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.


  • $\begingroup$ I never heard of this before, but I guess it's en.wikipedia.org/wiki/Fragment_(logic) for axiomatizations of QM... $\endgroup$ – Mitchell Porter Jul 28 '18 at 20:06
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    $\begingroup$ 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. $\endgroup$ – Ryan Thorngren Jul 28 '18 at 20:37
  • $\begingroup$ 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). $\endgroup$ – 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.

Please see here:



Spekkens is referring to the subset of quantum computation where you only use the Clifford gate set (also called "stabilizer circuits"). A gate set that is universal for this subset is the CNOT + S + H + Mz gate set. If it can't be done with controlled-nots, quarter-phasing, Hadamards, and single-qubit computational basis measurement (with classically controlled feed forward operations), then it's not part of the "Clifford fragment".

Stabilizer circuits are very useful theoretically because they exhibit many quantum behaviors but can be simulated efficiently classically. In particular, you can make quantum error correcting codes that are stabilizer circuits.


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