Matrix exponentiation in Python

问题

I'm trying to exponentiate a complex matrix in Python and am running into some trouble. I'm using the scipy.linalg.expm function, and am having a rather strange error message when I try the following code:

import numpy as np
from scipy import linalg

hamiltonian = np.mat('[1,0,0,0;0,-1,0,0;0,0,-1,0;0,0,0,1]')

# This works
t_list = np.linspace(0,1,10)
unitary = [linalg.expm(-(1j)*t*hamiltonian) for t in t_list]

# This doesn't
t_list = np.linspace(0,10,100)
unitary = [linalg.expm(-(1j)*t*hamiltonian) for t in t_list]

The error when the second experiment is run is:

This works!
Traceback (most recent call last):
  File "matrix_exp.py", line 11, in <module>
    unitary_t = [linalg.expm(-1*t*(1j)*hamiltonian) for t in t_list]
  File "/usr/lib/python2.7/dist-packages/scipy/linalg/matfuncs.py",     line 105, in expm
    return scipy.sparse.linalg.expm(A)
  File "/usr/lib/python2.7/dist- packages/scipy/sparse/linalg/matfuncs.py", line 344, in expm
    X = _fragment_2_1(X, A, s)
  File "/usr/lib/python2.7/dist-  packages/scipy/sparse/linalg/matfuncs.py", line 462, in _fragment_2_1
    X[k, k] = exp_diag[k]
TypeError: only length-1 arrays can be converted to Python scalars

This seems really strange since all I changed was the range of t I was using. Is it because the Hamiltonian is diagonal? In general, the Hamiltonians won't be, but I also want it to work for diagonal ones. I don't really know the mechanics of expm, so any help would be greatly appreciated.

回答1:

That is interesting. One thing I can say is that the problem is specific to the np.matrix subclass. For example, the following works fine:

h = np.array(hamiltonian)
unitary = [linalg.expm(-(1j)*t*h) for t in t_list]

Digging a little deeper into the traceback, the exception is being raised in _fragment_2_1 in scipy.sparse.linalg.matfuncs.py, specifically these lines:

n = X.shape[0]
diag_T = T.diagonal().copy()

# Replace diag(X) by exp(2^-s diag(T)).
scale = 2 ** -s
exp_diag = np.exp(scale * diag_T)
for k in range(n):
    X[k, k] = exp_diag[k]

The error message

    X[k, k] = exp_diag[k]
TypeError: only length-1 arrays can be converted to Python scalars

suggests to me that exp_diag[k] ought to be a scalar, but is instead returning a vector (and you can't assign a vector to X[k, k], which is a scalar).

Setting a breakpoint and examining the shapes of these variables confirms this:

ipdb> l
    751     # Replace diag(X) by exp(2^-s diag(T)).
    752     scale = 2 ** -s
    753     exp_diag = np.exp(scale * diag_T)
    754     for k in range(n):
    755         import ipdb; ipdb.set_trace()  # breakpoint e86ebbd4 //
--> 756         X[k, k] = exp_diag[k]
    757 
    758     for i in range(s-1, -1, -1):
    759         X = X.dot(X)
    760 
    761         # Replace diag(X) by exp(2^-i diag(T)).

ipdb> exp_diag.shape
(1, 4)
ipdb> exp_diag[k].shape
(1, 4)
ipdb> X[k, k].shape
()

The underlying problem is that exp_diag is assumed to be either 1D or a column vector, but the diagonal of an np.matrix object is a row vector. This highlights a more general point that np.matrix is generally less well-supported than np.ndarray, so in most cases it's better to use the latter.

One possible solution would be to use np.ravel() to flatten diag_T into a 1D np.ndarray: