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Fast Quantum Amplitude Encoding of Typical Classical Data

Vittorio Pagni, Sigurd Huber, Michael Epping, Michael Felderer

TL;DR

By analysing the data density, it is proved that the average runtime is $\mathcal{O}(\log^{1.5} N)$ for uniformly random inputs and numerical evidence that this favourable runtime behaviour also holds for real-world data, such as radar satellite images.

Abstract

We present an improved version of a quantum amplitude encoding scheme that encodes the $N$ entries of a unit classical vector $\vec{v}=(v_1,..,v_N)$ into the amplitudes of a quantum state. Our approach has a quadratic speed-up with respect to the original one. We also describe several generalizations, including to complex entries of the input vector and a parameter $M$ that determines the parallelization. The number of qubits required for the state preparation scales as $\mathcal{O}(M\log N)$. The runtime, which depends on the data density $ρ$ and on the parallelization paramater $M$, scales as $\mathcal{O}(\frac{1}{\sqrtρ}\frac{N}{M}\log (M+1))$, which in the most parallel version ($M=N$) is always less than $\mathcal{O}(\sqrt{N}\log N)$. By analysing the data density, we prove that the average runtime is $\mathcal{O}(\log^{1.5} N)$ for uniformly random inputs. We present numerical evidence that this favourable runtime behaviour also holds for real-world data, such as radar satellite images. This is promising as it allows for an input-to-output advantage of the quantum Fourier transform.

Fast Quantum Amplitude Encoding of Typical Classical Data

TL;DR

By analysing the data density, it is proved that the average runtime is for uniformly random inputs and numerical evidence that this favourable runtime behaviour also holds for real-world data, such as radar satellite images.

Abstract

We present an improved version of a quantum amplitude encoding scheme that encodes the entries of a unit classical vector into the amplitudes of a quantum state. Our approach has a quadratic speed-up with respect to the original one. We also describe several generalizations, including to complex entries of the input vector and a parameter that determines the parallelization. The number of qubits required for the state preparation scales as . The runtime, which depends on the data density and on the parallelization paramater , scales as , which in the most parallel version () is always less than . By analysing the data density, we prove that the average runtime is for uniformly random inputs. We present numerical evidence that this favourable runtime behaviour also holds for real-world data, such as radar satellite images. This is promising as it allows for an input-to-output advantage of the quantum Fourier transform.

Paper Structure

This paper contains 38 sections, 2 theorems, 133 equations, 15 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Results
  3. An improved state preparation algorithm that uses amplitude amplification for post-processing
  4. Classical pre-processing
  5. Quantum encoding circuit E
  6. Structure
  7. Registers
  8. Gates
  9. State Evolution
  10. Theoretical scaling of runtime and memory cost
  11. Generalization to complex numbers
  12. Case Study - Analysis of the data density rho over satellite images
  13. Scaling of $\rho$ over a satellite image of Germany
  14. Example - Application to QFT
  15. Discussion
  16. ...and 23 more sections

Key Result

Lemma S1.1

The expectation value of the square of the infinity norm over a normal distribution with $\sigma=1$ behaves for large values of N as

Figures (15)

  • Figure 1: Rescaled image of a part of Germany taken by the satellite Sentinel-1A using a SAR interferometric radar instrument CopernicusSentinel1. Sentinel-1 is a European radar mission developed for the Copernicus program, a joint initiative of the European Commission (EC) and the European Space Agency (ESA).
  • Figure 2: Visual representation of the encoder circuit $\mathcal{E}$. It is divided into the initial step, which is performed only once, the encoding step that is repeated an amount of times $\frac{N}{M}$, and the final step.
  • Figure 3: Heat maps displaying local density for different grid sizes. We notice how the average density increases as the sector size decreases.
  • Figure 4: Examples of different sectors with varying sizes from gray scale version of Fig \ref{['initial_image']}, each corresponding to a specific grid size.
  • Figure 5: Behavior of the average data density as a function of the size of the flattened and normalized versions of the 2D arrays of the sectors compared to the one expected when sampling from a uniform distribution over the N-sphere and to $\propto \frac{1}{\sqrt{x}}$.
  • ...and 10 more figures

Theorems & Definitions (4)

  • Lemma S1.1
  • proof
  • Lemma S1.2
  • proof