# Explicit Models Tutorial¶

This tutorial will show you how to create and use ExplicitOpModel objects. Model objects are fundamental to pyGSTi, as each represents a set of quantum gates along with state preparation and measurement (i.e. POVM) operations. In pyGSTi, a "state space" refers to a Hilbert space of pure quantum states (often thought of as length-$d$ vectors, where $d=2^N$ for $N$ qubits). A "density matrix space" refers to a Hilbert space of density matrices, which while often thought of as $d \times d$ matrices can also be represented by length $d^2$ vectors. Mathematically, these vectors live in Hilbert-Schmidt space, the space of linear operators on the original $d\times d$ density matrix space. pyGSTi uses the "Liouville" vector-representation for density matrices and POVM effects, since this allows quantum gates to be represented by $d^2 \times d^2$ matrices which act on Hilbert-Schmidt vectors.

ExplicitOpModel objects are the simplest type of Model objects in pyGSTi. They have the look and feel of Python dictionaries which hold $d^2\times d^2$ operation matrices, length-$d^2$ state preparation vectors, and sets of length-$d^2$ effect vectors which encode positive operator value measures (POVMs). State preparation and POVM effect vectors are both generically referred to as "SPAM" (state preparation and measurement) vectors.

In [ ]:
import pygsti
import pygsti.construction as pc


## Creating models¶

Before getting to ExplicitOpModels in particular, lets explain two quantites that all Model objects posess: a basis and state space labels:

• A model's .state_space_labels member (a StateSpaceLabels object) describes the model's state space as the direct sum and tensor product of labelled factors. Typically, this is just a tensor product of one or more 2-dimensional qubit spaces labelled by the integers 0 to $N_{qubits}-1$ or "Q0", "Q1", etc. We specify a 1-qubit state space using ["Q0"] below (the "Q" tells pyGSTi it's a 2-dimensional qubit space). If you had two qubits you could use ["Q0","Q1"] or [0,1] to describe the tensor product of two qubit spaces, as pyGSTi assumes integer labels stand for qubit spaces too. To learn more about the StateSpaceLabels object, see the state space labelling tutorial.
• A model's .basis member (a Basis object) describes how any dense representations (matrices or vectors) of the the operations in a Model should be interpreted. We'll be using the "Pauli product" basis, which is named "pp" in pyGSTi and consists of the tensor products of Pauli matrices (since our example has just a 1-qubit state space the "pp" basis is just the 4 Pauli matrices $\{\sigma_0,\sigma_X,\sigma_Y,\sigma_Z\}$). To learn more about Basis objects see the Basis object tutorial).

## Creating explicit models¶

There are more or less four ways to create ExpicitOpModel objects in pyGSTi:

• By creating an empty ExpicitOpModel and setting its elements directly, possibly with the help of pygsti.construction's build_operation and build_vector functions.
• By a single call to create_explicit_model, which automates the above approach.
• By loading from a text-format model file using pygsti.io.load_model.
• By copying one from one of the pygsti.construction.std* modules.

### Creating a ExplicitOpModel from scratch¶

Layer operations (often called "gates" in a 1- or even 2-qubit context) and SPAM vectors can be assigned to a ExplicitOpModel object as to an ordinary python dictionary. Internally a ExpicitOpModel holds these quantities as LinearOperator- and SPAMVec- and POVM-derived objects, but you may assign lists, Numpy arrays, or other types of Python iterables to a ExplicitOpModel key and a conversion will be performed automatically. To keep gates, state preparations, and POVMs separate, the ExplicitOpModel object looks at the beginning of the dictionary key being assigned: keys beginning with rho, M, and G are categorized as state preparations, POVMs, and gates, respectively. To avoid ambiguity, each key must begin with one of these three prefixes.

To separately access (set or get) the state preparations, POVMs, and operations contained in a ExplicitOpModel use the preps, povms, and operations members respectively. Each one provides dictionary-like access to the underlying objects. For example, myModel.operations['Gx'] accesses the same underlying LinearOperator object as myModel['Gx'], and similarly for myModel.preps['rho0'] and myModel['rho0']. The values of operations and state preparation vectors can be read and written in this way.

A POVM object acts similarly to dictionary of SPAMVec-derived effect vectors, but typically requires all such vectors to be initialized at once, that is, you cannot assign individual effect vectors to a POVM. The string-valued keys of a POVM label the outcome associated with each effect vector, and are therefore termed effect labels or outcome labels. The outcome labels also designate data within a DataSet object (see the DataSet tutorial), and thereby associate modeled POVMs with experimental measurements.

The below cell illustrates how to create a ExplicitOpModel from scratch.

In [ ]:
from math import sqrt
import numpy as np

#Initialize an empty Model object
#Designate the basis being used for the matrices and vectors below
# as the "Pauli product" basis of dimension 2 - i.e. the four 2x2 Pauli matrices I,X,Y,Z
model1 = pygsti.objects.ExplicitOpModel(['Q0'],'pp')

#Populate the Model object with states, effects, gates,
# all in the *normalized* Pauli basis: { I/sqrt(2), X/sqrt(2), Y/sqrt(2), Z/sqrt(2) }
# where I, X, Y, and Z are the standard Pauli matrices.
model1['rho0'] = [ 1/sqrt(2), 0, 0, 1/sqrt(2) ] # density matrix [[1, 0], [0, 0]] in Pauli basis
model1['Mdefault'] = pygsti.objects.UnconstrainedPOVM(
{'0': [ 1/sqrt(2), 0, 0, 1/sqrt(2) ],   # projector onto [[1, 0], [0, 0]] in Pauli basis
'1': [ 1/sqrt(2), 0, 0, -1/sqrt(2) ] }) # projector onto [[0, 0], [0, 1]] in Pauli basis

model1['Gi'] = np.identity(4,'d') # 4x4 identity matrix
model1['Gx'] = [[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 0,-1],
[0, 0, 1, 0]] # pi/2 X-rotation in Pauli basis

model1['Gy'] = [[1, 0, 0, 0],
[0, 0, 0, 1],
[0, 0, 1, 0],
[0,-1, 0, 0]] # pi/2 Y-rotation in Pauli basis

pygsti.io.write_model(model1, "../tutorial_files/Example_gatesetFromScratch.txt", title="My Model")


Check out the model file that was written here.

### Creating a ExplicitOpModel from scratch using modelconstruction._basis_create_operation and modelconstruction._basis_create_spam_vector¶

The modelconstruction._basis_create_operation and modelconstruction._basis_create_spam_vector functions take a human-readable string representation of a gate or SPAM vector, and return a LinearOperator or SPAMVector object that gets stored in a dictionary-like ExplicitOpModel or POVM object. To use these functions, you must specify what state space you're working with, and the basis for that space - so the .state_space_labels and .basis member of your Model object, as described above.

build_vector currently only understands strings which are integers (e.g. "1"), for which it creates a vector performing state preparation of (or, equivalently, a state projection onto) the $i^{th}$ state of the Hilbert space, that is, the state corresponding to the $i^{th}$ row and column of the $d\times d$ density matrix.

build_operation accepts a wider range of descriptor strings, which take the form of functionName(args) and include:

• I(label0, label1, ...) : the identity on the spaces labeled by label0, label1, etc.
• X(theta,Qlabel), Y(theta,Qlabel), Z(theta,Qlabel) : single qubit X-, Y-, and Z-axis rotations by angle theta (in radians) on the qubit labeled by Qlabel. Note that pi can be used within an expression for theta, e.g. X(pi/2,Q0).
• CX(theta, Qlabel1, Qlabel2), CY(theta, Qlabel1, Qlabel2), CZ(theta, Qlabel1, Qlabel2) : two-qubit controlled rotations by angle theta (in radians) on qubits Qlabel1 (the control) and Qlabel2 (the target).
In [ ]:
#Initialize an empty Model object
model2 = pygsti.objects.ExplicitOpModel(['Q0'],'pp') # single qubit labelled 'Q0'; Pauli basis
spaceLabels = model2.state_space_labels
basis = model2.basis

#Populate the Model object with states, effects, and gates using
# build_vector, build_operation, and create_identity_vec.
model2['rho0'] = pc.modelconstruction._basis_create_spam_vector("0",basis)
model2['Mdefault'] = pygsti.objects.UnconstrainedPOVM(
{ '0': pc.modelconstruction._basis_create_spam_vector("0",basis),
'1': pc.modelconstruction._basis_create_spam_vector("1",basis) } )
model2['Gi'] = pc.modelconstruction._basis_create_operation(spaceLabels,"I(Q0)",basis)
model2['Gx'] = pc.modelconstruction._basis_create_operation(spaceLabels,"X(pi/2,Q0)",basis)
model2['Gy'] = pc.modelconstruction._basis_create_operation(spaceLabels,"Y(pi/2,Q0)",basis)


### Create a ExplicitOpModel in a single call to create_explicit_model¶

The approach illustrated above using calls to build_vector and build_operation can be performed in a single call to create_explicit_model. You will notice that all of the arguments to create_explicit_model corrspond to those used to construct a model using build_vector and build_operation; the create_explicit_model function is merely a convenience function which allows you to specify everything at once. These arguments are:

• Arg 1 : the state-space-labels, as described above.
• Args 2 & 3 : list-of-gate-labels, list-of-gate-expressions (labels must begin with 'G'; "expressions" being the descriptor strings passed to build_operation)
• Args 4 & 5 : list-of-prep-labels, list-of-prep-expressions (labels must begin with 'rho'; "expressions" being the descriptor strings passed to build_vector)
• Args 6 & 7 : list-of-effect-labels, list-of-effect-expressions (labels can be anything; "expressions" being the descriptor strings passed to build_vector). These effect vectors will comprise a single POVM named "Mdefault" by default, but which can be changed via the povmLabels argument (see doc string for details).

The optional argument basis can be set to any of the known built-in basis names (e.g. "gm", "pp", "qt", or "std") to select the basis for the Model as described above. By default, "pp" is used when possible (if the state space corresponds to an integer number of qubits), "qt" if the state space has dimension 3, and "gm" otherwise. The optional argument parameterization is used to specify the parameterization used for the created gates (see below).

In [ ]:
model3 = pc.create_explicit_model(['Q0'],
['Gi','Gx','Gy'], [ "I(Q0)","X(pi/2,Q0)", "Y(pi/2,Q0)"],
prep_labels=['rho0'], prep_expressions=["0"],
effect_labels=['0','1'], effect_expressions=["0","1"] )


In this case, the parameters to create_explicit_model, specify:

• The state space has dimension 2 and is interpreted as that of a single qubit labeled "Q0" (label must begin with 'Q' or be an integer if we don't want to create a full StateSpaceLabels object that contains the dimensions too.)

• there are three gates: Idle, $\pi/2$ x-rotation, $\pi/2$ y-rotation, labeled Gi, Gx, and Gy.

• there is one state prep operation, labeled rho0, which prepares the 0-state (the first basis element of the 2D state space)

• there is one POVM (~ measurement), named Mdefault with two effect vectors: '0' projects onto the 0-state (the first basis element of the 2D state space) and '1' projects onto the 1-state.

Note that by default, there is a single state prep, "rho0", that prepares the 0-state and a single POVM, "Mdefault", which consists of projectors onto each standard basis state that are labelled by their integer indices (so just '0' and '1' in the case of 1-qubit). Thus, all but the first four arguments used above just specify the default behavior and can be omitted:

In [ ]:
model4 = pc.create_explicit_model( ['Q0'], ['Gi','Gx','Gy'], [ "I(Q0)","X(pi/2,Q0)", "Y(pi/2,Q0)"] )


### Load a ExplicitOpModel from a file¶

You can also construct a ExplicitOpModel object from a file using pygsti.io.load_model. The format of the text file should be fairly self-evident given the above discussion. Note that vector and matrix elements need not be simple numbers, but can be any mathematical expression parseable by the Python interpreter, and in addition to numbers can include "sqrt" and "pi".

In [ ]:
#3) Write a text-format model file and read it in.
model5_txt = \
"""
# Example text file describing a model

PREP: rho0
LiouvilleVec
1/sqrt(2) 0 0 1/sqrt(2)

POVM: Mdefault

EFFECT: 0
LiouvilleVec
1/sqrt(2) 0 0 1/sqrt(2)

EFFECT: 1
LiouvilleVec
1/sqrt(2) 0 0 -1/sqrt(2)

END POVM

GATE: Gi
LiouvilleMx
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

GATE: Gx
LiouvilleMx
1 0 0 0
0 1 0 0
0 0 0 -1
0 0 1 0

GATE: Gy
LiouvilleMx
1 0 0 0
0 0 0 1
0 0 1 0
0 -1 0 0

STATESPACE: Q0(4)
BASIS: pp
"""
with open("../tutorial_files/Example_ExplicitModel.txt","w") as gsetfile:
gsetfile.write(model5_txt)


In [ ]:
#All four of the above models are identical.  See this by taking the frobenius differences between them:
assert(model1.frobeniusdist(model2) < 1e-8)
assert(model1.frobeniusdist(model3) < 1e-8)
assert(model1.frobeniusdist(model4) < 1e-8)
assert(model1.frobeniusdist(model5) < 1e-8)


## Viewing models¶

In the cells below, we demonstrate how to print and access information within a ExplicitOpModel.

In [ ]:
#Printing the contents of a Model is easy
print("Model 1:\n", model1)

In [ ]:
#You can also access individual gates like they're numpy arrays:
Gx = model1['Gx'] # a LinearOperator object, but behaves like a numpy array

#By printing a gate, you can see that it's not just a numpy array
print("Gx = ", Gx)

#But can be accessed as one:
print("Array-like printout\n", Gx[:,:],"\n")
print("First row\n", Gx[0,:],"\n")
print("Element [2,3] = ",Gx[2,3], "\n")

Id = np.identity(4,'d')
Id_dot_Gx = np.dot(Id,Gx)
print("Id_dot_Gx\n", Id_dot_Gx, "\n")


## Basic Operations with Explicit Models¶

ExplicitOpModel objects have a number of methods that support a variety of operations, including:

• Depolarizing or rotating every gate
• Writing the model to a file
• Computing products of operation matrices
In [ ]:
#Add 10% depolarization noise to the gates
depol_model3 = model3.depolarize(op_noise=0.1)

#Add a Y-axis rotation uniformly to all the gates
rot_model3 = model3.rotate(rotate=(0,0.1,0))

In [ ]:
#Writing a model as a text file
pygsti.io.write_model(depol_model3, "../tutorial_files/Example_depolarizedModel.txt", title="My Model")

In [ ]:
print("Probabilities of outcomes of the gate\n sequence GxGx (rho0 and Mdefault assumed)= ",
depol_model3.probabilities( ("Gx", "Gx")))
print("Probabilities of outcomes of the \"complete\" gate\n sequence rho0+GxGx+Mdefault = ",
depol_model3.probabilities( ("rho0", "Gx", "Gx", "Mdefault")))

In [ ]:
#Printing more detailed information about a model
depol_model3.print_info()


It is also possible to manipulate the underlying operations by accessing the forward simulator in the Model class. For example, using the matrix simulator type, one can compute the product of two gate operations. For more details, see the ForwardSimulationTypes tutorial.

In [ ]:
# Computing the product of operation matrices (only allowed with the matrix simulator type)
print("Product of Gx * Gx = \n",depol_model3.sim.product(("Gx", "Gx")), end='\n\n')


## Explicit Model Parameterizations¶

In addition to specifying a set of $d^2 \times d^2$ operation matrices and length-$d^2$ SPAM vectors, every Model encapsulates a parametrization, that is, a function mapping a set of real-valued parameters to its set of operation matrices and SPAM vectors. A Model's contents must always correspond to a valid set of parameters, which can be obtained by its to_vector method, and can always be initialized from a vector of parameters via its from_vector method. The number of parameters (obtained via num_params) is independent (and need not equal!) the total number of gate-matrix and SPAM-vector elements comprising the Model. For example, in a "TP-parameterized" model, the first row of each operation matrix is fixed at [1,0,...0], regardless to what the Model's underlying parameters are. When pyGSTi generates Model estimates the parameters of an initial Model (often times the "target" model) supplied by the caller are optimized. Thus, by its parameterization a single Model can determine the space of possible Models that are searched for a best-fit estimate.

Each gate and SPAM vector within a ExplicitOpModel have independent paramterizations, so that each pygsti.objects.LinearOperator-derived gate object and pygsti.objects.SPAMVec-derived SPAM vector has its own to_vector, from_vector, and num_params method. A Model's parameter vector is just the concatenation of the parameter vectors of its contents, in the order: 1) state preparation vectors, 2) measurement vectors, 3) gates.

Users are able to create their own gate parameterizations by deriving from pygsti.objects.LinearOperator or pygsti.objects.DenseOperator (which itself derives from LinearOperator). Included in pyGSTi are several convenient gate parameterizations which are worth knowing about:

• The FullDenseOp class defines a gate which has a parameter for every element, and thus optimizations using this gate class allow the operation matrix to be completely arbitrary.
• The TPDenseOp class defines a gate whose first row must be [1,0,...0]. This corresponds to a trace-preserving (TP) gate in the Gell-Mann and Pauli-product bases. Each element in the remaining rows is a separate parameter, similar to a fully parameterized gate. Optimizations using this gate type are used to constrain the estimated gate to being trace preserving.
• The LindbladOp (or LindbladDenseOp) class defines a gate whose logarithm take a particular Lindblad form. This class is fairly flexible, but is predominantly used to constrain optimizations to the set of infinitesimally-generated CPTP maps.

Similarly, there are FullSPAMVec and TPSPAMVec classes, the latter which fixes its first element to $\sqrt{d}$, where $d^2$ is the vector length, as this is the appropriate value for a unit-trace state preparation.

We now illustrate how one map specify the type of paramterization in create_explicit_model, and change the parameterizations of all of a ExplicitOpModel's contents using its set_all_parameterizaions method.

In [ ]:
# Speciy basis as 'gm' for Gell-Mann (could also be 'pp' for Pauli-Product)
# and parameterization to 'TP', so that gates are TPParameterizedGates
model6 = pc.create_explicit_model(['Q0'], ['Gi',], [ "I(Q0)"],
basis='pp', parameterization="TP")

#See that gates and prep vectors are TP, whereas previous Model's have
# fully parameterized elements
print("model6 gate type = ", type(model6['Gi']))
print("model6 prep type = ", type(model6['rho0']))
print("model5 gate type = ", type(model5['Gi']))
print("model5 prep type = ", type(model5['rho0']))

#Switch parameterization to CPTP gates
model6.set_all_parameterizations('CPTP')
print("\nAfter setting all parameterizations to CPTP:")
print("model6 gate type = ", type(model6['Gi']))
print("model6 prep type = ", type(model6['rho0']))


To alter an individual gate or SPAM vector's parameterization, one can simply construct a LinearOperator or SPAMVec object with the desired parameterization and assign it to the Model.

In [ ]:
newOp = pygsti.objects.TPDenseOp(model6['Gi'])
model6['Gi'] = newOp
print("model6['Gi'] =",model6['Gi'])


NOTE: When a LinearOperator or SPAMVec-derived object is assigned as an element of an ExplicitOpModel (as above), the object replaces any existing object with the given key. However, if any other type of object is assigned to an ExplicitOpModel element, an attempt is made to initialize or update the existing existing gate using the assigned data (using its set_matrix function internally). For example:

In [ ]:
import numpy as np
numpy_array = np.array( [[1, 0, 0, 0],
[0, 0.5, 0, 0],
[0, 0, 0.5, 0],
[0, 0, 0, 0.5]], 'd')
model6['Gi'] = numpy_array # after assignment with a numpy array...
print("model6['Gi'] =",model6['Gi']) # this is STILL a TPDenseOp object

#If you try to assign a gate to something that is either invalid or it doesn't know how
# to deal with, it will raise an exception
invalid_TP_array = np.array( [[2, 1, 3, 0],
[0, 0.5, 0, 0],
[0, 0, 0.5, 0],
[0, 0, 0, 0.5]], 'd')
try:
model6['Gi'] = invalid_TP_array
except ValueError as e:
print("ERROR!! " + str(e))

In [ ]: