Create some abstract quantum functions with tequila

Installation

if you are on Linux or Mac, you can install tequila directly from the PyPi cloud via

pip install tequila-basic

this installs you the tequila package with only the absolutely necessary dependencies.
Using windows as an operating systems will lead to errors here as the jax library is not supported.
In the future there will be a separate blog entry for windows users, meanwhile see the github readme.

Not necessary but recommended it the installation of a capable simulator for you quantum circuits. Tequila supports the usual suspects, the by far fastest option is however the qualcs simulator that can conveniently be installed as

pip install qulacs

Define an Expectation Value

An expectation value \(E = \langle H \rangle_U\) consists of two parts: 1. a quantum circuit U (a unitary operation expressed as a sequence of unitaries) that defines the wavefunction 2. a qubit Hamiltonian H (an hermitian operator exporessed in Pauli-strings)

here is a small example with the Hamiltonian \[ H = X(0)X(1) + \frac{1}{2} Z(0)Z(1) \] and the quantum circuit

<frozen importlib._bootstrap>:228: RuntimeWarning: scipy._lib.messagestream.MessageStream size changed, may indicate binary incompatibility. Expected 56 from C header, got 64 from PyObject

note that we have parametrized one of the gates (marked in pink), so that our expectation value becomes a function of this paramteter \[E=f(a).\]

Evaluate the Expectation Value

We can evaluate the expectation value by compiling it to a backend (a simulator or interface to some quantum hardware)

f = tq.compile(E)evaluated = f({"a":1.0})print("f(1.0) = {:+2.4f}".format(evaluated))

f(1.0) = +1.3415

if you have installed qulacs before then tq.compile translated the underlying objects to qulacs. Otherwise another simulator on your system, or the horrible tequila debug simulator was used. You can figure out which simulators you have installed by calling tq.show_available_simualtors. When passing backend="qulacs" to tq.compile you can speficy which backend you want to compile to.

If ever in doubt if the current object is already compiled, just print it

print("abstract expectation value is:")print(E)print("compiled function is:")print(f)

abstract expectation value is:
Objective with 1 unique expectation values
total measurements = 2
variables = [a]
types = not compiled
compiled function is:
Objective with 1 unique expectation values
total measurements = 2
variables = [a]
types = [<class 'tequila.simulators.simulator_qulacs.BackendExpectationValueQulacs'>]

The printput states that our objects are tq.Objectives with a given number of unique expectation values. More about this now.

Differentiate and Manipulate Expectation Values

The expecation value can be used to define more complicated objects. Here are some examples:

The gradient \[ \frac{\partial }{\partial a } E(a) \] with respect to variable a can be computed as

dE = tq.grad(E,"a")print(dE)

Objective with 2 unique expectation values
total measurements = 4
variables = [a]
types = not compiled

and the corresponding object can then be compiled and evaluated in the same way

apart from differentiating we can also combine the tq.Objective objects. The final example illustrates most of these possibilities.

Example

Let’s create the function \[ L(a) = \frac{\partial E}{\partial a}(a) E(a)^3 + e^{-\left(\frac{\partial E }{\partial a}(a) \right)^2} \]

L = E**3*dE + (-dE**2).apply(tq.numpy.exp)l = tq.compile(L)

and now let’s have a look how this function and it’s gradient \(\frac{\partial L}{\partial a}\) looks like

Code

import numpyimport matplotlib.pyplot as plt# get the abstract gradient and compiledL = tq.grad(L, "a")dl = tq.compile(dL)x =list(numpy.linspace(-numpy.pi, numpy.pi, 100))y0 = [ l({"a":xx}) for xx in x]y1 = [ dl({"a":xx}) for xx in x]plt.plot(x,y0, label="$L(a)$", color="navy")plt.plot(x,y1, label=r"$\frac{\partial L}{\partial a}(a)$", color="tab:red")plt.legend()plt.savefig("fig.png")plt.show()

Source Code

---title: "Tequila Basics: Getting Started"author: "Jakob Kottmann"date: "2022-11-03"categories: [tequila, code]image: tq-basics-pic.pngformat: html: code-fold: false eval: truejupyter: tq-pypi---Create some abstract quantum functions with tequila<center>![](tq-basics-pic.png){width=400}</center>## Installationif you are on Linux or Mac, you can install tequila directly from the PyPi cloud via```{bash}pip install tequila-basic```this installs you the tequila package with only the absolutely necessary dependencies. Using windows as an operating systems will lead to errors here as the jax library is not supported. In the future there will be a separate blog entry for windows users, meanwhile see the github [readme](https://github.com/tequilahub/tequila/). Not necessary but recommended it the installation of a capable simulator for you quantum circuits. Tequila supports the usual suspects, the by far fastest option is however the [qualcs](https://github.com/qualcs) simulator that can conveniently be installed as```{bash}pip install qulacs```## Define an Expectation ValueAn expectation value $E = \langle H \rangle_U$ consists of two parts: 1. a quantum circuit `U` (a unitary operation expressed as a sequence of unitaries) that defines the wavefunction2. a qubit Hamiltonian `H` (an hermitian operator exporessed in Pauli-strings)here is a small example with the Hamiltonian$$ H = X(0)X(1) + \frac{1}{2} Z(0)Z(1) $$and the quantum circuit <center><imgsrc="circuit.png"width=30%fig-align="center"></center>```{python}import tequila as tqH = tq.paulis.X([0,1]) +0.5*tq.paulis.Z([0,1])U = tq.gates.Ry("a", 0) + tq.gates.X(1,control=0)E = tq.ExpectationValue(H=H, U=U)```note that we have parametrized one of the gates (marked in pink), so that our expectation value becomes a function of this paramteter $$E=f(a).$$## Evaluate the Expectation ValueWe can evaluate the expectation value by compiling it to a backend (a simulator or interface to some quantum hardware)```{python}f = tq.compile(E)evaluated = f({"a":1.0})print("f(1.0) = {:+2.4f}".format(evaluated))```if you have installed qulacs before then `tq.compile` translated the underlying objects to qulacs. Otherwise another simulator on your system, or the horrible tequila debug simulator was used. You can figure out which simulators you have installed by calling `tq.show_available_simualtors`. When passing `backend="qulacs"` to `tq.compile` you can speficy which backend you want to compile to. If ever in doubt if the current object is already compiled, just print it```{python}print("abstract expectation value is:")print(E)print("compiled function is:")print(f)```The printput states that our objects are `tq.Objectives` with a given number of unique expectation values. More about this now.## Differentiate and Manipulate Expectation ValuesThe expecation value can be used to define more complicated objects. Here are some examples: The gradient$$ \frac{\partial }{\partial a } E(a) $$with respect to variable `a` can be computed as```{python}dE = tq.grad(E,"a")print(dE)```and the corresponding object can then be compiled and evaluated in the same way```{python}df = tq.compile(dE)evaluated = df({"a":1.0})print("df/da(1.0 = {:+2.4f}".format(evaluated))```apart from differentiating we can also combine the `tq.Objective` objects. The final example illustrates most of these possibilities.## ExampleLet's create the function$$ L(a) = \frac{\partial E}{\partial a}(a) E(a)^3 + e^{-\left(\frac{\partial E }{\partial a}(a) \right)^2} $$```{python}L = E**3*dE + (-dE**2).apply(tq.numpy.exp)l = tq.compile(L)```and now let's have a look how this function and it's gradient $\frac{\partial L}{\partial a}$ looks like ```{python}#| code-fold: trueimport numpyimport matplotlib.pyplot as plt# get the abstract gradient and compiledL = tq.grad(L, "a")dl = tq.compile(dL)x =list(numpy.linspace(-numpy.pi, numpy.pi, 100))y0 = [ l({"a":xx}) for xx in x]y1 = [ dl({"a":xx}) for xx in x]plt.plot(x,y0, label="$L(a)$", color="navy")plt.plot(x,y1, label=r"$\frac{\partial L}{\partial a}(a)$", color="tab:red")plt.legend()plt.savefig("fig.png")plt.show()```