Predefined States (MPS) and Tensor Operations#

The SeeMPS library provides exact MPS representations of common functions that can be constructed analytically with low bond dimension. These building blocks enable the efficient composition of more complex functions through tensor operations.

Elementary functions#

Several elementary functions have exact MPS representations with bond dimension that does not grow with the number of qubits:

  • Exponentials: \(\exp(kx)\) can be represented exactly with bond dimension 1, since the exponential factorizes over the binary digits of \(x\).

  • Trigonometric functions: \(\sin(kx)\) and \(\cos(kx)\) are constructed as sums of complex exponentials and have bond dimension 2.

  • Heaviside step function: \(\Theta(x - x_0)\) has bond dimension 2 and is useful for constructing piecewise functions.

Example usage:

from seemps.analysis.mesh import RegularInterval
from seemps.analysis.factories import mps_exponential, mps_sin, mps_cos

# Define a grid with 2^10 = 1024 points on [0, 2π)
interval = RegularInterval(0, 2 * 3.14159, 2**10)

# Create MPS representations
exp_x = mps_exponential(interval, k=-1.0)  # exp(-x)
sin_x = mps_sin(interval, k=2.0)           # sin(2x)
cos_x = mps_cos(interval, k=1.0)           # cos(x)

Interval representations#

The mps_interval() function creates an MPS representing the coordinate variable \(x\) itself on a given interval. This is useful as a building block for polynomial constructions:

from seemps.analysis.mesh import RegularInterval, ChebyshevInterval
from seemps.analysis.factories import mps_interval

# Regular (equispaced) grid
regular = RegularInterval(-1, 1, 2**10)
x_regular = mps_interval(regular)

# Chebyshev (non-uniform) grid
chebyshev = ChebyshevInterval(-1, 1, 2**10)
x_chebyshev = mps_interval(chebyshev)

Tensor operations#

MPS can be combined to represent multivariate functions using tensor products and sums:

  • Tensor product: mps_tensor_product() constructs \(f(x) \cdot g(y)\) from separate MPS for \(f\) and \(g\).

  • Tensor sum: mps_tensor_sum() constructs \(f(x) + g(y)\) as a sum of separable terms.

These operations preserve the MPS structure and enable efficient representation of multivariate functions with separable structure.

API reference#

mps_exponential(interval[, k])

Returns an MPS encoding of \(exp(k x)\).

mps_sin(interval[, k])

Returns an MPS encoding of \(sin(k x)\).

mps_cos(interval[, k])

Returns an MPS encoding of \(cos(k x)\).

mps_affine(mps, orig, dest)

Applies an affine transformation to an MPS, mapping it from one interval [x0, x1] to another [u0, u1].

mps_heaviside(interval[, x0])

MPS quantization of the Heaviside function \(\Theta(x)\).

mps_interval(interval)

Returns an MPS corresponding to a specific type of interval.

mps_tensor_product(mps_list[, mps_order, ...])

Returns the tensor product of a list of MPS, with the sites arranged according to the specified MPS order.

mps_tensor_sum(mps_list[, mps_order, ...])

Returns the tensor sum of a list of MPS, with the sites arranged according to the specified MPS order.