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skill-033 Quantum mechanics simulations and analysis using QuTiP (Quantum Toolbox in Python). Use when working with quantum systems including: (1) quantum states (kets, bras, density matrices), (2) quantum operators and gates, (3) time evolution and dynamics (Schrödinger, master equations, Monte Carlo), (4) open quantum systems with dissipation, (5) quantum measurements and entanglement, (6) visualization (Bloch sphere, Wigner functions), (7) steady states and correlation functions, or (8) advanced methods (Floquet theory, HEOM, stochastic solvers). Handles both closed and open quantum systems across various domains including quantum optics, quantum computing, and condensed matter physics.
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name skill-033 description Quantum mechanics simulations and analysis using QuTiP (Quantum Toolbox in Python). Use when working with quantum systems including: (1) quantum states (kets, bras, density matrices), (2) quantum operators and gates, (3) time evolution and dynamics (Schrödinger, master equations, Monte Carlo), (4) open quantum systems with dissipation, (5) quantum measurements and entanglement, (6) visualization (Bloch sphere, Wigner functions), (7) steady states and correlation functions, or (8) advanced methods (Floquet theory, HEOM, stochastic solvers). Handles both closed and open quantum systems across various domains including quantum optics, quantum computing, and condensed matter physics.
QuTiP: Quantum Toolbox in Python
Overview
QuTiP provides comprehensive tools for simulating and analyzing quantum mechanical systems. It handles both closed (unitary) and open (dissipative) quantum systems with multiple solvers optimized for different scenarios.
Installation
uv pip install qutip
Optional packages for additional functionality:
uv pip install qutip-qip
uv pip install qutip-qtrl
Quick Start
from qutip import *
import numpy as np
import matplotlib.pyplot as plt
psi = basis(2 , 0 )
H = sigmaz()
0
10
100
0
'Time'
'⟨σz⟩'
Core Capabilities
1. Quantum Objects and States Create and manipulate quantum states and operators:
psi = basis(N, n)
psi = coherent(N, alpha)
rho = thermal_dm(N, n_avg)
a = destroy(N)
H = num(N)
sx, sy, sz = sigmax(), sigmay(), sigmaz()
psi_AB = tensor(psi_A, psi_B)
See references/core_concepts.md for comprehensive coverage of quantum objects, states, operators, and tensor products.
2. Time Evolution and Dynamics Multiple solvers for different scenarios:
result = sesolve(H, psi0, tlist, e_ops=[num(N)])
c_ops = [np.sqrt(0.1 ) * destroy(N)]
result = mesolve(H, psi0, tlist, c_ops, e_ops=[num(N)])
result = mcsolve(H, psi0, tlist, c_ops, ntraj=500 , e_ops=[num(N)])
sesolve: Pure states, unitary evolutionmesolve: Mixed states, dissipation, general open systemsmcsolve: Quantum jumps, photon counting, individual trajectoriesbrmesolve: Weak system-bath couplingfmmesolve: Time-periodic Hamiltonians (Floquet) See references/time_evolution.md for detailed solver documentation, time-dependent Hamiltonians, and advanced options.
3. Analysis and Measurement Compute physical quantities:
n_avg = expect(num(N), psi)
S = entropy_vn(rho)
C = concurrence(rho)
F = fidelity(psi1, psi2)
D = tracedist(rho1, rho2)
corr = correlation_2op_1t(H, rho0, taulist, c_ops, A, B)
w, S = spectrum_correlation_fft(taulist, corr)
rho_ss = steadystate(H, c_ops)
See references/analysis.md for entropy, fidelity, measurements, correlation functions, and steady state calculations.
4. Visualization Visualize quantum states and dynamics:
b = Bloch()
b.add_states(psi)
b.show()
xvec = np.linspace(-5 , 5 , 200 )
W = wigner(psi, xvec, xvec)
plt.contourf(xvec, xvec, W, 100 , cmap='RdBu' )
plot_fock_distribution(psi)
hinton(rho)
matrix_histogram(H.full())
See references/visualization.md for Bloch sphere animations, Wigner functions, Q-functions, and matrix visualizations.
5. Advanced Methods Specialized techniques for complex scenarios:
T = 2 * np.pi / w_drive
f_modes, f_energies = floquet_modes(H, T, args)
result = fmmesolve(H, psi0, tlist, c_ops, T=T, args=args)
from qutip.nonmarkov.heom import HEOMSolver, BosonicBath
bath = BosonicBath(Q, ck_real, vk_real)
hsolver = HEOMSolver(H_sys, [bath], max_depth=5 )
result = hsolver.run(rho0, tlist)
psi = dicke(N, j, m)
Jz = jspin(N, 'z' )
See references/advanced.md for Floquet theory, HEOM, permutational invariance, stochastic solvers, superoperators, and performance optimization.
Common Workflows
Simulating a Damped Harmonic Oscillator
N = 20
omega = 1.0
kappa = 0.1
H = omega * num(N)
c_ops = [np.sqrt(kappa) * destroy(N)]
psi0 = coherent(N, 3.0 )
tlist = np.linspace(0 , 50 , 200 )
result = mesolve(H, psi0, tlist, c_ops, e_ops=[num(N)])
plt.plot(tlist, result.expect[0 ])
plt.xlabel('Time' )
plt.ylabel('⟨n⟩' )
plt.title('Photon Number Decay' )
plt.show()
Two-Qubit Entanglement Dynamics
psi0 = bell_state('00' )
gamma = 0.1
c_ops = [
np.sqrt(gamma) * tensor(sigmaz(), qeye(2 )),
np.sqrt(gamma) * tensor(qeye(2 ), sigmaz())
]
def compute_concurrence (t, psi ):
rho = ket2dm(psi) if psi.isket else psi
return concurrence(rho)
tlist = np.linspace(0 , 10 , 100 )
result = mesolve(qeye([2 , 2 ]), psi0, tlist, c_ops)
C_t = [concurrence(state.proj()) for state in result.states]
plt.plot(tlist, C_t)
plt.xlabel('Time' )
plt.ylabel('Concurrence' )
plt.title('Entanglement Decay' )
plt.show()
Jaynes-Cummings Model
N = 10
wc = 1.0
wa = 1.0
g = 0.05
a = tensor(destroy(N), qeye(2 ))
sm = tensor(qeye(N), sigmam())
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() * sm + a * sm.dag())
psi0 = tensor(coherent(N, 2 ), basis(2 , 0 ))
kappa = 0.1
gamma = 0.05
c_ops = [np.sqrt(kappa) * a, np.sqrt(gamma) * sm]
n_cav = a.dag() * a
n_atom = sm.dag() * sm
tlist = np.linspace(0 , 50 , 200 )
result = mesolve(H, psi0, tlist, c_ops, e_ops=[n_cav, n_atom])
fig, axes = plt.subplots(2 , 1 , figsize=(8 , 6 ), sharex=True )
axes[0 ].plot(tlist, result.expect[0 ])
axes[0 ].set_ylabel('⟨n_cavity⟩' )
axes[1 ].plot(tlist, result.expect[1 ])
axes[1 ].set_ylabel('⟨n_atom⟩' )
axes[1 ].set_xlabel('Time' )
plt.tight_layout()
plt.show()
Tips for Efficient Simulations
Truncate Hilbert spaces : Use smallest dimension that captures dynamicsChoose appropriate solver : sesolve for pure states is faster than mesolveTime-dependent terms : String format (e.g., 'cos(w*t)') is fastestStore only needed data : Use e_ops instead of storing all statesAdjust tolerances : Balance accuracy with computation time via OptionsParallel trajectories : mcsolve automatically uses multiple CPUsCheck convergence : Vary ntraj, Hilbert space size, and tolerances
Troubleshooting Memory issues : Reduce Hilbert space dimension, use store_final_state option, or consider Krylov methods
Slow simulations : Use string-based time-dependence, increase tolerances slightly, or try method='bdf' for stiff problems
Numerical instabilities : Decrease time steps (nsteps option), increase tolerances, or check Hamiltonian/operators are properly defined
Import errors : Ensure QuTiP is installed correctly; quantum gates require qutip-qip package
References This skill includes detailed reference documentation:
references/core_concepts.mdreferences/time_evolution.mdreferences/visualization.mdreferences/analysis.mdreferences/advanced.md
External Resources 