Operator spreading in random circuits with orthogonal or symplectic symmetry

This paper investigates operator spreading in random quantum circuits with orthogonal or symplectic symmetry, revealing distinct features such as ternary-valued weight relaxation, finite-width domain walls, and a fundamental dichotomy in butterfly velocity behavior that differs significantly from the well-studied unitary-invariant case.

The Gist

Imagine a quantum computer not as a super-fast calculator, but as a giant, chaotic game of "telephone" played with information. In this game, a piece of information (an "operator") starts at one specific spot. As the game progresses, this information gets scrambled and spreads out across the entire system, becoming entangled with everything else. This process is called operator spreading.

Scientists usually study this using "random circuits," where the rules of the game (the "gates") are chosen completely at random from a massive library of possibilities. This paper investigates what happens when we change the library. Instead of picking from the standard "Unitary" library, the authors look at two other specific libraries: the Orthogonal and Symplectic libraries. These libraries represent systems with specific symmetries, like time-reversal or particle-hole symmetry.

Here is what they found, explained through everyday analogies:

1. The "Ternary" vs. "Binary" Switch

In the standard "Unitary" game, the information spreading looks like a simple on/off switch. A piece of information is either "trivial" (it hasn't changed much) or "scrambled" (it's fully mixed up). It's a binary world: 0 or 1.

However, in the Orthogonal and Symplectic games, the world is ternary (three-valued). The information doesn't just switch between "off" and "on." It has a third state: it can be "even" or "odd" (symmetric or antisymmetric).

• The Analogy: Imagine a standard light switch (On/Off). In the new games, the switch has a middle position. The light can be Off, On, or "Dim/Flashing" (the third state). The system takes time to settle into this three-state pattern, whereas the old system settled into a two-state pattern immediately.

2. The Foggy Wall vs. The Sharp Edge

When information spreads, it creates a "front" or a "wall" separating the area where nothing has happened (trivial) from the area where everything is scrambled.

• In the old (Unitary) games: This wall is razor-sharp. It's like a cliff edge. You are either in the calm zone or the chaos zone.
• In the new (Orthogonal/Symplectic) games: The wall is fuzzy. Even if the rules are chosen completely at random (Haar-random), there is a "fog" or a transition zone where the information is neither fully calm nor fully scrambled.
• The Analogy: The old system is like a sharp drop-off from a cliff. The new system is like a sandy beach slope. You can't pinpoint exactly where the "calm" ends and the "chaos" begins; there is a blurry middle ground that always exists.

3. The Speed Limit Surprise (The Butterfly Velocity)

Scientists measure how fast this information spreads using a speed called the "butterfly velocity" (named after the butterfly effect).

• The Expectation: Usually, the fastest speed is set by the most random, chaotic rules (the Haar-random limit).
• The Surprise: The authors found that in the Orthogonal world, there are two different "sectors" (like two different teams playing by slightly different rules).
  • Team A (Special Orthogonal): Their speed is normal. It's somewhere between doing nothing and the maximum speed.
  • Team B (Negative Determinant): This team behaves strangely. They have a minimum speed that is strictly greater than zero, no matter how you tune the rules. You can't slow them down to a crawl.
  • The Super-Speed: Even more surprisingly, for small systems (specifically with 2-dimensional units), Team B can actually run faster than the maximum speed limit of the standard Unitary game.
• The Analogy: Imagine a race. The standard rules say the fastest anyone can run is 10 mph. The "Special Orthogonal" team runs between 0 and 10 mph. But the "Negative Determinant" team has a rule that says "You must run at least 2 mph," and in some cases, they can actually sprint at 12 mph, breaking the usual speed limit.

4. Why This Matters (According to the Paper)

The paper doesn't talk about building better computers or medical applications yet. Instead, it focuses on the fundamental physics of how information moves.

• It shows that symmetry matters. The specific mathematical "shape" of the rules (Orthogonal vs. Unitary) changes the texture of the chaos.
• It reveals that randomness isn't always the same. Even if you pick rules completely at random, if you pick them from the "Orthogonal" library instead of the "Unitary" library, the information spreads differently, with a fuzzy front and a three-state structure.

Summary

This paper is like discovering that while everyone thought the universe scrambled information like a sharp, binary switch with a clear edge, there are actually other ways to scramble it. In these other ways, the switch has three positions, the edge is fuzzy, and sometimes, the information spreads faster than anyone thought possible, simply because of the hidden symmetry rules governing the game.

Technical Summary

Technical Summary: Operator Spreading in Random Circuits with Orthogonal or Symplectic Symmetry

Problem Statement
Random unitary circuits serve as minimal models for investigating fundamental aspects of many-body quantum dynamics, including information spreading, quantum chaos, and thermalization. While the behavior of these circuits under Haar-random unitary gates (maximal randomness) is well-established, the dynamics of operator spreading in circuits constrained by orthogonal or symplectic symmetries remain less understood. Specifically, it is unclear how the invariance of gate distributions under orthogonal or symplectic transformations—rather than general unitary transformations—affects the propagation of information, the structure of the operator front, and the resulting "butterfly velocity."

Methodology
The authors investigate random quantum circuits with a brickwork structure, where two-qudit gates are drawn from ensembles satisfying an invariance property $P(U) = P(VUV^\dagger)$, where $V$ is restricted to the orthogonal or symplectic group. The study focuses on the time evolution of an initially local operator $O(t)$ expanded in a basis of generalized Pauli strings.

The core methodology involves:

1. Correlation Function Evolution: The dynamics are mapped to the evolution of correlation functions $\rho_{pq}(t) = \langle \gamma_p(t)\gamma_q^*(t) \rangle$ of the Pauli-string coefficients.
2. Stochastic Growth Mapping: By exploiting the invariance properties, the quantum evolution is mapped to a classical stochastic growth model. This involves analyzing the transition probabilities $\langle |W_{ap}|^2 \rangle$ between Pauli strings.
3. Projection to Ternary Subspaces: Unlike the unitary-invariant case, which allows a reduction to a binary (trivial vs. non-trivial) description, the orthogonal and symplectic cases require a projection onto a "ternary" subspace. This accounts for the parity of generalized Pauli operators under transposition (orthogonal case) or duality (symplectic case). The authors define a projected ternary-string distribution $\bar{\rho}_{\bar{p}}$ where the string index $\bar{p}_x \in \{-1, 0, 1\}$ represents the trivial, even, or odd parity of the operator at site $x$.
4. Drift-Diffusion Analysis: The evolution of the right-propagating density of these ternary strings is analyzed. By truncating the hierarchy of $n$-point density equations and approximating higher-order terms with their maximally random steady-state values, the authors derive drift-diffusion equations. These yield analytical expressions for the butterfly velocity $v_B$ and the diffusion constant $D$.
5. Validation: The analytical results, derived via a truncation scheme (orders $n=0, 2, 4, 6$), are compared against direct numerical simulations of the classical stochastic growth process.

Key Contributions and Results

• Ternary vs. Binary Structure: The most significant qualitative distinction found is that for orthogonal- or symplectic-invariant circuits, the ensemble-averaged Pauli-string weights relax to a ternary-valued structure (trivial, symmetric, antisymmetric) rather than the binary structure (trivial vs. non-trivial) observed in unitary-invariant circuits. This ternary structure emerges after a finite thermalization time.
• Finite Front Width: In Haar-random unitary circuits, the domain wall separating trivial and scrambled regions is sharp. In contrast, the authors find that for orthogonal- or symplectic-invariant circuits, the domain wall possesses a finite width even when the gates are drawn from a Haar distribution over the respective groups.
• Butterfly Velocity Dichotomy in Orthogonal Ensembles: The paper reveals a fundamental dichotomy within the orthogonal group based on its disconnected components:
  • For the special orthogonal ensemble ($SO$), the butterfly velocity $v_B$ lies between zero and the value for the Haar-random unitary case.
  • For the negative-determinant sector ($SO^-$), there is a non-zero lower bound for $v_B$ for any gate distribution. Furthermore, for qubit size $q=2$, the butterfly velocity in the $SO^-$ sector can exceed the value of the Haar-random ensemble. This indicates that the determinant structure of the gate ensemble plays a crucial role in determining the speed of information spreading, potentially overriding the typical upper bound set by Haar randomness.
• Generalization to Arbitrary $q$: While the initial derivation focuses on qudit dimensions $q=2m$ (where a basis with well-defined parity exists), the authors generalize the formalism to arbitrary $q$ by defining "transpose" and "dual" indices for generalized Pauli matrices, establishing a well-defined stochastic growth model for the projected ternary weights in all cases.
• Specific Ensembles: The authors provide explicit calculations for:
  • Haar-orthogonal circuits: Showing convergence of the truncation scheme and confirming the finite front width.
  • Brownian special orthogonal circuits: Interpolating between trivial and Haar limits, where the large-$q$ limit recovers unitary results.
  • $SO^-$ ensembles: Demonstrating the parameter dependence of $v_B$ and its potential to surpass Haar limits for $q=2$.

Significance
The paper establishes that discrete symmetries (orthogonal and symplectic) fundamentally alter the mechanism of operator spreading compared to the standard unitary-invariant case. The findings highlight that:

1. The "front" of operator spreading is intrinsically broadened by these symmetries, even in maximally random settings.
2. The classification of gate ensembles must consider topological components (like the determinant sign in orthogonal groups), as these can lead to qualitatively different dynamical behaviors, such as non-zero lower bounds on spreading velocity or velocities exceeding the Haar limit.
3. The relaxation to a ternary distribution provides a more nuanced picture of operator ergodicity in symmetric systems, distinguishing between symmetric and antisymmetric operator components.

The work extends the theoretical framework for random circuits to include these symmetry classes, offering a more complete understanding of how symmetry constraints influence quantum chaos and thermalization.