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Preconditioned Multivariate Quantum Solution Extraction

Gumaro Rendon, Stepan Smid

TL;DR

This paper tackles the explicit extraction of a function encoded in quantum amplitudes when solving differential equations with quantum algorithms, addressing the bottleneck where direct amplitude readout negates potential speedups. It introduces preconditioning by shifting the target function with a constant distribution to bound the effective condition number, enabling extraction at the Heisenberg limit, and extends the method to multivariate functions using cumulative distribution sampling and Chebyshev interpolation. The approach combines integral estimation via quantum amplitude estimation with tensor-product Chebyshev interpolation to reconstruct the full function from the encoded amplitudes, achieving favorable scaling in dimension and avoiding dependence on a vanishing minimum amplitude. Numerical simulations in one dimension using Grover-Rudolph encoding illustrate the pipeline from encoding to integral estimation and function extraction, while highlighting practical challenges and the need for additional qubits to realize accurate results. The work paves the way for robust quantum solution extraction in PDE tasks, offering a concrete route toward practical quantum advantages in high-dimensional settings.

Abstract

Numerically solving partial differential equations is a ubiquitous computational task with broad applications in many fields of science. Quantum computers can potentially provide high-degree polynomial speed-ups for solving PDEs, however many algorithms simply end with preparing the quantum state encoding the solution in its amplitudes. Trying to access explicit properties of the solution naively with quantum amplitude estimation can subsequently diminish the potential speed-up. In this work, we present a technique for extracting a smooth positive function encoded in the amplitudes of a quantum state, which achieves the Heisenberg limit scaling. We improve upon previous methods by allowing higher dimensional functions, by significantly reducing the quantum complexity with respect to the number of qubits encoding the function, and by removing the dependency on the minimum of the function using preconditioning. Our technique works by sampling the cumulative distribution of the given function, fitting it with Chebyshev polynomials, and subsequently extracting a representation of the whole encoded function. Finally, we trial our method by carrying out small scale numerical simulations.

Preconditioned Multivariate Quantum Solution Extraction

TL;DR

This paper tackles the explicit extraction of a function encoded in quantum amplitudes when solving differential equations with quantum algorithms, addressing the bottleneck where direct amplitude readout negates potential speedups. It introduces preconditioning by shifting the target function with a constant distribution to bound the effective condition number, enabling extraction at the Heisenberg limit, and extends the method to multivariate functions using cumulative distribution sampling and Chebyshev interpolation. The approach combines integral estimation via quantum amplitude estimation with tensor-product Chebyshev interpolation to reconstruct the full function from the encoded amplitudes, achieving favorable scaling in dimension and avoiding dependence on a vanishing minimum amplitude. Numerical simulations in one dimension using Grover-Rudolph encoding illustrate the pipeline from encoding to integral estimation and function extraction, while highlighting practical challenges and the need for additional qubits to realize accurate results. The work paves the way for robust quantum solution extraction in PDE tasks, offering a concrete route toward practical quantum advantages in high-dimensional settings.

Abstract

Numerically solving partial differential equations is a ubiquitous computational task with broad applications in many fields of science. Quantum computers can potentially provide high-degree polynomial speed-ups for solving PDEs, however many algorithms simply end with preparing the quantum state encoding the solution in its amplitudes. Trying to access explicit properties of the solution naively with quantum amplitude estimation can subsequently diminish the potential speed-up. In this work, we present a technique for extracting a smooth positive function encoded in the amplitudes of a quantum state, which achieves the Heisenberg limit scaling. We improve upon previous methods by allowing higher dimensional functions, by significantly reducing the quantum complexity with respect to the number of qubits encoding the function, and by removing the dependency on the minimum of the function using preconditioning. Our technique works by sampling the cumulative distribution of the given function, fitting it with Chebyshev polynomials, and subsequently extracting a representation of the whole encoded function. Finally, we trial our method by carrying out small scale numerical simulations.
Paper Structure (15 sections, 2 theorems, 82 equations, 5 figures)

This paper contains 15 sections, 2 theorems, 82 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preconditioning
  3. Shift by a constant
  4. Estimation of $a_{\rm shift}$
  5. Bounds on the condition number $\kappa(\widetilde{\psi})$
  6. Multivariate solution extraction
  7. Integral estimation
  8. Chebyshev interpolation of the integral
  9. Function extraction
  10. Numerical simulations
  11. Grover-Rudolph technique for function encoding
  12. Integral estimation
  13. Function extraction
  14. Illustrative function extraction with controlled errors
  15. Conclusions and outlook

Key Result

Lemma 1.1

Provided an unknown analytic function function $\psi(x)$ for $x \in [-1,1]$ that is normalized: and whose derivatives are bounded by $\left| \frac{{\rm d}^j\psi(x)}{d x^j} \right| \leq \Lambda^{j+1}$, which can be stored in a quantum memory with $n$ qubits the following way: where $\mathcal{N} = \sum_j |\psi(x_j)|^2$, and $x_j = 2 j/2^n -1$, one can estimate it at a quantum gate cost that goes

Figures (5)

  • Figure 1: Pseudo-code for the Solution Extraction Algorithm
  • Figure 2: Encoding the function $\psi(x)\propto (\sin(5x)+2)e^x$ into a quantum state with $n=5$ qubits using the Grover–Rudolph method with different numbers of ancilla qubits controlling the precision, and comparing the rescaled amplitudes with the exact functional values.
  • Figure 3: Measuring and interpolating the integral $\Psi(x) \propto \int_{-1}^x (\sin(5y)+2)e^y\ \mathrm{d} y$ with the described technique, using the encoded function $\psi(x) \propto (\sin(5x)+2)e^x$ in $n=5$ qubits with $m=6$ ancillas, and QPE with $K=5$ qubits for precision, and $M=17$ Chebyshev nodes (or their closest $n$-bit approximations). The figure shows individual measured points, and compares the function interpolated from them with the exact integral $\Psi(x)$.
  • Figure 4: Function extracted from the measured integral values on Figure \ref{['fig:Psi_integral_measured']}. This step incurs errors proportional to the number of points $M$ cubed. The precision in the measured values of $\Psi(x)$ in this small scale simulation wasn't sufficient in order to yield a reasonable estimate for $\psi(x)$.
  • Figure 5: Demonstration of extracting the function $\psi(x)$ from the values of $\Psi(x)$ sampled at the same $M=17$ Chebyshev nodes as in Figure \ref{['fig:Psi_integral_measured']}, but instead of a quantum circuit simulation, here we sample $\Psi(x)$ exactly and add random Gaussian noise to it, which we control to be roughly $100\times$ or $20\times$ smaller than the one we measured respectively. These precisions would require only a several more qubits in each register, which we, however, can't simulate.

Theorems & Definitions (5)

  • Lemma 1.1
  • proof
  • proof
  • Remark 3.1
  • Theorem 3.2