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self.dims

If other is Qobj:ManipulatinAddition

SubtractionMultiplication

DivisionDaggerDiagonal

ExponentiationFull

NormSqrtTraceUnit

QuTiP

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fsesolvefloquet_master_equation_rates

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QuTiPv2.2

The Quantum Toolbox in Python J.R. Johansson (RIKEN), P.D. Nation (Korea University), F. Nori (RIKEN, U. Michigan)

QuTiP is an open-source software framework for simulating the dynamics of closed and open quantum systems. The QuTiP library depends on the excellent Numpy and Scipy numerical packages. In addition, graphical output is provided by Matplotlib. QuTiP aims to provide user-friendly and efficient numerical simulations of a wide variety of quantum mechanical problems, including those with Hamiltonians and/or collapse operators with arbitrary time-dependence, commonly found in a wide range of physics applications, such as quantum optics, trapped ions, superconducting circuits, semiconducting quantum dots, and quantum nanomechanical resonators. QuTiP is freely available for use and/or modification on all Unix-based platforms and on Windows.

About QuTiP

1. J. R. Johansson, P. D. Nation, and F. Nori, QuTiP 2: A Python framework for the dynamics of open quantum systems. Comp. Phys. Comm. 184, 1234 (2013) 2. J. R. Johansson, P. D. Nation, and F. Nori, QuTiP: An open-source Python framework for the dynamics of open quantum systems. Comp. Phys. Comm. 183, 1760–1772 (2012)

More information and download at: http://www.qutip.org

v. Publications

i - Quantum Objects iii - Time evolution of quantum systems

iv. Visualization

ii. States, operators, gates

A core feature in QuTiP, the Qobj class, provides an representation of quantum objects, such as operators, density matrices, state vectors (bras and kets). In addition, a large number of operations that one commonly need to perform on quantum objects are implemented as methods in the Qobj class. For example:

- artihmetic operations - conjugate and transpose - groundstate, eigenstates and energies - basis transformations - matrix elements - partial trace and partial transpose - extract and eliminate states - matrix exponentiation - norm and normalization

Time evolution of quantum systems is one of the main application areas of QuTiP, which includes solvers for:

- Schrödinger equation: sesolve - Lindblad ME: mesolve - Monte Carlo: mcsolve - Bloch-Redfield ME: brmesolve - Floquet formalism: fsesolve, fmmesolve - Stochastic: ssesolve, smesolve - Steady state: steady, steadystate - Correlation functions: correlation

Dacaying Rabi vacuum oscillations

from qutip import *

N = 10a = tensor(destroy(N), qeye(2))sz = tensor(qeye(N), sigmaz())s = tensor(qeye(N), destroy(2))wc = wq = 1.0 * 2 * pig = 0.5 * 2 * pikappa = 0.25

H = wc * a.dag() * a - 0.5 * wq * sz + 0.5 * g * (a * s.dag() + a.dag() * s)psi0 = tensor(basis(N,1), basis(2,0))result = mesolve(H, psi0, linspace(0, 10, 100), [sqrt(kappa) * a], [a.dag() * a])

Driven qubit, time-dependent Hamiltonian

from qutip import *

epsilon = delta = 1.0psi0 = basis(2,0) # Initial state: ground stateH0 = - delta * sigmaz() - epsilon * sigmax() H1 = - sigmaz()h_t = [H0, [H1, 'A * cos(w*t)']] # Hamiltonianargs = {'A': 10.017, 'w': 0.025*2*pi} e_ops = [sigmax(), sigmay(), sigmaz()] # Expectation values

output = sesolve(h_t, psi0, linspace(0, 160, 500), e_ops, args)

Dissipative evolution, two-qubit iSWAP gate

from qutip import *

g = 1.0 * 2 * pig1, g2, n_th = 0.75, 0.25, 1.5 # dissipation parametersc_ops = []H = g * (tensor(sigmax(), sigmax()) + tensor(sigmay(), sigmay()))

sm1 = tensor(sigmam(), qeye(2)) # qubit 1 collapse operatorssz1 = tensor(sigmaz(), qeye(2))c_ops.append(sqrt(g1 * (1+n_th)) * sm1)c_ops.append(sqrt(g1 * n_th) * sm1.dag())c_ops.append(sqrt(g2) * sz1)

sm2 = tensor(qeye(2), sigmam()) # qubit 2 collapse operatorssz2 = tensor(qeye(2), sigmaz())c_ops.append(sqrt(g1 * (1+n_th)) * sm2)c_ops.append(sqrt(g1 * n_th) * sm2.dag())c_ops.append(sqrt(g2) * sz2)

U = propagator(H, pi/(4*g), c_ops)

Stochastic decay of a single Fock state

from qutip import *

N = 4 # number of basis stateskappa = 1.0/0.129 # coupling to heat bathnth = 0.063 # temperature with <n>=0.063

a = destroy(N) # cavity annihilation operatorH = a.dag() * a # harmonic oscillator Hamiltonianpsi0 = basis(N,1) # initial state

c_op_list = []c_op_list.append(sqrt(kappa * (1 + nth)) * a)c_op_list.append(sqrt(kappa * nth) * a.dag())

ntraj = [1, 5, 15, 904]tlist = linspace(0,0.6,100)

mc = mcsolve(H, psi0, tlist, c_op_list, [a.dag()*a], ntraj)me = mesolve(H, psi0, tlist, c_op_list, [a.dag()*a])

# calculate steady statefinal_state = steadystate(H, c_op_list) fexpt = expect(a.dag()*a, final_state)

# plot results ...

Visualization of quantum states and processes is an important part of QuTiP, which includes a large number of classes and functions for visualization techniques commonly used in quantum mechanics. For example:

Bloch sphere

from qutip import *

b = Bloch()b.add_points([x_vec, y_vec, z_vec])...

Wigner functions

from qutip import *

psi = sum([basis(10, n) for n in [0, 3, 6]]).unit()

w = WignerDistribition(psi)w.visualize()

Process tomography

from qutip import *

op_basis = [[qeye(2), sigmax(), sigmay(), sigmaz()] for i in range(2)]op_labels = [["i", "x", "y", "z"] for i in range(2)]

chi = qpt(U, op_basis)qpt_plot_combined(chi, op_labels)

Quantum objects

>>> psi = Qobj([1, 0]) # create a bra state vector Quantum object: dims = [[1], [2]], shape = [1, 2], type = ket Qobj data = [[ 1. 0.]]

>>> psi.dag() Quantum object: dims = [[2], [1]], shape = [2, 1], type = ket Qobj data = [[ 1.] [ 0.]]

Composite systems

>>> H = tensor(sigmax(), sigmax()) Quantum object: dims = [[2, 2], [2, 2]], shape = [4, 4], type = oper, isherm = True Qobj data = [[ 0. 0. 0. 1.] [ 0. 0. 1. 0.] [ 0. 1. 0. 0.] [ 1. 0. 0. 0.]]

>>> psi = tensor([fock(2, 1), fock(2, 0)])>>> ptrace(psi, 0) Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True Qobj data = [[ 0. 0.] [ 0. 1.]]

QuTiP includes a large library of functions thatcreats Qobj representations of commonquantum states, operators and superoperators:

States: - Coherent states: coherent(N, alpha) - Fock states: fock(N, n) - Thermal states: thermal_dm(N, n_th)

Operators: - Idenity: identity(N) - Annihilation: destroy(N) - Creation: create(N) - Number: num(N) - Displacement: displace(N, alpha) - Pauli operators: sigmax(), sigmay() sigmaz(), sigmam(), sigmap()

- Spin operators: jmat(k, s)

Steady states

H = ... # Hamiltonianc_ops = ... # collapse operatorsrho_ss = steadystate(H, c_ops)

Superoperators: - Liouvillian: liouvillian(H, c_ops) - Pre- and post: spre(op), spost(op)

Floquet energies

from qutip import *

H0 = delta/2.0 * sigmaz() - eps0/2.0 * sigmax()H1 = A/2.0 * sigmax()for idx, A in enumerate(A_vec): H = [H0, [H1, lambda t, w: sin(w*t)]] f_modes, f_energies = \ floquet_modes(H, T, args=omega, True) q_energies[idx,:] = f_energies

http://www.qutip.org

Gates: - Not: cnot(), snot() - Swap: iswap(), swap(), sqrtiswap()

- Phase: phasegate(theta) - 3-qubit gates: fredkin(), toffoli()