⚛️ quantum physics

KPZ Superdiffusion of Local Correlators in Diffusive Random Quantum Circuits

This paper demonstrates that single-particle Green's functions in one-dimensional diffusive random quantum circuits coupled to an external bath exhibit Kardar-Parisi-Zhang (KPZ) superdiffusion, characterized by $t^{2/3}$ wandering of the probability center and $t^{1/3}$ free energy fluctuations, a behavior confirmed numerically across both strong and weak noise regimes.

Original authors: Ewan McCulloch

Published 2026-08-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ewan McCulloch

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crowded dance floor where everyone is moving to a chaotic, unpredictable beat. In the world of quantum physics, this dance floor is a collection of tiny particles, and the "beat" is the complex rules they follow. Usually, scientists assume that if you wait long enough, this chaotic dance will smooth out. The particles will forget their individual moves, mix thoroughly, and settle into a predictable, average rhythm, much like a cup of hot coffee eventually cooling down to room temperature. This process is called thermalization, and it's the reason why we can use simple laws of fluid dynamics to describe things like water flowing or heat spreading.

However, there's a special kind of dance move called "quantum coherence." Think of it as a dancer trying to remember a specific, intricate step sequence perfectly. In a noisy room, you'd expect this dancer to get confused and forget the steps almost instantly. But what if the noise isn't just random static, but a specific kind of chaos that actually helps the dancer remember? This is the puzzle scientists are trying to solve: How do quantum systems lose their "memory" (decohere), and does the environment always destroy that memory, or can it sometimes preserve it in strange, unexpected ways? Understanding this is crucial because it tells us how long quantum information can survive in a real-world computer or how materials conduct electricity when they are messy and disordered.

In this paper, the author, Ewan McCulloch, investigates exactly this question using a model of a one-dimensional quantum system—a line of qubits (the quantum dancers)—that is constantly being jostled by an external "bath" or noise. The goal was to track a specific type of quantum memory: a single-particle Green's function. You can think of this as a "ghost" of a particle that was created at one spot and is trying to travel to another. The paper asks: How does this ghost move and fade away when the system is noisy and disordered?

The findings are surprisingly counterintuitive. The author argues that this quantum ghost doesn't just wander randomly like a drunk person stumbling home (which would be standard diffusion). Instead, it behaves like a surfer riding a chaotic wave. The paper finds that the path of this ghost is governed by a phenomenon known as KPZ scaling (named after Kardar, Parisi, and Zhang). In everyday terms, imagine a drop of ink spreading on a piece of paper. In a normal, calm world, it spreads out in a perfect circle, and the distance it travels grows steadily with time. But in this quantum world, the ink drop doesn't just spread; it wanders wildly. The center of the drop moves back and forth in a way that grows much faster than expected—specifically, the distance it wanders scales with time raised to the power of 2/3 ( $t^{2/3}$ ). This is "superdiffusion," a wilder, more energetic spread than normal.

The paper distinguishes between two types of noise: "strong" noise and "weak" noise.

In the strong-noise limit (where the jostling is intense), the author shows that the system maps perfectly onto a mathematical model called "Directed Waves in a Random Medium." Here, the quantum ghost's path is determined by a competition between different possible routes, and the fluctuations in its energy follow the famous KPZ rules. The paper confirms this using computer simulations (tensor networks) that mimic the behavior of individual circuits.
In the weak-noise limit (where the jostling is gentle), the story gets even more interesting. Here, the quantum ghost gets stuck in a "void"—a region where the surrounding particles have cleared out to let the ghost pass. This void acts like a slow-moving bubble. The ghost rides inside this bubble. The paper finds that the bubble itself moves slowly, but the time it takes for the system to switch from behaving like a normal diffuser to this wild KPZ superdiffuser is incredibly long. Specifically, the crossover happens at a time scale proportional to $O(\gamma^{-3/2})$ , where $\gamma$ is the noise strength. This means if the noise is very weak, you have to wait a very long time to see the wild KPZ behavior emerge.

The author uses a clever trick called "phase-annealing" to simulate this. Imagine taking a photo of the quantum dance, but blurring out the exact timing of the steps (the phases) while keeping the speed of the moves (the hopping rates) sharp. This allows the complex quantum interference to be treated more simply, revealing that the system behaves like a "Directed Polymer in a Random Medium"—essentially, a flexible string trying to find the easiest path through a forest of obstacles. The paper shows that even with this simplification, the wild KPZ scaling remains.

Crucially, the paper argues against the idea that quantum systems in these conditions simply decay exponentially (fading away quickly and predictably). Instead, it suggests that the decay is "stretched," meaning the quantum memory survives longer than expected, supported by rare, low-entropy regions (the voids) where the chaos is temporarily tamed. The paper confirms these predictions through numerical simulations of individual circuits, showing that the wandering of the particle's center and the fluctuations in its energy match the $t^{2/3}$ and $t^{1/3}$ scaling laws predicted by KPZ theory.

In summary, this paper reveals that in a disordered, noisy quantum world, quantum information doesn't just fade away; it surfs on chaotic waves, wandering much further and faster than classical physics would predict. It shows that the "noise" of the environment can actually create a structure—a void—that protects and guides the quantum ghost, leading to a universal, wild behavior known as KPZ superdiffusion. While the results are based on simulations and theoretical mapping rather than a physical experiment in a lab, the consistency between the math and the computer models provides a strong argument that this is how nature behaves in these specific quantum circuits.

Technical Summary: KPZ Superdiffusion of Local Correlators in Diffusive Random Quantum Circuits

Problem Statement
The paper addresses the thermalization dynamics of isolated quantum many-body systems, specifically focusing on the behavior of single-particle Green's functions, $G(x, t) = \langle \sigma^-_x(0)\sigma^+_0(t) \rangle$ , in one-dimensional particle-number-conserving random unitary circuits coupled to an external bath. While hydrodynamic theories successfully describe the evolution of conserved densities (classical variables), they fail to capture the dynamics of intrinsically quantum, phase-sensitive observables. Previous work suggested that chaotic systems act as their own baths, leading to exponential decay of Green's functions; however, recent findings indicated a slower, stretched-exponential decay in rare low-entropy regions. The central problem is to determine the universal long-time behavior of these Green's functions in typical circuit realizations (fixed disorder) rather than ensemble-averaged (annealed) cases, particularly in the presence of both weak and strong noise.

Methodology
The author employs a combination of analytical mapping techniques and numerical simulations using tensor networks (specifically Time-Evolving Block Decimation, TEBD).

Model Definition: The system consists of a 1D brickwork circuit of qubits with $U(1)$ -conserving Haar-random two-site gates. After each unitary layer, sites are coupled to an external bath at rate $\gamma$ with filling $\nu$ . The study focuses on the infinite-temperature Green's function.
Strong-Noise Limit ( $\gamma \gg 1$ ): In this regime, the dynamics are mapped to a discrete-time non-Hermitian Schrödinger evolution for a single coherence. The operator histories collapse onto single-particle dynamics due to strong damping of multi-particle components. This is shown to be exactly equivalent to Directed Waves in a Random Medium (DWRM).
Weak-Noise Limit ( $\gamma \ll 1$ ): Here, coherent motion couples to slow diffusive modes. The author utilizes a "projected operator dynamics" ansatz (source-manifold) previously developed for annealed observables.
- A phase-annealed proxy is derived by averaging over gate phases while retaining exact hopping disorder. This maps the problem to a Directed Polymer in a Random Medium (DPRM).
- A replica treatment is applied to the typical (quenched) Green's function, showing that interference effects restore the DWRM description, albeit with the same scaling exponents as the DPRM proxy.
Macroscopic Fluctuation Theory (MFT): To analyze the weak-noise regime, the author derives an effective action for the "void" (a low-entropy region surrounding the coherence) using MFT. This involves a point-absorber approximation where the coherence acts as a sink for the surrounding fluid.
Numerical Verification: Tensor-network simulations are performed on individual circuit realizations at moderate noise ( $\gamma \approx 0.67$ ) and on the full two-copy transfer matrix dynamics for weak noise ( $\gamma \in [0.02, 0.06]$ ) to test the predicted scaling laws.

Key Contributions and Results

KPZ Scaling in Strong Noise: The paper establishes that in the strong-noise limit, the wandering statistics of the normalized spatial distribution $p(x, t) \propto |G(x, t)|^2$ and the associated free energy follow Kardar–Parisi–Zhang (KPZ) universality.
- The center of the distribution, $\langle x(t) \rangle$ , wanders on a length scale of $O(t^{2/3})$ .
- Sample-to-sample fluctuations of the free energy, $-\log \sum_x |G(x, t)|^2$ , scale as $t^{1/3}$ .
- The width of the distribution within a fixed sample remains diffusive ( $O(t)$ ).
- These predictions are confirmed numerically for individual circuits.
Weak-Noise Crossover and DPRM/DWRM Mapping: In the weak-noise limit, the paper identifies a parametrically long crossover time $t_{cross} \sim O(\gamma^{-3/2})$ before the system reaches the strong-disorder fixed point.
- The phase-annealed norm is shown to map to a DPRM.
- Crucially, the full typical Green's function (including quenched phase disorder) maps to a DWRM.
- Despite the different microscopic descriptions (polymer vs. wave), both exhibit the same KPZ scaling exponents ( $t^{2/3}$ for wandering, $t^{1/3}$ for free energy fluctuations) and the same crossover time scale.
Relevance of Hopping Disorder: A significant finding is that while microscopic hopping disorder is irrelevant for classical densities and currents at the diffusive fixed point, it becomes relevant for intrinsically quantum, phase-sensitive observables. This disorder drives the system to a strong-disorder fixed point characterized by KPZ superdiffusion.
Void Formation and Dynamics: The analysis reveals the formation of a "void" (a low-entropy region) around the coherence. The dynamics of this void are governed by a slow diffusion constant $D_{void} \sim \sqrt{D_0 \gamma}$ , which is much slower than the bare coherence diffusion. The interplay between the void relaxation time and the coherence motion justifies the static-void approximation used in the MFT derivation.

Significance and Claims
The paper claims to resolve the universal long-time behavior of Green's functions in typical realizations of diffusive random quantum circuits. It demonstrates that quantum coherence persists longer than expected, supported by rare low-entropy regions, leading to KPZ superdiffusion rather than simple diffusion or exponential decay.

The work highlights a fundamental distinction between classical and quantum correlators in the same system: classical densities are governed by diffusive hydrodynamics, whereas phase-sensitive quantum correlators are controlled by a strong-disorder fixed point (DWRM/DPRM). The paper provides a rigorous analytical framework (via DWRM and DPRM mappings) and numerical benchmarks (via tensor networks) that confirm these predictions. It modestly notes that while the phase-annealed proxy is fully verified, direct verification of the weak-noise DWRM scaling for genuine single-sample Green's functions remains a challenge due to the parametrically large crossover time $O(\gamma^{-3/2})$ and competing pre-asymptotic contributions. The author suggests future extensions to higher dimensions and higher-order observables as open directions.

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