Uniform Grids and Affine Transformations#

Numerical analysis problems are defined on a discretized space. Given a one-dimensional problem with $x \in [a,b), L=|b-a|$, the discretized grid is defined for the points

\[x_s = a + s\Delta x, \quad \Delta x = \frac{L}{N},\]

where $s=0,\dots,N-1$.

Ref. García-Molina et al. [GMRMGR22] presents a set of quantum Fourier analysis techniques to efficiently represent functions and operators in an n-qubit quantum register. By setting N=2^n, each of the points x_s of the grid maps to a quantum state s of the quantum register. This leads to a binary encoding of the coordinates

\[x_s^{(n)} = x_{(s_0s_1\dots s_{N-1})}^{(n)} = a + \Delta{x}^{(n)} \sum_{k=0}^{n-1}s_k 2^{n-k-1}.\]

Since MPS are representations of quantum states, they are also suitable for this quantum register representation, by mapping the qubits to the physical indices of the MPS.

It is also possible to use a symmetric definition of the position space interval, by setting Delta x = L/(N-1).

Multidimensional functions#

Let us consider a d dimensional function $f(\mathbf{x})=f(x_1, x_2,\dots,x_d)$ with a definition interval $x_i \in [a_i,b_i)$ of size $L_{x_i}=|b_i-a_i|$$ for each dimension $i=(1,\dots,d)$. An n-qubit quantum register encodes this function, using $n_i`$ qubits to represent each coordinate over $2^{n_i}$ points as

\[x_{i,s_i}^{(n_i)}=a_i+s_i \Delta x_i^{(n_i)}.\]

The integer $s_i\in\lbrace0,1, \dots, 2^{n_i}-1\rbrace$ labels the grid coordinate $x_{i,s_i}^{(n_i)}$ of the i-th dimension. Then, a set of labels $\mathbf{s}=(s_1,s_2,\dots,s_d)$ represents the d-dimensional state for the coordinate $\mathbf{x}_\mathbf{s}$$. For a multidimensional function the set of labels $\mathbf{s}=(s_{1}, s_{2}, \dots, s_{d})$ results from grouping the states of the $n=\sum n_i$ qubits in some order. They may be labeled sequentially, $s_1:=s_{11},s_2:=s_{12},\ldots$, or by significance, $s_1:=s_{11},\,s_2:=s_{21},\ldots$.

The :class::Space creates an object to define the problem’s coordinates for a multidimensional problem.

`Space`(qubits_per_dimension, L[, closed, order])	Coordinate grid class.
`mpo_flip`(operator)	Swap the qubits in the quantum register, to fix the reversal suffered during the quantum Fourier transform.

Implicit representations#

Alternatively, the multivariate spaces can be represented implicitly using the Interval and Mesh classes.

Essentially, the Interval class represents an univariate discretization implicitly, and can be indexed similarly as an explicit array. Then, the Mesh class represents a multivariate space as a collection of Interval objects. These objects can be indexed using multidimensional indices similarly as explicit multivariate arrays, without explicitly containing them and avoiding an exponential memory overhead.

Currently, there are three types of Interval implemented:

RegularInterval: An interval representing a regular discretization.
ChebyshevInterval: An interval representing an irregular discretization on the Chebyshev zeros or extrema.
IntegerInterval: An interval representing a regular discretization with integers.

`Mesh`(intervals)	Multidimensional mesh object.
`Interval`(start, stop, size)	Interval Abstract Base Class.
`RegularInterval`(start, stop, size[, ...])	Equispaced discretization between start and stop with size points.
`ChebyshevInterval`(start, stop, size[, endpoints])	Irregular discretization between start and stop given by the zeros or extrema of a Chebyshev polynomial of order size or size-1 respectively.
`IntegerInterval`(start, stop[, step])	Equispaced integer discretization between start and stop with given step.