Optimal sampling of tensor networks targeting wave function's fast decaying tails

Marco Ballarin; Pietro Silvi; Simone Montangero; Daniel Jaschke

doi:10.22331/q-2025-04-18-1714

Optimal sampling of tensor networks targeting wave function’s fast decaying tails

Marco Ballarin^1,2, Pietro Silvi^1,2, Simone Montangero^1,2,3, and Daniel Jaschke^3,1,2

¹Dipartimento di Fisica e Astronomia "G. Galilei" & Padua Quantum Technologies Research Center, Università degli Studi di Padova, Italy I-35131, Padova, Italy
²Dipartimento di Fisica e Astronomia "G. Galilei" & Padua Quantum Technologies Research Center, Università degli Studi di Padova, Italy I-35131, Padova, Italy
³Institute for Complex Quantum Systems, Ulm University, Albert-Einstein-Allee 11, 89069 Ulm, Germany

Published:	2025-04-18, volume 9, page 1714
Editor:	Luca Tagliacozzo
Eprint:	arXiv:2401.10330v2
Doi:	https://doi.org/10.22331/q-2025-04-18-1714
Citation:	Quantum 9, 1714 (2025).

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Abstract

We introduce an optimal strategy to sample quantum outcomes of local measurement strings for isometric tensor network states. Our method generates samples based on an exact cumulative bounding function, without prior knowledge, in the minimal amount of tensor network contractions. The algorithm avoids sample repetition and, thus, is efficient at sampling distribution with exponentially decaying tails. We illustrate the computational advantage provided by our optimal sampling method through various numerical examples, involving condensed matter, optimization problems, and quantum circuit scenarios. Theory predicts up to an exponential speedup reducing the scaling for sampling the space up to an accumulated unknown probability $\epsilon$ from $\mathcal{O}(\epsilon^{-1})$ to $\mathcal{O}(\log(\epsilon^{-1}))$ for a decaying probability distribution. We confirm this in practice with over one order of magnitude speedup or multiple orders improvement in the error depending on the application. Our sampling strategy extends beyond local observables, e.g., to quantum magic.

Featured image: Different ways of sampling a given probability distribution function encoded in an $n=3$ qubit state.
a) The motivation to sample by shots are experiments, where the probabilities are proportional to the normalized number of shots for a given state and converge in the limit of large numbers. Thus, the probability bar in green is proportional to the shots shown as red circles.
b) The actual probability distribution function is the reference distribution for any sampling method.
c) One can rewrite the exact probabilities into a cumulative probability function which highlights that each state $\alpha$ can be associated with an interval $\delta_{\alpha}$, where the intervals for the subsystem state $|{0}\rangle_{0}$ and state $|{100}\rangle$ are indicated as $\delta_{0}$ and $\delta_{100}$. The intervals are defined via the left and right boundaries of the intervals, i.e., $\delta_{\alpha} = (b_{\alpha}^{l}, b_{\alpha}^{r})$. The probability stemming from the previous states in the order is shown in black, the probability of the state itself is shown in cyan.
d) The standard sampling follows a shot-by-shot approach to sample states $\alpha$, but has access to the exact probabilities $p_{\alpha}$. Additionally, probabilities of states differing only in the last qubit of the bitstring are known at no additional cost, see non-zero bars without shot (red circle). The probability of the green bars do not sum to one, which accounts for the unknown probability space.
e) Using OPES, we avoid resampling and thus all shots are for different states; in detail, red circles fall into different bars. Moreover, pairs of probabilities are discovered together analog to d) and even the substates of the first $(n – 1)$ qubits of the shots in red are unique for each shot. Four steps are sufficient to sample a space of eight states, which generalizes to an upper bound of $\mathcal{O}(2^{n – 1})$ samples to scan a full Hilbert space. The distribution of the shots sampled in this way does not converge to the true distribution anymore, as there is either zero or one shot per bin; one always uses the exact probabilities.
f) The OPES scaling is achieved by removing intervals from the space where the algorithm is allowed to sample; only previously not sampled states remain as potential next samples. While running the iterations top-down in this illustration, we gain partial knowledge of the cumulative probability distribution of b), i.e., of the intervals of the full state when reaching the green checkmark as well as subsystem intervals as indicated. For example in step $n_r = 1$, we identify the intervals $\delta_{0}$ and $\delta_{11}$ for subsystems. The intervals with green checkmarks are then removed for all the following iterations, see red crosses.

► BibTeX data

@article{Ballarin2025optimalsamplingof,
  doi = {10.22331/q-2025-04-18-1714},
  url = {https://doi.org/10.22331/q-2025-04-18-1714},
  title = {Optimal sampling of tensor networks targeting wave function's fast decaying tails},
  author = {Ballarin, Marco and Silvi, Pietro and Montangero, Simone and Jaschke, Daniel},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1714},
  month = apr,
  year = {2025}
}

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[3] Davide Rattacaso, Daniel Jaschke, Marco Ballarin, Ilaria Siloi, and Simone Montangero, "Quantum circuit compilation with quantum computers", Physical Review Research 7 3, 033268 (2025).

[4] Pedro R. Nicácio Falcão, Poetri Sonya Tarabunga, Martina Frau, Emanuele Tirrito, Jakub Zakrzewski, and Marcello Dalmonte, "Nonstabilizerness in U(1) lattice gauge theory", Physical Review B 111 8, L081102 (2025).

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