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Estimating shots and variance on noisy quantum circuits

Manav Seksaria, Anil Prabhakar

TL;DR

This work tackles the practical problem of budgeting shots in noisy quantum devices by combining a bottom-up, physics-based analysis of single-qubit noise with a top-down empirical model for multi-qubit expectation-value circuits. It employs the Central Limit Theorem to relate noise sources (SPAM, $T_1$, $T_2$, and gate noise) to a variance budget, decomposing the estimator variance into a quantum term and a bias floor, which can be estimated from minimal additional measurements. The authors validate the approach first on single-qubit coins and then on a hydrogen VQE (H2) test case, showing that a target variance can be achieved with a calculable number of shots (e.g., about 7000 on IBM Pittsburgh for $\sigma^2=0.01$). The work provides a practical framework for shot planning on NISQ devices, highlighting both its utility and the open questions surrounding the origin of the bias floor in multi-qubit systems. It lays groundwork for adaptive shot budgeting and confidence quantification in near-term quantum experiments.

Abstract

We present a method for estimating the number of shots required to achieve a desired variance in the results of a quantum circuit. First, we establish a baseline for single-qubit characterisation of individual noise sources. We then move on to multi-qubit circuits, focusing on expectation-value circuits. We decompose the variance of the estimator into a sum of a statistical term and a bias floor. These are independently estimated with one additional run of the circuit. We test our method on a Variational Quantum Eigensolver for $H_2$ and show that we can predict the variance to within known error bounds. We go on to show that for IBM Pittsburgh's noise characteristics, at that instant, 7000 shots for the given circuit would have achieved a $σ^2 \approx 0.01$

Estimating shots and variance on noisy quantum circuits

TL;DR

This work tackles the practical problem of budgeting shots in noisy quantum devices by combining a bottom-up, physics-based analysis of single-qubit noise with a top-down empirical model for multi-qubit expectation-value circuits. It employs the Central Limit Theorem to relate noise sources (SPAM, , , and gate noise) to a variance budget, decomposing the estimator variance into a quantum term and a bias floor, which can be estimated from minimal additional measurements. The authors validate the approach first on single-qubit coins and then on a hydrogen VQE (H2) test case, showing that a target variance can be achieved with a calculable number of shots (e.g., about 7000 on IBM Pittsburgh for ). The work provides a practical framework for shot planning on NISQ devices, highlighting both its utility and the open questions surrounding the origin of the bias floor in multi-qubit systems. It lays groundwork for adaptive shot budgeting and confidence quantification in near-term quantum experiments.

Abstract

We present a method for estimating the number of shots required to achieve a desired variance in the results of a quantum circuit. First, we establish a baseline for single-qubit characterisation of individual noise sources. We then move on to multi-qubit circuits, focusing on expectation-value circuits. We decompose the variance of the estimator into a sum of a statistical term and a bias floor. These are independently estimated with one additional run of the circuit. We test our method on a Variational Quantum Eigensolver for and show that we can predict the variance to within known error bounds. We go on to show that for IBM Pittsburgh's noise characteristics, at that instant, 7000 shots for the given circuit would have achieved a
Paper Structure (17 sections, 30 equations, 13 figures, 1 table, 1 algorithm)

This paper contains 17 sections, 30 equations, 13 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Single Qubit Experiments
  3. Central Limit Theorem and Relative Standard Deviation
  4. SPAM Noise
  5. Simulation
  6. Hardware
  7. $T_1$ noise
  8. $T_2$ noise
  9. Gate Noise
  10. Circuit Experiments
  11. Expectation Value Circuits
  12. Application
  13. $H_2$ VQE
  14. Conclusion
  15. Appendix
  16. ...and 2 more sections

Figures (13)

  • Figure 1: Circuit baseline fair coin toss.
  • Figure 2: As more shots are taken, the width of the distribution falls off as $1/\sqrt{n}$.
  • Figure 3: When we add asymmetric noise to the coin toss, the distribution shifts downward. However, for symmetric noise, the mean is independent of the amount of noise.
  • Figure 4: The standard deviation varies as $1/\sqrt{n}$ for each of the three cases simulated, noiseless, IBM noisy simulator and CERN noisy simulator.
  • Figure 5: Circuit for $T_1$ noise with variable wait.
  • ...and 8 more figures