# qml.metric_tensor¶

metric_tensor(tape, argnum=None, approx=None, allow_nonunitary=True, aux_wire=None, device_wires=None)[source]

Returns a function that computes the metric tensor of a given QNode or quantum tape.

The metric tensor convention we employ here has the following form:

$\text{metric_tensor}_{i, j} = \text{Re}\left[ \langle \partial_i \psi(\bm{\theta}) | \partial_j \psi(\bm{\theta}) \rangle - \langle \partial_i \psi(\bm{\theta}) | \psi(\bm{\theta}) \rangle \langle \psi(\bm{\theta}) | \partial_j \psi(\bm{\theta}) \rangle \right]$

with short notation $$| \partial_j \psi(\bm{\theta}) \rangle := \frac{\partial}{\partial \theta_j}| \psi(\bm{\theta}) \rangle$$. It is closely related to the quantum fisher information matrix, see quantum_fisher() and eq. (27) in arxiv:2103.15191.

Note

Only gates that have a single parameter and define a generator are supported. All other parametrized gates will be decomposed if possible.

The generator of all parametrized operations, with respect to which the tensor is computed, are assumed to be Hermitian. This is the case for unitary single-parameter operations.

Parameters
• tape (pennylane.QNode or QuantumTape) – quantum tape or QNode to find the metric tensor of

• argnum (int or Sequence[int] or None) – Trainable tape-parameter indices with respect to which the metric tensor is computed. If argnum=None, the metric tensor with respect to all trainable parameters is returned. Excluding tape-parameter indices from this list reduces the computational cost and the corresponding metric-tensor elements will be set to 0.

• approx (str) –

Which approximation of the metric tensor to compute.

• If None, the full metric tensor is computed

• If "block-diag", the block-diagonal approximation is computed, reducing the number of evaluated circuits significantly.

• If "diag", only the diagonal approximation is computed, slightly reducing the classical overhead but not the quantum resources (compared to "block-diag").

• allow_nonunitary (bool) – Whether non-unitary operations are allowed in circuits created by the transform. Only relevant if approx is None. Should be set to True if possible to reduce cost.

• aux_wire (int or str or pennylane.wires.Wires) – Auxiliary wire to be used for Hadamard tests. If None (the default), a suitable wire is inferred from the (number of) used wires in the original circuit and device_wires.

• device_wires (wires.Wires) – Wires of the device that is going to be used for the metric tensor. Facilitates finding a default for aux_wire if aux_wire is None.

• hybrid (bool) –

Specifies whether classical processing inside a QNode should be taken into account when transforming a QNode.

• If True, and classical processing is detected, the Jacobian of the classical processing will be computed and included. When evaluated, the returned metric tensor will be with respect to the QNode arguments. The output shape can vary widely.

• If False, any internal QNode classical processing will be ignored. When evaluated, the returned metric tensor will be with respect to the gate arguments, and not the QNode arguments. The output shape is a single two-dimensional tensor.

Returns

• If the input is a QNode, an object representing the metric tensor (function) of the QNode that takes the same arguments as the QNode and can be executed to obtain the metric tensor (matrix).

• If the input is a tape, a tuple containing a list of generated tapes, together with a post-processing function to be applied to the results of the evaluated tapes in order to obtain the metric tensor.

Return type

function or tuple[list[QuantumTape], function]

The block-diagonal part of the metric tensor always is computed using the covariance-based approach. If no approximation is selected, the off block-diagonal is computed using Hadamard tests.

Warning

Performing the Hadamard tests requires a device that has an additional wire as compared to the wires on which the original circuit was defined. This wire may be specified via aux_wire. The available wires on the device may be specified via device_wires.

By default (that is, if device_wires=None ), contiguous wire numbering and usage is assumed and the additional wire is set to the next wire of the device after the circuit wires.

If the given or inferred aux_wire does not exist on the device, a warning is raised and the block-diagonal approximation is computed instead. It is significantly cheaper in this case to explicitly set approx="block-diag" .

The flag allow_nonunitary should be set to True whenever the device with which the metric tensor is computed supports non-unitary operations. This will avoid additional decompositions of gates, in turn avoiding a potentially large number of additional Hadamard test circuits to be run. State vector simulators, for example, often allow applying operations that are not unitary. On a real QPU, setting this flag to True may cause exceptions because the computation of the metric tensor will request invalid operations on a quantum device.

Example

Consider the following QNode:

dev = qml.device("default.qubit", wires=3)

def circuit(weights):
qml.RX(weights[0], wires=0)
qml.RY(weights[1], wires=0)
qml.CNOT(wires=[0, 1])
qml.RZ(weights[2], wires=1)
qml.RZ(weights[3], wires=0)
return qml.expval(qml.PauliZ(0) @ qml.PauliZ(1)), qml.expval(qml.PauliY(1))


We can use the metric_tensor transform to generate a new function that returns the metric tensor of this QNode:

>>> mt_fn = qml.metric_tensor(circuit)
>>> weights = np.array([0.1, 0.2, 0.4, 0.5], requires_grad=True)
>>> mt_fn(weights)
tensor([[ 0.25  ,  0.    , -0.0497, -0.0497],
[ 0.    ,  0.2475,  0.0243,  0.0243],
[-0.0497,  0.0243,  0.0123,  0.0123],


In order to save cost, one might want to compute only the block-diagonal part of the metric tensor, which requires significantly fewer executions of quantum functions and does not need an auxiliary wire on the device. This can be done using the approx keyword:

>>> mt_fn = qml.metric_tensor(circuit, approx="block-diag")
>>> weights = np.array([0.1, 0.2, 0.4, 0.5], requires_grad=True)
>>> mt_fn(weights)
tensor([[0.25  , 0.    , 0.    , 0.    ],
[0.    , 0.2475, 0.    , 0.    ],
[0.    , 0.    , 0.0123, 0.0123],
[0.    , 0.    , 0.0123, 0.0123]], requires_grad=True)


These blocks are given by parameter groups that belong to groups of commuting gates.

The tensor can be further restricted to the diagonal via approx="diag". However, this will not save further quantum function evolutions but only classical postprocessing.

The returned metric tensor is also fully differentiable in all interfaces. For example, we can compute the gradient of the Frobenius norm of the metric tensor with respect to the QNode weights :

>>> norm_fn = lambda x: qml.math.linalg.norm(mt_fn(x), ord="fro")
array([-0.0282246 ,  0.01340413,  0.        ,  0.        ])


This transform can also be applied to low-level QuantumTape objects. This will result in no implicit quantum device evaluation. Instead, the processed tapes, and post-processing function, which together define the metric tensor are directly returned:

>>> params = np.array([1.7, 1.0, 0.5], requires_grad=True)
>>> with qml.tape.QuantumTape() as tape:
...     qml.RX(params[0], wires=0)
...     qml.RY(params[1], wires=0)
...     qml.CNOT(wires=[0, 1])
...     qml.PhaseShift(params[2], wires=1)
...     qml.expval(qml.PauliX(0))
>>> tapes, fn = qml.metric_tensor(tape)
>>> tapes
[<QuantumTape: wires=[0, 1], params=0>,
<QuantumTape: wires=[0, 1], params=1>,
<QuantumTape: wires=[0, 1], params=3>,
<QuantumTape: wires=[2, 0], params=1>,
<QuantumTape: wires=[2, 0, 1], params=2>,
<QuantumTape: wires=[2, 0, 1], params=2>]


This can be useful if the underlying circuits representing the metric tensor computation need to be analyzed. We clearly can distinguish the first three tapes used for the block-diagonal from the last three tapes that use the auxiliary wire 2 , which was not used by the original tape.

The output tapes can then be evaluated and post-processed to retrieve the metric tensor:

>>> dev = qml.device("default.qubit", wires=3)
>>> fn(qml.execute(tapes, dev, None))
array([[ 0.25      ,  0.        ,  0.42073549],
[ 0.        ,  0.00415023, -0.26517488],
[ 0.42073549, -0.26517488,  0.24878844]])


The first term of the off block-diagonal entries of the full metric tensor are computed with Hadamard tests. This first term reads

$\mathfrak{Re}\left\{\langle \partial_i\psi|\partial_j\psi\rangle\right\}$

and can be computed using an augmented circuit with an additional qubit. See for example the appendix of McArdle et al. (2019) for details. The block-diagonal of the tensor is computed using the covariance matrix approach.

In addition, we may extract the factors for the second terms $$\langle \psi|\partial_j\psi\rangle$$ of the off block-diagonal tensor from the quantum function output for the covariance matrix!

This means that in total only the tapes for the first terms of the off block-diagonal are required in addition to the circuits for the block diagonal.

Warning

The argnum argument can be used to restrict the parameters which are taken into account for computing the metric tensor. When the metric tensor of a QNode is computed, the ordering of the parameters has to be specified as they appear in the corresponding QuantumTape.

Example

Consider the following QNode in which parameters are used out of order:

>>> dev = qml.device("default.qubit", wires=3)
>>> def circuit(weights):  # , extra_weight):
...     qml.RX(weights[1], wires=0)
...     qml.RY(weights[0], wires=0)
...     qml.CNOT(wires=[0, 1])
...     qml.RZ(weights[2], wires=1)
...     qml.RZ(weights[3], wires=0)
...     return qml.expval(qml.PauliZ(0))

>>> weights = np.array([0.1, 0.2, 0.4, 0.5], requires_grad=True)
>>> mt = qml.metric_tensor(circuit, argnum=(0, 2, 3))(weights)
>>> print(mt)
[[ 0.          0.          0.          0.        ]
[ 0.          0.25       -0.02495835 -0.02495835]
[ 0.         -0.02495835  0.01226071  0.01226071]
[ 0.         -0.02495835  0.01226071  0.01226071]]


Because the 0-th element of weights appears second in the QNode and therefore in the underlying tape, it is the 1st tape parameter. By setting argnum = (0, 2, 3) we exclude the 0-th element of weights from the computation of the metric tensor and not the 1st element, as one might expect.