qml.math.cov_matrix¶
-
cov_matrix
(prob, obs, wires=None, diag_approx=False)[source]¶ Calculate the covariance matrix of a list of commuting observables, given the joint probability distribution of the system in the shared eigenbasis.
Note
This method only works for commuting observables. If the probability distribution is the result of a quantum circuit, the quantum state must be rotated into the shared eigenbasis of the list of observables before measurement.
- Parameters
prob (tensor_like) – probability distribution
obs (list[Observable]) – a list of observables for which to compute the covariance matrix
diag_approx (bool) – if True, return the diagonal approximation
wires (Wires) – The wire register of the system. If not provided, it is assumed that the wires are labelled with consecutive integers.
- Returns
the covariance matrix of size
(len(obs), len(obs))
- Return type
tensor_like
Example
Consider the following ansatz and observable list:
>>> obs_list = [qml.X(0) @ qml.Z(1), qml.Y(2)] >>> ansatz = qml.templates.StronglyEntanglingLayers
We can construct a QNode to output the probability distribution in the shared eigenbasis of the observables:
dev = qml.device("default.qubit", wires=3) @qml.qnode(dev, interface="autograd") def circuit(weights): ansatz(weights, wires=[0, 1, 2]) # rotate into the basis of the observables for o in obs_list: o.diagonalizing_gates() return qml.probs(wires=[0, 1, 2])
We can now compute the covariance matrix:
>>> shape = qml.templates.StronglyEntanglingLayers.shape(n_layers=2, n_wires=3) >>> weights = pnp.random.random(shape, requires_grad=True) >>> cov = qml.math.cov_matrix(circuit(weights), obs_list) >>> cov tensor([[0.98125435, 0.4905541 ], [0.4905541 , 0.99920878]], requires_grad=True)
Autodifferentiation is fully supported using all interfaces. Here we use autograd:
>>> cost_fn = lambda weights: qml.math.cov_matrix(circuit(weights), obs_list)[0, 1] >>> qml.grad(cost_fn)(weights) array([[[ 4.94240914e-17, -2.33786398e-01, -1.54193959e-01], [-3.05414996e-17, 8.40072236e-04, 5.57884080e-04], [ 3.01859411e-17, 8.60411436e-03, 6.15745204e-04]], [[ 6.80309533e-04, -1.23162742e-03, 1.08729813e-03], [-1.53863193e-01, -1.38700657e-02, -1.36243323e-01], [-1.54665054e-01, -1.89018172e-02, -1.56415558e-01]]])