krylov-complexity-analog-simulator

name	krylov-complexity-analog-simulator
description	Bridging Krylov complexity theory with universal analog quantum simulation — using Lanczos algorithm and Krylov subspace growth to characterize computational power of analog quantum simulators. Activation: Krylov complexity, analog quantum simulator, Lanczos algorithm quantum, operator growth complexity.

Krylov Complexity for Analog Quantum Simulation

Description

Methodology connecting Krylov complexity theory with universal analog quantum simulation. Uses Lanczos algorithm to track operator growth in Krylov space as a measure of computational complexity in analog quantum simulators. Applicable to quantum chaos characterization, simulator benchmarking, and complexity phase transitions.

Activation Keywords

• Krylov complexity analog simulator
• Lanczos algorithm quantum complexity
• operator growth Krylov space
• quantum simulator benchmarking complexity
• Krylov basis quantum dynamics
• analog quantum simulation complexity
• Krylov复杂性量子模拟

Tools Used

• terminal: Run Krylov subspace calculations
• execute_code: Implement Lanczos algorithm and Krylov complexity measures
• web_search: Research Krylov complexity and analog quantum simulation

Core Concepts

Krylov Complexity

• Operator Growth: Heisenberg evolution of operators expands in operator space
• Krylov Basis: Lanczos algorithm generates orthonormal basis {O_n} from repeated commutation [H, O]
• b_n Coefficients: Lanczos coefficients determine growth rate of operator complexity
• Krylov Complexity: K(t) = Σ n |φ_n(t)|² measures spread in Krylov basis

Lanczos Algorithm for Quantum Operators

Input: Hamiltonian H, initial operator O_0
1. b_1 O_1 = [H, O_0] - a_0 O_0    (a_0 = ⟨O_0|[H,O_0]|O_0⟩)
2. b_{n+1} O_{n+1} = [H, O_n] - a_n O_n - b_n O_{n-1}
3. Repeat until convergence or dimension limit
Output: Lanczos coefficients {b_n}, Krylov basis {O_n}

Complexity Phases

• Integrable Systems: b_n saturates or grows slowly — low complexity
• Chaotic Systems: b_n ~ n (linear growth) — maximal complexity
• Many-Body Localization: b_n decays — complexity freezes

Analog Quantum Simulator Benchmarking

1. Map Simulator to Model: Identify the effective Hamiltonian being simulated
2. Compute Lanczos Coefficients: Track b_n growth from experimentally accessible operators
3. Compare to Theory: Match measured b_n profile to expected complexity phase
4. Validate Universality: Check if simulator can access different complexity regimes

Implementation Pattern

Step 1: Lanczos Iteration

import numpy as np
from scipy.linalg import commutator

def lanczos_operator_growth(H, O0, max_iter=50, tol=1e-10):
    """Compute Lanczos coefficients for operator growth."""
    # Normalize initial operator
    O0 = O0 / np.sqrt(np.trace(O0.conj().T @ O0))
    
    b_coeffs = []
    a_coeffs = []
    basis = [O0]
    
    for n in range(max_iter):
        # Compute commutator
        comm = 1j * (H @ basis[-1] - basis[-1] @ H)
        
        # Project onto previous basis elements
        a_n = np.real(np.trace(comm.conj().T @ basis[-1]))
        comm -= a_n * basis[-1]
        
        if n > 0:
            comm -= b_coeffs[-1] * basis[-2]
        
        # Norm gives next b coefficient
        b_n = np.sqrt(np.real(np.trace(comm.conj().T @ comm)))
        
        if b_n < tol:
            break
            
        b_coeffs.append(b_n)
        a_coeffs.append(a_n)
        basis.append(comm / b_n)
    
    return np.array(b_coeffs), np.array(a_coeffs), basis

Step 2: Krylov Complexity Evolution

def krylov_complexity(b_coeffs, t_values):
    """Compute Krylov complexity K(t) from Lanczos coefficients."""
    # Tridiagonal Hamiltonian in Krylov basis
    n = len(b_coeffs)
    H_krylov = np.zeros((n+1, n+1))
    for i in range(n):
        H_krylov[i, i+1] = b_coeffs[i]
        H_krylov[i+1, i] = b_coeffs[i]
    
    # Initial state: all weight in |0⟩
    psi_0 = np.zeros(n+1)
    psi_0[0] = 1.0
    
    complexities = []
    for t in t_values:
        # Time evolution
        U = scipy.linalg.expm(-1j * H_krylov * t)
        psi_t = U @ psi_0
        
        # K(t) = Σ n |φ_n(t)|²
        K_t = sum(n * abs(psi_t[n])**2 for n in range(n+1))
        complexities.append(K_t)
    
    return np.array(complexities)

Step 3: Complexity Phase Classification

def classify_complexity_phase(b_coeffs):
    """Classify complexity phase from Lanczos coefficient growth."""
    # Fit b_n to different models
    n_vals = np.arange(1, len(b_coeffs)+1)
    
    # Linear fit (chaotic): b_n ~ α*n
    lin_fit = np.polyfit(n_vals, b_coeffs, 1)
    lin_error = np.sum((b_coeffs - np.polyval(lin_fit, n_vals))**2)
    
    # Saturated fit (integrable): b_n → const
    const_fit = np.mean(b_coeffs)
    const_error = np.sum((b_coeffs - const_fit)**2)
    
    if lin_error < const_error:
        return "chaotic", lin_fit[0]  # slope
    else:
        return "integrable", const_fit

Applications

• Quantum Simulator Validation: Verify simulator reaches expected complexity regime
• Chaos Detection: Identify quantum chaos through Lanczos coefficient growth
• Benchmarking: Compare different quantum simulator platforms
• Resource Estimation: Predict computational resources needed for simulation

Pitfalls

• Dimension Truncation: Krylov space dimension is bounded by Hilbert space dimension — may saturate artificially
• Numerical Stability: Lanczos algorithm suffers from loss of orthogonality — use reorthogonalization for large spaces
• Initial Operator Choice: Different O_0 lead to different Krylov spaces — use physically relevant operators
• Finite Size Effects: Small systems may not show asymptotic b_n behavior

Verification

• Check b_n growth rate matches theoretical prediction for known models (SYK, random matrix)
• Verify K(t) shows expected early-time exponential growth for chaotic systems
• Cross-validate with out-of-time-order correlators (OTOCs) for chaos detection

References

• arXiv:2605.07668 — Bridging Krylov Complexity and Universal Analog Quantum Simulator
• Related: operator growth, quantum chaos, Lanczos algorithm, complexity geometry

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