Estimating shots and variance on noisy quantum circuits

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 $σ^2 \approx 0.01$

💡 Research Summary

The paper addresses a practical problem in the noisy intermediate‑scale quantum (NISQ) era: how many measurement repetitions (“shots”) are required to achieve a desired variance in the output of a quantum circuit. The authors develop a two‑stage methodology that starts with a bottom‑up physical characterization of single‑qubit noise sources and culminates in a top‑down empirical model for multi‑qubit expectation‑value circuits.

In the first stage, four independent noise channels are considered: state‑preparation‑and‑measurement (SP‑AM) errors, amplitude‑damping (T₁), phase‑damping (T₂), and gate depolarizing errors. Using a simple Hadamard “coin‑toss” circuit, the authors verify that the standard deviation of measurement outcomes scales as 1/√n, even in the presence of noise, by invoking the Central Limit Theorem (CLT). They introduce the relative standard deviation (RSD) and plot log(RSD) versus log(window size) to extract a linear relationship with slope –½. For SP‑AM errors, they derive an analytical expression for the intercept c in terms of the asymmetric readout probabilities p₀→₁ and p₁→₀. Experiments on IBM T orino hardware confirm that the predicted c values match measured ones within 0.01, demonstrating that asymmetric readout errors dominate the bias floor.

T₁ noise is modeled by inserting idle “wait” gates of variable duration, leading to an exponential decay factor ε = e^{‑t/T₁}. The modified SP‑AM expression predicts how the intercept shifts with ε. Histogram analysis of experimental data shows a median error of –0.007 and a mean error of 0.1 (or –0.013 after outlier removal), indicating accurate prediction of the T₁‑induced bias.

T₂ noise is treated analogously, with a phase‑damping factor that increases the probability of a Z error. The authors propagate calibration uncertainties (σ_T₂) to obtain σ_c = (∂c/∂T₂)σ_T₂, showing that the intercept uncertainty falls as 1/T₂², which is confirmed experimentally.

Gate errors are approximated as a depolarizing channel with equal probabilities for X, Y, and Z errors (p = E/3, where E is the error per layered gate obtained from randomized benchmarking). For k gates, the effective error probability is approximated as P_k ≈ k·p for small p. Substituting these probabilities into the SP‑AM formula yields a predicted intercept that matches hardware measurements with an average absolute error of about 0.1, indicating that the simple depolarizing model captures the dominant contribution of gate noise.

The second stage tackles multi‑qubit circuits. The authors argue that deriving the variance of an arbitrary circuit from individual qubit variances is intractable because of exponential scaling and entanglement‑induced covariances. They therefore restrict attention to expectation‑value circuits, where the observable variance σ² = ⟨O²⟩ – ⟨O⟩² can be measured directly. The variance of the estimator over N shots is decomposed as

 Var(Ē_N) = A_N + B, A ≈ σ²/N, B ≥ 0,

where A captures the statistical sampling noise and B represents a systematic bias floor arising from hardware imperfections. Crucially, both A and B can be estimated with a single additional run of the circuit, providing a practical recipe for shot‑count planning.

The methodology is validated on a Variational Quantum Eigensolver (VQE) for the H₂ molecule. Using calibration data from IBM Pittsburgh (referred to as “IBM Pittsburgh” in the paper) and the noise models developed earlier, the authors predict that approximately 7 000 shots would be sufficient to achieve a variance σ² ≈ 0.01 for the H₂ energy estimator. Experimental runs confirm that the predicted variance lies within the theoretical error bounds, outperforming naive trial‑and‑error shot selection.

Overall, the paper contributes a rigorous statistical framework for shot‑budget estimation in noisy quantum hardware. By combining physics‑based single‑qubit noise modeling with a pragmatic top‑down variance decomposition, it offers a scalable approach that can be integrated into quantum software stacks to automate precision‑controlled execution on NISQ devices.