# 1. Defining a quantum circuit¶

The first step in running a Quantum Machine Learning algorithm consists in defining a parametric quantum circuit. A quantum circuit is a sequence of instructions, or gates, that we apply to one or more qubits, the computational units of a quantum computer. A circuit being parametric means that the gates depend on some parameters, which in our case will be real valued angles.

Example of a quantum circuit.

The figure gives a graphical representation of a quantum circuit. Each line represents a qubit, to which gates are applied from the left to right. We use rectangles to identify single-qubit gates and vertical lines, terminating with a specific marker, to represent entangling gates, which affect two qubits. The leftmost symbol of each line gives the initial state of the corresponding qubit, which we always assume to be .

## 1.1. The circuitBuilder class¶

Since different quantum computer providers use different functions to implement the same operations, we resort to a circuit builder, which abstracts away the provider’s syntax.

The circuitBuilder class defines some high level methods to address qubit operations:

• input(idx, theta): Add input gate to the qubit corresponding to index idx. This consists of a qubit rotation of . Both idx and theta can be iterables; in this case, for each element of theta, a single input gate is added to the corresponding index in idx.

Input gate representation.

• allin(theta): This operation takes a list of angles and, for each element, add an input gate to the corresponding qubit. It has the same effect as input(range(nbqbits), theta).

• cz(a, b): Add CZ gate between qubits a and b. This is a symmetric entangling gate.

CZ gate representation.

• measure_all(): Add measurement to all qubits.

Measurement gate representation.

• circuit(): This method returns the circuit completed with the provided instructions. The specific return type varies across the builder implementations, but, as we will see when discussing the make_circuit function and the circuitML class, we never manipulate circuits directly.

### 1.1.1. Example¶

We can now build our first quantum circuit. As circuitBuilder is an abstract class, we use for this example the Qiskit implementation, but note that any builder would produce the same circuit. The Qiskit implementation is presented in depth in The Qiskit/IBMQ interface.

We use a vector of size seven to define the three-qubits circuit in figure.

from polyadicqml.qiskit import qkBuilder
import numpy as np

# We create the vector [-pi/2, -pi/3, -pi/6, 0, pi/6, pi/3, pi/2]
theta = np.linspace(-np.pi/2, np.pi/2, 7)

# We instantiate the builder

bdr = qkBuilder(3)  # 3 is the number of qubits

# We start by inputting the first three elements of theta

bdr.allin(theta[:3])

# Then we entangle the first two qubits

bdr.cz(0,1)

# Then input el. 3 and 4 from theta in qubits 0 and 1

bdr.input([0,1], theta[[3,4]])

# Then again entanglement and input
# Note that we can concatenate instructions

bdr.cz(1,2).input([1,2], theta[[5,6]])

# Finally, we measure.
bdr.measure_all()


Now that the builder is complete, we can retrieve the built circuit with bdr.circuit(). Furthermore, since we are using the Qiskit implementation, the return type is a qiskit.circuit.QuantumCircuit. We can print it and verify that it corresponds indeed to our desired circuit:

>>> print(bdr.circuit())
┌──────────┐┌───────────┐┌──────────┐   ┌──────────┐ ┌───────┐  ┌──────────┐                                        ░ ┌─┐
qr_0: ┤ RX(pi/2) ├┤ RZ(-pi/2) ├┤ RX(pi/2) ├─■─┤ RX(pi/2) ├─┤ RZ(0) ├──┤ RX(pi/2) ├────────────────────────────────────────░─┤M├──────
├──────────┤├───────────┤├──────────┤ │ ├──────────┤┌┴───────┴─┐├──────────┤   ┌──────────┐┌──────────┐┌──────────┐ ░ └╥┘┌─┐
qr_1: ┤ RX(pi/2) ├┤ RZ(-pi/3) ├┤ RX(pi/2) ├─■─┤ RX(pi/2) ├┤ RZ(pi/6) ├┤ RX(pi/2) ├─■─┤ RX(pi/2) ├┤ RZ(pi/3) ├┤ RX(pi/2) ├─░──╫─┤M├───
├──────────┤├───────────┤├──────────┤   └──────────┘└──────────┘└──────────┘ │ ├──────────┤├──────────┤├──────────┤ ░  ║ └╥┘┌─┐
qr_2: ┤ RX(pi/2) ├┤ RZ(-pi/6) ├┤ RX(pi/2) ├────────────────────────────────────────■─┤ RX(pi/2) ├┤ RZ(pi/2) ├┤ RX(pi/2) ├─░──╫──╫─┤M├
└──────────┘└───────────┘└──────────┘                                          └──────────┘└──────────┘└──────────┘ ░  ║  ║ └╥┘
meas_0: ═══════════════════════════════════════════════════════════════════════════════════════════════════════════════════════╩══╬══╬═
║  ║
meas_1: ══════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════╩══╬═
║
meas_2: ═════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════╩═


## 1.2. The make_circuit function¶

We just learned how to use a circuitBuilder to build a circuit. Nonetheless, in machine learning, we are interested in running the same circuit architecture with many different inputs and many different parameters. It is obvious that writing a new builder for each change in numerical values would be a tedious and useless task. Although, if we recall our previous circuit example, we notice how with can easily substitute the vector theta with a variable, as in the builder methods we only refer to it through its indices.

This idea motivates the introduction of the make_circuit function, whose documentation we report here:

As we see, this function makes an explicit distinction between input and model parameters, respectively x and params. Furthermore, the function signature has to respect the precised one, by taking a builder, alongside the input and parameters vectors, and returning an instructed builder.

Therefore, the implementation of a make_circuit function proceed as follows, where we see highlighted the important parts.

def make_circuit(bdr, x, params):

... # Series of instruction for the builder, e.g:
bdr.input([0,1], x[[0,1]]).cz(0,1)
bdr.input([0,1], params[[0,1]])
...

# Return the instructed builder
return bdr


This function is the only place in which we talk to quantum circuits – even if indirectly. In fact, we give this function to a circuitML, which will handle the backend, without having to worry about the shape of the internal representation.

Note

We usually avoid adding measure gates – measure_all() – in the make_circuit function, as we do not know in advance if we are working with the actual circuit output (i.e. the measurement) or with the simulated quantum state.

In fact, measurement would collapse the quantum state, so its amplitudes, which we are interested in, would not be the bitstring probabilities anymore, but a mere sample.

Again, this is handled by the circuitML class, explained next.

### 1.2.1. Example 1¶

As first example, we just translate the circuitBuilder example, so that will now be any possible parameter vector params, as shown in the figure.

def make_circuit(bdr, x, params):

# We pipeline some of the previous instructions
# Note that theta has become params
bdr.allin(params[:3])
bdr.cz(0,1).input([0,1], params[[3,4]])
bdr.cz(1,2).input([1,2], params[[5,6]])


Note that this circuit is independent from the input vector x, as it is not used, but it is still required in the function signature. We can obtain the same circuit as in previous example by calling

make_circuit(qkBuilder(3), [], theta).measure_all().circuit()


### 1.2.2. Example 2¶

For the second example, we define a circuit on two qubits, for four-dimensional data. The circuit has eight parameters and its structure is repetitive, which we can leverage with the functional nature of make_circuit and the builder. The figure represents the desired circuit, where represents the input vector x; the red rectangles highlight the repeated block.

# We define a block function, as the structure is repetitive
def block(bdr, x, p):
bdr.allin(x[[0,1]])
bdr.cz(0,1).allin(p[[0,1]])

bdr.cz(0,1).allin(x[[2,3]])
bdr.cz(0,1).allin(p[[2,3]])

def make_circuit(bdr, x, params):
# The fist block uses all x, but
# only the first 4 elements of params
block(bdr, x, params[:4])

bdr.cz(0,1)

# The block repeats with the other parameters
block(bdr, x, params[4:])

return bdr


## 1.3. The circuitML class¶

The circuitML class provides the interface to run multiple parametric circuits with different inputs and model parameters. We report its documentation:

This is an abstract class, so we cannot directly instaciate it, but it fixes the interface. As we see, the constructor requires four arguments: make_circuit, nbqbits, nbparams, and cbuilder. The first argument, make_circuit, corresponds to the function from previous section, which circuitML will access as if it was an internal method; thus the documentation provides the signature. The latter, cbuilder, is the class (and not an instance) of the circuit builder to be used; as each child class knows its own backend implementation, this argument is provided by default.

The main method of this class is run, which takes as input a design matrix X and a parameter vector params and runs each corresponding circuit.

People in quantum computing know that, given a circuit, we transform a quantum state into another. Nonetheless, the final quantum state is not accessible in an actual quantum computer; we measure each qubit, which reads either 0 or 1. The output of each mesurement is thus a binary sequence of length nbqbits, a bitstring, which follows a probability distribution given by the quantum state. For each run, the quantum computer executes the circuit many times, in what we call shots. The run output is thus a sequence of nbshots bitstrings, which gives a statistic of the underlying distribution.

Therefore, the output of a run call will be an array. Each row corresponds to the run of the circuit given by the input sample. Each row entry – i.e. each column – gives the number of times the corresponding bitstring appeared in the shots, i.e. its counts.

That said, with enough computing power, it is possible to simulate quantum computers with a small number of qubits and retrieve the actual bitstring distribution. In that case, we can exactly compute the bitstring distribution. When possible, this corresponds to calling run with nbshots=None, which is also the default value.

### 1.3.1. Example¶

We define a simple circuit, then, we run it for two different input points. The 2-qubit circuit in figure takes a bidimensional input and two parameters, which will be fixed.

We use the manyq implementation, which relies on a simulator and is explained in detail in The manyq simulator. We need to define the make_circuit and then we can instantiate the circuitML.

# make_circuit can have any name
def simple_circuit(bdr, x, params):
return bdr.allin(x).cz(0,1).allin(params)

# We instantiate the circuitML

circuit = mqCircuitML(simple_circuit, nbqbits=2, nbparams=2)


Now that we have a circuit, we can run it. As we are using a simulator, we have acces to the probability distribution, from which we get counts realizations.

# We create an input matrix and a param.s vector
import numpy as np

X = np.array([[-np.pi/4, np.pi/4],
[np.pi/4, -np.pi/4]])
params = np.array([np.pi/2, -np.pi/2])

# We retrieve the probabilities and a realization of the counts
probs = circuit.run(X, params)
counts = circuit.run(X, params, nbshots=100)


We show in the figures the probability distribution for the two samples we considered and a count realization over 100 samples that we obtain by specifying nbshots=100.

Bitstring probabilities

Bitstring counts over 100 shots