# What are Clifford fragments?

In his article/lecture on "What quantum physics can learn from Egyptian hieroglyphs"", researcher Robert Spekkens talks about Clifford fragments. He describes them as "containing only a subset of the full set of quantum states and measurements – which admit of an interpretation where every quantum state can be understood as a mathematical encoding of a probability distribution over a set of deeper physical states."

What are Clifford fragments, and is there any literature available on them? Particularly, are they connected to Clifford Algebra? A fast google search of the term "Clifford fragments" returns only what appear to be disconnected results.

https://insidetheperimeter.ca/quantum-physics-egyptian-hieroglyphs/

• I never heard of this before, but I guess it's en.wikipedia.org/wiki/Fragment_(logic) for axiomatizations of QM... – Mitchell Porter Jul 28 '18 at 20:06
• It seems like it has to do with the Gottesman-Knill theorem: en.wikipedia.org/wiki/Gottesman%E2%80%93Knill_theorem . There is a subset of quantum gates called the Clifford gates which can be classically simulated but which if you add any other gate to it you get a universal set. It tempts one to try and define a maximal classical "subset" (fragment?) of a quantum system. – Ryan Thorngren Jul 28 '18 at 20:37
• See first page of arxiv.org/abs/1506.03055 for some "fragments" distinguished by the fact that allowed operations include increasingly refined phase rotations of a qubit (rotation by multiples of pi, of pi/2, of pi/4). – Mitchell Porter Jul 29 '18 at 0:19

The ZX calculus is a graphical language for pure state qubit QM. It is universal, any quantum evolution can be represented by a ZX diagram. These are parameterized by angles, and various fragments of the lagnuage have been proposed. These are based on some restrictions on the angles.

The π/p-fragment consist of diagrams only made with angles multiple of π/p.

The π/2 fragment is the stabilizer QM and is not universal for QM.

The π/4 fragment is the Clifford fragment and is for Clifford QM, and is apprix. universal. Any quantum evolution can be approximated in this fragment with arbitrary accuracy.