ZX-calculus for the working quantum computer scientist
arXiv:2012.13966
Abstract
The ZX-calculus is a graphical language for reasoning about quantum computation that has recently seen an increased usage in a variety of areas such as quantum circuit optimisation, surface codes and lattice surgery, measurement-based quantum computation, and quantum foundations. The first half of this review gives a gentle introduction to the ZX-calculus suitable for those familiar with the basics of quantum computing. The aim here is to make the reader comfortable enough with the ZX-calculus that they could use it in their daily work for small computations on quantum circuits and states. The latter sections give a condensed overview of the literature on the ZX-calculus. We discuss Clifford computation and graphically prove the Gottesman-Knill theorem, we discuss a recently introduced extension of the ZX-calculus that allows for convenient reasoning about Toffoli gates, and we discuss the recent completeness theorems for the ZX-calculus that show that, in principle, all reasoning about quantum computation can be done using ZX-diagrams. Additionally, we discuss the categorical and algebraic origins of the ZX-calculus and we discuss several extensions of the language which can represent mixed states, measurement, classical control and higher-dimensional qudits.
About 75 pages of text + 8 pages of appendices
References in corpus (18)
- Restrictions on Transversal Encoded Quantum Gate Sets
- Quantum circuits of T-depth one
- Novel constructions for the fault-tolerant Toffoli gate
- Generalized Flow and Determinism in Measurement-based Quantum Computation
- A Simple Proof that Toffoli and Hadamard are Quantum Universal
- Both Toffoli and Controlled-NOT need little help to do universal quantum computation
- The ZX-calculus is incomplete for quantum mechanics
- A universal completion of the ZX-calculus
- Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing
- The ZX-calculus is complete for the single-qubit Clifford+T group
- The algebra of entanglement and the geometry of composition
- Graphical description of the action of Clifford operators on stabilizer states
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- Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics
- Graphical Fourier Theory and the Cost of Quantum Addition
- Completeness of the Phase-free ZH-calculus