seemps.analysis.chebyshev.cheb2mps

seemps.analysis.chebyshev.cheb2mps(coefficients, initial_mps=None, domain=None, strategy=<seemps.state.core.Strategy object>, clenshaw=True, rescale=True)

Composes a function on an initial MPS by expanding it on the basis of Chebyshev polynomials. Allows to load functions on MPS by providing a suitable initial MPS for a given interval. Takes as input the Chebyshev coefficients of a function f(x) defined in an interval [a, b] and, optionally, an initial MPS representing a function g(x) that is taken as the first order polynomial of the expansion. With this information, it constructs the MPS that approximates f(g(x)).

Parameters:

coefficientsnp.polynomial.Chebyshev

The Chebyshev expansion coefficients representing the target function that is defined on a given interval [a, b].

initial_mpsMPS, optional

The initial MPS on which to apply the expansion. By default (if rescale is True), it must have a support inside the domain of definition of the function [a, b]. If rescale is False, it must have a support inside [-1, 1].

domainInterval, optional

An alternative way to specify the initial MPS by constructing it from the given Interval.

strategyStrategy, default=DEFAULT_STRATEGY

The simplification strategy for operations between MPS.

clenshawbool, default=True

Whether to use the Clenshaw algorithm for polynomial evaluation.

rescalebool, default=True

Whether to perform an affine transformation of the initial MPS from the domain [a, b] of the Chebyshev coefficients to the canonical Chebyshev interval [-1, 1].

Returns:

f_mpsMPS

MPS representation of the polynomial expansion.

Notes

• The computational complexity of the expansion depends on bond dimensions of the intermediate

states. For the case of loading univariate functions on RegularInterval domains, these are bounded by the polynomial order of each intermediate step. In general, these are determined by the function, the bond dimensions of the initial state and the simplification strategy used.

• The Clenshaw evaluation method has a better performance overall, but performs worse when

the expansion order is overestimated. This can be avoided using the estimate_order method.

Examples

# Load an univariate Gaussian in an equispaced domain.
start, stop = -1, 1
n_qubits = 10
func = lambda x: np.exp(-x**2)
coefficients = interpolation_coefficients(func, start=start, stop=stop)
domain = RegularInterval(start, stop, 2**n_qubits)
mps = cheb2mps(coefficients, domain=domain)