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
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- The MQT Handbook: A Summary of Design Automation Tools and Software for Quantum Computing
- Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions
- Equivalence Checking of Quantum Circuits with the ZX-Calculus
- Constructing quantum circuits with global gates
- Hybrid quantum cycle generative adversarial network for small molecule generation
- Exact solution of long-range stabilizer Rényi entropy in the dual-unitary XXZ model
- Quantum Picturalism: Learning Quantum Theory in High School
- Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus
- Operator dynamics and entanglement in space-time dual Hadamard lattices
- Hybrid quantum-classical circuit simplification with the ZX-calculus
- Optimizing ZX-Diagrams with Deep Reinforcement Learning
- AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states
- Two imaginarity monotones induced by unified -relative entropy
- Kindergarden quantum mechanics graduates (...or how I learned to stop gluing LEGO together and love the ZX-calculus)
- A recursively partitioned approach to architecture-aware ZX Polynomial synthesis and optimization
- Non-stabilizerness and entanglement from cat-state injection