Choosing a model

The following sections provide more information on the various models.


A NumpyModel uses the unitary and density matrix simulators in the low-level lambeq.backend, which convert quantum circuits into a tensor network. The resulting tensor network is efficiently contracted using opt_einsum.

Circuits containing only Bra, Ket and unitary gates are evaluated using a unitary simulator, while circuits containing Encode, Measure or Discard are evaluated using a density matrix simulator.


Note that the unitary simulator converts a circuit with n output qubits into a tensor of shape (2, ) * n, while the density matrix simulator converts a circuit with n output qubits and m output bits into a tensor of shape (2, ) * (2 * n + m).

In the common use case of using a stairs_reader or a TreeReader with discarding for binary classification, the process involves measuring (Measure) one of the “open” qubits, and discarding (Discard) the rest of them.

One advantage that the NumpyModel has over the TketModel is that it supports the just-in-time (jit) compilation provided by the library jax. This speeds up the model’s diagram evaluation by an order of magnitude. The NumpyModel with jit mode enabled can be instantiated with the following command:

from lambeq import NumpyModel

model = NumpyModel.from_diagrams(circuits, use_jit=True)


Using the NumpyModel with jit mode enabled is not recommended for large models, as it requires a large amount of memory to store the pre-compiled functions for each circuit.

To use the NumpyModel with jit mode, you need to install lambeq with the extra packages by running the following command:

pip install lambeq[extras]


To enable GPU support for jax, follow the installation instructions on the JAX GitHub repository.

NumpyModel should be used with the QuantumTrainer.

See also the following use cases:


PennyLaneModel uses PennyLane and PyTorch to allow classical-quantum machine learning experiments. With probabilities=False, PennyLaneModel performs a state vector simulation, while with probabilties=True it performs a probability simulation. The state vector and probability simulations correspond to unitary and density matrix simulations.

To run the model on real quantum hardware, probabilities=True must be used, so that the lambeq circuits are optimized using the parameter-shift rule to calculate the gradients.

PennyLaneModel can be used to optimize simulated circuits using exact backpropagation with PyTorch, which may give improved results over using NumpyModel with SPSAOptimizer. However, this optimization process is not possible on real quantum hardware, so for more realistic results the parameter-shift rule should be preferred.

To construct a hybrid model that passes the output of a circuit through a classical neural network, it is only necessary to subclass PennyLaneModel and modify the __init__() method to store the classical PyTorch parameters, and the forward() method to pass the result of get_diagram_output() to the neural network. For example:

import torch
from lambeq import PennyLaneModel

class MyCustomModel(PennyLaneModel):
   def __init__(self, **kwargs):
      super().__init__(**kwargs) = torch.nn.Linear(2, 2)

   def forward(self, input):
      preds = self.get_diagram_output(input)

This neural net can be real- or complex-valued, though this affects the non-linearities that can be used.

PennyLaneModel can be used with the PytorchTrainer, or a standard PyTorch training loop.

By using different backend configurations, PennyLaneModel can be used for several different use-cases, listed below:

Table 2 Backend configurations for different use cases.

Use case


Exact non shot-based simulation with state outputs

{'backend': 'default.qubit', 'probabilities'=False}

Exact non shot-based simulation with probability outputs

{'backend': 'default.qubit', 'probabilities'=True}

Noiseless shot-based simulation

{'backend': 'default.qubit', 'shots'=1000, 'probabilities'=True}

Noisy shot-based simulation on local hardware

{'backend': 'qiskit.aer', noise_model=my_noise_model, 'shots'=1000, 'probabilities'=True}, where my_noise_model is an AER NoiseModel.

Noisy shot-based simulation on cloud-based emulators

{'backend': 'qiskit.ibmq', 'device'='ibmq_qasm_simulator', 'shots'=1000, 'probabilities'=True}
{'backend': 'honeywell.hqs', device=('H1-1E' or 'H1-2E'), 'shots'=1000, 'probabilities'=True}

Evaluation of quantum circuits on a quantum computer

{'backend': 'qiskit.ibmq', 'device'='ibmq_hardware_device', 'shots'=1000, 'probabilities'=True}, where ibmq_hardware_device is one that you have access to via your IBMQ account.
{'backend': 'honeywell.hqs', device=('H1' or 'H1-1' or 'H1-2'), 'shots'=1000, 'probabilities'=True}

All of these backends are compatible with hybrid quantum-classical models. Note that using quantum hardware or cloud-based emulators are much slower than local simulations.

See also the following use cases:


PytorchModel is the right choice for classical experiments. Here, string diagrams are treated as tensor networks, where boxes represent tensors and edges define the specific tensor contractions. Tensor contractions are optimised by the python package opt_einsum.

To prepare the diagrams for the computation, we use a TensorAnsatz that converts a pregroup diagram into a tensor diagram. Subclasses of TensorAnsatz include the SpiderAnsatz and the MPSAnsatz, which reduce the size of large tensors by spliting them into chains of many smaller boxes. To prepare a tensor diagram for a sentence, for example:

from lambeq import AtomicType, BobcatParser, TensorAnsatz
from lambeq.backend.tensor import Dim

parser = BobcatParser()
pregroup_diagram = parser.sentence2diagram('This is a tensor network.')

ansatz = TensorAnsatz({AtomicType.NOUN: Dim(2), AtomicType.SENTENCE: Dim(4)})
tensor_diagram = ansatz(pregroup_diagram)

After preparing a list of tensor diagrams, we can initialise the model through:

from lambeq import PytorchModel

model = PytorchModel.from_diagrams(tensor_diagrams)

The PytorchModel is capable of combining tensor networks and neural network architectures. For example, it is possible to feed the output of a tensor diagram into a neural network, by subclassing and modifying the forward() method:

import torch
from lambeq import PytorchModel

class MyCustomModel(PytorchModel):
   def __init__(self):
      super().__init__() = torch.nn.Linear(2, 2)

   def forward(self, input):
      """define a custom forward pass here"""
      preds = self.get_diagram_output(input)  # performs tensor contraction

To simplify training, the PytorchModel can be used with the PytorchTrainer. A comprehensive tutorial can be found here.


The loss function and the accuracy metric in the tutorial are defined for two-dimensional binary labels: [[1,0], [0,1], ...]. If your data has a different structure, you must implement your custom loss function and evaluation metrics.

See also the following use cases:


TketModel uses pytket to retrieve shot-based results from a quantum computer, then uses the shot counts to build the resulting tensor.

The AerBackend can be used with TketModel to perform a noisy, architecture-aware simulation of an IBM machine. Other backends supported by pytket can also be used. To run an experiment on a real quantum computer, for example:

from lambeq import TketModel
from pytket.extensions.quantinuum import QuantinuumBackend

machine = 'H1-1E'
backend = QuantinuumBackend(device_name=machine)

backend_config = {
 'backend': backend,
 'compilation': backend.default_compilation_pass(2),
 'shots': 2048

model = TketModel.from_diagrams(all_circuits, backend_config=backend_config)


Note that you need user accounts and allocated resources to run experiments on real machines. However, IBM Quantum provides some limited resources for free.

For initial experiments we recommend using a NumpyModel, as it performs noiseless simulations and is orders of magnitude faster.

TketModel should be used with the QuantumTrainer.

See also the following use cases: