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Efficient explicit circuit for quantum state preparation of piecewise continuous functions

Nikita Guseynov, Nana Liu

TL;DR

This work addresses efficient quantum data-loading by encoding a function $f(x)$ on the interval $[-1,1]$ as amplitudes of a quantum state, with discretization $x_k=-1+k\Delta x$ and $N=2^n$. It develops a four-parity real polynomial decomposition and leverages Quantum Singular Value Transformation (QSVT), Alternating Phase Modulation Sequence (APTS), and Linear Combination of Unitaries (LCU) to construct explicit, hardware-conscious state-preparation circuits that realize $f(\hat{x})$ via a coordinate-polynomial oracle. The main results show $U_f$ achieves $\mathcal{O}(Q\,n\log n)$ gate complexity (plus $\lceil\log_2 n\rceil+1$ pure ancillas) for a single polynomial and extends to piecewise polynomials with $G$ segments at $\mathcal{O}(Q_{\max}\,n\log n + G n + Q_{\max} G)$. These explicit circuits enable uploading high-degree polynomial amplitudes (up to $10^4$) and provide practical, scalable data-loading capabilities for quantum algorithms relying on polynomial representations and segmented data.

Abstract

Efficiently uploading data into quantum states is essential for many quantum algorithms to achieve advantage across various applications. In this paper, we address this challenge by developing a method to upload a polynomial function $f(x)$ on the interval $x \in [-1,1]$ into a pure quantum state consisting of qubits, where a discretized $f(x)$ is the amplitude of this state. The preparation cost has $\mathcal{O}(n\log n)$ scaling in the number of qubits $n$ and linear scaling with the degree of the polynomial $Q$. This efficiency allows the preparation of states whose amplitudes correspond to high-degree polynomials (up to $10^4$), enabling accurate approximation of functions that admit efficient polynomial series representations and whose amplitude profiles are not extremely localized. We provide a fully explicit circuit realization, based on four real polynomials that meet specific parity and boundedness conditions. We extend this construction to cover piece-wise polynomial functions, a case not previously addressed explicitly in the literature, the algorithm scaling linearly with the number of piecewise parts. Our method achieves efficient quantum circuit implementation and we present detailed gate counting and resource analysis.

Efficient explicit circuit for quantum state preparation of piecewise continuous functions

TL;DR

This work addresses efficient quantum data-loading by encoding a function on the interval as amplitudes of a quantum state, with discretization and . It develops a four-parity real polynomial decomposition and leverages Quantum Singular Value Transformation (QSVT), Alternating Phase Modulation Sequence (APTS), and Linear Combination of Unitaries (LCU) to construct explicit, hardware-conscious state-preparation circuits that realize via a coordinate-polynomial oracle. The main results show achieves gate complexity (plus pure ancillas) for a single polynomial and extends to piecewise polynomials with segments at . These explicit circuits enable uploading high-degree polynomial amplitudes (up to ) and provide practical, scalable data-loading capabilities for quantum algorithms relying on polynomial representations and segmented data.

Abstract

Efficiently uploading data into quantum states is essential for many quantum algorithms to achieve advantage across various applications. In this paper, we address this challenge by developing a method to upload a polynomial function on the interval into a pure quantum state consisting of qubits, where a discretized is the amplitude of this state. The preparation cost has scaling in the number of qubits and linear scaling with the degree of the polynomial . This efficiency allows the preparation of states whose amplitudes correspond to high-degree polynomials (up to ), enabling accurate approximation of functions that admit efficient polynomial series representations and whose amplitude profiles are not extremely localized. We provide a fully explicit circuit realization, based on four real polynomials that meet specific parity and boundedness conditions. We extend this construction to cover piece-wise polynomial functions, a case not previously addressed explicitly in the literature, the algorithm scaling linearly with the number of piecewise parts. Our method achieves efficient quantum circuit implementation and we present detailed gate counting and resource analysis.

Paper Structure

This paper contains 9 sections, 6 theorems, 35 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results
  4. Related Work
  5. Explicit quantum circuits for the multi-control unitaries
  6. Efficient Series Expansions and Error Terms
  7. Coordinate-polynomial-oracle
  8. From $x$ to a polynomial
  9. Piece-wise polynomial function

Key Result

Theorem 1

Let $f(x)$ be a continuous function $f: \mathbb{R} \rightarrow \mathbb{C}$ with polynomial decomposition where polynomials $P_s(x)$ satisfy the conditions cond1, cond2, cond3. Then we can construct a unitary operation $U_f$ that prepares a $2^n$-discrete version of $f(x)$ with probability $P\sim \mathcal{F}$ (see Definition definition: filling ratio) and resources no greater than:

Figures (9)

  • Figure 1: The general scheme of block-encoding for matrix $A$. The last operation on the second register indicates measurement of zero, projecting the main $n$-qubit register onto the desired state. Wires for pure ancilla (Definition \ref{['def:pure ancilla']}) are depicted as green dash-dotted lines; workspace ancilla (Definition \ref{['definition:workspace_ancilla']}) are shown as red dotted wires; and the system register is represented by a solid black wire.
  • Figure 2: The geometrical interpretation of the filling ratio. The filling ratio is the area under the function $\abs{f(x)}^2$ (blue area) relative to its doubled absolute maximum value $2\abs{f_{\max}}^2$ (area in the dashed rectangle).
  • Figure 3: (a) The general view of multi-control operator $C_U^b$ from the definition \ref{['def:multiconrol operator']}. Wires for pure ancilla are depicted as green dash-dotted lines; and the system register is represented by a solid black wire. Select operation means that the operator $U$ is applied only if the state on the upper register is $\ket{b}^n$ (b) Explicit quantum circuit for implementation $C_X^{00\dots 0}$. (c) Explicit quantum circuit for implementation $C^{11\dots1}_X$ which we use everywhere in this paper. We use $2m-3$ Toffoli application and $m-2$ pure ancillas.
  • Figure 4: A construction of the Amplitude-oracle for $2^n\times 2^n$ matrix $\hat{x}$ which is $(1,\lceil \log_2n\rceil,0)$-block-encoding. The construction using LCU technique as in Eq. (\ref{['eq: LCU for x']}). The unitary $U_x$ is defined in the Eq. (\ref{['eq:aux superposition for O_x']}) with its transposed version $U_x^T$. Workspace ancilla are shown as red dotted wires; and the system register is represented by a solid black wire.
  • Figure 5: Alternating phase modulation sequence $U_\Phi$ for $\hat{O}_x$ with odd $q$. Workspace ancilla are shown as red dotted wires; and the system register is represented by a solid black wire.
  • ...and 4 more figures

Theorems & Definitions (16)

  • Definition 1: Multi-control operator
  • Definition 2: Pure ancilla
  • Remark 1
  • Definition 3: Block-encoding(modified Definition 43 from gilyen2019quantum)
  • Definition 4: Workspace ancilla
  • Definition 5: Filling ratio rattew2022preparing
  • Theorem 1: Linear combination of polynomials
  • Remark 2
  • Theorem 2: Piece-wise polynomial function
  • Lemma 1: Amplitude-oracle for coordinate operator (Appendix A from guseynov2023depth)
  • ...and 6 more