# Kindergarden quantum mechanics graduates (...or how I learned to stop gluing LEGO together and love the ZX-calculus)

arXiv: https://arxiv.org/abs/2102.10984 · HTML: https://mlsanity.com/paper/2102.10984/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 119 references that are in the map.
- First posted on arXiv 22 Feb 2021.

## Builds on (most cited first, 28 of 28)

- [Foundations for Near-Term Quantum Natural Language Processing](https://mlsanity.com/paper/2012.03755/) (2020)
- [Quantum Natural Language Processing on Near-Term Quantum Computers](https://mlsanity.com/paper/2005.04147/) (2020)
- [Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus](https://mlsanity.com/paper/2102.01828/) (2021)
- [The Frobenius anatomy of word meanings I: subject and object relative pronouns](https://mlsanity.com/paper/1404.5278/) (2014)
- [Quantum Algorithms for Compositional Natural Language Processing](https://mlsanity.com/paper/1608.01406/) (2016)
- [Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus](https://mlsanity.com/paper/1902.03178/) (2019)
- [A Study of Entanglement in a Categorical Framework of Natural Language](https://mlsanity.com/paper/1405.2874/) (2014)
- [Interacting Hopf Algebras](https://mlsanity.com/paper/1403.7048/) (2014)
- [Integrated Information in Process Theories](https://mlsanity.com/paper/2002.07654/) (2020)
- [Diagrammatic Design and Study of Ans"atze for Quantum Machine Learning](https://mlsanity.com/paper/2011.11073/) (2020)
- [A Diagrammatic Axiomatisation for Qubit Entanglement](https://mlsanity.com/paper/1501.07082/) (2015)
- [The ZX calculus is a language for surface code lattice surgery](https://mlsanity.com/paper/1704.08670/) (2017)
- [A Compositional Model of Consciousness based on Consciousness-Only](https://mlsanity.com/paper/2007.16138/) (2020)
- [A categorical semantics for causal structure](https://mlsanity.com/paper/1701.04732/) (2017)
- [Quantomatic: A Proof Assistant for Diagrammatic Reasoning](https://mlsanity.com/paper/1503.01034/) (2015)
- [Generalised Mermin-type non-locality arguments](https://mlsanity.com/paper/1702.01772/) (2017)
- [A universal completion of the ZX-calculus](https://mlsanity.com/paper/1706.09877/) (2017)
- [String Diagram Rewrite Theory I: Rewriting with Frobenius Structure](https://mlsanity.com/paper/2012.01847/) (2020)
- [Pivoting makes the ZX-calculus complete for real stabilizers](https://mlsanity.com/paper/1307.7048/) (2013)
- [The ZX-calculus is incomplete for quantum mechanics](https://mlsanity.com/paper/1404.3633/) (2014)
- [Interacting Frobenius Algebras are Hopf](https://mlsanity.com/paper/1601.04964/) (2016)
- [Interacting Conceptual Spaces](https://mlsanity.com/paper/1608.01402/) (2016)
- [A Finite Presentation of CNOT-Dihedral Operators](https://mlsanity.com/paper/1701.00140/) (2016)
- [The algebra of entanglement and the geometry of composition](https://mlsanity.com/paper/1709.08086/) (2017)
- [Towards a Minimal Stabilizer ZX-calculus](https://mlsanity.com/paper/1709.08903/) (2017)
- [A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond](https://mlsanity.com/paper/1804.03084/) (2018)
- [Contextual Equivalence for Signal Flow Graphs](https://mlsanity.com/paper/2002.08874/) (2020)
- [Giving Operational Meaning to the Superposition of Causal Orders](https://mlsanity.com/paper/2003.13306/) (2020)

## Built on by (most cited first, 4 of 4)

- [Compositionality as we see it, everywhere around us](https://mlsanity.com/paper/2110.05327/) (2021)
- [Differentiating and Integrating ZX Diagrams with Applications to Quantum Machine Learning](https://mlsanity.com/paper/2201.13250/) (2022)
- [Representing and Implementing Matrices Using Algebraic ZX-calculus](https://mlsanity.com/paper/2110.06898/) (2021)
- [Quantum Concept Music Score from Quantum Picturalism: Musical Incarnation of a Bell-Pair under Measurements](https://mlsanity.com/paper/2510.05391/) (2025)

## Abstract (arXiv)

This paper is a `spiritual child' of the 2005 lecture notes Kindergarten Quantum Mechanics, which showed how a simple, pictorial extension of Dirac notation allowed several quantum features to be easily expressed and derived, using language even a kindergartner can understand. Central to that approach was the use of pictures and pictorial transformation rules to understand and derive features of quantum theory and computation. However, this approach left many wondering `where's the beef?' In other words, was this new approach capable of producing new results, or was it simply an aesthetically pleasing way to restate stuff we already know? The aim of this sequel paper is to say `here's the beef!', and highlight some of the major results of the approach advocated in Kindergarten Quantum Mechanics, and how they are being applied to tackle practical problems on real quantum computers. We will focus mainly on what has become the Swiss army knife of the pictorial formalism: the ZX-calculus. First we look at some of the ideas behind the ZX-calculus, comparing and contrasting it with the usual quantum circuit formalism. We then survey results from the past 2 years falling into three categories: (1) completeness of the rules of the ZX-calculus, (2) state-of-the-art quantum circuit optimisation results in commercial and open-source quantum compilers relying on ZX, and (3) the use of ZX in translating real-world stuff like natural language into quantum circuits that can be run on today's (very limited) quantum hardware. We also take the title literally, and outline an ongoing experiment aiming to show that ZX-calculus enables children to do cutting-edge quantum computing stuff. If anything, this would truly confirm that `kindergarten quantum mechanics' wasn't just a joke.