qml.devices.default_qubit.DefaultQubit¶

class DefaultQubit(wires=None, shots=None, seed='global', max_workers=None)[source]¶

Bases: pennylane.devices.device_api.Device

A PennyLane device written in Python and capable of backpropagation derivatives.

Parameters

wires (int, Iterable[Number, str]) – Number of wires present on the device, or iterable that contains unique labels for the wires as numbers (i.e., [-1, 0, 2]) or strings (['ancilla', 'q1', 'q2']). Default None if not specified.
shots (int, Sequence[int], Sequence[Union[int, Sequence[int]]]) – The default number of shots to use in executions involving this device.
seed (Union[str, None, int, array_like[int], SeedSequence, BitGenerator, Generator, jax.random.PRNGKey]) – A seed-like parameter matching that of seed for numpy.random.default_rng, or a request to seed from numpy’s global random number generator. The default, seed="global" pulls a seed from NumPy’s global generator. seed=None will pull a seed from the OS entropy. If a jax.random.PRNGKey is passed as the seed, a JAX-specific sampling function using jax.random.choice and the PRNGKey will be used for sampling rather than numpy.random.default_rng.
max_workers (int) – A ProcessPoolExecutor executes tapes asynchronously using a pool of at most max_workers processes. If max_workers is None, only the current process executes tapes. If you experience any issue, say using JAX, TensorFlow, Torch, try setting max_workers to None.

Example:

n_layers = 5
n_wires = 10
num_qscripts = 5

shape = qml.StronglyEntanglingLayers.shape(n_layers=n_layers, n_wires=n_wires)
rng = qml.numpy.random.default_rng(seed=42)

qscripts = []
for i in range(num_qscripts):
    params = rng.random(shape)
    op = qml.StronglyEntanglingLayers(params, wires=range(n_wires))
    qs = qml.tape.QuantumScript([op], [qml.expval(qml.Z(0))])
    qscripts.append(qs)

>>> dev = DefaultQubit()
>>> program, execution_config = dev.preprocess()
>>> new_batch, post_processing_fn = program(qscripts)
>>> results = dev.execute(new_batch, execution_config=execution_config)
>>> post_processing_fn(results)
[-0.0006888975950537501,
0.025576307134457577,
-0.0038567269892757494,
0.1339705146860149,
-0.03780669772690448]

This device currently supports backpropagation derivatives:

>>> from pennylane.devices import ExecutionConfig
>>> dev.supports_derivatives(ExecutionConfig(gradient_method="backprop"))
True

For example, we can use jax to jit computing the derivative:

import jax

@jax.jit
def f(x):
    qs = qml.tape.QuantumScript([qml.RX(x, 0)], [qml.expval(qml.Z(0))])
    program, execution_config = dev.preprocess()
    new_batch, post_processing_fn = program([qs])
    results = dev.execute(new_batch, execution_config=execution_config)
    return post_processing_fn(results)

>>> f(jax.numpy.array(1.2))
DeviceArray(0.36235774, dtype=float32)
>>> jax.grad(f)(jax.numpy.array(1.2))
DeviceArray(-0.93203914, dtype=float32, weak_type=True)

Tracking

DefaultQubit tracks:

executions: the number of unique circuits that would be required on quantum hardware
shots: the number of shots
resources: the Resources for the executed circuit.
simulations: the number of simulations performed. One simulation can cover multiple QPU executions, such as for non-commuting measurements and batched parameters.
batches: The number of times execute() is called.
results: The results of each call of execute()
derivative_batches: How many times compute_derivatives() is called.
execute_and_derivative_batches: How many times execute_and_compute_derivatives() is called
vjp_batches: How many times compute_vjp() is called
execute_and_vjp_batches: How many times execute_and_compute_vjp() is called
jvp_batches: How many times compute_jvp() is called
execute_and_jvp_batches: How many times execute_and_compute_jvp() is called
derivatives: How many circuits are submitted to compute_derivatives() or execute_and_compute_derivatives().
vjps: How many circuits are submitted to compute_vjp() or execute_and_compute_vjp()
jvps: How many circuits are submitted to compute_jvp() or execute_and_compute_jvp()

Accelerate calculations with multiprocessing

Suppose one has a processor with 5 cores or more, these scripts can be executed in parallel as follows

>>> dev = DefaultQubit(max_workers=5)
>>> program, execution_config = dev.preprocess()
>>> new_batch, post_processing_fn = program(qscripts)
>>> results = dev.execute(new_batch, execution_config=execution_config)
>>> post_processing_fn(results)

If you monitor your CPU usage, you should see 5 new Python processes pop up to crunch through those QuantumScript’s. Beware not oversubscribing your machine. This may happen if a single device already uses many cores, if NumPy uses a multi- threaded BLAS library like MKL or OpenBLAS for example. The number of threads per process times the number of processes should not exceed the number of cores on your machine. You can control the number of threads per process with the environment variables:

OMP_NUM_THREADS
MKL_NUM_THREADS
OPENBLAS_NUM_THREADS

where the last two are specific to the MKL and OpenBLAS libraries specifically.

Warning

Multiprocessing may fail depending on your platform and environment (Python shell, script with a protected entry point, Jupyter notebook, etc.) This may be solved changing the so-called start method. The supported start methods are the following:

Windows (win32): spawn (default).
macOS (darwin): spawn (default), fork, forkserver.
Linux (unix): spawn, fork (default), forkserver.

which can be changed with multiprocessing.set_start_method(). For example, if multiprocessing fails on macOS in your Jupyter notebook environment, try restarting the session and adding the following at the beginning of the file:

import multiprocessing
multiprocessing.set_start_method("fork")

Additional information can be found in the multiprocessing doc.

Attributes

`capabilities`	A `DeviceCapabilities` object describing the capabilities of the backend device.
`config_filepath`	A device can use a toml file to specify the capabilities of the backend device.
`name`	The name of the device.
`shots`	Default shots for execution workflows containing this device.
`tracker`	A `Tracker` that can store information about device executions, shots, batches, intermediate results, or any additional device dependent information.
`wires`	The device wires.

capabilities = None¶: A DeviceCapabilities object describing the capabilities of the backend device.

config_filepath = None¶: A device can use a toml file to specify the capabilities of the backend device. If this is provided, the file will be loaded into a DeviceCapabilities object assigned to the capabilities attribute.

name¶: The name of the device.

shots¶

Default shots for execution workflows containing this device.

Note that the device itself should always pull shots from the provided QuantumTape and its shots, not from this property. This property is used to provide a default at the start of a workflow.

tracker = <pennylane.tracker.Tracker object>¶

A Tracker that can store information about device executions, shots, batches, intermediate results, or any additional device dependent information.

A plugin developer can store information in the tracker by:

# querying if the tracker is active
if self.tracker.active:

    # store any keyword: value pairs of information
    self.tracker.update(executions=1, shots=self._shots, results=results)

    # Calling a user-provided callback function
    self.tracker.record()

wires¶

The device wires.

Note that wires are optional, and the default value of None means any wires can be used. If a device has wires defined, they will only be used for certain features. This includes:

Validation of tapes being executed on the device
Defining the wires used when evaluating a state() measurement

Methods

`compute_derivatives`(circuits[, execution_config])	Calculate the jacobian of either a single or a batch of circuits on the device.
`compute_jvp`(circuits, tangents[, ...])	The jacobian vector product used in forward mode calculation of derivatives.
`compute_vjp`(circuits, cotangents[, ...])	The vector jacobian product used in reverse-mode differentiation.
`eval_jaxpr`(jaxpr, consts, *args)	An experimental method for natively evaluating PLXPR.
`execute`(circuits[, execution_config])	Execute a circuit or a batch of circuits and turn it into results.
`execute_and_compute_derivatives`(circuits[, ...])	Compute the results and jacobians of circuits at the same time.
`execute_and_compute_jvp`(circuits, tangents)	Execute a batch of circuits and compute their jacobian vector products.
`execute_and_compute_vjp`(circuits, cotangents)	Calculate both the results and the vector jacobian product used in reverse-mode differentiation.
`get_prng_keys`([num])	Get `num` new keys with `jax.random.split`.
`preprocess`([execution_config])	This function defines the device transform program to be applied and an updated device configuration.
`preprocess_transforms`([execution_config])	Returns the transform program to preprocess a circuit for execution.
`reset_prng_key`()	Reset the RNG key to its initial value.
`setup_execution_config`([config, circuit])	Sets up an `ExecutionConfig` that configures the execution behaviour.
`supports_derivatives`([execution_config, circuit])	Check whether or not derivatives are available for a given configuration and circuit.
`supports_jvp`([execution_config, circuit])	Whether or not this device defines a custom jacobian vector product.
`supports_vjp`([execution_config, circuit])	Whether or not this device defines a custom vector jacobian product.

compute_derivatives(circuits, execution_config=ExecutionConfig(grad_on_execution=None, use_device_gradient=None, use_device_jacobian_product=None, gradient_method=None, gradient_keyword_arguments={}, device_options={}, interface=<Interface.NUMPY: 'numpy'>, derivative_order=1, mcm_config=MCMConfig(mcm_method=None, postselect_mode=None)))[source]¶

Calculate the jacobian of either a single or a batch of circuits on the device.

Parameters

circuits (Union[QuantumTape, Sequence[QuantumTape]]) – the circuits to calculate derivatives for
execution_config (ExecutionConfig) – a datastructure with all additional information required for execution

Returns

The jacobian for each trainable parameter

Return type

Tuple

Execution Config:

The execution config has gradient_method and order property that describes the order of differentiation requested. If the requested method or order of gradient is not provided, the device should raise a NotImplementedError. The supports_derivatives() method can pre-validate supported orders and gradient methods.

Return Shape:

If a batch of quantum scripts is provided, this method should return a tuple with each entry being the gradient of each individual quantum script. If the batch is of length 1, then the return tuple should still be of length 1, not squeezed.

compute_jvp(circuits, tangents, execution_config=ExecutionConfig(grad_on_execution=None, use_device_gradient=None, use_device_jacobian_product=None, gradient_method=None, gradient_keyword_arguments={}, device_options={}, interface=<Interface.NUMPY: 'numpy'>, derivative_order=1, mcm_config=MCMConfig(mcm_method=None, postselect_mode=None)))[source]¶

The jacobian vector product used in forward mode calculation of derivatives.

Parameters

circuits (Union[QuantumTape, Sequence[QuantumTape]]) – the circuit or batch of circuits
tangents (tensor-like) – Gradient vector for input parameters.
execution_config (ExecutionConfig) – a datastructure with all additional information required for execution

Returns

A numeric result of computing the jacobian vector product

Return type

Tuple

Definition of jvp:

If we have a function with jacobian:

$\vec{y} = f(\vec{x}) \qquad J_{i,j} = \frac{\partial y_i}{\partial x_j}$

The Jacobian vector product is the inner product with the derivatives of $x$ , yielding only the derivatives of the output $y$ :

$\text{d}y_i = \Sigma_{j} J_{i,j} \text{d}x_j$

Shape of tangents:

The tangents tuple should be the same length as circuit.get_parameters() and have a single number per parameter. If a number is zero, then the gradient with respect to that parameter does not need to be computed.

compute_vjp(circuits, cotangents, execution_config=ExecutionConfig(grad_on_execution=None, use_device_gradient=None, use_device_jacobian_product=None, gradient_method=None, gradient_keyword_arguments={}, device_options={}, interface=<Interface.NUMPY: 'numpy'>, derivative_order=1, mcm_config=MCMConfig(mcm_method=None, postselect_mode=None)))[source]¶

The vector jacobian product used in reverse-mode differentiation. DefaultQubit uses the adjoint differentiation method to compute the VJP.

Parameters

circuits (Union[QuantumTape, Sequence[QuantumTape]]) – the circuit or batch of circuits
cotangents (Tuple[Number, Tuple[Number]]) – Gradient-output vector. Must have shape matching the output shape of the corresponding circuit. If the circuit has a single output, cotangents may be a single number, not an iterable of numbers.
execution_config (ExecutionConfig) – a datastructure with all additional information required for execution

Returns

A numeric result of computing the vector jacobian product

Return type

tensor-like

Definition of vjp:

If we have a function with jacobian:

$\vec{y} = f(\vec{x}) \qquad J_{i,j} = \frac{\partial y_i}{\partial x_j}$

The vector jacobian product is the inner product of the derivatives of the output y with the Jacobian matrix. The derivatives of the output vector are sometimes called the cotangents.

$\text{d}x_i = \Sigma_{i} \text{d}y_i J_{i,j}$

Shape of cotangents:

The value provided to cotangents should match the output of execute(). For computing the full Jacobian, the cotangents can be batched to vectorize the computation. In this case, the cotangents can have the following shapes. batch_size below refers to the number of entries in the Jacobian:

For a state measurement, the cotangents must have shape (batch_size, 2 ** n_wires)
For n expectation values, the cotangents must have shape (n, batch_size). If n = 1, then the shape must be (batch_size,).

eval_jaxpr(jaxpr, consts, *args)[source]¶

An experimental method for natively evaluating PLXPR. See the capture module for more details.

Parameters

jaxpr (jax.core.Jaxpr) – Pennylane variant jaxpr containing quantum operations and measurements
consts (list[TensorLike]) – the closure variables consts corresponding to the jaxpr
*args (TensorLike) – the variables to use with the jaxpr’.

Returns

the result of evaluating the jaxpr with the given parameters.

Return type

list[TensorLike]

execute(circuits, execution_config=ExecutionConfig(grad_on_execution=None, use_device_gradient=None, use_device_jacobian_product=None, gradient_method=None, gradient_keyword_arguments={}, device_options={}, interface=<Interface.NUMPY: 'numpy'>, derivative_order=1, mcm_config=MCMConfig(mcm_method=None, postselect_mode=None)))[source]¶

Execute a circuit or a batch of circuits and turn it into results.

Parameters

circuits (Union[QuantumTape, Sequence[QuantumTape]]) – the quantum circuits to be executed
execution_config (ExecutionConfig) – a datastructure with additional information required for execution

Returns

A numeric result of the computation.

Return type

TensorLike, tuple[TensorLike], tuple[tuple[TensorLike]]

Interface parameters:

The provided circuits may contain interface specific data-types like torch.Tensor or jax.Array when gradient_method of "backprop" is requested. If the gradient method is not backpropagation, then only vanilla numpy parameters or builtins will be present in the circuits.

Return Shape

See Return Type Specification for more detailed information.

The result for each QuantumTape must match the shape specified by shape.

The level of priority for dimensions from outer dimension to inner dimension is:

Quantum Script in batch
Shot choice in a shot vector
Measurement in the quantum script
Parameter broadcasting
Measurement shape for array-valued measurements like probabilities

For a batch of quantum scripts with multiple measurements, a shot vector, and parameter broadcasting:

result[0]: the results for the first script
result[0][0]: the first shot number in the shot vector
result[0][0][0]: the first measurement in the quantum script
result[0][0][0][0]: the first parameter broadcasting choice
result[0][0][0][0][0]: the first value for an array-valued measurement

With the exception of quantum script batches, dimensions with only a single component should be eliminated.

For example:

With a single script and a single measurement process, execute should return just the measurement value in a numpy array. shape currently accepts a device, as historically devices stored shot information. In the future, this method will accept an ExecutionConfig instead.

>>> tape = qml.tape.QuantumTape(measurements=qml.expval(qml.Z(0))])
>>> tape.shape(dev)
()
>>> dev.execute(tape)
array(1.0)

If execute recieves a batch of scripts, then it should return a tuple of results:

>>> dev.execute([tape, tape])
(array(1.0), array(1.0))
>>> dev.execute([tape])
(array(1.0),)

If the script has multiple measurements, then the device should return a tuple of measurements.

>>> tape = qml.tape.QuantumTape(measurements=[qml.expval(qml.Z(0)), qml.probs(wires=(0,1))])
>>> tape.shape(dev)
((), (4,))
>>> dev.execute(tape)
(array(1.0), array([1., 0., 0., 0.]))

execute_and_compute_derivatives(circuits, execution_config=ExecutionConfig(grad_on_execution=None, use_device_gradient=None, use_device_jacobian_product=None, gradient_method=None, gradient_keyword_arguments={}, device_options={}, interface=<Interface.NUMPY: 'numpy'>, derivative_order=1, mcm_config=MCMConfig(mcm_method=None, postselect_mode=None)))[source]¶

Compute the results and jacobians of circuits at the same time.

Parameters

circuits (Union[QuantumTape, Sequence[QuantumTape]]) – the circuits or batch of circuits
execution_config (ExecutionConfig) – a datastructure with all additional information required for execution

Returns

A numeric result of the computation and the gradient.

Return type

tuple

See execute() and compute_derivatives() for more information about return shapes and behaviour. If compute_derivatives() is defined, this method should be as well.

This method can be used when the result and execution need to be computed at the same time, such as during a forward mode calculation of gradients. For certain gradient methods, such as adjoint diff gradients, calculating the result and gradient at the same can save computational work.

execute_and_compute_jvp(circuits, tangents, execution_config=ExecutionConfig(grad_on_execution=None, use_device_gradient=None, use_device_jacobian_product=None, gradient_method=None, gradient_keyword_arguments={}, device_options={}, interface=<Interface.NUMPY: 'numpy'>, derivative_order=1, mcm_config=MCMConfig(mcm_method=None, postselect_mode=None)))[source]¶

Execute a batch of circuits and compute their jacobian vector products.

Parameters

circuits (Union[QuantumTape, Sequence[QuantumTape]]) – circuit or batch of circuits
tangents (tensor-like) – Gradient vector for input parameters.
execution_config (ExecutionConfig) – a datastructure with all additional information required for execution

Returns

A numeric result of execution and of computing the jacobian vector product

Return type

Tuple, Tuple

See also

execute() and compute_jvp()

execute_and_compute_vjp(circuits, cotangents, execution_config=ExecutionConfig(grad_on_execution=None, use_device_gradient=None, use_device_jacobian_product=None, gradient_method=None, gradient_keyword_arguments={}, device_options={}, interface=<Interface.NUMPY: 'numpy'>, derivative_order=1, mcm_config=MCMConfig(mcm_method=None, postselect_mode=None)))[source]¶

Calculate both the results and the vector jacobian product used in reverse-mode differentiation.

Parameters

circuits (Union[QuantumTape, Sequence[QuantumTape]]) – the circuit or batch of circuits to be executed
cotangents (Tuple[Number, Tuple[Number]]) – Gradient-output vector. Must have shape matching the output shape of the corresponding circuit. If the circuit has a single output, cotangents may be a single number, not an iterable of numbers.
execution_config (ExecutionConfig) – a datastructure with all additional information required for execution

Returns

the result of executing the scripts and the numeric result of computing the vector jacobian product

Return type

Tuple, Tuple

See also

execute() and compute_vjp()

get_prng_keys(num=1)[source]¶

Get num new keys with jax.random.split.

A user may provide a jax.random.PRNGKey as a random seed. It will be used by the device when executing circuits with finite shots. The JAX RNG is notably different than the NumPy RNG as highlighted in the JAX documentation. JAX does not keep track of a global seed or key, but needs one anytime it draws from a random number distribution. Generating randomness therefore requires changing the key every time, which is done by “splitting” the key. For example, when executing n circuits, the PRNGkey is split n times into 2 new keys using jax.random.split to simulate a non-deterministic behaviour. The device seed is modified in-place using the first key, and the second key is fed to the circuit, and hence can be discarded after returning the results. This same key may be split further down the stack if necessary so that no one key is ever reused.

preprocess(execution_config=ExecutionConfig(grad_on_execution=None, use_device_gradient=None, use_device_jacobian_product=None, gradient_method=None, gradient_keyword_arguments={}, device_options={}, interface=<Interface.NUMPY: 'numpy'>, derivative_order=1, mcm_config=MCMConfig(mcm_method=None, postselect_mode=None)))[source]¶

This function defines the device transform program to be applied and an updated device configuration.

Parameters: execution_config (Union[ExecutionConfig, Sequence[ExecutionConfig]]) – A data structure describing the parameters needed to fully describe the execution.
Returns: A transform program that when called returns QuantumTapes that the device can natively execute as well as a postprocessing function to be called after execution, and a configuration with unset specifications filled in.
Return type: TransformProgram, ExecutionConfig

This device supports any qubit operations that provide a matrix

preprocess_transforms(execution_config=None)¶

Returns the transform program to preprocess a circuit for execution.

Parameters: execution_config (ExecutionConfig) – The execution configuration object
Returns: A transform program that is called before execution
Return type: TransformProgram

The transform program is composed of a list of individual transforms, which may include:

Decomposition of operations and measurements to what is supported by the device.
Splitting a circuit with measurements of non-commuting observables or Hamiltonians into multiple executions.
Splitting a circuit with batched parameters into multiple executions.
Validation of wires, measurements, and observables.
Gradient specific preprocessing, such as making sure trainable operators have generators.

Example

All transforms that are part of the preprocessing transform program need to respect the transform contract defined in pennylane.transform().

from pennylane.tape import QuantumScriptBatch
from pennylane.typing import PostprocessingFn

@qml.transform
def my_preprocessing_transform(tape: qml.tape.QuantumScript) -> tuple[QuantumScriptBatch, PostprocessingFn]:
    # e.g. valid the measurements, expand the tape for the hardware execution, ...

    def blank_processing_fn(results):
        return results[0]

    return [tape], processing_fn

A transform program can hold an arbitrary number of individual transforms:

def preprocess(self, config):
    program = TransformProgram()
    program.add_transform(my_preprocessing_transform)
    return program

See also

transform() and TransformProgram

Post processing function and derivatives

Derivatives and Jacobian products will be bound to the machine learning library before the postprocessing function is called on the results. Therefore, the machine learning library will be responsible for combining and post-processing derivatives returned from the device.

from pennylane.interfaces.jax import execute as jax_boundary

def f(x):
    circuit = qml.tape.QuantumScript([qml.Rot(*x, wires=0)], [qml.expval(qml.Z(0))])
    config = ExecutionConfig(gradient_method="adjoint")
    config = dev.setup_execution_config(config)
    program = dev.preprocess_transforms(config)
    circuit_batch, postprocessing = program((circuit, ))

    def execute_fn(tapes):
        return dev.execute_and_compute_derivatives(tapes, config)

    results = jax_boundary(circuit_batch, dev, execute_fn, None, {})
    return postprocessing(results)

x = jax.numpy.array([1.0, 2.0, 3.0])
jax.grad(f)(x)

In the above code, the quantum derivatives are registered with jax in the jax_boundary function. Only then is the classical postprocessing called on the result object.

reset_prng_key()[source]¶: Reset the RNG key to its initial value.

setup_execution_config(config=None, circuit=None)¶

Sets up an ExecutionConfig that configures the execution behaviour.

The execution config stores information on how the device should perform the execution, as well as how PennyLane should interact with the device. See ExecutionConfig for all available options and what they mean.

An ExecutionConfig is constructed from arguments passed to the QNode, and this method allows the device to update the config object based on device-specific requirements or preferences. See Execution Config for more details.

Parameters

config (ExecutionConfig) – The initial ExecutionConfig object that describes the parameters needed to configure the execution behaviour.
circuit (QuantumScript) – The quantum circuit to customize the execution config for.

Returns

The updated ExecutionConfig object

Return type

ExecutionConfig

supports_derivatives(execution_config=None, circuit=None)[source]¶

Check whether or not derivatives are available for a given configuration and circuit.

DefaultQubit supports backpropagation derivatives with analytic results, as well as adjoint differentiation.

Parameters

execution_config (ExecutionConfig) – The configuration of the desired derivative calculation
circuit (QuantumTape) – An optional circuit to check derivatives support for.

Returns

Whether or not a derivative can be calculated provided the given information

Return type

Bool

supports_jvp(execution_config=None, circuit=None)[source]¶

Whether or not this device defines a custom jacobian vector product.

DefaultQubit supports backpropagation derivatives with analytic results, as well as adjoint differentiation.

Parameters

execution_config (ExecutionConfig) – The configuration of the desired derivative calculation
circuit (QuantumTape) – An optional circuit to check derivatives support for.

Returns

Whether or not a derivative can be calculated provided the given information

Return type

bool

supports_vjp(execution_config=None, circuit=None)[source]¶

Whether or not this device defines a custom vector jacobian product.

DefaultQubit supports backpropagation derivatives with analytic results, as well as adjoint differentiation.

Parameters

execution_config (ExecutionConfig) – A description of the hyperparameters for the desired computation.
circuit (None, QuantumTape) – A specific circuit to check differentation for.

Returns

Whether or not a derivative can be calculated provided the given information

Return type

bool

qml.devices.default_qubit.DefaultQubit¶

Tracking

Accelerate calculations with multiprocessing

Attributes

Methods

Return Shape

Post processing function and derivatives

Contents

Downloads

qml.devices.default_qubit.DefaultQubit¶

Tracking

Accelerate calculations with multiprocessing

Attributes

Methods

Return Shape

Post processing function and derivatives

Contents

Downloads

Related