Difference Kernel MPO

A difference kernel is a matrix whose entries depend only on the coordinate difference:

A[x, x'] = f(x - x').

This tutorial starts from a one-dimensional QTT for f(z) and builds the periodic MPO for A[x, x'] = f((x - x') mod 2^R).

Runnable source: docs/tutorial-code/src/bin/qtt_difference_kernel.rs

Key API Pieces

difference_kernel_mpo takes a binary QTT over the difference coordinate and returns an MPO with one fused local index per bit. The local value is encoded as x_bit * 2 + xprime_bit.

fn main() -> anyhow::Result<()> {
use num_complex::Complex64;
use tensor4all_quanticstransform::{difference_kernel_mpo, BoundaryCondition};
use tensor4all_simplett::{AbstractTensorTrain, TensorTrain};
let bits = 6;
let site_dims = vec![2; bits];

// A real-valued QTT can be converted to Complex64 before calling the transform.
// This compact example uses a constant kernel QTT.
let f = TensorTrain::constant(&site_dims, Complex64::new(1.0, 0.0));

let mpo = difference_kernel_mpo(&f, BoundaryCondition::Periodic)?;
assert_eq!(mpo.len(), bits);
assert_eq!(mpo.site_dims(), vec![4; bits]);
Ok(())
}

For BoundaryCondition::AntiPeriodic, the same API multiplies entries with x < x' by -1. The checked-in tutorial data uses the periodic case only.

What It Computes

The example builds a smooth periodic source kernel

f(z) = exp(2(cos(2πz/N) - 1))

with N = 2^R, converts the real QTT cores to Complex64, and calls difference_kernel_mpo. The resulting MPO is sampled densely and compared with the direct reference f((x - x') mod N).

The MPO bonds combine the two-state carry network for x - x' with the source kernel QTT bonds, so the practical bond dimensions are bounded by twice the kernel QTT bond dimensions before any later compression.