Cyclotomic Structure and Exact Recurrences in Floquet Many-Body Quantum Dynamics

Technical Summary: Cyclotomic Structure and Exact Recurrences in Floquet Many-Body Quantum Dynamics

Problem Statement
Periodically driven (Floquet) quantum systems typically exhibit complex dynamics, including chaos and thermalization, where stroboscopic evolution rarely closes exactly. While approximate quantum recurrences (Poincaré recurrences) are guaranteed for finite-dimensional unitary systems, they occur on timescales that scale exponentially or doubly exponentially with system size, rendering them impractical for analysis or application. However, at special parameter values, Floquet systems can exhibit exact recurrences, where the evolution operator $U_F^n$ returns to the identity (up to a global phase) after a finite number of drive cycles $n$.

Determining whether such an exact recurrence exists is challenging. A direct search over successive powers of the Floquet operator is computationally expensive and offers no clear way to rule out recurrence. Furthermore, the spectral condition for recurrence (that all eigenvalue ratios are $n$-th roots of unity) is difficult to apply when the full spectrum of interacting many-body systems is unknown. The authors aim to establish a rigorous, necessary, and sufficient arithmetic criterion to determine the existence of exact recurrences and to calculate the minimal recurrence period without requiring explicit diagonalization of the Hamiltonian.

Methodology
The authors develop a framework based on algebraic number theory, specifically the arithmetic structure of cyclotomic fields. The method assumes the entries of the one-period Floquet unitary $U$ lie within a number field $K$ (typically a cyclotomic field $\mathbb{Q}(\zeta_m)$ determined by the Hamiltonian parameters).

The core procedure involves the following steps:

1. Square-Free Polynomial Construction: Instead of using the characteristic polynomial $f_U(x)$, which may have repeated roots due to spectral degeneracies, the authors construct its square-free part, $f_{sf}(x)$. This polynomial contains each distinct eigenvalue exactly once and serves as the minimal polynomial of $U$ (since $U$ is unitary and thus diagonalizable).
2. Ramification Analysis: The authors analyze the field extensions required for the eigenvalues to align. If $U^n = \tau I$, the splitting field of $f_{sf}(x)$ must contain the cyclotomic field $\mathbb{Q}(\zeta_n)$. By studying the ramification of primes in the tower of number fields $K \subseteq L$ (where $L$ is the splitting field), the authors identify a finite set of rational primes $S$ that can possibly divide the recurrence period $n$. These primes are derived from:
   • The discriminant of the base field $K$.
   • The integer $c$ used to clear denominators in the coefficients of $f_{sf}(x)$.
   • The norm of the discriminant of the polynomial $f_{sf}(x)$.
3. Universal Exponent Construction: Using the set of admissible primes $S$, the authors construct a "universal exponent" $E$. This integer is defined such that if a recurrence exists, the minimal period $n$ must divide $E$. The exponent is calculated using Euler's totient function constraints: $\phi(n) \mid [K:\mathbb{Q}] \cdot d_{sf}!$, where $d_{sf}$ is the degree of the square-free polynomial.
4. Periodicity Testing: The existence of a recurrence is tested by computing the polynomial remainder $\rho_E(x) = x^E \pmod{f_{sf}(x)}$.
   • If $\rho_E(x)$ is a constant, $U^E$ is a scalar matrix, confirming periodicity.
   • If $\rho_E(x)$ is non-constant, the system is exactly aperiodic (no finite $n$ exists).
5. Minimal Period Determination: If periodicity is confirmed, the minimal period is found by testing the divisors of $E$ to find the smallest $n$ such that $x^n \pmod{f_{sf}(x)}$ is constant.

Key Contributions and Results
The authors apply this arithmetic framework to the kicked Ising chain, a standard model for studying quantum chaos and many-body dynamics.

• Clifford Regime: For parameter sets where the Floquet operator belongs to the Clifford group, the authors confirm that exact recurrences always exist (as the Clifford group modulo phases is finite). They demonstrate that the minimal recurrence period depends on the system size $N$ in a non-monotonic manner. For example, the period decreases from 48 at $N=5$ to 18 at $N=6$ before increasing again, challenging the assumption that recurrence times simply scale with system size.
• Non-Clifford Regime: The framework rigorously distinguishes between recurrent and aperiodic non-Clifford dynamics.
  • For a specific grid of parameters (rational multiples of $\pi$), the method proves that many non-Clifford circuits are exactly aperiodic (no finite $n$ satisfies $U_F^n \propto I$), demonstrating that rational parameters alone do not guarantee recurrence.
  • Conversely, the authors identify an infinite family of non-Clifford parameters (specifically with transverse kick $h=\pi/2$ and interaction $J=\pi/2s$) that exhibit exact recurrence. For this family, the minimal period is $n_{rec} = 2s$, independent of the system size $N$.
• Magic Generation: In the recurrent non-Clifford regime, the authors analyze the generation of "magic" (non-stabilizer resources) using the second stabilizer Rényi entropy ($M_2$). They show that starting from a stabilizer state, the system generates magic at intermediate times ($M_2 > 0$) but returns exactly to a stabilizer state ($M_2 = 0$) at the recurrence period. This confirms that exact recurrence is compatible with non-Clifford evolution and that magic can be generated and subsequently erased in a controlled, periodic fashion.

Significance and Claims
The paper claims to provide a necessary and sufficient arithmetic criterion for exact recurrence that applies to any finite-dimensional Floquet unitary with entries in a number field, regardless of whether the system is integrable or non-integrable, or Clifford or non-Clifford.

Key significance points highlighted by the authors include:

• Extension Beyond Clifford Circuits: The work demonstrates that exact recurrences are not exclusive to Clifford circuits. It establishes a class of non-Clifford Floquet unitaries that recur exactly, expanding the known landscape of predictable quantum dynamics.
• Resource Generation: The identification of recurrent non-Clifford circuits that generate and revive magic suggests a new mechanism for controlled magic generation, a crucial resource for universal quantum computation.
• Hardware Benchmarking: The authors propose that these exact recurrences can serve as benchmarks for quantum hardware. Because the recurrence times are mathematically determined and can be tuned via Hamiltonian parameters, they provide exact reference points to probe error accumulation in long quantum circuits.
• Methodological Efficiency: The approach avoids the computational bottleneck of full spectral analysis or brute-force search, relying instead on polynomial arithmetic and number-theoretic properties of the Hamiltonian parameters.

The authors emphasize that their results are derived strictly from the arithmetic structure of the Floquet unitary and do not rely on specific circuit decompositions, making the framework broadly applicable to various driven quantum systems.