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Nonlinear transformation of complex amplitudes via quantum singular value transformation

Naixu Guo, Kosuke Mitarai, Keisuke Fujii

TL;DR

This work defines Nonlinear Transformation of Complex Amplitudes (NTCA) and shows how to implement nonlinear edits to quantum state amplitudes by encoding amplitudes into singular values via block-encoding and applying Quantum Singular Value Transformation (QSVT). The authors introduce a block-encoding of amplitudes that yields Hermitian matrices whose eigenvalues reproduce the real and imaginary parts of the state amplitudes, and then construct a NTCA algorithm that approximates polynomial activations $P'$ and $Q'$ to transform amplitudes as $P'(x_k)+Q'(y_k)$. They establish a complexity scaling of $\mathcal{O}\big(d\gamma\sqrt{N/\sum_k|P'(x_k)+Q'(y_k)|^2}\big)$ uses of controlled-$U$ (with $d$ the polynomial degree and $\gamma$ a normalization factor), and show an exponential speedup in precision and a favorable $\sqrt{N}$ dependence on input dimension compared to prior work. The method enables quantum neural-network–style nonlinear processing of data encoded in complex amplitudes and opens avenues for quantum machine learning with nonlinear activation functions on quantum hardware. The results highlight a principled route to introduce substantial nonlinearity into quantum state evolution, potentially impacting quantum learning and data analysis tasks on quantum devices.

Abstract

Due to the linearity of quantum operations, it is not straightforward to implement nonlinear transformations on a quantum computer, making some practical tasks like a neural network hard to be achieved. In this work, we define a task called nonlinear transformation of complex amplitudes and provide an algorithm to achieve this task. Specifically, we construct a block-encoding of complex amplitudes from a state preparation unitary. This allows us to transform the complex amplitudes by using quantum singular value transformation. We evaluate the required overhead in terms of input dimension and precision, which reveals that the algorithm depends on the roughly square root of input dimension and achieves an exponential speedup on precision compared with previous work. We also discuss its possible applications to quantum machine learning, where complex amplitudes encoding classical or quantum data are processed by the proposed method. This paper provides a promising way to introduce highly complex nonlinearity of the quantum states, which is essentially missing in quantum mechanics.

Nonlinear transformation of complex amplitudes via quantum singular value transformation

TL;DR

This work defines Nonlinear Transformation of Complex Amplitudes (NTCA) and shows how to implement nonlinear edits to quantum state amplitudes by encoding amplitudes into singular values via block-encoding and applying Quantum Singular Value Transformation (QSVT). The authors introduce a block-encoding of amplitudes that yields Hermitian matrices whose eigenvalues reproduce the real and imaginary parts of the state amplitudes, and then construct a NTCA algorithm that approximates polynomial activations and to transform amplitudes as . They establish a complexity scaling of uses of controlled- (with the polynomial degree and a normalization factor), and show an exponential speedup in precision and a favorable dependence on input dimension compared to prior work. The method enables quantum neural-network–style nonlinear processing of data encoded in complex amplitudes and opens avenues for quantum machine learning with nonlinear activation functions on quantum hardware. The results highlight a principled route to introduce substantial nonlinearity into quantum state evolution, potentially impacting quantum learning and data analysis tasks on quantum devices.

Abstract

Due to the linearity of quantum operations, it is not straightforward to implement nonlinear transformations on a quantum computer, making some practical tasks like a neural network hard to be achieved. In this work, we define a task called nonlinear transformation of complex amplitudes and provide an algorithm to achieve this task. Specifically, we construct a block-encoding of complex amplitudes from a state preparation unitary. This allows us to transform the complex amplitudes by using quantum singular value transformation. We evaluate the required overhead in terms of input dimension and precision, which reveals that the algorithm depends on the roughly square root of input dimension and achieves an exponential speedup on precision compared with previous work. We also discuss its possible applications to quantum machine learning, where complex amplitudes encoding classical or quantum data are processed by the proposed method. This paper provides a promising way to introduce highly complex nonlinearity of the quantum states, which is essentially missing in quantum mechanics.

Paper Structure

This paper contains 15 sections, 3 theorems, 50 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Result
  4. Block-encoding of amplitudes
  5. Nonlinear transformation of complex amplitudes
  6. Comparison with previous work
  7. Application to quantum machine learning
  8. Conclusion
  9. Details for Theorem \ref{['BCA']}
  10. Derivation of Eq. (\ref{['eq:op_W']})
  11. Eigenvalues and eigenvectors of $G$
  12. One- and two-qubit gate counts
  13. Detailed error analysis for Theorem \ref{['NTCAalg']}
  14. Polynomial approximation of tanh
  15. Generalization to multi-layer neural network

Key Result

Lemma 3

Suppose that $U$ is an $(\alpha, a, \varepsilon)$-block-encoding of a Hermitian matrix $A$. If $\delta \geq 0$ and $P:\mathbb{R}\rightarrow \mathbb{C}$ is a d-degree polynomial satisfying that then there is a quantum circuit $\tilde{U}$, which is an $(1, a+3, 4d \sqrt{\varepsilon / \alpha}+N\delta)$-block-encoding of $P(A / \alpha)$, and consists of d applications of $U$ and $U^{\dagger}$ gates,

Figures (4)

  • Figure 1: Definition of operation $W$ in Theorem \ref{['BCA']}
  • Figure 2: Definition of operation $G$ in Theorem \ref{['BCA']}
  • Figure 3: Definition of operation $\tilde{G}$ in Theorem \ref{['BCA']}
  • Figure 4: Quantum circuit for the nonlinear transformation of part of complex amplitudes

Theorems & Definitions (7)

  • Definition 1: Nonlinear transformation of complex amplitudes (NTCA)
  • Definition 2: Block-encoding Low_2019chakraborty_et_al:LIPIcs:2019:10609
  • Lemma 3: Polynomial eigenvalue transformation of arbitrary parity 10.1145/3313276.3316366
  • Theorem 4: Block-encoding of real part amplitudes
  • proof
  • Theorem 5: Nonlinear transformation of complex amplitudes
  • proof