nonstabilizerness-diffusive-dynamics

name	nonstabilizerness-diffusive-dynamics
description	Nonstabilizerness diffusion dynamics methodology for analyzing magic resource generation in many-body quantum systems using stabilizer Renyi entropy and tensor network methods.

Nonstabilizerness Diffusive Dynamics

Description

Methodology for analyzing nonstabilizerness (magic) generation and dynamics in many-body quantum systems. Computes stabilizer Renyi entropy using four-replica tensor networks evaluated by S4-adapted iTEBD in the thermodynamic limit. Identifies diffusive universality class with 1/t gap closing for late-time approach to random-state value.

Activation Keywords

• nonstabilizerness dynamics
• stabilizer Renyi entropy
• magic state generation
• tensor network quantum dynamics
• hydrodynamic quantum information
• diffusive magic dynamics
• iTEBD quantum circuits
• 非稳定化性动力学
• 稳定化Renyi熵

Tools Used

• exec: Run tensor network simulations
• write: Save entropy calculations
• terminal: Execute iTEBD computations

Usage Patterns

Pattern 1: Stabilizer Renyi Entropy Computation

For U(1)-symmetric random circuits:

1. Construct four-replica tensor network representation
2. Apply S4-symmetric iTEBD in thermodynamic limit
3. Compute stabilizer Renyi entropy M2 = -log(E[sum p_i^2])
4. Track convergence to random-state value

Pattern 2: Hydrodynamic Analysis

For identifying universality class:

1. Map nonstabilizerness dynamics to hydrodynamic equation
2. Identify diffusive scaling: gap ~ 1/t
3. Verify scaling across different circuit families
4. Connect to random-state ensemble predictions

Pattern 3: Energy-Conserving System Verification

For nonintegrable Ising chains:

1. Simulate time evolution under energy-conserving Hamiltonian
2. Compute stabilizer Renyi entropy at multiple times
3. Verify 1/t scaling matches random circuit prediction
4. Establish universality across model classes

Instructions for Agents

Step 1: System Setup

• Define the quantum circuit/Hamiltonian
• Identify symmetries (U(1), Z2, etc.)
• Choose initial state (product state, GHZ, etc.)

Step 2: Tensor Network Construction

def build_four_replica_tn(circuit, symmetries):
    """Build four-replica tensor network for stabilizer Renyi entropy."""
    # Each physical site has 4 replica indices
    # S4 symmetry permutes replicas
    tn = TensorNetwork(replicas=4, symmetry='S4')
    for gate in circuit:
        tn.apply_gate(gate, replica_structure='diagonal')
    return tn

Step 3: iTEBD Evaluation

def compute_stabilizer_renyi(tn, max_bond_dim=128):
    """Compute M2 via iTEBD."""
    state = tn.initialize_infinite_mps(bond_dim=max_bond_dim)
    for step in range(max_steps):
        state = iTEBD_step(state, tn, symmetrize='S4')
        if converged(state):
            break
    return stabilizer_renyi_entropy(state)

Step 4: Scaling Analysis

• Plot M2(t) vs time
• Fit to 1/t + const form
• Extract diffusion coefficient
• Compare across system sizes

Error Handling

Bond Dimension Truncation

If convergence issues:

• Increase max_bond_dim
• Check truncation error threshold
• Use extrapolation in bond dimension

Symmetry Enforcement

If S4 symmetry violated:

• Project onto symmetric subspace
• Use symmetry-adapted tensor format
• Verify symmetry at each iTEBD step

Mathematical Framework

Stabilizer Renyi Entropy

M2(rho) = -log( sum_{P in P_n} |tr(P * rho)|^2 / 2^n ) where P_n is the n-qubit Pauli group

Diffusive Scaling

M2(infinity) - M2(t) ~ c/t (late-time approach) where c depends on circuit details and dimension

Hydrodynamic Description

dm/dt = D * nabla^2 m + noise where m is nonstabilizerness density, D is diffusion constant

Resources

• arXiv: 2606.13606 (Xiao & Ryu, 2026)
• Related: Stabilizer formalism (Gottesman 1998)
• iTEBD: Infinite Time-Evolving Block Decimation

Related Skills

• tensor-network-quantum-electromechanics
• quantum-statistical-mechanics-gauge
• statistical-mechanics-quantum-decoding