Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Assumption-free fidelity bounds for hardware noise characterization

Nicolo Colombo

TL;DR

This work uses Machine Learning data-driven approaches and Conformal Prediction, a Machine Learning uncertainty quantification tool known for its mild assumptions and finite-sample validity, to find theoretically valid upper bounds of the fidelity between noiseless and noisy outputs of quantum devices.

Abstract

In the Quantum Supremacy regime, quantum computers may overcome classical machines on several tasks if we can estimate, mitigate, or correct unavoidable hardware noise. Estimating the error requires classical simulations, which become unfeasible in the Quantum Supremacy regime. We leverage Machine Learning data-driven approaches and Conformal Prediction, a Machine Learning uncertainty quantification tool known for its mild assumptions and finite-sample validity, to find theoretically valid upper bounds of the fidelity between noiseless and noisy outputs of quantum devices. Under reasonable extrapolation assumptions, the proposed scheme applies to any Quantum Computing hardware, does not require modeling the device's noise sources, and can be used when classical simulations are unavailable, e.g. in the Quantum Supremacy regime.

Assumption-free fidelity bounds for hardware noise characterization

TL;DR

This work uses Machine Learning data-driven approaches and Conformal Prediction, a Machine Learning uncertainty quantification tool known for its mild assumptions and finite-sample validity, to find theoretically valid upper bounds of the fidelity between noiseless and noisy outputs of quantum devices.

Abstract

In the Quantum Supremacy regime, quantum computers may overcome classical machines on several tasks if we can estimate, mitigate, or correct unavoidable hardware noise. Estimating the error requires classical simulations, which become unfeasible in the Quantum Supremacy regime. We leverage Machine Learning data-driven approaches and Conformal Prediction, a Machine Learning uncertainty quantification tool known for its mild assumptions and finite-sample validity, to find theoretically valid upper bounds of the fidelity between noiseless and noisy outputs of quantum devices. Under reasonable extrapolation assumptions, the proposed scheme applies to any Quantum Computing hardware, does not require modeling the device's noise sources, and can be used when classical simulations are unavailable, e.g. in the Quantum Supremacy regime.

Paper Structure

This paper contains 31 sections, 3 theorems, 28 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contribution
  3. CP for physical systems.
  4. Unpaired samples.
  5. Non-exchangeability.
  6. Sample reweighting.
  7. Calibration training.
  8. A general scheme.
  9. Related Work
  10. CP for generative models
  11. Paired and unpaired samples
  12. CP for QC
  13. Quantum Error Mitigation
  14. Methods
  15. The Bhattacharyya Coefficient (BC)
  16. ...and 16 more sections

Key Result

Lemma 2.3

Let $S_n = \phi(\hat{Y}_n)$, $n=1, \dots, N+1$ be defined as in Assumption assumption ordinal and assume $BC(Y_n, \hat{Y}_n) \sim {\cal N}(S_n, 1)$ for all $n=1, \dots, N+1$ with $S_{n} = s_1$ for $n=1, \dots, \bar{n}$, $S_{n} = s_2 > s_1$ for $n=\bar{n}+1, \dots, N$, and $S_{N+1} = s_3 > s_2$ Then, the validity gap defined in approximate validity is reduced by discarding the first $\bar{n}$ sampl

Figures (3)

  • Figure 1: The Bloch sphere represents a qubit, $|\psi \rangle$ as the complex superposition of two states, $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$, with $\alpha = \cos(\theta/2)$, $\beta = e^{i\phi}\sin(\theta/2)$, ${|0\rangle ={\bigl [}{10}{\bigr ]}}$, and ${|1\rangle ={\bigl [}{10}{\bigr ]}}$. Measuring a qubit can only produce two outcomes, $0$ with probability $\alpha^2$ and $1$ with probability $\beta^2$, and destroys the superposition (by setting either $\alpha$ or $\beta$ to 0). If the qubit is entangled with another, measuring the first may affect the (unmeasured) state of the second. A quantum circuit consists of a series of matrix operators acting on $|\psi\langle$ before it is measured.
  • Figure 2: The modular 4-qubit quantum circuit of martina2022learning. The 3-gate structure is repeated 3 times to increase the circuit depth. The top horizontal lines and blocks represent the device's interacting qubits and logical gates. The bottom line represents the device output. In this case, only 2 qubits are measured, $q2$ and $q3$, making each run produce a 2-digit binary string, e.g. $Y_{m}, \hat{Y}_{m} \in \{ {\tt 00}, {\tt 01},{\tt 10}, {\tt 11} \}$. The first two blue 2-qubit links represent CNOT gates and the third 3-qubit is a Toffoli gate. See barenco1995elementary for a formal definition. The red boxes are Hadamard gates, used to prepare the unobserved qubits into an equal superposition state $\frac{1}{2}(|0\rangle + |1\rangle)$. By convention, the initial state of all qubits is $|0\rangle$. The grey vertical lines represent 4-qubit synchronization.
  • Figure 3: Scatter plot between the theoretical Bhattacharyya distance, $\bar{d}_{\rm BC}$ defined in \ref{['Bhattacharyya distance']}, and the density-ratio estimation of $d_{\rm BC}$, $d_{\rm KL}$, and $d_{\rm TV}$ obtained for all weights in the first of 5 simulations. All distances were min-max normalized before computing the correlation. The legend reports the average correlation and corresponding standard deviations on the 5 equivalent runs.

Theorems & Definitions (3)

  • Lemma 2.3
  • Theorem 2.5
  • Lemma 2.6