key distribution (BB84)
A 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.
4 resources · IntermediateRunning on real hardware
Several vendors give free or cheap cloud access to real quantum processors, where you meet queueing, calibration data, connectivity constraints and genuinely noisy results.
4 resources · IntermediateError correction, complexity, and the research frontier.
Quantum error correction
Encoding 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.
11 resources · AdvancedSurface codes
A two-dimensional topological code needing only nearest-neighbour connectivity, with a relatively forgiving error threshold near 1% — the leading candidate for real fault-tolerant hardware.
5 resources · AdvancedFault tolerance & the threshold theorem
The result that if physical error rates are below a threshold, arbitrarily long quantum computations become possible with only polylogarithmic overhead — plus the machinery (transversal gates, magic-state distillation) that gets you there.
5 resources · AdvancedError mitigation
Post-processing techniques — zero-noise extrapolation, probabilistic error cancellation, readout correction — that recover accurate expectation values from noisy hardware without full error correction, at the cost of extra shots.
3 resources · AdvancedQuantum complexity theory
The study of what quantum computers can and cannot do efficiently: the class BQP, its relationship to P, NP and PSPACE, the quantum analogue QMA, and the oracle separations that justify the field.
5 resources · AdvancedHamiltonian simulation
Simulating the time evolution of a quantum system — Feynman's original motivation — via Trotter-Suzuki product formulas, linear combination of unitaries, or qubitization.
7 resources · AdvancedVariational algorithms (VQE & QAOA)
Hybrid loops where a shallow parameterised quantum circuit produces expectation values and a classical optimiser tunes the parameters — the dominant paradigm for near-term hardware.
9 resources · AdvancedQuantum machine learning
Using quantum circuits as trainable models or kernels — plus the sober literature on data loading bottlenecks and classical "dequantization" results that erase many claimed speedups.
5 resources · AdvancedHardware modalities
The competing physical platforms — superconducting transmons, trapped ions, neutral atoms, photonics, spin qubits — each with different gate speeds, fidelities, connectivity and scaling problems.
15 resources · AdvancedQuantum advantage & benchmarking
Experiments claiming a quantum computer did something no classical computer feasibly can — random circuit sampling, boson sampling — and the classical simulation efforts that keep pushing back.
8 resources · AdvancedPost-quantum cryptography
Classical cryptography designed to resist quantum attack — the NIST-standardised lattice and hash-based schemes now being deployed, entirely separate from QKD.
4 resources · AdvancedQuantum information theory
The rigorous framework beyond pure states and unitaries: density matrices, quantum channels, POVMs, entropy, distance measures and channel capacities.
7 resources · AdvancedCompilation & transpilation
Turning an abstract circuit into one a specific device can execute: decomposing into native gates, mapping logical to physical qubits, and inserting SWAPs to satisfy connectivity — while minimising depth.
4 resources · AdvancedResearch & careers
How to move from learning to contributing: where papers appear, which open-source projects take contributions, and which mentorship and internship programmes exist.
13 resources · AdvancedOr browse by level: Beginner · Intermediate · Advanced