Anomalous transport in U(1)-symmetric quantum circuits

Technical Summary: Anomalous Transport in U(1)-Symmetric Quantum Circuits

Problem Statement
The advent of high-fidelity digital quantum devices has enabled the realization of "digital phases of matter," such as discrete time crystals and measurement-induced phase transitions. However, the transport properties of these discrete-time systems remain partially mapped, particularly in the presence of disorder. Unlike continuous-time models, discrete-time (Floquet) systems do not conserve energy but can sustain other conserved quantities, such as magnetization. The central problem addressed is how to probe phase transitions and information spreading in such digital matter, specifically investigating whether unique transport regimes exist that have no analog in continuous-time counterparts. The authors focus on a generic, disordered, U(1)-symmetric model tuned across various dynamical regimes to identify these features.

Methodology
The authors construct a generic gate-based model consisting of $N$ qubits undergoing discrete-time evolution via a Floquet unitary $\hat{U}$.

• Model Construction: The unitary is decomposed into a product of nearest-neighbor gates $\hat{U}_{n,n+1}$, ordered by a random permutation to introduce disorder. Each gate comprises single-qubit rotations and a nearest-neighbor XXZ-like interaction generated by a Hamiltonian with parameters $\{J, J_z\}$ and random phase fields $\{h_n, h'_n, \phi_n\}$. The model conserves total magnetization $\hat{M} = \sum \hat{S}^z_n$.
• Phase Space Exploration: The authors restrict the parameter space to $J, J_z \in [0, \pi]$. They identify three integrable boundaries: the free-fermion line ($J_z=0$), the Ising line ($J=0$), and the dual-unitary (DU) line ($J=\pi$). The primary analysis focuses on the $J_z = \pi$ line, varying $J$ to traverse localized, ergodic, and intermediate regimes.
• Static Diagnostics: To characterize equilibrium properties, the authors employ exact diagonalization (using the POLFED method) to compute the mean eigenstate entanglement entropy (normalized by the Page value for fixed magnetization sectors) and the mean adjacent gap ratio. These metrics distinguish between ergodic (Wigner-Dyson statistics) and integrable/localized (Poisson statistics) phases.
• Dynamic Diagnostics: To quantify transport, the authors map the local magnetization profile $M_n(t)$ to a quasi-probability distribution $p_n(t)$. They introduce a novel aggregate quantity: the circular statistical moment $R(t)$, defined as the complex circular mean of the distribution on the unit circle. From $R(t)$, they extract:
  • The spread parameter $\sigma(t)$ (related to the magnitude $|R(t)|$).
  • The drift position $\mu(t)$ (related to the argument $\text{Arg}[R(t)]$).
  • The instantaneous drift speed $\nu(t) = |d\mu/dt|$.
  • The peak decay exponent $\alpha_p$ derived from the maximum value of the distribution.

Key Contributions

1. Circular Statistical Moment ($R$): The authors develop $R(t)$ as a concise, experimentally accessible aggregate quantity composed solely of local Z-basis expectation values. This quantity elegantly captures both the spatial spread and coherent drift of excitations in periodic systems, visualizing transport dynamics on the complex unit circle.
2. Discovery of the "Swappy" Regime: The paper identifies a distinct prethermal regime unique to discrete-time systems located between the ergodic phase and the dual-unitary line. In this regime, excitations propagate coherently and ballistically for extended timescales, even in the presence of strong disorder.
3. Prethermal Scaling: The authors demonstrate that the thermalization time in the swappy regime diverges as a power law with the distance from the integrable SWAP point ($J=\pi$), specifically $t_{th} \propto (\pi - J)^{-\gamma}$ with $\gamma \approx 2.4\text{--}2.6$.

Results

• Static Properties: Spectral analysis confirms the expected crossover from localized (low entanglement, Poisson statistics) to ergodic (Page entropy, Wigner-Dyson statistics) regimes as $J$ increases. Near the SWAP point ($J \to \pi$), deviations from ergodicity are observed, attributed to the emergence of near-conserved operators splitting the spectrum.
• Transport Dynamics:
  • Localized Regime ($J \approx 0.4$): Excitations remain confined; transport is suppressed, exhibiting logarithmic or sub-diffusive spreading.
  • Ergodic Regime ($J \approx 1.4$): The system exhibits standard diffusive transport ($\sigma(t) \sim t^{1/2}$) and rapid thermalization.
  • Swappy Regime ($J \approx 2.55$): A unique regime where excitations propagate ballistically ($\sigma(t) \sim t$) with a non-zero drift speed before eventually thermalizing. This behavior is characterized by "twin peaks" in the averaged magnetization profile, indicating coherent propagation in both directions.
  • Near-SWAP Regime ($J \approx 3.14$): Coherent ballistic transport persists for very long times with minimal spreading, though finite-size effects eventually lead to apparent thermalization in averages.
• Exponents: The transport exponents extracted from the peak decay ($\alpha_p$) and spread ($\alpha_\sigma$) clearly distinguish the regimes: $\alpha \approx 0$ (localized), $\alpha \approx 1/2$ (diffusive), and $\alpha > 1/2$ (super-diffusive/ballistic) in the swappy regime.

Significance and Claims
The paper claims to have identified a genuinely discrete-time transport phenomenon—the "swappy" regime—which has no counterpart in continuous-time XXZ models. This regime arises from the finite, non-scaled unitary kicks inherent to Floquet dynamics, which create interference effects and a nearby integrable point that vanish in the continuous-time (Trotterized) limit.

The authors emphasize that the swappy regime is:

1. Distinct: It exhibits a clear crossover from ergodic behavior.
2. Dynamical: It is invisible to static spectral analysis but prominent in time-evolution.
3. Prethermal: It delays thermalization over timescales that diverge as the system approaches the integrable SWAP point.
4. Stable: It persists across system sizes and parameter variations near the DU line.
5. Unique to Digital Matter: It reflects a transport mechanism intrinsic to discrete-time systems.

The work concludes that the circular moment $R(t)$ provides a robust, experimentally feasible tool for analyzing transport in digital matter, paving the way for future experimental verification of these Floquet-specific phenomena on current quantum hardware. The authors remain modest regarding the thermodynamic limit of the localized regime, noting that their claims are limited to phenomenological signatures at finite system sizes ($N \le 22$) and that the stability of many-body localization in discrete-time systems requires further study.