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Quantum data encoding as a distinct abstraction layer in the design of quantum circuits

Gabriele Agliardi, Enrico Prati

TL;DR

The paper confronts the lack of a systematic formalization for quantum data encoding by introducing quantum data encoding as a distinct abstraction layer that separates encoding from data loading, processing, and extraction. It develops a taxonomy of encodings for single data points, data sets, and multiple data sets, and presents exact and approximate loading methods, including conversions between encodings via mechanisms like the Quantum Fourier Transform. It reinterprets key quantum algorithms—QFT as an encoding converter and Quantum Amplitude Estimation as a data-extraction routine—and demonstrates their utility in quantum-based Monte Carlo simulations, highlighting how encoding choices shape circuit design and performance. The framework aims to improve modularity and clarity in complex quantum circuits, enabling high-level quantum programming abstractions and more efficient, scalable quantum algorithm design.

Abstract

Complex quantum circuits are constituted by combinations of quantum subroutines. The computation is possible as long as the quantum data encoding is consistent throughout the circuit. Despite its fundamental importance, the formalization of quantum data encoding has never been addressed systematically so far. We formalize the concept of quantum data encoding, namely the format providing a representation of a data set through a quantum state, as a distinct abstract layer with respect to the associated data loading circuit. We survey existing encoding methods and their respective strategies for classical-to-quantum exact and approximate data loading, for the quantum-to-classical extraction of information from states, and for quantum-to-quantum encoding conversion. Next, we show how major quantum algorithms find a natural interpretation in terms of data loading. For instance, the Quantum Fourier Transform is described as a quantum encoding converter, while the Quantum Amplitude Estimation as an extraction routine. The new conceptual framework is exemplified by considering its application to quantum-based Monte Carlo simulations, thus showcasing the power of the proposed formalism for the description of complex quantum circuits. Indeed, the approach clarifies the structure of complex quantum circuits and enables their efficient design.

Quantum data encoding as a distinct abstraction layer in the design of quantum circuits

TL;DR

The paper confronts the lack of a systematic formalization for quantum data encoding by introducing quantum data encoding as a distinct abstraction layer that separates encoding from data loading, processing, and extraction. It develops a taxonomy of encodings for single data points, data sets, and multiple data sets, and presents exact and approximate loading methods, including conversions between encodings via mechanisms like the Quantum Fourier Transform. It reinterprets key quantum algorithms—QFT as an encoding converter and Quantum Amplitude Estimation as a data-extraction routine—and demonstrates their utility in quantum-based Monte Carlo simulations, highlighting how encoding choices shape circuit design and performance. The framework aims to improve modularity and clarity in complex quantum circuits, enabling high-level quantum programming abstractions and more efficient, scalable quantum algorithm design.

Abstract

Complex quantum circuits are constituted by combinations of quantum subroutines. The computation is possible as long as the quantum data encoding is consistent throughout the circuit. Despite its fundamental importance, the formalization of quantum data encoding has never been addressed systematically so far. We formalize the concept of quantum data encoding, namely the format providing a representation of a data set through a quantum state, as a distinct abstract layer with respect to the associated data loading circuit. We survey existing encoding methods and their respective strategies for classical-to-quantum exact and approximate data loading, for the quantum-to-classical extraction of information from states, and for quantum-to-quantum encoding conversion. Next, we show how major quantum algorithms find a natural interpretation in terms of data loading. For instance, the Quantum Fourier Transform is described as a quantum encoding converter, while the Quantum Amplitude Estimation as an extraction routine. The new conceptual framework is exemplified by considering its application to quantum-based Monte Carlo simulations, thus showcasing the power of the proposed formalism for the description of complex quantum circuits. Indeed, the approach clarifies the structure of complex quantum circuits and enables their efficient design.
Paper Structure (53 sections, 1 theorem, 38 equations, 12 figures)

This paper contains 53 sections, 1 theorem, 38 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Quantum encoding methods
  3. Encoding of a data point
  4. Basis encoding.
  5. Angle encoding (or qubit encoding).
  6. Angle encoding (or qubit encoding).
  7. Fourier encoding.
  8. Encoding of a data set
  9. Multi-register encoding.
  10. Equally-weighted encoding (also called digital encoding or quantum associative memory QuAM).
  11. Angle encoding.
  12. Amplitude encoding (also called analog encoding).
  13. Divide & conquer encoding (D&C encoding).
  14. Bidirectional encoding.
  15. Generalized amplitude encoding.
  16. ...and 38 more sections

Key Result

Proposition 4.2

Suppose a unitary operator $F$ is given, such that $\ket{\psi} = F \ket{0}$, where $\ket{\psi}$ is the state of interest in Eq. eq:est-from-ampl. Then, the Quantum Amplitude Estimation method builds a circuit whose measurement, once post-processed, provides an estimator $\hat{\mu}$ for $\mu := \abs{ The algorithm employs $m$ qubits in addition to those necessary for $\ket{\psi}$, and one shot requ

Figures (12)

  • Figure 1: A quantum circuit embodied for the gate-model architecture can be interpreted as a quantum data loading procedure, followed by a collection of quantum subroutines, and finally by a data extraction process. A data loading routine takes as input information encoded in a classical structure, such as for instance a classical variable, an array, or a binary tree, and it produces a state that represents the same information in a given quantum encoding. A quantum subroutine takes an input state representing some information in a given encoding, and produces an output state representing new information in another given encoding. A data extraction routine takes an input state in a given encoding, and returns part of the information as an output register of bits, after a measurement process. Typically, a classical post-processing of multiple shots is needed to retrieve a significant amount of information from the quantum state.
  • Figure 2: An encoding conversion routine is a special quantum subroutine that takes some information $x$ in input as a state in a given encoding, and returns the same information in output, as a state in a different encoding.
  • Figure 3: Width and depth of the loading circuit for the bidirectional encoding araujo2021dividearaujo_configurable_2022, as a function of the split level $s$ and number of qubits $n$.
  • Figure 4: An example of a bijection $g$ realizing an encoding: here, $\ket{j}$ is used to represent the value $d_j=g^{-1}(j)$.
  • Figure 5: The circuit for the swap test. The probability of the measurement to be $0$ is $\frac{1}{2} + \frac{1}{2} \abs{ \sum_j a_j b_j }^2$. Here, $H$ is the Hadamard gate.
  • ...and 7 more figures

Theorems & Definitions (7)

  • Remark 2.1: Quantum parallel execution
  • Remark 4.1
  • Proposition 4.2: QAE
  • Remark 5.1: Time comparison against direct summation
  • Remark 5.2: Time comparison against classical Monte Carlo
  • Remark 5.3: Time comparison summary
  • Remark 5.4