Resources

“My [algebraic] methods are really methods of working and thinking; this is why they have crept in everywhere anonymously.” — Emmy Noether.

Quantum things move fast...too fast. If you're new and feeling overwhelmed, start here! First, a set of mathematical subjects (tools) useful specifically for quantum computing. Second, a list of some of my favorite textbook references. These shaped the mathematical/quantum foundations for me as well as explain the intuition, which is just as important.

Mathematical subject	Why it matters
Linear algebra	Foundational: quantum states are vectors in a complex (Hilbert) vector space, and quantum gates are matrices (unitaries) acting on those states. Understanding eigenvalues, tensor (Kronecker) products, and unitary transformations is necessary for analyzing and designing quantum circuits.
Probability and statistics	Measurement outcomes are inherently probabilistic. This intrinsic randomness is central to quantum mechanics. Concepts like expectation values, variance, statistical estimation, and sampling underpin algorithms, tomography, and error analysis.
Group theory and Lie algebras	Quantum gates form continuous symmetry groups (e.g., special unitary groups). Lie algebras describe the infinitesimal generators of these transformations, enabling structured decompositions and optimization of gate sequences.
Functional analysis	Extends linear algebra to infinite-dimensional spaces. Many realistic models require such spaces to treat observables, operators, and quantum dynamics rigorously. A fundamental example is the position–momentum pair, which has continuous spectra that finite-dimensional spaces cannot accommodate.
Information theory	Quantities like entropy and mutual information (and related notions such as channel capacity) quantify how much information can be stored, transmitted, or lost, which is essential for quantum communication and error correction.
Graph theory (opinion)	Many quantum algorithms can be framed as graph problems, with adjacency matrices linked to Hamiltonians and interaction/connectivity patterns.
Geometry	Characterizing quantum constraints and optimization landscapes is central to algorithm design. The geometry of state space clarifies entanglement structure, which no one (still) fully understands.
Topology	Arguably, topology appears throughout quantum theory. Physical pure states live in projective Hilbert space; modding out the global phase is a topological operation that shapes the geometry (e.g., geodesics) of state space. Topological quantum computing and homology-based error-correcting codes exploit invariants (e.g., Chern numbers, K-theory) to achieve robustness.
Scientific computing	In quantum computing, there is quite a bit of simulations, numerically solving Schrodinger equations, solving for eigenvalues/eigenvectors, etc.

So as you can see, quantum computing is the all-you-can-eat math buffet: every flavor of math is on the menu.

Reference	Author	Level	Notes
Introduction to Classical and Quantum Computing	Thomas Wong	Advanced high school and early undergraduate level text	Suitable for self study get some elementary level coding.
Quantum Theory for Mathematicians	Brian Hall	Advanced graduate text	Does not assume physics knowledge. Builds from the ground up. (personal favorite)
Geometry of Quantum States	Ingemar Bengtsson and Karol Zyczkowski	Senior undergraduate and early graduate level text	Assumed basic knowledge of quantum mechanics.
From Classical to Quantum Shannon Theory	Mark Wilde	Early graduate level text	Does not assume information theory knowledge. Builds on intuition. Calculation heavy.
Quantum Computation and Quantum Information	Michael A. Nielsen and Isaac L. Chuang	Early graduate level text	Very popular reference
Quantum Computing: A Gentle Introduction	Eleanor G. Rieffel and Wolfgang H. Polak	Advanced senior undergrad level text	Comprehensive introduction to the field. Many examples for intuition.
Supervised Learning with Quantum Computers	Maria Schuld and Francesco Petruccione	Graduate level text	Assumes some knowledge of machine learning. Clear connections between machine learning and quantum information. Explanations builds intuition.