Optimal expansion in unitary groups I: Golden adelic groups and cutoff

Technical Summary: Optimal Expansion in Unitary Groups I: Golden Adelic Groups and Cutoff

Problem Statement
The paper addresses two interconnected problems in the intersection of arithmetic geometry, representation theory, and quantum computation. First, it seeks to construct $p$-arithmetic lattices in projective unitary groups ($PU_n$) that act transitively on their corresponding Bruhat-Tits buildings. Such groups are crucial for constructing "Golden Gate sets"—generating sets for quantum gates that allow for efficient, near-optimal compilation of quantum circuits. Second, the paper investigates the "cutoff phenomenon" (a sharp transition in mixing time) for random walks on quotients of Bruhat-Tits buildings. While the Ramanujan property (Naïve Ramanujan Conjecture, NRC) guarantees cutoff for certain groups (like $U_2$), it is known to fail for matrix unitary groups $U_n$ with $n \ge 3$. The authors aim to establish cutoff under the weaker Sarnak-Xue Density Hypothesis (SXDH), which bounds the density of non-tempered representations.

Methodology
The authors employ a combination of arithmetic group theory, the theory of Bruhat-Tits buildings, and spectral analysis of Hecke operators.

1. Arithmetic Constructions: To prove the existence of class number one $p$-arithmetic groups (which act transitively on the building), the authors utilize two distinct techniques:

   • Siegel-Prasad Volume Formula: They compute the mass of the arithmetic group using the Tamagawa measure and compare it to the size of the group over the integers. If the mass equals the reciprocal of the group size, the class number is one.
   • Strong Approximation and "One-for-All": They use the strong approximation theorem to show that if a subgroup acts transitively on the vertices of the building, the group has class number one. This involves counting elements in specific congruence classes and relating them to the degree of the building.

2. Bruhat-Tits Theory: The paper relies heavily on the structure of Bruhat-Tits buildings for semisimple $p$-adic groups. It utilizes the Cartan and Iwahori decompositions, the structure of parahoric and Iwahori subgroups, and the action of these groups on the building's chambers and vertices. The authors specifically analyze the stabilizers of faces and the transitivity of group actions on these structures.

3. Spectral Bounds on Hecke Operators: A core technical contribution is the derivation of upper bounds for the operator norms of Hecke operators acting on unitary irreducible representations. The authors bound the norm $\|1_{PSP} | \pi_P \|_{op}$ in terms of the size of the set $S$ and the rate of decay $\sigma(\pi)$ of the matrix coefficients of the representation $\pi$. This generalizes previous results to broader classes of compact open subgroups $P$ and sets $S$.

4. Cutoff Analysis: The authors analyze the mixing time of geometric operators (such as powers of adjacency operators) on finite quotients of the building. They decompose the $L^2$ space of the quotient into irreducible representations. By combining the spectral bounds on Hecke operators with the SXDH (which limits the contribution of non-tempered representations) and a spectral gap assumption, they prove that the total variation distance to the stationary distribution drops sharply at a specific time.

Key Contributions and Results

• Construction of Golden Adelic Groups:

  • The authors construct explicit examples of $p$-arithmetic subgroups in $PU_4$ and $PU_8$ that have class number one. These groups act simply transitively on the vertices of the Bruhat-Tits building of $PU_4(\mathbb{Q}_p)$ (for specific primes $p$).
  • These constructions extend the seminal works of Lubotzky, Phillips, and Sarnak (LPS) to higher dimensions. Specifically, the current paper provides the necessary arithmetic foundations (class number one groups in $PU_4$ and $PU_8$) to construct "Golden Gate sets" for 2-qubit ($PU_4$) and 3-qubit ($PU_8$) systems.
  • The paper presents new examples based on the $E_8$ lattice realized over quadratic imaginary fields ($\mathbb{Q}[\sqrt{-1}], \mathbb{Q}[\sqrt{-2}], \mathbb{Q}[\sqrt{-3}], \mathbb{Q}[\sqrt{-7}]$), as well as constructions involving non-quasisplit groups where the building is a tree. The full construction of the Golden Gate sets for 2 and 3 qubits, utilizing these arithmetic groups alongside SXDH results, is completed in the companion paper [DEP].

• Cutoff under SXDH:

  • The paper proves that a family of finite quotients of a Bruhat-Tits building exhibits the cutoff phenomenon if the family satisfies the Sarnak-Xue Density Hypothesis (SXDH) and a spectral gap property.
  • This result generalizes previous cutoff theorems for Ramanujan complexes (which require the stronger NRC) to a broader class of groups where NRC fails.
  • The mixing time is shown to be $t(n) = \log_d(n)$, where $d$ is the exponential growth rate of the geometric operators, with a cutoff window $w(n) = o(t(n))$.

• Spectral Bounds:

  • Theorem 5.3 provides a general bound for the operator norm of Hecke operators: $\|1_{PSP} | \pi_P \|_{op} \ll (\log |S|)^{C+d} |S|^{\frac{\sigma-1}{\sigma}}$. This bound is crucial for controlling the error terms in the mixing time analysis when the Ramanujan property does not hold.

Significance
The paper claims significance in two primary areas:

1. Quantum Computation: The constructed $p$-arithmetic groups in $PU_4$ and $PU_8$ provide the essential arithmetic components required to construct "Golden Gate sets" for 2 and 3 qubits. These sets are optimal in terms of circuit length and possess efficient compiling algorithms. This extends the applicability of arithmetic group theory to universal quantum computation in higher dimensions, a field previously limited largely to $PU(2)$ constructions. The actual construction of these sets is finalized in the companion paper [DEP] by combining the results of this paper with SXDH proofs for unitary groups.

2. Mathematical Theory of Expansion: The paper establishes that the cutoff phenomenon is robust, persisting even when the Naïve Ramanujan Conjecture fails. By replacing NRC with the SXDH, the authors provide a framework for understanding the expansion properties of quotients of Bruhat-Tits buildings for general semisimple $p$-adic groups. This bridges the gap between the theory of Ramanujan graphs/complexes and the more general landscape of arithmetic groups where non-tempered representations are unavoidable.

The authors note that the companion paper [DEP] utilizes these results to prove the SXDH for definite algebraic unitary groups in dimensions 4 and 8, thereby completing the construction of the Golden Gate sets and proving the almost-optimal covering property for the associated real Lie groups.