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arxiv: 2306.01043 · v2 · submitted 2023-06-01 · 🪐 quant-ph · hep-th· math.GR

Clifford Orbits from Cayley Graph Quotients

Cynthia Keeler , William Munizzi , Jason Pollack This is my paper

Pith reviewed 2026-05-24 08:23 UTC · model grok-4.3

classification 🪐 quant-ph hep-thmath.GR
keywords Clifford groupCayley graphquotient graphstabilizer subgroupClifford orbitquantum state evolutionW stateDicke state
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The pith

Quotienting the Cayley graph of the Clifford group by a state's stabilizer subgroup yields a reduced graph of the state's orbit under Clifford action.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a quotient construction on Cayley graphs to track how Clifford gates act on quantum states. The full Cayley graph has vertices as Clifford group elements and edges as generators; dividing by the stabilizer subgroup of any chosen state collapses equivalent elements and produces a smaller graph whose vertices are the distinct states in the orbit and whose edges show the Clifford gates connecting them. The same procedure works for stabilizer states and for non-stabilizer states such as the W state and Dicke states, and it recovers earlier reachability graphs for two qubits. A reader would care because Clifford circuits are the backbone of many fault-tolerant protocols, and an explicit map of reachable states clarifies what any given Clifford circuit can accomplish from a fixed starting point.

Core claim

Quotienting the Cayley graph by the stabilizer subgroup of a state gives a reduced graph which depicts the state's Clifford orbit. The procedure is state-independent, reproduces and generalizes known reachability graphs for C_2, and extends without change to non-stabilizer states including the W and Dicke states.

What carries the argument

The quotient of the Clifford group's Cayley graph by the stabilizer subgroup of a given quantum state, which encodes the orbit under left or right multiplication by Clifford elements.

If this is right

  • The construction reproduces and generalizes the reachability graphs previously obtained for two-qubit states.
  • The same quotient applies unchanged to non-stabilizer states such as the W state and Dicke states.
  • The resulting graphs supply a state-independent description of how Clifford circuits evolve an arbitrary initial state.
  • The method yields both an algebraic and a graphical record of reachability within any Clifford orbit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Orbit graphs obtained this way could be searched to find shortest Clifford sequences that map one state to another.
  • The quotient technique may apply to other finite gate groups or to approximate versions of the Clifford group.
  • Explicit computation of these graphs for n greater than 2 could expose patterns in orbit size or connectivity not visible from the abstract group action alone.

Load-bearing premise

The algebraic quotient must faithfully capture the orbit without introducing extra identifications or omitting edges that would alter which states are reachable by Clifford multiplication.

What would settle it

Build the quotient graph for a concrete state and set of generators, then verify that every path in the quotient corresponds to a valid sequence of Clifford gates reaching the claimed target state and that no such sequence is missing; any mismatch between the graph and explicit gate action would falsify the construction.

read the original abstract

We describe the structure of the $n$-qubit Clifford group $C_n$ via Cayley graphs, whose vertices represent group elements and edges represent generators. In order to obtain the action of Clifford gates on a given quantum state, we introduce a quotient procedure. Quotienting the Cayley graph by the stabilizer subgroup of a state gives a reduced graph which depicts the state's Clifford orbit. Using this protocol for $C_2$, we reproduce and generalize the reachability graphs introduced in arXiv:2204.07593. Since the procedure is state-independent, we extend our study to non-stabilizer states, including the W and Dicke states. Our new construction provides a more precise understanding of state evolution under Clifford circuit action.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper describes the n-qubit Clifford group C_n via its Cayley graphs and introduces a quotient construction that divides the graph by the stabilizer subgroup of a chosen quantum state; the resulting reduced graph is claimed to depict the orbit of that state under left multiplication by Clifford elements. For n=2 the construction reproduces the reachability graphs of arXiv:2204.07593 and is then applied to non-stabilizer states including the W and Dicke states.

Significance. If the quotient map is correctly realized as the Schreier graph of the Clifford action, the method supplies a uniform, state-independent graphical representation of Clifford orbits that extends existing reachability studies to arbitrary states. This could facilitate systematic enumeration of reachable states and circuit-depth analysis in Clifford circuits.

minor comments (3)
  1. [§2.2] §2.2: the precise definition of the quotient map (vertices as right cosets, edges induced by generators) is stated only informally; an explicit formula or pseudocode would remove ambiguity about edge multiplicity.
  2. [Figure 3] Figure 3 caption: the coloring convention for edges corresponding to different generators is not restated, making the figure harder to read without returning to the text.
  3. [§4.3] §4.3: when extending to the W state, the stabilizer subgroup is given only by its generators; listing the full order or a reference to its known structure would help readers verify the quotient size.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the accurate summary of our construction, and the recommendation to accept. There are no major comments to address.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central construction quotients the Cayley graph of the Clifford group by a state's stabilizer subgroup to produce a graph on the orbit. This follows directly from the standard definitions of Cayley graphs, coset actions, and the orbit-stabilizer theorem (yielding the Schreier graph on right cosets), with no fitted parameters, no self-referential equations, and no load-bearing self-citations required for the core claim. The result is self-contained as a faithful encoding of the group action on states, independent of any prior fitted results or author-specific uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction relies on standard definitions from group theory and quantum information; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption The n-qubit Clifford group is generated by a known finite set of gates (Hadamard, phase, CNOT) whose Cayley graph can be defined in the usual way.
    Standard background fact in quantum computing invoked to set up the Cayley graph.
  • domain assumption Every quantum state possesses a well-defined stabilizer subgroup inside the Clifford group.
    Required for the quotient construction; standard for stabilizer states and assumed to extend to the non-stabilizer cases mentioned.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Magic and Non-Clifford Gates in Topological Quantum Field Theory

    hep-th 2026-04 unverdicted novelty 5.0

    Non-Clifford gates including Ising, Toffoli, and T arise as exact path integrals in Chern-Simons and Dijkgraaf-Witten topological quantum field theories.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · cited by 1 Pith paper · 9 internal anchors

  1. [1]

    Keeler, W

    C. Keeler, W. Munizzi, and J. Pollack, Entropic lens on stabilizer states , Phys. Rev. A 106 (2022), no. 6 062418, [ arXiv:2204.07593]

    work page arXiv 2022

  2. [2]

    A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum error correction and orthogonal geometry, Phys. Rev. Lett. 78 (1997) 405–408, [quant-ph/9605005]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  3. [3]

    A Theory of Fault-Tolerant Quantum Computation

    D. Gottesman, A theory of fault tolerant quantum computation , Phys. Rev. A 57 (1998) 127, [ quant-ph/9702029]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  4. [4]

    Stabilizer Codes and Quantum Error Correction

    D. Gottesman, Stabilizer codes and quantum error correction , arXiv preprint arXiv:quant-ph/9705052 (5, 1997) [ quant-ph/9705052]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  5. [5]

    The Heisenberg Representation of Quantum Computers

    D. Gottesman, The Heisenberg representation of quantum computers , in 22nd International Colloquium on Group Theoretical Methods in Physics , pp. 32–43, International Press of Boston, Inc., 7, 1998. quant-ph/9807006

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  6. [6]

    Improved Simulation of Stabilizer Circuits

    S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits , Phys. Rev. A 70 (2004), no. 5 052328, [ quant-ph/0406196v5]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  7. [7]

    Cayley, Desiderata and Suggestions: No

    A. Cayley, Desiderata and Suggestions: No. 2. The Theory of Groups: Graphical Representation, American Journal of Mathematics 1 (1878), no. 2 174–176

    work page

  8. [8]

    Generators and relations for n-qubit Clifford operators

    P. Selinger, Generators and relations for n-qubit Clifford operators , Logical Methods in Computer Science, Volume 11, Issue 2 (June 19, 2015) lmcs:1570 (Oct., 2013) [arXiv:1310.6813]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  9. [9]

    Keeler, W

    C. Keeler, W. Munizzi, and J. Pollack, Entanglement Bounds from Cayley Graphs , (forthcoming) (2023)

    work page 2023

  10. [10]

    N. Bao, S. Nezami, H. Ooguri, B. Stoica, J. Sully, and M. Walter, The Holographic Entropy Cone, JHEP 09 (2015) 130, [ arXiv:1505.07839]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  11. [11]

    Cayley-Graphs

    W. Munizzi, “Cayley-Graphs.” https://github.com/WMunizzi/Cayley-Graphs, 2023

    work page 2023

  12. [12]

    J. L. Alperin and R. B. Bell, Groups and representations, vol. 162 of Grad. Texts Math. New York, NY: Springer-Verlag, 1995

    work page 1995

  13. [13]

    H. J. Garc´ ıa, I. L. Markov, and A. W. Cross, On the geometry of stabilizer states , Quantum Information and Computation (QIC) 14 (2014), no. 7-8 683 – 720, [arXiv:1711.07848]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  14. [14]

    Gross and M

    D. Gross and M. Walter, Stabilizer information inequalities from phase space distributions, J. Math. Phys. 54 (2013), no. 8 082201, [https://doi.org/10.1063/1.4818950]

    work page doi:10.1063/1.4818950 2013

  15. [15]

    Bosma, J

    W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4 235–265. Computational algebra and number theory (London, 1993). – 41 –

    work page 1997

  16. [16]

    W. Dur, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62 (2000) 062314, [ quant-ph/0005115]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  17. [17]

    H. J. Schnitzer, The entropy cones of WN states, arXiv preprint arXiv:2204.04532v3 (4, 2022) [ arXiv:2204.04532]

    work page arXiv 2022

  18. [18]

    R. H. Dicke, Coherence in Spontaneous Radiation Processes, Phys. Rev. 93 (1954) 99–110

    work page 1954

  19. [19]

    B¨ artschi and S

    A. B¨ artschi and S. Eidenbenz,Deterministic Preparation of Dicke States , 22nd International Symposium on Fundamentals of Computation Theory, FCT’19, 126-139, 2019 (Apr., 2019) [ arXiv:1904.07358]

    work page arXiv 2019

  20. [20]

    Munizzi and H

    W. Munizzi and H. J. Schnitzer, Entropy Cone and Clifford Orbits of Dicke States , (forthcoming) (2023)

    work page 2023

  21. [21]

    Establishing Maximal Entanglement Speed via Branchial Lightcones

    W. Munizzi, “Establishing Maximal Entanglement Speed via Branchial Lightcones.” http://notebookarchive.org/2022-07-9p7c7ej, Nov, 2022

    work page 2022

  22. [22]

    Balasubramanian, M

    V. Balasubramanian, M. Decross, A. Kar, and O. Parrikar, Quantum Complexity of Time Evolution with Chaotic Hamiltonians , JHEP 01 (2020) 134, [arXiv:1905.05765]

    work page arXiv 2020

  23. [23]

    Bravyi and A

    S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas , Phys. Rev. A 71 (Feb, 2005) 022316

    work page 2005

  24. [24]

    C. D. White, C. Cao, and B. Swingle, Conformal field theories are magical , Phys. Rev. B 103 (2021), no. 7 075145, [ arXiv:2007.01303]. – 42 –

    work page arXiv 2021