A brief introduction to tequila circuits

1: initialization
2: parametriation
3: tricks (re-parametrize, map qubits, decompose gates, get generators)

Initialize Tequila Circuits

Quantum circuits can be initialized in an intuitive way by assembling tq.gates objects (see here for a list of all gates). Take for example:

import tequila as tq
U = tq.gates.Ry(target=0, angle=1.0) + tq.gates.X(control=0,target=1)

which intializes a two-qubit circuit with a Y-rotation and a controlled-NOT operation in sequence. All tequila gates can be initialized with the following keywords

target: the target qubit(s) | integer or list of integers
control: the control qubits(s) | integer or list of integers
angle: a parameter for the gate (see below) | float or variable

the angle is optional for some gates that have a defined default value - like X, Y, Z and H.

We can simulate the circuit and get the corresponding wavefunction back. This is convenient, although not the main purpose of tequila

wfn = tq.simulate(U)
print(wfn)

+0.8776|00> +0.4794|11>

In the code block above, we did not specify the simulator to be used, tequila determines this automatically. You can however specify the similator via the backend keyword. If you are not sure which simulators are installed on your system, use tq.show_available_simulators()

Code

wfn = tq.simulate(U, backend="cirq")
tq.show_available_simulators()

Parametrization

All tequila gates are parametrized as

\[ U(\theta) = e^{-i\frac{\theta}{2} G} \]

were $G$ is the so called generator of the gate. With $G=X$ we obtain the standard $R_x(\theta)$ rotation gate while with $G=\left(X-1\right)$ and $\theta=\pi$ we obtain the standard Pauli-$X$ gate. In the same way, initializing tq.gates.X(target=0, angle="a") defines the gate \[ e^{-i\frac{a}{2} \left(X-1\right)} = X^{\frac{a}{\pi}}.\]
Usually it is convenient to leave gates like $X,Y,Z,H$ and $CNOT$ unparametrized and use parametrized gates generated from single tensor products of Pauli operators. The latter can be viewed as the native gate set of tequila. An example is

U = tq.gates.Rp(paulistring="X(0)Y(2)", angle="a")

which initialize the gate $e^{-i\frac{a}{2} \left(X(0)Y(2)\right)}$.

Parameters that are passed with the angle keyword can be three datatypes

floats
strings or any hashable type
tq.Variable
any callable type that returns a float when given a dictionary of variables

we have already seen an example for the first type. Let’s do it again for the other two. Let’s start with strings

U = tq.gates.Ry("a",0) + tq.gates.CNOT(0,1)
wfn = tq.simulate(U, variables={"a":1.0})
print(wfn)

+0.8776|00> +0.4794|11>

note that we need to specify the values of all parameters when we are simulating the circuit. The same example with the tq.Variable type

a = tq.Variable("a")
U = tq.gates.Ry(a,0) + tq.gates.CNOT(0,1)
wfn = tq.simulate(U, variables={"a":1.0})
print(wfn)

+0.8776|00> +0.4794|11>

note here, that the variables dictionary does not need to be initialized with the variable type, the plain names (as hashable types like strings) are enough - in the background this will all be converted to tq.Variable.

For the last datatype, we already saw an example, that is tq.Variable itself, as it can be evaluated

variable = tq.Variable("name")
evaluated = variable(variables={"name":1.0})
print(evaluated)

1.0

as this alone would be boring, we can manipulate the variable object, here is some example in which we initialize a circuit parameterized by the function \[ f(a) = \sin^2\left(\frac{a}{2}\pi\right) \]

from numpy import pi,sin

a = tq.Variable("a")
f = a*pi*0.5
f = f.apply(sin)
f = f**2
U = tq.gates.Ry(f,0) + tq.gates.CNOT(0,1)
wfn = tq.simulate(U, variables={"a":1.0})
print(wfn)

+0.8776|00> +0.4794|11>

Tricks

Add Controls

You can take one circuit and add control-options to it

cU = U.add_controls([3,4])

Re-Parametrize

UX = U.map_variables({"a":"b"})
UY = UX.map_variables({"b":1.0})
print("angles in U :",U.extract_variables())
print("angles in UX:",UX.extract_variables())
print("angles in UY:",UY.extract_variables())

angles in U : [a]
angles in UX: [b]
angles in UY: []

Map-Qubits

UZ = U.map_qubits({0:1,1:2}) 
print(UZ)

circuit: 
Ry(target=(1,), parameter=f([a]))
X(target=(2,), control=(1,))

Decompose Gates

Tequila can decompose gates automatically (e.g. for differentiation or translation to backend), you can however also do this manually with tq.compile_circuit. The default compiles down to the lowest lever (only standard single qubit gates and controlled-nots). You can specifiy the compile options with keywords like controlled_rotation=False. See here for all keywords.

U1 = tq.gates.H(0)
U1+= tq.gates.Ry("a",0,control=1) 
U1+= tq.gates.ExpPauli(angle="b", paulistring="X(0)X(2)")
# let's compile to different levels
U2 = tq.compile_circuit(U1)
U3 = tq.compile_circuit(U1, controlled_rotation=False)
U4 = tq.compile_circuit(U1, exponential_pauli=False)

the four circuits are

Get Generators

The generators of individual gates can be constructed with the make_generator function. Here we can include the control qubits in the definition of the generator. Take for example

U = tq.gates.Ry(angle="a", target=0) + tq.gates.X(control=0, target=1) 
for gate in U.gates:
    generator = gate.make_generator()
    print("gate generated by:", generator)
    if gate.is_controlled():
        cgenerator = gate.make_generator(include_controls=True)
        print("including control qubits:", cgenerator)

gate generated by: +1.0000Y(0)
gate generated by: +1.0000X(1)-1.0000
including control qubits: +0.5000X(1)-0.5000-0.5000Z(0)X(1)+0.5000Z(0)

--- title: "Tequila Basics: Circuits" author: "Jakob Kottmann" date: "2022-11-04" categories: [tequila, code] image: "tq-circuits-pic.png" image-width: "1cm" image-height: "1cm" format: html: code-fold: false eval: true jupyter: tq-pypi --- A brief introduction to tequila circuits - [-@sec-init]: initialization - [-@sec-param]: parametriation - [-@sec-tricks]: tricks (re-parametrize, map qubits, decompose gates, get generators) <center> ![](tq-circuits-pic.png){width=400} </center> # Initialize Tequila Circuits {#sec-init} Quantum circuits can be initialized in an intuitive way by assembling `tq.gates` objects (see [here](https://github.com/tequilahub/tequila/blob/master/src/tequila/circuit/gates.py) for a list of all gates). Take for example: ```{python} import tequila as tq U = tq.gates.Ry(target=0, angle=1.0) + tq.gates.X(control=0,target=1) ``` which intializes a two-qubit circuit with a Y-rotation and a controlled-NOT operation in sequence. All tequila gates can be initialized with the following keywords - `target`: the target qubit(s) | integer or list of integers - `control`: the control qubits(s) | integer or list of integers - `angle`: a parameter for the gate (see below) | float or variable the `angle` is optional for some gates that have a defined default value - like `X`, `Y`, `Z` and `H`. We can simulate the circuit and get the corresponding wavefunction back. This is convenient, although not the main purpose of tequila ```{python} wfn = tq.simulate(U) print(wfn) ``` In the code block above, we did not specify the simulator to be used, tequila determines this automatically. You can however specify the similator via the `backend` keyword. If you are not sure which simulators are installed on your system, use `tq.show_available_simulators()` ```{python} #| code-fold: true #| eval: false #| output: false wfn = tq.simulate(U, backend="cirq") tq.show_available_simulators() ``` # Parametrization {#sec-param} All tequila gates are parametrized as $$ U(\theta) = e^{-i\frac{\theta}{2} G} $$ were $G$ is the so called **generator** of the gate. With $G=X$ we obtain the standard $R_x(\theta)$ rotation gate while with $G=\left(X-1\right)$ and $\theta=\pi$ we obtain the standard Pauli-$X$ gate. In the same way, initializing `tq.gates.X(target=0, angle="a")` defines the gate $$ e^{-i\frac{a}{2} \left(X-1\right)} = X^{\frac{a}{\pi}}.$$ Usually it is convenient to leave gates like $X,Y,Z,H$ and $CNOT$ unparametrized and use parametrized gates generated from single tensor products of Pauli operators. The latter can be viewed as the native gate set of tequila. An example is ```{python} #| code-fold: false U = tq.gates.Rp(paulistring="X(0)Y(2)", angle="a") ``` which initialize the gate $e^{-i\frac{a}{2} \left(X(0)Y(2)\right)}$. Parameters that are passed with the `angle` keyword can be three datatypes - floats - strings or any hashable type - `tq.Variable` - any callable type that returns a float when given a dictionary of variables we have already seen an example for the first type. Let's do it again for the other two. Let's start with strings ```{python} U = tq.gates.Ry("a",0) + tq.gates.CNOT(0,1) wfn = tq.simulate(U, variables={"a":1.0}) print(wfn) ``` note that we need to specify the values of all parameters when we are simulating the circuit. The same example with the `tq.Variable` type ```{python} a = tq.Variable("a") U = tq.gates.Ry(a,0) + tq.gates.CNOT(0,1) wfn = tq.simulate(U, variables={"a":1.0}) print(wfn) ``` note here, that the variables dictionary does not need to be initialized with the variable type, the plain names (as hashable types like strings) are enough - in the background this will all be converted to `tq.Variable`. For the last datatype, we already saw an example, that is `tq.Variable` itself, as it can be evaluated ```{python} variable = tq.Variable("name") evaluated = variable(variables={"name":1.0}) print(evaluated) ``` as this alone would be boring, we can manipulate the variable object, here is some example in which we initialize a circuit parameterized by the function $$ f(a) = \sin^2\left(\frac{a}{2}\pi\right) $$ ```{python} from numpy import pi,sin a = tq.Variable("a") f = a*pi*0.5 f = f.apply(sin) f = f**2 U = tq.gates.Ry(f,0) + tq.gates.CNOT(0,1) wfn = tq.simulate(U, variables={"a":1.0}) print(wfn) ``` # Tricks {#sec-tricks} ## Add Controls You can take one circuit and add control-options to it ```{python} cU = U.add_controls([3,4]) ``` ![original](circuit.png){width=30%} ![added controls](cU.png){width=30%} ## Re-Parametrize ```{python} UX = U.map_variables({"a":"b"}) UY = UX.map_variables({"b":1.0}) print("angles in U :",U.extract_variables()) print("angles in UX:",UX.extract_variables()) print("angles in UY:",UY.extract_variables()) ``` ## Map-Qubits ```{python} UZ = U.map_qubits({0:1,1:2}) print(UZ) ``` ## Decompose Gates Tequila can decompose gates automatically (e.g. for differentiation or translation to backend), you can however also do this manually with `tq.compile_circuit`. The default compiles down to the lowest lever (only standard single qubit gates and controlled-nots). You can specifiy the compile options with keywords like `controlled_rotation=False`. See [here](https://github.com/tequilahub/tequila/blob/master/src/tequila/circuit/compiler.py) for all keywords. ```{python} U1 = tq.gates.H(0) U1+= tq.gates.Ry("a",0,control=1) U1+= tq.gates.ExpPauli(angle="b", paulistring="X(0)X(2)") # let's compile to different levels U2 = tq.compile_circuit(U1) U3 = tq.compile_circuit(U1, controlled_rotation=False) U4 = tq.compile_circuit(U1, exponential_pauli=False) ``` the four circuits are ![U1](U1.png){width=30%} ![U2 (full compile)](U2.png){width=70%} ![U3 (conrolled_rotation=False)](U3.png){width=50%} ![U4 (exponential_pauli=False)](U4.png){width=50%} ## Get Generators The generators of individual gates can be constructed with the `make_generator` function. Here we can include the control qubits in the definition of the generator. Take for example ```{python} #| code-fold: false U = tq.gates.Ry(angle="a", target=0) + tq.gates.X(control=0, target=1) for gate in U.gates: generator = gate.make_generator() print("gate generated by:", generator) if gate.is_controlled(): cgenerator = gate.make_generator(include_controls=True) print("including control qubits:", cgenerator) ```