Parametric Programs

Qadence base Parameter type is a subtype of sympy.Symbol. There are three kinds of parameter subtypes used:

• Fixed Parameter: A constant with a fixed, non-trainable value (e.g. \(\dfrac{\pi}{2}\)).
• Variational Parameter: A trainable parameter which can be be optimized.
• Feature Parameter: A non-trainable parameter which can be used to encode classical data into a quantum state.

Fixed Parameters

To pass a fixed parameter to a gate (or any parametrizable block), one can simply use either Python numeric types or wrapped in a torch.Tensor.

from torch import pi
from qadence import RX, run

# Let's use a torch type.
block = RX(0, pi)
wf = run(block)

# Let's pass a simple float.
block = RX(0, 1.)
wf = run(block)

wf = tensor([[6.1232e-17+0.j, 0.0000e+00-1.j]])
wf = tensor([[0.8776+0.0000j, 0.0000-0.4794j]])

Variational Parameters

To parametrize a block by an angle theta, either a Python string or an instance of VariationalParameter can be passed instead of a numeric type to the gate constructor:

from qadence import RX, run, VariationalParameter

block = RX(0, "theta")
# This is equivalent to:
block = RX(0, VariationalParameter("theta"))

wf = run(block)

wf = tensor([[0.9575+0.0000j, 0.0000-0.2883j]])

In the first case in the above example, theta is automatically inferred as a VariationalParameter (i.e. trainable). It is initialized to a random value for the purposes of execution. In the context of a QuantumModel, there is no need to pass a value for theta to the run method since it is stored within the underlying model parameter dictionary.

Feature Parameters

FeatureParameter types (i.e. inputs), always need to be provided with a value or a batch of values as a dictionary:

from torch import tensor
from qadence import RX, run, FeatureParameter

block = RX(0, FeatureParameter("phi"))

wf = run(block, values={"phi": tensor([1., 2.])})

wf = tensor([[0.8776+0.0000j, 0.0000-0.4794j],
        [0.5403+0.0000j, 0.0000-0.8415j]])

Now, run returns a batch of states, one for every provided angle which coincides with the value of the particular FeatureParameter.

Multiparameter Expressions

However, an angle can itself be an expression Parameter types of any kind. As such, any sympy expression expr: sympy.Basic consisting of a combination of free symbols (i.e. sympy types) and Qadence Parameter can be passed to a block, including trigonometric functions.

from torch import tensor
from qadence import RX, Parameter, run, FeatureParameter
from sympy import sin

theta, phi = Parameter("theta"), FeatureParameter("phi")
block = RX(0, sin(theta+phi))

# Remember, to run the block, only FeatureParameter values have to be provided:
values = {"phi": tensor([1.0, 2.0])}
wf = run(block, values=values)

wf = tensor([[0.8818+0.0000j, 0.0000-0.4716j],
        [0.9827+0.0000j, 0.0000-0.1854j]])

Parameters Redundancy

Parameters are uniquely defined by their name and redundancy is allowed in composite blocks to assign the same value to different blocks.

import torch
from qadence import RX, RY, run, chain, kron

block = chain(
    kron(RX(0, "phi"), RY(1, "theta")),
    kron(RX(0, "phi"), RY(1, "theta")),
)

wf = run(block)  # Same random initialization for all instances of phi and theta.

wf = tensor([[0.8666+0.0000j, 0.4489+0.0000j, 0.0000-0.1937j, 0.0000-0.1003j]])

Parametrized Circuits

Now, let's have a look at the construction of a variational ansatz which composes FeatureParameter and VariationalParameter types:

import sympy
from qadence import RX, RY, RZ, CNOT, Z, run, chain, kron, FeatureParameter, VariationalParameter

phi = FeatureParameter("phi")
theta = VariationalParameter("theta")

block = chain(
    kron(
        RX(0, phi/theta),
        RY(1, theta*2),
        RZ(2, sympy.cos(phi)),
    ),
    kron(
        RX(0, phi),
        RY(1, theta),
        RZ(2, phi),
    ),
    kron(
        RX(0, phi),
        RY(1, theta),
        RZ(2, phi),
    ),
    kron(
        RX(0, phi + theta),
        RY(1, theta**2),
        RZ(2, sympy.cos(phi)),
    ),
    chain(CNOT(0,1), CNOT(1,2))
)
block.tag = "Rotations"

obs = 2*kron(*map(Z, range(3)))
block = chain(block, obs)

Please note the different colors for the parametrization with different types. The default palette assigns light blue for VariationalParameter, light green for FeatureParameter and shaded red for observables.

Parametrized QuantumModels

As a quick reminder: FeatureParameter are used for data input and data encoding into a quantum state. VariationalParameter are trainable parameters in a variational ansatz. When used within a QuantumModel, an abstract quantum circuit is made differentiable with respect to both variational and feature parameters which are uniquely identified by their name.

from qadence import FeatureParameter, Parameter, VariationalParameter

# Feature parameters are non-trainable parameters.
# Their primary use is input data encoding.
fp = FeatureParameter("x")
assert fp == Parameter("x", trainable=False)

# Variational parameters are trainable parameters.
# Their primary use is for optimization.
vp = VariationalParameter("y")
assert vp == Parameter("y", trainable=True)

Let's construct a parametric quantum circuit.

from qadence import QuantumCircuit, RX, RY, chain, kron

theta = VariationalParameter("theta")
phi = FeatureParameter("phi")

block = chain(
    kron(RX(0, theta), RY(1, theta)),
    kron(RX(0, phi), RY(1, phi)),
)

circuit = QuantumCircuit(2, block)
unique_params = circuit.unique_parameters

unique_params = [theta, phi]

In the circuit above, four parameters are defined but only two unique names. Therefore, there will be only one variational parameter to be optimized.

The QuantumModel class also provides convenience methods to manipulate parameters.

from qadence import QuantumModel, BackendName, DiffMode

model = QuantumModel(circuit, backend=BackendName.PYQTORCH, diff_mode=DiffMode.AD)
num_vparams = model.num_vparams
vparams_values = model.vparams

num_vparams = 1
vparams_values = OrderedDict([('theta', tensor([0.5877]))])

Only provide feature parameter values to the quantum model

In order to run the variational circuit only feature parameter values have to be provided. Variational parameters are stored in the model itself. If multiple feature parameters are present, values must be provided in batches of same length.

import torch

values = {"phi": torch.rand(3)} # theta does not appear here
wf = model.run(values)

wf = tensor([[0.5120+0.0000j, 0.4999+0.0000j, 0.0000-0.4999j, 0.0000-0.4880j],
        [0.5607+0.0000j, 0.4963+0.0000j, 0.0000-0.4963j, 0.0000-0.4393j],
        [0.6286+0.0000j, 0.4832+0.0000j, 0.0000-0.4832j, 0.0000-0.3714j]],
       grad_fn=<TBackward0>)

Standard constructors

The unique parameter identification is relevant when using built-in Qadence block constructors in the qadence.constructors module such as feature maps and hardware efficient ansatze (HEA).

from qadence import QuantumCircuit, hea

n_qubits = 4
depth = 2

hea1 = hea(n_qubits=n_qubits, depth=depth)
circuit = QuantumCircuit(n_qubits, hea1)
num_unique_parameters = circuit.num_unique_parameters

Unique parameters with a single HEA: 24

A new circuit can be created by adding another identical HEA. As expected, the number of unique parameters is the same.

hea2 = hea(n_qubits=n_qubits, depth=depth)

circuit = QuantumCircuit(n_qubits, hea1, hea2)
num_unique_params_two_heas = circuit.num_unique_parameters

Unique parameters with two stacked HEAs: 24

Avoid non-unique names by prefixing

A parameter prefix for each HEA can be passed as follows:

hea1 = hea(n_qubits=n_qubits, depth=depth, param_prefix="p1")
hea2 = hea(n_qubits=n_qubits, depth=depth, param_prefix="p2")

circuit = QuantumCircuit(n_qubits, hea1, hea2)
n_params_two_heas = circuit.num_unique_parameters

Unique parameters with two stacked HEAs: 48

The hea function will be further explored in the QML Constructors tutorial.

Parametric observables

In Qadence, one can define quantum observables with classical optimizable parameters to improve the convergence of QML calculations. This is particularly useful for differentiable quantum circuits.

from qadence import VariationalParameter, Z, add, tag

s = VariationalParameter("s")
observable = add(s * Z(i) for i in range(n_qubits))

Now, a quantum model can be created with the parametric observable. The observable variational parameters are included among the model ones.

from qadence import QuantumModel, QuantumCircuit

circuit = QuantumCircuit(n_qubits, hea(n_qubits, depth))
model = QuantumModel(circuit, observable=observable)

Variational parameters = OrderedDict([('s', tensor([0.3775])), ('theta_0', tensor([0.4898])), ('theta_1', tensor([0.2495])), ('theta_10', tensor([0.7195])), ('theta_11', tensor([0.9956])), ('theta_12', tensor([0.9429])), ('theta_13', tensor([0.2319])), ('theta_14', tensor([0.9940])), ('theta_15', tensor([0.9964])), ('theta_16', tensor([0.6051])), ('theta_17', tensor([0.7252])), ('theta_18', tensor([0.3309])), ('theta_19', tensor([0.5177])), ('theta_2', tensor([0.2404])), ('theta_20', tensor([0.1400])), ('theta_21', tensor([0.8528])), ('theta_22', tensor([0.3188])), ('theta_23', tensor([0.2410])), ('theta_3', tensor([0.8505])), ('theta_4', tensor([0.2859])), ('theta_5', tensor([0.2913])), ('theta_6', tensor([0.4297])), ('theta_7', tensor([0.0077])), ('theta_8', tensor([0.4412])), ('theta_9', tensor([0.8012]))])

One optimization step (forward and backward pass) can be performed using built-in torch functionalities. Variational parameters can be checked to have been updated accordingly:

import torch

mse_loss = torch.nn.MSELoss()
optimizer = torch.optim.Adam(model.parameters())

# Compute forward & backward pass
optimizer.zero_grad()
loss = mse_loss(model.expectation({}), torch.zeros(1))
loss.backward()

# Update the parameters and check the parameters.
optimizer.step()

Variational parameters = OrderedDict([('s', tensor([0.3765])), ('theta_0', tensor([0.4908])), ('theta_1', tensor([0.2485])), ('theta_10', tensor([0.7205])), ('theta_11', tensor([0.9966])), ('theta_12', tensor([0.9439])), ('theta_13', tensor([0.2329])), ('theta_14', tensor([0.9930])), ('theta_15', tensor([0.9974])), ('theta_16', tensor([0.6061])), ('theta_17', tensor([0.7262])), ('theta_18', tensor([0.3319])), ('theta_19', tensor([0.5167])), ('theta_2', tensor([0.2414])), ('theta_20', tensor([0.1410])), ('theta_21', tensor([0.8538])), ('theta_22', tensor([0.3178])), ('theta_23', tensor([0.2420])), ('theta_3', tensor([0.8515])), ('theta_4', tensor([0.2869])), ('theta_5', tensor([0.2923])), ('theta_6', tensor([0.4307])), ('theta_7', tensor([0.0067])), ('theta_8', tensor([0.4422])), ('theta_9', tensor([0.8002]))])

Non-unitary circuits

Qadence allows to compose with non-unitary blocks. Here is an example of a non-unitary block as a sum of Pauli operators with complex coefficients.

Currently, only the PyQTorch backend fully supports execution with non-unitary circuits.

from qadence import QuantumModel, QuantumCircuit, Z, X
c1 = 2.0
c2 = 2.0 + 2.0j

block = c1 * Z(0) + c2 * X(1) + c1 * c2 * (Z(2) + X(3))
circuit = QuantumCircuit(4, block)

model = QuantumModel(circuit)  # BackendName.PYQTORCH and DiffMode.AD by default.

wf = tensor([[6.+4.j, 4.+4.j, 0.+0.j, 0.+0.j, 2.+2.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j,
         0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]])