# Matrix Basis Tutorial¶

Consider the space of density matrices corresponding to a Hilbert space $\mathcal{H}$ of dimension $d$. The basis used for this Hilbert-Schmidt space, $B(\mathcal{H})$, can be any set of $d\times d$ matrices which span the density matrix space. pyGSTi supports arbitrary bases by deriving from the pygsti.tools.Basis class, and constains built-in support for the following basis sets:

• the matrix unit, or "standard" basis, consisting of the matrices with a single unit (1.0) element and otherwise zero. This basis is selected by passing "std" to appropriate function arguments.
• the Pauli-product basis, consisting of tensor products of the four Pauli matrices {I, X, Y, Z} normalized so that $Tr(B_i B_j) = \delta_{ij}$. All of these matrices are Hermitian, so that Hilbert-Schmidt vectors and matrices are real when this basis is used. This basis can only be used when the $d = 4^i$ for integer $i$, and is selected using the string "pp".
• the Gell-Mann basis, consisting of the normalized Gell-Mann matrices (see Wikipedia if you don't know what these are). Similar to the Pauli-product case, these matrices are also Hermitian, so that Hilbert-Schmidt vectors and matrices are real when this basis is used. Unlike the Pauli-product case, since Gell-Mann matrices are well defined in any dimension, the Gell-Mann basis is not restricted to cases when $d=4^i$. This basis is selected using the string "gm".
• a special basis of $3 \times 3$ matricies designed for Qutrit systems formed by taking the symmetric subspace of a 2-qubit system. This basis is selected using the string "qt".

Various functions and objects within pyGSTi require knowledge of what Hilbert-Schmidt basis is being used. The pygsti.objects.Basis object encapsulates a basis, and is the most flexible way of specifying a basis in pyGSTi. Alternatively, many functions also accept the short strings "std", "gm", "pp", and "qt" to select one of the standard bases. In this tutorial, we'll demonstrate how to create a Basis object and use it and related functions to obtain and change the basis of the operation matrices and SPAM vectors stored in a Model.

The most straightforward way to create a Basis object is to provide its short name and dimension to the Basis.cast function, which "casts" various things as a basis object. PyGSTi contains built-in support for bases consisting of the tensor product of Pauli matrices (or just the Pauli matrices in the case of 1 qubit), named "pp", as well as the Gell-Mann matrices, named "gm". It also contains a special "qutrit" basis, named "qt", for the case of 3-level quantum systems. In cases when there are an integral number of qubits, and the dimension equals $4^N$, the "pp" basis is usually preferred since it is more intuitive. In other cases, where the Hilbert space includes non-qubit (e.g. environmental) degrees of freedom, the Gell-Mann basis may be useful since it can be used in any dimension. Note that both the Gell-Mann and Pauli-Product bases reduce to the usual Pauli matrices plus identity in when the dimension equals 4 (1 qubit).

Here are some examples:

In [ ]:
import pygsti
from pygsti import Basis
pp  = Basis.cast('pp',  4) # Pauli-product (in this dim=4 case, just the Paulis)
std = Basis.cast('std', 4) # "standard" basis of matrix units
gm  = Basis.cast('gm',  4) # Gell-Mann
qt  = Basis.cast('qt',  9) # qutrit - must be dim 9
bases = [pp, std, gm, qt]


Each of the pp, std, and gm bases created will have $4$ $2x2$ matrices each. The qt basis has $9$ $3x3$ matrices instead:

In [ ]:
for basis in bases:
print('\n{} basis (dim {}):'.format(basis.name, basis.dim))
print('{} elements:'.format(len(basis)))
for element in basis:
pygsti.tools.print_mx(element)


However, custom "explicit" bases, which expicitly hold a set of basis-element matrices, can be easily created by supplying a list of the elements:

In [ ]:
import numpy as np
from pygsti.objects import ExplicitBasis
std2x2Matrices = [
np.array([[1, 0],
[0, 0]]),
np.array([[0, 1],
[0, 0]]),
np.array([[0, 0],
[1, 0]]),
np.array([[0, 0],
[0, 1]])]

alt_standard = ExplicitBasis(std2x2Matrices, ["myElement%d" % i for i in range(4)],
name='std', longname='Standard')
print(alt_standard)


More complex bases can be created by chaining the elements of other bases together along the diagonal. This yields a basis for the direct-sum of the spaces spanned by the original basis. For example, a composition of the $2x2$ std basis with the $1x1$ std basis leads to a basis with state vectors of length $5$, or $5x5$ matrices:

In [ ]:
from pygsti.objects import DirectSumBasis
comp = Basis.cast('std', [4, 1])
comp = Basis.cast([('std', 4), ('std', 1)])
comp = DirectSumBasis([ Basis.cast('std', 4), Basis.cast('std', 1)])

#All three comps above give the same final basis
print(comp)
for element in comp.elements:
print(element)


### Basis usage¶

Once created, bases are used to manipulate matrices and vectors within pygsti:

In [ ]:
from pygsti.tools import change_basis, flexible_change_basis

mx = np.array([[1, 0, 0, 1],
[0, 1, 2, 0],
[0, 2, 1, 0],
[1, 0, 0, 1]])

change_basis(mx, 'std', 'gm') # shortname lookup
change_basis(mx, std, gm)     # object only
change_basis(mx, std, 'gm')   # combination


Composite bases can be converted between expanded and contracted forms:

In [ ]:
mxInStdBasis = np.array([[1,0,0,2],
[0,0,0,0],
[0,0,0,0],
[3,0,0,4]],'d')

begin = Basis.cast('std', 4)
end   = Basis.cast('std', [1,1])
mxInReducedBasis = flexible_change_basis(mxInStdBasis, begin, end)
print(mxInReducedBasis)
original         = flexible_change_basis(mxInReducedBasis, end, begin)

In [ ]: