Is it possible to do linalg.multi_dot for an ndarray along an axis?

Question

First of all, I have a group of 12 (2x2) matrices.

II = np.identity(2, dtype=complex)
X = np.array([[0, 1], [1, 0]], dtype=complex)
Y = np.array([[0, -1j], [1j, 0]], dtype=complex)
Z = np.array([[1, 0], [0, -1]], dtype=complex)
PPP = (-II + 1j*X + 1j*Y + 1j*Z)/2
PPM = (-II + 1j*X + 1j*Y - 1j*Z)/2
PMM = (-II + 1j*X - 1j*Y - 1j*Z)/2
MMM = (-II - 1j*X - 1j*Y - 1j*Z)/2
MMP = (-II - 1j*X - 1j*Y + 1j*Z)/2
MPP = (-II - 1j*X + 1j*Y + 1j*Z)/2
PMP = (-II + 1j*X - 1j*Y + 1j*Z)/2
MPM = (-II - 1j*X + 1j*Y - 1j*Z)/2

Currently I have a function operator_groups that draws a random matrix from this group for every j loop and it gets appended into a list sequence . The random matrix drawn inbetween all the individual j loops are then used to do some calculations, irrelevant to our discussion here. At the end of the j loop, the sequences of the elements of the list sequence are reversed, then linalg.multi_dot is performed and then its hermitian conjugate is being taken (hence the .conj().T )

def operator_groups():
    return random.choice([II, X, Y, Z, PPP, PPM, PMM, MMM, MMP, MPP, PMP, MPM])


for i in range(1, sample_size+1, 1):
    sequence = []
    for j in range(1, some_number, 1):
        noise = operator_groups()
        """some matrix calculations here"""
        sequence.append(noise)
    sequence_inverse = np.linalg.multi_dot(sequence[::-1]).conj().T

Now I wish to vectorize the i loop , by just doing the j loop in one big matrix. The noise is now an ndarray of N matrices(instead of just 1 matrix) randomly sampled from the group, with each matrix representing the iterations of j , but parallelized. The code now looks something like this.

def operator_groups(sample_size):
    return random.sample([II, X, Y, Z], sample_size)


sequence = []
for j in range(1, some_number, 1):
    noise = operator_groups(sample_size)
    sequence.append(noise)
sequence_inverse = np.linalg.multi_dot(sequence[::-1]).conj().T

Now that sequence is a multi-dimensional array, I'm having trouble with appending the multidimensional noise into the right order within sequence , and then subsequently also problem with performing linalg.multidot for the inverse of sequence and taking its Hermitian conjugate. In this case I'd want to multi_dot the inverse of all the stored up noise for each j row corresponding to each of the j loop. How can this be done?

I'll provide some "pseudo-examples" below to further demonstrate my problem, using j = 3. For simplicity, here I'll only "randomly draw" X, Y, Z .

Non-vectorised case:

i = 1
sequence = []
    j = 1
    noise = X (randomised)
    sequence.append(noise)
    sequence = [X]
    j = 2
    noise = Y (randomised)
    sequence.append(noise)
    sequence = [X, Y]
    j = 3
    noise = Z (randomised)
    sequence.append(noise)
    sequence = [X, Y, Z]

    end of j loop
take reverse order: [Z, Y, X]
do multi_dot: [ZYX] (Note: dot products, not element-wise multiplication)
take conjugate and tranpose(to get Hermitian): [ZYX].conj().T = [ZYX.conj().T]

Vectorized case(say if I was doing sample_size = 3):

sequence = []
    j = 1
    noise = [X,Z,Y](randomised)
    sequence.append(noise)
    sequence = [[X,Z,Y]]
    j = 2
    noise = [Z,Y,X] (randomised)
    sequence.append(noise)
    sequence = [[X,Z,Y],
                [Z,Y,X]]
    j = 3
    noise = [Z,Z,X] (randomised)
    sequence.append(noise)
    sequence = [[X,Z,Y],
                [Z,Y,X],
                [Z,Z,X]]
    end of j loop
take reverse order: [[Z,Z,X],
                     [Z,Y,X],
                     [X,Z,Y]]
do multi_dot(along an axis,
which is what I have trouble with): [ZZX,ZYZ,XXY]
take conjugate and tranpose(to get Hermitian): 
[ZZX,ZYZ,XXY].conj().T = [ZZX.conj().T, ZYZ.conj().T, XXY.conj().T]

I hope these examples demonstrate my problem

Answer 1

With your two random selectors:

In [13]: operator_groups()        # returns one (2,2) array                                                                                   
Out[13]: 
array([[-0.5+0.5j,  0.5-0.5j],
       [-0.5-0.5j, -0.5-0.5j]])
In [14]: operator_groups1(4)      # returns a list of (2,2) arrays                                                                           
Out[14]: 
[array([[0.+0.j, 1.+0.j],
        [1.+0.j, 0.+0.j]]), array([[ 0.+0.j, -0.-1.j],
        [ 0.+1.j,  0.+0.j]]), array([[ 1.+0.j,  0.+0.j],
        [ 0.+0.j, -1.+0.j]]), array([[1.+0.j, 0.+0.j],
        [0.+0.j, 1.+0.j]])]

Your loop creates a list of arrays:

In [15]: seq=[] 
    ...: for j in range(4): 
    ...:     seq.append(operator_groups()) 
    ...:                                                                                                     
In [16]: seq                                                                                                 
Out[16]: 
[array([[-0.5-0.5j, -0.5+0.5j],
        [ 0.5+0.5j, -0.5+0.5j]]), array([[1.+0.j, 0.+0.j],
        [0.+0.j, 1.+0.j]]), array([[-0.5+0.5j, -0.5-0.5j],
        [ 0.5-0.5j, -0.5-0.5j]]), array([[-0.5-0.5j,  0.5-0.5j],
        [-0.5-0.5j, -0.5+0.5j]])]

which can be given to multi_dot for sequential dotting:

In [17]: np.linalg.multi_dot(seq)                                                                            
Out[17]: 
array([[0.-1.j, 0.+0.j],
       [0.+0.j, 0.+1.j]])

If we build the sequence with the groups selector, we get a list of lists:

In [18]: seq=[] 
    ...: for j in range(4): 
    ...:     seq.append(operator_groups1(3)) 
    ...:                                                                                                     
In [19]: seq                                                                                                 
Out[19]: 
[[array([[ 0.+0.j, -0.-1.j],
         [ 0.+1.j,  0.+0.j]]), array([[ 1.+0.j,  0.+0.j],
         [ 0.+0.j, -1.+0.j]]), array([[0.+0.j, 1.+0.j],
         [1.+0.j, 0.+0.j]])], [array([[ 0.+0.j, -0.-1.j],
         [ 0.+1.j,  0.+0.j]]), array([[ 1.+0.j,  0.+0.j],
         [ 0.+0.j, -1.+0.j]]), array([[0.+0.j, 1.+0.j],
         [1.+0.j, 0.+0.j]])], [array([[1.+0.j, 0.+0.j],
         [0.+0.j, 1.+0.j]]), array([[ 1.+0.j,  0.+0.j],
         [ 0.+0.j, -1.+0.j]]), array([[ 0.+0.j, -0.-1.j],
         [ 0.+1.j,  0.+0.j]])], [array([[1.+0.j, 0.+0.j],
         [0.+0.j, 1.+0.j]]), array([[ 1.+0.j,  0.+0.j],
         [ 0.+0.j, -1.+0.j]]), array([[ 0.+0.j, -0.-1.j],
         [ 0.+1.j,  0.+0.j]])]]
In [20]: len(seq)                                                                                            
Out[20]: 4
In [21]: len(seq[0])                                                                                         
Out[21]: 3

We can 'stack' the inner lists, creating a list of (n,2,2) arrays:

In [22]: seq1 = [np.stack(el) for el in seq]                                                                 
In [23]: seq1                                                                                                
Out[23]: 
[array([[[ 0.+0.j, -0.-1.j],
         [ 0.+1.j,  0.+0.j]],

        [[ 1.+0.j,  0.+0.j],
         [ 0.+0.j, -1.+0.j]],

        [[ 0.+0.j,  1.+0.j],
         [ 1.+0.j,  0.+0.j]]]), array([[[ 0.+0.j, -0.-1.j],
         [ 0.+1.j,  0.+0.j]],

        [[ 1.+0.j,  0.+0.j],
         [ 0.+0.j, -1.+0.j]],

        [[ 0.+0.j,  1.+0.j],
         [ 1.+0.j,  0.+0.j]]]), array([[[ 1.+0.j,  0.+0.j],
         [ 0.+0.j,  1.+0.j]],

        [[ 1.+0.j,  0.+0.j],
         [ 0.+0.j, -1.+0.j]],

        [[ 0.+0.j, -0.-1.j],
         [ 0.+1.j,  0.+0.j]]]), array([[[ 1.+0.j,  0.+0.j],
         [ 0.+0.j,  1.+0.j]],

        [[ 1.+0.j,  0.+0.j],
         [ 0.+0.j, -1.+0.j]],

        [[ 0.+0.j, -0.-1.j],
         [ 0.+1.j,  0.+0.j]]])]

we can then apply matmul repeatedly on this list:

In [25]: res = seq1[0] 
    ...: for el in seq1[1:]: 
    ...:     res = res@el 
    ...:      
    ...:                                                                                                     
In [26]: res                                                                                                 
Out[26]: 
array([[[1.+0.j, 0.+0.j],
        [0.+0.j, 1.+0.j]],

       [[1.+0.j, 0.+0.j],
        [0.+0.j, 1.+0.j]],

       [[1.+0.j, 0.+0.j],
        [0.+0.j, 1.+0.j]]])

In effect matmul is like dot , but it treats the leading dimension(s) as a 'batch' dimension.

With random selection it's a pain to compare different results (unless I set the seed), so I leave the verification up to you.