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SPAM Tolerance for Pauli Error Estimation

Ryan O'Donnell, Samvitti Sharma

TL;DR

The paper addresses the problem of learning Pauli error rates for an unknown $n$-qubit Pauli channel in a setting with SPAM errors, aiming for subexponential resource use while using only unentangled state preparation and measurements. It builds a SPAM-tolerant reduction to Population Recovery by modeling SPAM as a combined $\mathrm{BSC}_{δ/2} \circ Z_{1/3}$ channel (via a $Z$-flip framework) and extending complex-analytic Population Recovery techniques to this composite channel. The main contributions are: (i) a SPAM-robust, entanglement-free Pauli error estimation algorithm with sample complexity $m = \exp\left(O\big((δn)^{1/3}\ln^{2/3}(1/ε)\big)\right)$ in the relevant regime, (ii) a near-SPAM-free bound of $O(n\log(n/ε))$ factors in the low-SPAM regime, and (iii) a nontrivial lower bound showing that any SPAM-tolerant method must use at least on the order of $\exp(n^{1/3})$ channel uses; the results extend to general Pauli-twirled channels. Overall, the work demonstrates subexponential, SPAM-robust Pauli error estimation with explicit generating-function and complex-analysis machinery, significantly advancing practical error characterization for quantum devices.

Abstract

The Pauli channel is a fundamental model of noise in quantum systems, motivating the task of Pauli error estimation. We present an algorithm that builds on the reduction to Population Recovery introduced in [FO21]. Addressing an open question from that work, our algorithm has the key advantage of robustness against even severe state preparation and measurement (SPAM) errors. To tolerate SPAM, we must analyze Population Recovery on a combined erasure/bit-flip channel, which necessitates extending the complex analysis techniques from [PSW17, DOS17]. For $n$-qubit channels, our Pauli error estimation algorithm requires only $\exp(n^{1/3})$ unentangled state preparations and measurements, improving on previous SPAM-tolerant algorithms that had $2^n$-dependence even for restricted families of Pauli channels. We also give evidence that no SPAM-tolerant method can make asymptotically fewer than $\exp(n^{1/3})$ uses of the channel.

SPAM Tolerance for Pauli Error Estimation

TL;DR

The paper addresses the problem of learning Pauli error rates for an unknown -qubit Pauli channel in a setting with SPAM errors, aiming for subexponential resource use while using only unentangled state preparation and measurements. It builds a SPAM-tolerant reduction to Population Recovery by modeling SPAM as a combined channel (via a -flip framework) and extending complex-analytic Population Recovery techniques to this composite channel. The main contributions are: (i) a SPAM-robust, entanglement-free Pauli error estimation algorithm with sample complexity in the relevant regime, (ii) a near-SPAM-free bound of factors in the low-SPAM regime, and (iii) a nontrivial lower bound showing that any SPAM-tolerant method must use at least on the order of channel uses; the results extend to general Pauli-twirled channels. Overall, the work demonstrates subexponential, SPAM-robust Pauli error estimation with explicit generating-function and complex-analysis machinery, significantly advancing practical error characterization for quantum devices.

Abstract

The Pauli channel is a fundamental model of noise in quantum systems, motivating the task of Pauli error estimation. We present an algorithm that builds on the reduction to Population Recovery introduced in [FO21]. Addressing an open question from that work, our algorithm has the key advantage of robustness against even severe state preparation and measurement (SPAM) errors. To tolerate SPAM, we must analyze Population Recovery on a combined erasure/bit-flip channel, which necessitates extending the complex analysis techniques from [PSW17, DOS17]. For -qubit channels, our Pauli error estimation algorithm requires only unentangled state preparations and measurements, improving on previous SPAM-tolerant algorithms that had -dependence even for restricted families of Pauli channels. We also give evidence that no SPAM-tolerant method can make asymptotically fewer than uses of the channel.

Paper Structure

This paper contains 19 sections, 15 theorems, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our result
  3. Setup: the channel.
  4. Setup: SPAM.
  5. The connection with Population Recovery, and the difficulty of SPAM
  6. Prior work, and our improvements
  7. Preliminaries
  8. Quantum preliminaries
  9. Population Recovery and classical channels
  10. Recapping FO21paulierror
  11. Modeling SPAM
  12. The algorithm under SPAM.
  13. Individual recovery for combined noise channels
  14. Generating functions
  15. Lower-bounding $\eta(\epsilon)$
  16. ...and 4 more sections

Key Result

theorem 1.1

In the above setup, let $r = r_{\textnormal{prep}} \cdot r_{\textnormal{meas}}$ be the overall SPAM retention parameter, and write $r = 1 - \delta$, which may be thought of as the overall rate of SPAM. Let a precision parameter $0 < \epsilon \leq 1/2$ be given, and assume $\delta \leq .99$. Then the

Figures (2)

  • Figure 1: Chord $I_0$ and arc $I_{\theta/2}$ on circle $\partial D_{\mathfrak{r}_1}(1 - \mathfrak{r}_1)$.
  • Figure 2: Arc $I_{\theta/4}$ on circle $\partial D_{2\mathfrak{r}_1\cos(\theta/4)}(1 - 2\mathfrak{r}_1)$.

Theorems & Definitions (39)

  • theorem 1.1
  • definition 2.1
  • definition 2.2
  • definition 2.3
  • definition 2.4
  • definition 2.5
  • definition 2.6
  • definition 2.7
  • definition 2.8
  • proposition 2.9
  • ...and 29 more