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Quantum Computing Math Cheat Sheet: Qubits, Dirac Notation, Gates, Entanglement, Shor & Grover

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The mathematics behind quantum computing on four dense pages, from a single qubit to Shor's and Grover's algorithms. Every formula comes with a short note on what it is for, so you can follow a lecture, a textbook or a Qiskit notebook without stopping to look up the notation.

WHO IT IS FOR: Physics, computer science and engineering students taking a first course in quantum computing or quantum information, software engineers learning Qiskit or Cirq, and anyone who wants Dirac notation, the standard gates, entanglement and the main algorithms in one place instead of scattered across lecture notes.

WHAT IS INSIDE: Page 1: Cover, notation guide and how to use the sheet. Page 2: Qubits and Dirac notation. Complex amplitudes, kets and bras, inner and outer products, the Bloch sphere with the six cardinal states, the Born rule and expectation values, measuring in another basis, tensor products, how memory grows with qubit count, the linear algebra you need, Pauli eigenstates and commutators, and a worked single-qubit example Page 3: Gates and circuits. A table of seven single-qubit gates with matrix, action and Bloch-sphere rotation, rotation gates and the ZYZ decomposition, Pauli algebra, CNOT, CZ, SWAP, controlled-U and Toffoli, seven circuit identities, the Hadamard transform, oracles and phase kickback, universal gate sets, how a circuit's cost is measured, and the bit-ordering trap between textbooks and Qiskit Page 4: Entanglement, measurement and noise. The product-state test, the Schmidt decomposition, the four Bell states, partial and generalised measurement, density matrices, the partial trace with a worked example, entropy, no-cloning, teleportation, superdense coding, the Holevo bound, CHSH, noise channels in Kraus form, T1 and T2, fidelity and trace distance Page 5: Algorithms. Eleven algorithms with quantum cost against the best classical cost, the quantum Fourier transform and its circuit cost, phase estimation, Shor's algorithm in five steps, what Shor and Grover mean for RSA, elliptic curves and AES, Grover's geometry with real iteration counts, VQE, QAOA and Trotterisation, and error correction from the repetition code to the surface code

FORMATS: You receive 2 PDF files: A4 and US Letter. Print at actual size, or open the PDF in GoodNotes, Notability or any annotation app.

ABOUT THE AUTHOR: I am an Associate Professor of Computer Science and the author of a book on the mathematics of quantum computing. My research centres on optimisation and the design and analysis of algorithms. This sheet is a new, condensed reference to the mathematics every quantum computing course assumes you already know.

PLEASE NOTE: This is a digital download. No physical item will be shipped. Designed and written by the author. For personal use only. Please do not share, resell or upload the files. If you teach a class and want to use it with students, message me.

If a file does not open or something is unclear, message me, and I will fix it within 48 hours.

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Listed on 05 Oct, 2026
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