qml.fourier.coefficients

coefficients(f, n_inputs, degree, lowpass_filter=False, filter_threshold=None)[source]

Computes the first \(2d+1\) Fourier coefficients of a \(2\pi\) periodic function, where \(d\) is the highest desired frequency (the degree) of the Fourier spectrum.

While this function can be used to compute Fourier coefficients in general, the specific use case in PennyLane is to compute coefficients of the functions that result from measuring expectation values of parametrized quantum circuits, as described in Schuld, Sweke and Meyer (2020) and Vidal and Theis, 2019.

Details

Consider a quantum circuit that depends on a parameter vector \(x\) with length \(N\). The circuit involves application of some unitary operations \(U(x)\), and then measurement of an observable \(\langle \hat{O} \rangle\). Analytically, the expectation value is

\[\langle \hat{O} \rangle = \langle 0 \vert U^\dagger (x) \hat{O} U(x) \vert 0\rangle = \langle \psi(x) \vert \hat{O} \vert \psi (x)\rangle.\]

This output is simply a function \(f(x) = \langle \psi(x) \vert \hat{O} \vert \psi (x)\rangle\). Notably, it is a periodic function of the parameters, and it can thus be expressed as a multidimensional Fourier series:

\[f(x) = \sum \limits_{n_1\in \Omega_1} \dots \sum \limits_{n_N \in \Omega_N} c_{n_1,\dots, n_N} e^{-i x_1 n_1} \dots e^{-i x_N n_N},\]

where \(n_i\) are integer-valued frequencies, \(\Omega_i\) are the set of available values for the integer frequencies, and the \(c_{n_1,\ldots,n_N}\) are Fourier coefficients.

Parameters
  • f (callable) – Function that takes a 1D tensor of n_inputs scalar inputs.

  • n_inputs (int) – number of function inputs

  • degree (int) – max frequency of Fourier coeffs to be computed. For degree \(d\), the coefficients from frequencies \(-d, -d+1,...0,..., d-1, d\) will be computed.

  • lowpass_filter (bool) – If True, a simple low-pass filter is applied prior to computing the set of coefficients in order to filter out frequencies above the given degree. See examples below.

  • filter_threshold (None or int) – The integer frequency at which to filter. If lowpass_filter is set to True, but no value is specified, 2 * degree is used.

Returns

The Fourier coefficients of the function f up to the specified degree.

Return type

array[complex]

Example

Suppose we have the following quantum function and wish to compute its Fourier coefficients with respect to the variable inpt, which is an array with 2 values:

dev = qml.device('default.qubit', wires=['a'])

@qml.qnode(dev)
def circuit(weights, inpt):
    qml.RX(inpt[0], wires='a')
    qml.Rot(*weights[0], wires='a')

    qml.RY(inpt[1], wires='a')
    qml.Rot(*weights[1], wires='a')

    return qml.expval(qml.PauliZ(wires='a'))

Unless otherwise specified, the coefficients will be computed for all input values. To compute coefficients with respect to only a subset of the input values, it is necessary to use a wrapper function (e.g., functools.partial). We do this below, while fixing a value for weights:

>>> from functools import partial
>>> weights = np.array([[0.1, 0.2, 0.3], [-4.1, 3.2, 1.3]])
>>> partial_circuit = partial(circuit, weights)

Now we must specify the number of inputs, and the maximum desired degree. Based on the underlying theory, we expect the degree to be 1 (frequencies -1, 0, and 1).

>>> num_inputs = 2
>>> degree = 1

Then we can obtain the coefficients:

>>> coeffs = coefficients(partial_circuit, num_inputs, degree)
>>> print(coeffs)
[[ 0.    +0.j     -0.    +0.j     -0.    +0.j    ]
[-0.0014-0.022j  -0.3431-0.0408j -0.1493+0.0374j]
[-0.0014+0.022j  -0.1493-0.0374j -0.3431+0.0408j]]

If the specified degree is lower than the highest frequency of the function, aliasing may occur, and the resultant coefficients will be incorrect as they will include components of the series expansion from higher frequencies. In order to mitigate aliasing, setting lowpass_filter=True will apply a simple low-pass filter prior to computing the coefficients. Coefficients up to a specified value are computed, and then frequencies higher than the degree are simply removed. This ensures that the coefficients returned will have the correct values, though they may not be the full set of coefficients. If no threshold value is provided, the threshold will be set to 2 * degree.

Consider the circuit below:

@qml.qnode(dev)
def circuit(inpt):
    qml.RX(inpt[0], wires=0)
    qml.RY(inpt[0], wires=1)
    qml.CNOT(wires=[1, 0])
    return qml.expval(qml.PauliZ(0))

One can work out by hand that the Fourier coefficients are \(c_0 = 0.5, c_1 = c_{-1} = 0,\) and \(c_2 = c_{-2} = 0.25\). Suppose we would like only to obtain the coefficients \(c_0\) and \(c_1, c_{-1}\). If we simply ask for the coefficients of degree 1, we will obtain incorrect values due to aliasing:

>>> coefficients(circuit, 1, 1)
array([0.5 +0.j, 0.25+0.j, 0.25+0.j])

However if we enable the low-pass filter, we can still obtain the correct coefficients:

>>> coefficients(circuit, 1, 1, lowpass_filter=True)
array([0.5+0.j, 0. +0.j, 0. +0.j])

Note that in this case, 2 * degree gives us exactly the maximum coefficient; in other situations it may be desirable to set the threshold value explicitly.

The coefficients function can handle qnodes from all PennyLane interfaces.