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Sarnak's conjecture in quantum computing, cyclotomic unitary group coranks, and Shimura curves

Colin Ingalls, Bruce W. Jordan, Allan Keeton, Adam Logan, Yevgeny Zaytman

Abstract

Sarnak's conjecture in quantum computing concerns when the groups $\operatorname{PU}_2$ and $\operatorname{PSU}_2$ over cyclotomic rings $\mathbb{Z}[ζ_n, 1/2]$ with $ζ_n=e^{2πi/n}$, $4|n$, are generated by the Clifford-cyclotomic gate set. We previously settled this using Euler-Poincaré characteristics. A generalization of Sarnak's conjecture is to ask when these groups are generated by torsion elements. An obstruction to this is provided by the corank: a group $G$ has $\operatorname{corank} G >0$ only if $G$ is not generated by torsion elements. In this paper we study the corank of these cyclotomic unitary groups in the families $n=2^s$ and $n=3\cdot 2^s$, $n\geq 8$, by letting them act on Bruhat-Tits trees. The quotients by this action are finite graphs whose first Betti number is the corank of the group. Our main result is that for $n=2^s$ and $n=3\cdot 2^s$ the corank groups doubly exponentially in $s$ as $s\rightarrow \infty$; it is $0$ precisely when $n=8,12, 16,24$ and indeed the cyclotomic unitary groups are generated by torsion elements (in fact by the Clifford-cyclotomic gates) for these $n$. We give explicit lower bounds for the corank in two different ways. The first is to bound the isotropy subgroups in the action on the tree by explicit cyclotomy. The second is to relate our graphs to Shimura curves over $F_n=\mathbb{Q}(ζ_n)^+$ via interchanging local invariants and applying a result of Selberg and Zograf. We show that the cyclotomy arguments give the stronger bounds. In a final section we execute a program of Sarnak to show that our results for the $n=2^s$ and $n=3\cdot 2^s$ families are sufficient to give a second proof of Sarnak's conjecture.

Sarnak's conjecture in quantum computing, cyclotomic unitary group coranks, and Shimura curves

Abstract

Sarnak's conjecture in quantum computing concerns when the groups and over cyclotomic rings with , , are generated by the Clifford-cyclotomic gate set. We previously settled this using Euler-Poincaré characteristics. A generalization of Sarnak's conjecture is to ask when these groups are generated by torsion elements. An obstruction to this is provided by the corank: a group has only if is not generated by torsion elements. In this paper we study the corank of these cyclotomic unitary groups in the families and , , by letting them act on Bruhat-Tits trees. The quotients by this action are finite graphs whose first Betti number is the corank of the group. Our main result is that for and the corank groups doubly exponentially in as ; it is precisely when and indeed the cyclotomic unitary groups are generated by torsion elements (in fact by the Clifford-cyclotomic gates) for these . We give explicit lower bounds for the corank in two different ways. The first is to bound the isotropy subgroups in the action on the tree by explicit cyclotomy. The second is to relate our graphs to Shimura curves over via interchanging local invariants and applying a result of Selberg and Zograf. We show that the cyclotomy arguments give the stronger bounds. In a final section we execute a program of Sarnak to show that our results for the and families are sufficient to give a second proof of Sarnak's conjecture.

Paper Structure

This paper contains 42 sections, 97 theorems, 231 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Graphs and groups acting on trees
  3. Graph mass
  4. Unitary groups over Z["03B6n,1/2] for n=2"5Es and n=3"B72"5Es
  5. Classifying orders in quadratic extensions of Dedekind domains
  6. Cyclotomic rings
  7. The Hamilton quaternions
  8. Unitary groups
  9. Vertices of grn and maximal orders in Hn
  10. Optimal embeddings, mass, and a torsion invariant
  11. Mass
  12. A torsion invariant
  13. Naive bounds on the torsion invariant and Betti numbers
  14. Optimal embeddings
  15. The n=2"5Es family
  16. ...and 27 more sections

Key Result

Theorem 1.3

(IJKLZ) Suppose $4|n$, $n\geq 8$.

Figures (5)

  • Figure 1: The mass $M_n$ for $n=2^s$, $3\leq s\leq 8$.
  • Figure 2: The mass $M_n$ for $n=3\cdot2^s$, $2\leq s\leq 7$.
  • Figure 3: The minus part of the class number for the cyclotomic $n=2^s$ family
  • Figure 4: The minus part of the class number for the cyclotomic $n=3\cdot 2^s$ family
  • Figure 5: Comparing our upper bounds on $E_{1,2^s}$ at $\infty_1$ using Selberg-Zograf and at ${\mathfrak p}$ bounding class numbers

Theorems & Definitions (219)

  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • Definition 2.6
  • Theorem 2.7
  • proof
  • ...and 209 more