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Autonomous Hamiltonian certification and changepoint detection

Steven T. Flammia, Dmitrii Khitrin, Muzhou Ma, Jamie Sikora, Yu Tong, Alice Zheng

Abstract

Modern quantum devices require high-precision Hamiltonian dynamics, but environmental noise can cause calibrated Hamiltonian parameters to drift over time, necessitating expensive recalibration. Detecting when recalibration is needed is challenging, especially since the very gates required for sophisticated verification protocols may themselves be miscalibrated. While cloud quantum computing services implement heuristic routines for triggering recalibration, the fundamental limits of optimal recalibration are not yet known. We develop efficient Hamiltonian certification and changepoint detection protocols in the autonomous setting, where we cannot rely on an external noiseless device and use only single-qubit gates and measurements, making the protocols robust to the calibration issues for multi-qubit operations they aim to detect. For unknown $n$-qubit Hamiltonians $H$ and $H_0$ with operator norm bounded by $M$, our certification protocol distinguishes whether $\|H-H_0\|_F\geqε$ or $\|H-H_0\|_F\leq O(ε/\sqrt{n})$ with sample complexity $O(nM^2\ln(1/δ)/ε^2)$ and total evolution time $O(nM\ln(1/δ)/ε^2)$. We achieve this by evolving random stabilizer product states and performing adaptive single-qubit measurements based on a classically simulable hypothesis state. Extending this to continuous monitoring, we develop an online changepoint detection algorithm using the CUSUM procedure that achieves a detection delay time bound of $O(nM\ln(M\mathbb{E}_\infty[T])/ε^2)$, matching the known asymptotically optimal scaling with respect to false alarm run time $\mathbb{E}_\infty[T]$. Our approach enables quantum devices to autonomously monitor their own calibration status without requiring ancillary systems, entangling operations, or a trusted reference device, offering a practical solution for robust quantum computing with contemporary noisy devices.

Autonomous Hamiltonian certification and changepoint detection

Abstract

Modern quantum devices require high-precision Hamiltonian dynamics, but environmental noise can cause calibrated Hamiltonian parameters to drift over time, necessitating expensive recalibration. Detecting when recalibration is needed is challenging, especially since the very gates required for sophisticated verification protocols may themselves be miscalibrated. While cloud quantum computing services implement heuristic routines for triggering recalibration, the fundamental limits of optimal recalibration are not yet known. We develop efficient Hamiltonian certification and changepoint detection protocols in the autonomous setting, where we cannot rely on an external noiseless device and use only single-qubit gates and measurements, making the protocols robust to the calibration issues for multi-qubit operations they aim to detect. For unknown -qubit Hamiltonians and with operator norm bounded by , our certification protocol distinguishes whether or with sample complexity and total evolution time . We achieve this by evolving random stabilizer product states and performing adaptive single-qubit measurements based on a classically simulable hypothesis state. Extending this to continuous monitoring, we develop an online changepoint detection algorithm using the CUSUM procedure that achieves a detection delay time bound of , matching the known asymptotically optimal scaling with respect to false alarm run time . Our approach enables quantum devices to autonomously monitor their own calibration status without requiring ancillary systems, entangling operations, or a trusted reference device, offering a practical solution for robust quantum computing with contemporary noisy devices.

Paper Structure

This paper contains 18 sections, 8 theorems, 54 equations, 4 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Main results
  3. Certification:
  4. Changepoint detection:
  5. Comparison to alternative procedures
  6. Hamiltonian certification protocol
  7. Adaptive protocol
  8. Detailed analysis of the adaptive protocol
  9. Hamiltonian changepoint detection
  10. Cumulative sum (CUSUM)
  11. Performance and optimality:
  12. Robustness:
  13. Hamiltonian changepoint algorithm and analysis
  14. Numerical experiments
  15. Hamiltonian certification:
  16. ...and 3 more sections

Key Result

Theorem 2

The $\textnormal{HamCert}(\mathcal{O}(\epsilon/\sqrt{n}), \epsilon, \norm{\cdot}_F)$ problem can be solved with probability at least $1 - \delta$ by an autonomous quantum algorithm using $N$ product state inputs and product basis measurements, and total evolution time $T$, where

Figures (4)

  • Figure 1: An example application of our changepoint detection procedure to the recalibration problem. (a) The scheduling of a quantum device is divided into interleaved sections of certification (which run our change detection algorithm) and computing (which run other scheduled algorithms). The certification procedure can potentially trigger a recalibration procedure, which incurs a period of downtime. (b) Example Hamiltonian dynamics, with red regions indicating significant deviation from the target. The first shown certification section encounters this deviation and recalibrates; the second does not and proceeds as usual. (c) Same as the above in the frequency space. Lines within the red region indicate deviation above the upper threshold of our change detection procedure.
  • Figure 2: (left) Results of running \ref{['algo:adaptive_ham_certif']} given by the probability of accepting the Hamiltonian $H$ on logarithmic scale. (right) Corresponding certification verdicts. The Hamiltonian is rejected when the fraction of negative outcomes is greater than $10^{-4}$. The acceptance probabilities are calculated as the number of positive outcomes in the $2 \times10^4$-element array we obtain for each pair $(n, \norm{H-H_0}_F)$. The error bars correspond to one Wilson interval. $\norm{\cdot}_F$ denotes the normalized Frobenius norm.
  • Figure 3: Results of running \ref{['algo:ham_changepoint']} on a sequence of Hamiltonians that experiences (left) a significant change and (right) only small changes, with the blue dotted series showing Hamiltonian deviation over time. The red series is the cumulative sum $s_i$ aggregated by CUSUM (median and $95\%$ interpercentile range across $120$ trials).
  • Figure 4: Expected termination time of CUSUM when (left) run directly, and (right) used in \ref{['algo:ham_changepoint']} for Hamiltonian change detection. The right-hand plot shows the median and $95\%$ interpercentile range across $120$ trials.

Theorems & Definitions (18)

  • Definition 1: Hamiltonian certification problem
  • Theorem 2: Adaptive Hamiltonian certification
  • Definition 3: Hamiltonian changepoint problem
  • Theorem 4: Autonomous Hamiltonian change detection
  • Definition 5: Normal ordering
  • Definition 6
  • Lemma 7: CUSUM optimality lorden1971procedures
  • proof : Proof of \ref{['thm:online']}
  • Lemma 8: Following molloy2019minimax
  • Lemma 9
  • ...and 8 more