# Creating a 2-qubit model with a custom 2-qubit gate¶

While pyGSTi is able to support several common types of 2-qubit gates, the space of all possible 2-qubit gates is so large that some users will need to construct their own particular 2-qubit gate "from scratch". In this example, we look at how to construct a 2-qubit model manually. We'll use pygsti.construction.create_explicit_model to construct the single-qubit gates and we'll create a "non-standard" 2-qubit gate by specifying the unitary operation it performs as a $4\times 4$ matrix.

To perform GST on such a model, one may need to compute new sets of fiducials and/or germ sequences. Once these are obtained, 2-qubit GST can be run just as in the example which uses one of pyGSTi's built-in 2Q models.

In [1]:
import pygsti


## Step 1: Create a model with only single-qubit gates¶

Since the space of single-qubit gates is relatively small, we'll assume that the single-qubit gates in our model are able to be specified using pygsti.construction.create_explicit_model. So we'll start by construct a Model object containing all but the two-qubit gate(s) using create_explicit_model.

In [2]:
target_model = pygsti.construction.create_explicit_model(
[('Q0','Q1')],['Gii','Gix','Giy','Gxi','Gyi'],
[ "I(Q0):I(Q1)", "X(pi/2,Q1)", "Y(pi/2,Q1)", "X(pi/2,Q0)", "Y(pi/2,Q0)" ],
effectLabels=['00','01','10','11'], effectExpressions=["0","1","2","3"])


There are lots of arguments to this function, so let's review what they mean:

• [('Q0','Q1')] = interpret this 4-d space as that of two qubits 'Q0', and 'Q1' (note these labels must begin with 'Q'!)
• "Gix" = operation label; can be anything that begins with 'G' and is followed by lowercase letters
• "X(pi/2,Q1)" = pi/2 single-qubit x-rotation gate on the qubit labeled Q1
• "rho0" = prep label; can be anything that begins with "rho"
• '10' = designates a POVM effect label whose corresponding vector is given by the effectExpressions argument.
• "2" = a prep or effect expression indicating a projection/preparation of the 3rd (b/c 0-based) computational basis element

You can also explicity add identity operations, e.g. "I(Q0)", to the rotation gates to get the same model (see mdl_targetB below), and this same syntax can be used for non-entangling 2-qubit gates, e.g. "X(pi/2,Q0):X(pi/2,Q1)".

In [3]:
mdl_targetB = pygsti.construction.create_explicit_model(
[('Q0','Q1')],['Gii','Gix','Giy','Gxi','Gyi','Gcnot'],
[ "I(Q0):I(Q1)", "I(Q0):X(pi/2,Q1)", "I(Q0):Y(pi/2,Q1)", "X(pi/2,Q0):I(Q1)", "Y(pi/2,Q0):I(Q1)", "CNOT(Q0,Q1)" ],
effectLabels=['00','01','10','11'], effectExpressions=["0","1","2","3"])
assert(abs(target_model.frobeniusdist(mdl_targetB)) < 1e-6)


If our 2-qubit gate happens to be one that can be specified using create_explicit_model then we can just use it to construct the entire Model and be done. Currently, create_explicit_model can create any controlled $X$, $Y$, or $Z$ rotation using CX, CY and CZ, as well as the standard CNOT and CPHASE gates. Below we demonstrate creation with the CNOT gate. The resulting Model is then identical to pygsti.modelpacks.legacy.std2Q_XYCNOT.target_model().

In [4]:
mdl_withCNOT = pygsti.construction.create_explicit_model(
[('Q0','Q1')],['Gii','Gix','Giy','Gxi','Gyi','Gcnot'],
[ "I(Q0):I(Q1)", "I(Q0):X(pi/2,Q1)", "I(Q0):Y(pi/2,Q1)", "X(pi/2,Q0):I(Q1)", "Y(pi/2,Q0):I(Q1)", "CNOT(Q0,Q1)" ],
effectLabels=['00','01','10','11'], effectExpressions=["0","1","2","3"])

#Note this is the same model as one of pyGSTi's standard models:
from pygsti.modelpacks.legacy import std2Q_XYICNOT
assert(abs(mdl_withCNOT.frobeniusdist(std2Q_XYICNOT.target_model())) < 1e-6)


Thus, since our target_model just contains the 1-qubit gates of std2Q_XYCNOT.target_model, we could also create has all the 1-qubit gates are the same a third way to obtain target_model is to just remove Gcnot from the standard model:

In [5]:
mdl_targetC = std2Q_XYICNOT.target_model()
del mdl_targetC.operations['Gcnot']
assert(abs(target_model.frobeniusdist(mdl_targetC)) < 1e-6)


## Step 2: Create a custom 2-qubit gate¶

We're assuming that create_explicit_model can't make the 2-qubit gate we want, so we'll need to create our own. Below we demonstrate how to create a 2-qubit gate from a given unitary which acts on the 2-qubit, 4-dimensional, state space.

In [6]:
import numpy as np

#Unitary in acting on the state-space { |A>, |B>, |C>, |D> } == { |00>, |01>, |10>, |11> }.
# This unitary rotates the second qubit by pi/2 in either the (+) or (-) direction based on
# the state of the first qubit.
myUnitary = 1./np.sqrt(2) * np.array([[1,-1j,0,0],
[-1j,1,0,0],
[0,0,1,1j],
[0,0,1j,1]])

#Convert this unitary into a "superoperator", which acts on the
# space of vectorized density matrices instead of just the state space.
# These superoperators are what GST calls "gates".
mySuperOp_stdbasis = pygsti.unitary_to_process_mx(myUnitary)

#After the call to unitary_to_process_mx, the superoperator is a complex matrix
# in the "standard" or "matrix unit" basis given by { |A><A|, |A><B|, etc }.
# For use in GST, we want to work with a *real* matrix in either the
# Gell-Mann or Pauli-product basis. Here we choose the Pauli-product basis,
# which is typically more intuitive when working with 2 qubits.
mySuperOp_ppbasis = pygsti.change_basis(mySuperOp_stdbasis, "std", "pp")

#The resulting superoperator in the Pauli-product basis is exactly
# what goes into the Model object, which can be set using
# dictionary syntax.  The line below names our two-qubit gate 'Gtq'
target_model['Gtq'] = mySuperOp_ppbasis


## That's it!¶

We're done creating our 2-qubit model, target_model, printed below. To run 2-qubit GST with this model we ideally would generate fiducials and germs specifically for it. Actually, since the 1-qubit gates are the same as other standard 2Q models, and the fiducial seqeuences for these standard sets only contain 1-qubit gates, we can just use the fiducial sets from a standard set (e.g. std2Q_XYCNOT). The germs should be computed, though typically you can get away with just using the germ set of a standard model and replacing it's 2-qubit gate with the custom one - so it can be worth checking whether such a set is complete before running a full germ-selection procedure.

In [7]:
print(target_model)

rho0 = FullSPAMVec with dimension 16
0.50   0   0 0.50   0   0   0   0   0   0   0   0 0.50   0   0 0.50

Mdefault = UnconstrainedPOVM with effect vectors:
00: FullSPAMVec with dimension 16
0.50   0   0 0.50   0   0   0   0   0   0   0   0 0.50   0   0 0.50

01: FullSPAMVec with dimension 16
0.50   0   0-0.50   0   0   0   0   0   0   0   0 0.50   0   0-0.50

10: FullSPAMVec with dimension 16
0.50   0   0 0.50   0   0   0   0   0   0   0   0-0.50   0   0-0.50

11: FullSPAMVec with dimension 16
0.50   0   0-0.50   0   0   0   0   0   0   0   0-0.50   0   0 0.50

Gii =
FullDenseOp with shape (16, 16)
1.00   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0 1.00   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0 1.00   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0 1.00   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0 1.00   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0 1.00   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0 1.00   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0 1.00   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0 1.00   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0 1.00   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0 1.00   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0 1.00   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0 1.00   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0 1.00   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0 1.00   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0 1.00

Gix =
FullDenseOp with shape (16, 16)
1.00   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0 1.00   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0-1.00   0   0   0   0   0   0   0   0   0   0   0   0
0   0 1.00   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0 1.00   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0 1.00   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0-1.00   0   0   0   0   0   0   0   0
0   0   0   0   0   0 1.00   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0 1.00   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0 1.00   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0-1.00   0   0   0   0
0   0   0   0   0   0   0   0   0   0 1.00   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0 1.00   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0 1.00   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0-1.00
0   0   0   0   0   0   0   0   0   0   0   0   0   0 1.00   0

Giy =
FullDenseOp with shape (16, 16)
1.00   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0 1.00   0   0   0   0   0   0   0   0   0   0   0   0
0   0 1.00   0   0   0   0   0   0   0   0   0   0   0   0   0
0-1.00   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0 1.00   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0 1.00   0   0   0   0   0   0   0   0
0   0   0   0   0   0 1.00   0   0   0   0   0   0   0   0   0
0   0   0   0   0-1.00   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0 1.00   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0 1.00   0   0   0   0
0   0   0   0   0   0   0   0   0   0 1.00   0   0   0   0   0
0   0   0   0   0   0   0   0   0-1.00   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0 1.00   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0 1.00
0   0   0   0   0   0   0   0   0   0   0   0   0   0 1.00   0
0   0   0   0   0   0   0   0   0   0   0   0   0-1.00   0   0

Gxi =
FullDenseOp with shape (16, 16)
1.00   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0 1.00   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0 1.00   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0 1.00   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0 1.00   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0 1.00   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0 1.00   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0 1.00   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0-1.00   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0-1.00   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0-1.00   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0-1.00
0   0   0   0   0   0   0   0 1.00   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0 1.00   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0 1.00   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0 1.00   0   0   0   0

Gyi =
FullDenseOp with shape (16, 16)
1.00   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0 1.00   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0 1.00   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0 1.00   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0 1.00   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0 1.00   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0 1.00   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0 1.00
0   0   0   0   0   0   0   0 1.00   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0 1.00   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0 1.00   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0 1.00   0   0   0   0
0   0   0   0-1.00   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0-1.00   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0-1.00   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0-1.00   0   0   0   0   0   0   0   0

Gtq =
FullDenseOp with shape (16, 16)
1.00   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0 1.00   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0-1.00
0   0   0   0   0   0   0   0   0   0   0   0   0   0 1.00   0
0   0   0   0   0   0   0   0   0-1.00   0   0   0   0   0   0
0   0   0   0   0   0   0   0-1.00   0   0   0   0   0   0   0
0   0   0   0   0   0 1.00   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0 1.00   0   0   0   0   0   0   0   0
0   0   0   0   0 1.00   0   0   0   0   0   0   0   0   0   0
0   0   0   0 1.00   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0 1.00   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0 1.00   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0 1.00   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0 1.00   0   0
0   0   0-1.00   0   0   0   0   0   0   0   0   0   0   0   0
0   0 1.00   0   0   0   0   0   0   0   0   0   0   0   0   0


In [ ]: