Function Integration#

Functions encoded in MPS can be efficiently integrated, by contracting those quantum representations with representations of weights that implement some quadrature formula. For instance, a simple Riemann approximation results from the addition of all values of the functions, weighted by the interval size, which are equivalent to the contraction between the representation of \(f(x)\) and the identity function \(g(x)=1\)

\[\int f(x)\mathrm{d}x \simeq \sum_i f(x_i) \Delta{x} = \langle g | f\rangle.\]

In this scenario, the quadrature corresponding to the state \(\langle g |\) is given by the midpoint quadrature rule. More sophisticated quadrature rules result in more efficient convergence rates—i.e. requiring less nodes or tensor cores to compute an accurate estimation of the true integral.

In the following table we find both functions that construct the states associated to various quadratures—i.e. mps_* functions—and a function that implements the integral using any of those rules integrate_mps(). These quadrature rules divide in two families:

Newton-Côtes quadratures#

These are useful to integrate equispaced discretizations, each of increasing order. Compatible with discretizations stemming from RegularInterval objects. The larger the order, the better the convergence rate. However, large-order quadratures impose restrictions on the amounts of qubits they support:

mps_simpson() requires a number of qubits divisible by 2.
mps_fifth_order() requires a number of qubits divisible by 4.

`mps_midpoint`(start, stop, sites)	Returns the binary MPS representation of the midpoint quadrature on an interval.
`mps_trapezoidal`(start, stop, sites)	Returns the binary MPS representation of the trapezoidal quadrature on an interval.
`mps_simpson`(start, stop, sites)	Returns the binary MPS representation of the Simpson quadrature on an interval.
`mps_fifth_order`(start, stop, sites)	Returns the binary MPS representation of the fifth-order quadrature on an interval.

Clenshaw-Curtis quadratures#

These are useful to integrate irregular discretizations on either the Chebyshev zeros (Chebyshev-Gauss nodes) or the Chebyshev extrema (Chebyshev-Lobatto nodes). These have an exponentially better rate of convergence than the Newton-Côtes ones. Compatible with discretizations stemming from ChebyshevInterval objects.

`mps_fejer`(start, stop, sites[, strategy, ...])	Returns the binary MPS representation of the Fejér first quadrature rule on an interval.
`mps_clenshaw_curtis`(start, stop, sites[, ...])	Returns the binary MPS representation of the Clenshaw-Curtis quadrature rule on an interval.

Integration#

The standard method for integration consists in first constructing the multivariate quadrature rule using the previous routines, together with mps_tensor_product and mps_tensor_sum tensorized operations. Then, this quadrature is to be contracted with the desired MPS target using the scalar product routine scprod. However, for ease of use, a helper routine integrate_mps is given that automatically computes the best possible quadrature rule associated to a Mesh object, and contracts with the target MPS to compute the integral:

integrate_mps(mps, domain[, mps_order])

Returns the integral of a multivariate function represented as a MPS.

Note that this helper routine is only valid for standard function representations in MPS with binary quantization, while the former method is applicable in all cases.

An example on how to use these functions is shown in Integration.ipynb.