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Quantitative Universal Approximation for Noisy Quantum Neural Networks

Lukas Gonon, Antoine Jacquier, Marcel Mordarski

Abstract

We provide here a universal approximation theorem with precise quantitative error bounds for noisy quantum neural networks. We focus on applications to Quantitative Finance, where target functions are often given as expectations. We further provide a detailed numerical analysis, testing our results on actual noisy quantum hardware.

Quantitative Universal Approximation for Noisy Quantum Neural Networks

Abstract

We provide here a universal approximation theorem with precise quantitative error bounds for noisy quantum neural networks. We focus on applications to Quantitative Finance, where target functions are often given as expectations. We further provide a detailed numerical analysis, testing our results on actual noisy quantum hardware.

Paper Structure

This paper contains 27 sections, 13 theorems, 79 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quantum approximations of expectation functions
  3. Notations and background
  4. Extensions to expectation functions
  5. Examples
  6. Gaussian density
  7. Put option in the Bachelier model
  8. Black-Scholes
  9. Noisy Quantum neural networks
  10. Background on Quantum operations
  11. Noisy quantum circuit
  12. Noisy probabilities
  13. Noisy Quantitative Universal Approximation Theorem
  14. Depolarising noise channel
  15. Hardware-parameterised model for depolarising noise
  16. ...and 12 more sections

Key Result

Proposition 2.2

Let $L$ be an $\mathbb{R}^d$-valued random variable and $f(\boldsymbol{x}) := \mathbb{E}[\Phi_{1}({\boldsymbol{x}+L})]$ on $\mathbb{R}^d$. If $\Phi_{1} \in L^1(\mathbb{R}^d)$ and $\boldsymbol{\xi} \mapsto \mathbb{E}[\mathrm{e}^{\mathrm{i} L \cdot \boldsymbol{\xi} }]$ is integrable, then Statement st

Figures (10)

  • Figure 1: Fidelity computed in Proposition \ref{['prop:fidelity']} with $\mathfrak{n} \in \{1, 2, 8\}$ qubits.
  • Figure 2: Minimal number of required qubits according to \ref{['eq:nf_epsilon']}
  • Figure 3: Gaussian density approximations (Statement \ref{['st:state']}). (Part 1 of 2)
  • Figure 4: Gaussian density approximations (Statement \ref{['st:state']}). (Part 2 of 2)
  • Figure 5: Black--Scholes Put surface approximation (Method A, $n=8$, $\mathfrak{n}=5$ qubits, $40\times 40$ grid, $S_{0}=100$, $r=0.03$). Theoretical bound from Example \ref{['ex:TruncatedPutBound']}.
  • ...and 5 more figures

Theorems & Definitions (33)

  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Corollary 2.4
  • proof
  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3: Pure state decomposition
  • Proposition 3.4
  • proof
  • ...and 23 more