Local spreading of stabilizer Rényi entropy in a brickwork random Clifford circuit

This paper investigates the spreading of stabilizer Rényi entropy in brickwork random Clifford circuits, revealing that while the entropy typically exhibits a diffusive profile within a ballistic light cone for Haar-random gates, it transitions to superdiffusive behavior in restricted circuits, a non-ballistic phenomenon also observed in the robustness of magic.

The Gist

The Big Picture: What is "Magic" in Quantum Computers?

Imagine a quantum computer as a giant, complex kitchen. To cook a truly amazing meal (perform a quantum advantage), you need two main ingredients:

1. Entanglement: This is like the ingredients being mixed together so thoroughly that you can't tell where one ends and the other begins. It's essential, but it's not enough on its own. You can have a perfectly mixed soup that is still just "soup" (classical simulation).
2. Magic (Non-stabilizerness): This is the secret spice. In the world of quantum physics, this "magic" is what makes a computer truly powerful and impossible for classical computers to simulate. Without this spice, the quantum computer is just a very expensive, very fast calculator.

The paper asks a simple question: If we drop a single grain of this "magic spice" into a quantum system, how does it spread out over time?

The Experiment: The Brickwork Circuit

The researchers set up a simulation that looks like a brick wall.

• The Bricks: They use "Clifford gates," which are like standard, safe, predictable kitchen tools. They are great at mixing things up (creating entanglement) but they don't create new "magic" on their own.
• The Spice: They start with a system where almost everything is plain (a "stabilizer state"), but they inject one special "magic" qubit (a T-state) in the middle. Think of this as dropping a single drop of hot sauce into a bowl of plain water.
• The Action: They let the "brick wall" circuit run. The gates shuffle the qubits around, mixing the hot sauce into the water.

The Discovery 1: The "Light Cone" (The Speed Limit)

Just like a sound wave travels through air, the "magic" spreads through the quantum system. However, it has a speed limit.

• The Analogy: Imagine you drop a stone in a pond. The ripples move outward in a circle. You can't feel the ripple on the other side of the pond instantly; you have to wait for the wave to travel there.
• The Result: The magic spreads inside a "light cone." If you are far away from the drop point, the magic won't reach you until enough time has passed. This is expected and normal.

The Discovery 2: The Diffusive Spread (The "Muddy River")

Here is where it gets surprising. Usually, when things spread in a chaotic system without any rules (like gas molecules), they spread out in a straight line (ballistic). But the researchers found something different for the concentration of magic.

• The Analogy: Imagine pouring a drop of ink into a fast-flowing river.
  • Ballistic (Normal): The ink stays in a tight, fast-moving bullet shape.
  • Diffusive (What they found): The ink spreads out like a cloud of smoke or a muddy river. It doesn't just move forward; it spreads sideways and backward, getting wider and wider as it goes.
• The Surprise: Even though the quantum system has no "conserved charges" (no rules like "energy must stay in one place" that usually cause this kind of spreading), the "magic" still spreads out like a diffusing cloud. It behaves like a random walk, similar to how a drunk person stumbles down a street, rather than a sprinter running in a straight line.

The Discovery 3: The "Super" and "Sub" Variations

The researchers tested what happens if they change the rules of the kitchen slightly.

• Restricted Kitchen: They used a smaller, less random set of tools (restricted Clifford gates).
• The Result: The magic didn't spread like a normal diffuser anymore. It spread faster than normal diffusion (Super-diffusive), like a runner who is slightly faster than a sprinter but not quite ballistic.
• Another Measure: They also checked a different way to measure "magic" (called Robustness of Magic). It behaved similarly, confirming that this weird spreading pattern is a fundamental feature of how quantum magic moves, not just a quirk of one specific measurement.

The Real-World Meaning: Why Should We Care?

Why does this matter?

1. The "Lost" Magic: The paper shows that as the magic spreads, the amount of "pure" magic at any single spot drops off very quickly (exponentially). It's like the hot sauce getting so diluted in the river that you can't taste it anymore.
2. Recovering the Magic: The researchers found that the "magic" isn't truly gone; it's just hidden in the complex entanglement of the whole system. If you want to get that pure magic back at a specific spot, your chances of success drop the further away you are from the original drop point and the longer you wait.
3. Thermalization: This helps us understand how quantum systems "heat up" or settle into equilibrium. It tells us that even in systems that look chaotic, there are hidden, slow-moving patterns (like diffusion) governing how resources spread.

Summary in a Nutshell

Think of the quantum computer as a giant, chaotic dance floor.

• You drop one dancer with a special "magic" move in the center.
• The music (the circuit) makes everyone shuffle and mix.
• You might expect the special move to travel in a straight line to the edge of the room.
• Instead, the "vibe" of that special move spreads out like a slow-moving, widening cloud of fog. It moves forward, but it also spreads sideways and gets thinner as it goes.
• This happens even though the dancers are following strict, random rules.

The paper proves that quantum magic spreads like a diffusing gas, not a speeding bullet, and this behavior is robust even when we change the rules of the game. This gives us a new map for understanding how quantum information travels and how hard it is to recover specific quantum states.

Technical Summary

1. Problem Statement

Quantum "magic" (or non-stabilizerness) is a fundamental resource required for universal quantum computation, distinct from entanglement. While the global dynamics of magic in random quantum circuits have been studied, the local spreading dynamics of magic remain poorly understood. Specifically, it is unclear how non-stabilizer resources propagate through a many-body system under unitary evolution, particularly in Clifford circuits where the total global magic is conserved (as Clifford gates map stabilizer states to stabilizer states).

The authors investigate the spatial and temporal evolution of Stabilizer Rényi Entropy (SRE) in local subsystems (single qubits) starting from a product state containing localized magic, evolving under a brickwork random Clifford circuit.

2. Methodology

• System Setup:
  • A 1D chain of $L$ qubits initialized in a product state $|0\rangle^{\otimes L}$ with a single "magic" qubit at site $m$ prepared in the $|T\rangle$ state (a non-stabilizer state).
  • Time evolution is governed by a brickwork random Clifford circuit. The circuit consists of alternating layers of random two-qubit Clifford gates sampled uniformly from the full Clifford group $\mathcal{C}_2$ (Haar-random within the group).
• Measures of Magic:
  • Stabilizer Rényi Entropy (SRE): Specifically the second-order SRE ($M^{(2)}$), calculated for the reduced density matrix of each single qubit. For a single qubit, this is derived from the Pauli spectrum coefficients $c_P = \text{Tr}(\rho P)$.
  • Log-Free Robustness of Magic (LROM): A faithful magic monotone for mixed states, used to verify the robustness of the findings.
• Simulation Technique:
  • The authors employ an efficient Heisenberg picture simulation. Since Clifford gates map Pauli strings to Pauli strings, they track the time-evolved Pauli operators $P(t) = U^\dagger P U$.
  • The expectation values $\langle \psi_0 | P(t) | \psi_0 \rangle$ are computed exactly for the initial low-magic state, allowing for the exact calculation of the Pauli spectrum and SRE without approximations, even for large system sizes ($L \sim 30-40$) and deep circuits.
• Analysis:
  • The study analyzes the averaged single-qubit SRE ($M_i(t)$) over $10^6$ circuit realizations.
  • A normalized SRE ($a_i(t) = M_i(t) / \sum_j M_j(t)$) is introduced to isolate the spatial distribution profile from the global decay.
  • A restricted Clifford circuit (using only H, S, and CNOT gates) is tested to examine the robustness of the observed dynamics.

3. Key Contributions and Results

A. Exponential Decay of Total Local Magic

While the global pure state preserves total SRE, the sum of averaged single-qubit SREs ($M(t) = \sum_i M_i(t)$) decays exponentially in time:
$$M(t) \sim e^{-\Gamma t}$$
with a decay rate $\Gamma \approx 0.44$. This indicates that as the system evolves, the magic becomes increasingly encoded in non-local entanglement, making it inaccessible to local measurements.

B. Emergent Diffusive Spreading (Haar-Random Case)

The most significant finding is the spatial structure of the normalized single-qubit SRE ($a_i(t)$) within the causal light cone:

• Diffusive Profile: Despite the absence of explicit conserved charges (which typically drive diffusion in hydrodynamic systems), the normalized SRE spreads diffusively.
• Discrete Diffusion Equation: The numerical data satisfies the recurrence relation:
  $$a_i(t) = \frac{1}{2} [a_{i-1}(t-1) + a_{i+1}(t-1)]$$
  This converges to the continuous diffusion equation $\partial_t a = D \partial_x^2 a$ with diffusion constant $D=1/2$.
• Mechanism: The authors derive this behavior by analyzing the action of a single two-qubit Clifford gate. They show that the ratio of the sum of output SREs to input SREs is approximately constant ($e^{-\Gamma}$) and that the output SREs of the two qubits become equal on average. This symmetry leads to the diffusive update rule.

C. Superdiffusive Spreading (Restricted Circuit)

When the circuit is restricted to a generating set of gates (Hadamard, Phase, CNOT) rather than the full Haar-random Clifford group:

• The strict diffusive equation no longer holds because the symmetry ensuring equal output SREs is broken.
• However, the spreading remains non-ballistic. The width of the distribution scales as $\sigma(t) \sim t^\beta$ with $\beta \approx 0.65$, indicating superdiffusive behavior. This suggests the non-ballistic nature of magic spreading is robust across different Clifford ensembles.

D. Robustness of Magic (LROM)

The study extends to the Log-Free Robustness of Magic (LROM), a faithful measure for mixed states.

• LROM also exhibits exponential decay and a non-ballistic spreading profile within the light cone.
• The spreading is sub-diffusive ($\beta \approx 0.45$), showing that the intricate non-ballistic dynamics are a generic feature of magic spreading, independent of the specific magic monotone used.

E. Operational Interpretation

The authors provide a clear operational meaning for the single-qubit SRE:

• $M_i(t)$ provides an upper bound on the probability $p_i(t)$ of finding a pure magic state $|T\rangle$ at site $i$ (factorized from the rest of the system).
• The exponential decay and Gaussian spatial profile imply that the probability of recovering the initial localized magic state at a distant site is exponentially suppressed in both time and distance.

4. Significance

• New Hydrodynamic Phenomenon: The paper identifies a novel hydrodynamic-like behavior (diffusion) in a unitary system without conserved quantities. This challenges the intuition that diffusion requires conserved charges, suggesting that the structure of the Clifford group itself induces emergent hydrodynamics for magic.
• Resource Scrambling: It quantifies how "magic" scrambles into non-local correlations, providing a limit on the efficiency of magic-state distillation or local recovery of quantum resources in noisy or randomized environments.
• Universality: The observation of non-ballistic spreading (diffusive or superdiffusive) across different measures (SRE, LROM) and circuit ensembles (full vs. restricted) suggests a universal class of dynamics for the transport of quantum resources in chaotic systems.
• Experimental Relevance: Since SRE is experimentally accessible (via randomized measurements), these findings offer testable predictions for near-term quantum devices investigating the dynamics of quantum resources.

In conclusion, the work reveals that while Clifford circuits preserve global magic, they induce a complex, non-ballistic, and often diffusive redistribution of local magic, fundamentally altering the landscape of quantum resource availability in many-body systems.