# 7. Time Evolution¶

Time evolutions in quimb are handled by the class Evolution, which is initialized with a starting state and hamiltonian.

## 7.1. Basic Usage¶

Set up the Evolution object with a initial state and hamiltonian.

[1]:

from quimb import *

p0 = rand_ket(2**10)
h = ham_heis(10, sparse=True)
evo = Evolution(p0, h)


Update it in a single shot to a new time and get the state,

[2]:

evo.update_to(1)
evo.pt

[2]:

[[ 0.001538+0.012379j]
[-0.00541 +0.02611j ]
[-0.012026+0.015335j]
...
[-0.041381+0.02054j ]
[-0.004961-0.001546j]
[-0.019127+0.002967j]]


Lazily generate the state at multiple times:

[3]:

for pt in evo.at_times([2, 3, 4]):
print(expec(pt, p0))

0.0005106405568608293
0.009995197474010302
0.00525130789898347


## 7.2. Methods of Updating¶

There are three methods of updating the state:

• Evolution(..., method='integrate'): use definite integration. Get system at each time step, only need action of Hamiltonian on state. Generally efficient. For pure and mixed states. The additional option int_small_step={False, True} determines whether a low or high order adaptive stepping scheme is used, giving naturally smaller or larger times steps. See scipy.integrate.ode for details, False corresponds to "dop853", True to "dopri5".

• Evolution(..., method='solve'). Diagonalize the hamiltonian, which once done, allows quickly updating to arbitrary times. Supports pure and mixed states, recomended for small systems.

• Evolution(..., method='expm'): compute the evolved state using the action of the matrix exponential in a ‘single shot’ style. Only needs action of Hamiltonian, for very large systems can use distributed MPI. Only for pure states.

## 7.3. Computing on the fly¶

Sometimes, if integrating, it is best to just query the state at time-steps chosen dynamically by the adaptive scheme. This is achieved using the compute keyword supplied to Evolution. It can also just be a convenient way to set up calculations as well:

[4]:

p0 = rand_product_state(10)
h = ham_heis(10, sparse=True)

dims = [2] * 10
sysa, sysb = (0, 1), (2, 3)

def calc_t_and_logneg(t, pt):
ln = logneg_subsys(pt, dims, sysa, sysb)
return t, ln

evo = Evolution(p0, h, compute=calc_t_and_logneg, progbar=True)
evo.update_to(1)

ts, lns = zip(*evo.results)

100%|##########| 100/100 [00:00<00:00, 6697.06%/s]

[5]:

ts

[5]:

(0.0,
0.2312290367760555,
0.4377093533282588,
0.6399835221505082,
0.8549423071920399,
1.0)

[6]:

lns

[6]:

(0.0,
0.16992190183490394,
0.2865537529923966,
0.3808698480199881,
0.4716243573297176,
0.5295261025477127)


If a dict of callables is supplied to compute, (each should take two arguments, the time, and the state, as above), Evolution.results will itself be a dictionary containing the results of each function at each time step, under the respective key. This can be more convenient:

[7]:

def calc_t(t, _):
return t

def calc_logneg(_, pt):
return logneg_subsys(pt, [2]*10, 0, 1)

evo = Evolution(p0, h, compute={'t': calc_t, 'ln': calc_logneg}, progbar=True)
evo.update_to(1)

100%|##########| 100/100 [00:00<00:00, 4243.44%/s]

[8]:

evo.results

[8]:

{'t': [0.0,
0.2312290367760555,
0.4377093533282588,
0.6399835221505082,
0.8549423071920399,
1.0],
'ln': [0.0,
0.15268909733170494,
0.2827184207625697,
0.40032861740248604,
0.5081441062956807,
0.5669364519710873]}


## 7.4. Time-Dependent Evolutions¶

If you are using method='integrate' you can supply a callable to ham to evolve the state with a time dependent Hamiltonian. It should take a single argument t and return the Hamiltonian at the time. It probably makes sense to use a custom class here to avoid reconstructing as much of the Hamiltonian as possible at each step.

Here we’ll evolve the Neel state:

$| \psi(0) \rangle = | \uparrow \downarrow \uparrow \downarrow \uparrow \ldots \rangle$

with the Hamiltonian:

$H(t) = \sum_{i = 0}^{L - 1} S^Z_{i} S^Z_{i + 1} + \cos(t) \sum_{i}^{L} S^X_i$
[9]:

class MyTimeDepIsingHam:

def __init__(self, L):
self.h_interaction = ham_ising(L, sparse=True, jz=1.0, bx=0.0, cyclic=False)
self.h_field = ham_ising(L, sparse=True, jz=0.0, bx=1.0, cyclic=False)

def __call__(self, t):
return self.h_interaction + cos(t) * self.h_field

[10]:

L = 16

# our initial state
psi0 = neel_state(L)

# instantiate the ham object, it's __call__ method will be used by Evolution
fn_ham_t = MyTimeDepIsingHam(L)


We still want to compute some properties during the evolution:

[11]:

compute = {
'time': lambda t, p: t,
'entropy': lambda t, p: entropy_subsys(p, dims=[2] * L, sysa=range(L // 2))
}


Now we set up the evolution object again:

[12]:

evo = Evolution(psi0, fn_ham_t, progbar=True, compute=compute)

[13]:

evo.update_to(10)

100%|##########| 100/100 [00:11<00:00,  8.77%/s]


We can plot the half chain entropy that we computed on the fly:

[14]:

%matplotlib inline

from matplotlib import pyplot as plt

plt.plot(evo.results['time'], evo.results['entropy'])

[14]:

[<matplotlib.lines.Line2D at 0x7f2501956fd0>]


Or we can use the final state:

[15]:

fidelity(psi0, evo.pt)

[15]:

0.003302180752068526