QuTiP: Quantum Toolbox in Python

Core Capabilities

1. Quantum Objects and States

Create and manipulate quantum states and operators:

# States
psi = basis(N, n)  # Fock state |n⟩
psi = coherent(N, alpha)  # Coherent state |α⟩
rho = thermal_dm(N, n_avg)  # Thermal density matrix

# Operators
a = destroy(N)  # Annihilation operator
H = num(N)  # Number operator
sx, sy, sz = sigmax(), sigmay(), sigmaz()  # Pauli matrices

# Composite systems
psi_AB = tensor(psi_A, psi_B)  # Tensor product

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:

# Closed systems (unitary evolution)
result = sesolve(H, psi0, tlist, e_ops=[num(N)])

# Open systems (dissipation)
c_ops = [np.sqrt(0.1) * destroy(N)]  # Collapse operators
result = mesolve(H, psi0, tlist, c_ops, e_ops=[num(N)])

# Quantum trajectories (Monte Carlo)
result = mcsolve(H, psi0, tlist, c_ops, ntraj=500, e_ops=[num(N)])

Solver selection guide:

• sesolve: Pure states, unitary evolution
• mesolve: Mixed states, dissipation, general open systems
• mcsolve: Quantum jumps, photon counting, individual trajectories
• brmesolve: Weak system-bath coupling
• fmmesolve: 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:

# Expectation values
n_avg = expect(num(N), psi)

# Entropy measures
S = entropy_vn(rho)  # Von Neumann entropy
C = concurrence(rho)  # Entanglement (two qubits)

# Fidelity and distance
F = fidelity(psi1, psi2)
D = tracedist(rho1, rho2)

# Correlation functions
corr = correlation_2op_1t(H, rho0, taulist, c_ops, A, B)
w, S = spectrum_correlation_fft(taulist, corr)

# Steady states
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:

# Bloch sphere
b = Bloch()
b.add_states(psi)
b.show()

# Wigner function (phase space)
xvec = np.linspace(-5, 5, 200)
W = wigner(psi, xvec, xvec)
plt.contourf(xvec, xvec, W, 100, cmap='RdBu')

# Fock distribution
plot_fock_distribution(psi)

# Matrix visualization
hinton(rho)  # Hinton diagram
matrix_histogram(H.full())  # 3D bars

See references/visualization.md for Bloch sphere animations, Wigner functions, Q-functions, and matrix visualizations.

5. Advanced Methods

Specialized techniques for complex scenarios:

# Floquet theory (periodic Hamiltonians)
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)

# HEOM (non-Markovian, strong coupling)
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)

# Permutational invariance (identical particles)
psi = dicke(N, j, m)  # Dicke states
Jz = jspin(N, 'z')  # Collective operators

See references/advanced.md for Floquet theory, HEOM, permutational invariance, stochastic solvers, superoperators, and performance optimization.

Common Workflows

Simulating a Damped Harmonic Oscillator

# System parameters
N = 20  # Hilbert space dimension
omega = 1.0  # Oscillator frequency
kappa = 0.1  # Decay rate

# Hamiltonian and collapse operators
H = omega * num(N)
c_ops = [np.sqrt(kappa) * destroy(N)]

# Initial state
psi0 = coherent(N, 3.0)

# Time evolution
tlist = np.linspace(0, 50, 200)
result = mesolve(H, psi0, tlist, c_ops, e_ops=[num(N)])

# Visualize
plt.plot(tlist, result.expect[0])
plt.xlabel('Time')
plt.ylabel('⟨n⟩')
plt.title('Photon Number Decay')
plt.show()

Two-Qubit Entanglement Dynamics

# Create Bell state
psi0 = bell_state('00')

# Local dephasing on each qubit
gamma = 0.1
c_ops = [
    np.sqrt(gamma) * tensor(sigmaz(), qeye(2)),
    np.sqrt(gamma) * tensor(qeye(2), sigmaz())
]

# Track entanglement
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)

# Compute concurrence for each state
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

# System parameters
N = 10  # Cavity Fock space
wc = 1.0  # Cavity frequency
wa = 1.0  # Atom frequency
g = 0.05  # Coupling strength

# Operators
a = tensor(destroy(N), qeye(2))  # Cavity
sm = tensor(qeye(N), sigmam())  # Atom

# Hamiltonian (RWA)
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() * sm + a * sm.dag())

# Initial state: cavity in coherent state, atom in ground state
psi0 = tensor(coherent(N, 2), basis(2, 0))

# Dissipation
kappa = 0.1  # Cavity decay
gamma = 0.05  # Atomic decay
c_ops = [np.sqrt(kappa) * a, np.sqrt(gamma) * sm]

# Observables
n_cav = a.dag() * a
n_atom = sm.dag() * sm

# Evolve
tlist = np.linspace(0, 50, 200)
result = mesolve(H, psi0, tlist, c_ops, e_ops=[n_cav, n_atom])

# Plot
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

1. Truncate Hilbert spaces: Use smallest dimension that captures dynamics
2. Choose appropriate solver: sesolve for pure states is faster than mesolve
3. Time-dependent terms: String format (e.g., 'cos(w*t)') is fastest
4. Store only needed data: Use e_ops instead of storing all states
5. Adjust tolerances: Balance accuracy with computation time via Options
6. Parallel trajectories: mcsolve automatically uses multiple CPUs
7. Check 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.md: Quantum objects, states, operators, tensor products, composite systems
• references/time_evolution.md: All solvers (sesolve, mesolve, mcsolve, brmesolve, etc.), time-dependent Hamiltonians, solver options
• references/visualization.md: Bloch sphere, Wigner functions, Q-functions, Fock distributions, matrix plots
• references/analysis.md: Expectation values, entropy, fidelity, entanglement measures, correlation functions, steady states
• references/advanced.md: Floquet theory, HEOM, permutational invariance, stochastic methods, superoperators, performance tips

External Resources

• Documentation: https://qutip.readthedocs.io/
• Tutorials: https://qutip.org/qutip-tutorials/
• API Reference: https://qutip.readthedocs.io/en/stable/apidoc/apidoc.html
• GitHub: https://github.com/qutip/qutip

Converted and distributed by TomeVault | Claim this content