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Nonlinear quantum computation by amplified encodings

Matthias Deiml, Daniel Peterseim

TL;DR

This work develops a comprehensive quantum framework for high-dimensional nonlinear computation by encoding variables as amplitudes and employing block encodings to realize nonlinear operations. The authors introduce amplification techniques to preserve information efficiency and construct quantum variants of fixed-point iteration and Newton's method, deriving runtimes that are near-optimal in the tolerance parameter ε and can scale polylogarithmically with problem size in favorable cases. Key contributions include a modular set of encoding/operation tools, constructive encodings for multivariate polynomials and Jacobians, and a practical path to solving nonlinear systems on quantum hardware with demonstrated simulations and limited hardware experiments. The results suggest a potential quantum advantage for solving high-dimensional nonlinear problems, such as discretized nonlinear PDEs, while acknowledging substantial hardware and implementation challenges that remain to be overcome.

Abstract

This paper presents a novel framework for high-dimensional nonlinear quantum computation that exploits tensor products of amplified vector and matrix encodings to efficiently evaluate multivariate polynomials. The approach enables the solution of nonlinear equations by quantum implementations of the fixed-point iteration and Newton's method, with quantitative runtime bounds derived in terms of the error tolerance. These results show that a quantum advantage, characterized by a logarithmic scaling of complexity with the dimension of the problem, is preserved. While Newton's method attains near-optimal theoretical complexity, the fixed-point iteration may be better suited to near-term noisy hardware, as supported by our numerical experiments.

Nonlinear quantum computation by amplified encodings

TL;DR

This work develops a comprehensive quantum framework for high-dimensional nonlinear computation by encoding variables as amplitudes and employing block encodings to realize nonlinear operations. The authors introduce amplification techniques to preserve information efficiency and construct quantum variants of fixed-point iteration and Newton's method, deriving runtimes that are near-optimal in the tolerance parameter ε and can scale polylogarithmically with problem size in favorable cases. Key contributions include a modular set of encoding/operation tools, constructive encodings for multivariate polynomials and Jacobians, and a practical path to solving nonlinear systems on quantum hardware with demonstrated simulations and limited hardware experiments. The results suggest a potential quantum advantage for solving high-dimensional nonlinear problems, such as discretized nonlinear PDEs, while acknowledging substantial hardware and implementation challenges that remain to be overcome.

Abstract

This paper presents a novel framework for high-dimensional nonlinear quantum computation that exploits tensor products of amplified vector and matrix encodings to efficiently evaluate multivariate polynomials. The approach enables the solution of nonlinear equations by quantum implementations of the fixed-point iteration and Newton's method, with quantitative runtime bounds derived in terms of the error tolerance. These results show that a quantum advantage, characterized by a logarithmic scaling of complexity with the dimension of the problem, is preserved. While Newton's method attains near-optimal theoretical complexity, the fixed-point iteration may be better suited to near-term noisy hardware, as supported by our numerical experiments.

Paper Structure

This paper contains 11 sections, 11 theorems, 75 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Quantum building blocks for nonlinear computation
  4. Block encoding of nonlinear operations
  5. Amplification of block encodings
  6. Operations on amplified encodings
  7. Solution of nonlinear systems of equations
  8. Fixed-point iteration
  9. Newton's method
  10. Numerical experiments
  11. Conclusion

Key Result

Theorem 1

There is an algorithm that takes as input block encodings of the coefficients of a multivariate polynomial $f$ as well as a circuit that prepares a state $|x^{(0)}\rangle$ and outputs a given quantity of interest of the solution $x^*$ of eq:nonlinear-eq with arbitrarily small error tolerance $\varep for any $\mu >0$, provided that the classical Newton's method converges quadratically for the initi

Figures (3)

  • Figure 1: Circuit to prepare and amplify a state $|v^2\rangle$ representing the element-wise square of a vector $v$, where $U_v$ is a circuit preparing the state $|v\rangle$.
  • Figure 2: Constructive encoding implementing $g(x)$. The angle $\theta$ is related to the addition and depends on the norm of $x$, specifically $\theta = 2 \tan^{-1} |x|/\sqrt[4]{2}$.
  • Figure 3: First three fixed-point iterations of $g$ with starting value $[1\; 1]^T$ (left) and first two Newton iterations with starting value $[2\; \tfrac{1}{4}]^T$ (right). The plot includes classically computed reference values ($+$), values computed using a noiseless quantum simulator ($\times$) and the quantum computers ibm_fez and ibm_aachen ($\square$) available through IBM Cloud.

Theorems & Definitions (28)

  • Theorem 1: Main result: Quantum nonlinear solver for polynomials
  • Definition 2: Projection as $\operatorname{C}_{\Pi}\!\operatorname{NOT}$ gate
  • Definition 3: Block encoding of nonunitary operations DP25
  • Example 4: Block encoding of element-wise multiplication
  • Example 5: Block encoding of element-wise powers of a vector
  • Definition 6: Equivalence of projections
  • Lemma 7: Amplitude amplification
  • Theorem 8: Normalization of block encodings
  • proof
  • Remark 9: Practical improvements of the normalization algorithm
  • ...and 18 more