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Sampling Theorem01:15

Sampling Theorem

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In signal processing, the analysis of continuous-time signals, denoted as x(t), often involves sampling techniques to convert these signals into discrete-time signals. This process is essential for digital representation and manipulation. A critical component in sampling is the train of impulses, characterized by the sampling interval and the sampling frequency. The relationship between these parameters and the original signal's properties dictates the success of the sampling process.
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Sampling Methods: Overview01:06

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A sample refers to a smaller subset representative of a larger population. In analytical chemistry, studying or analyzing an entire population is often impractical or impossible. Therefore, samples are used to draw inferences and generalize the whole population. The sampling method selects individuals or items from a population to create a sample. Standard sampling methods include random, judgemental, systematic, stratified, and cluster sampling. 
In analytical chemistry, the choice of...
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Upsampling01:22

Upsampling

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Managing signal sampling rates is essential in digital signal processing to maintain signal integrity. A decimated signal, characterized by a reduced frequency range due to its lower sampling rate, can be upsampled by inserting zeros between each sample. This upsampling process expands the original spectrum and introduces repeated spectral replicas at intervals dictated by the new Nyquist frequency. To refine this zero-inserted sequence, it is passed through a lowpass filter with a cutoff...
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Sampling Continuous Time Signal01:11

Sampling Continuous Time Signal

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In signal processing, a continuous-time signal can be sampled using an impulse-train sampling technique, followed by the zero-order hold method. Impulse-train sampling involves the use of a periodic impulse train, which consists of a series of delta functions spaced at regular intervals determined by the sampling period. When a continuous-time signal is multiplied by this impulse train, it generates impulses with amplitudes corresponding to the signal's values at the sampling points.
In the...
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Sampling Plans01:23

Sampling Plans

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Sampling is a crucial step in analytical chemistry, allowing researchers to collect representative data from a large population. Common sampling methods include random, judgmental, systematic, stratified, and cluster sampling.
Random sampling is a method where each member of the population has an equal chance of being selected for the sample. It involves selecting individuals randomly, often using random number generators or lottery-type methods. For example, when analyzing the properties of a...
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Sampling Methods: Sample Types01:18

Sampling Methods: Sample Types

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Sampling materials are classified into three main types: solid, liquid, and gas.
Solid samples include a variety of substances, such as sediments from water bodies, soil, metals, and biological tissues. Two standard methods for extracting sediments from water bodies are grab sampling and piston coring. Grab sampling involves using a device to collect a discrete sediment sample from the bottom of a water body with minimal disturbance. Grab samples do not always represent the entire area due to...
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Related Experiment Video

Updated: Feb 22, 2026

Lensless Fluorescent Microscopy on a Chip
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Operator-level quantum acceleration of non-logconcave sampling.

Jiaqi Leng1,2, Zhiyan Ding2,3, Zherui Chen2

  • 1Simons Institute for the Theory of Computing, University of California, Berkeley, CA 94720.

Proceedings of the National Academy of Sciences of the United States of America
|February 20, 2026
PubMed
Summary

This study introduces a quantum algorithm to accelerate sampling from complex probability distributions, offering significant speedups for non-logconcave potentials where classical methods fail. It enables faster simulations in fields like physics and machine learning.

Keywords:
Gibbs samplingLangevin dynamicsWitten Laplacianquantum algorithmssingular value thresholding

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Area of Science:

  • Quantum computing
  • Computational physics
  • Statistical mechanics
  • Machine learning

Background:

  • Sampling from probability distributions is crucial in various scientific domains.
  • Classical methods like Langevin dynamics struggle with non-logconcave distributions, hindering performance.
  • Complex energy landscapes pose significant challenges for accurate and efficient sampling.

Purpose of the Study:

  • To develop a quantum algorithm for accelerating continuous-time sampling dynamics.
  • To address the limitations of classical sampling methods in non-logconcave settings.
  • To enable efficient sampling from complex, rugged energy landscapes.

Main Methods:

  • Encoding the target Gibbs measure into quantum state amplitudes.
  • Utilizing a block matrix factorization of the Witten Laplacian operator.
  • Implementing Gibbs sampling via singular value thresholding.
  • Developing a quantum algorithm for replica exchange Langevin diffusion.

Main Results:

  • A provable acceleration for a broad class of continuous-time sampling dynamics.
  • Up to a quartic quantum speedup over classical Langevin-based methods for non-logconcave distributions.
  • The first quantum algorithm to accelerate replica exchange Langevin diffusion.

Conclusions:

  • The developed quantum algorithm offers a significant advantage for sampling complex distributions.
  • This work provides a powerful new tool for simulating systems in physics, chemistry, and beyond.
  • Quantum computing can overcome fundamental limitations of classical sampling techniques.