Limitations for adaptive quantum state tomography in the presence of detector noise

TL;DR

The paper investigates adaptive quantum state tomography under detector noise and shows that any nonzero depolarizing readout noise eliminates the asymptotic $1/N$ advantage for pure-state targets, even with readout-error mitigation. By combining Fisher-information optimization with Bayesian inference, the authors demonstrate a fundamental bound on the recoverable information, leading to a transition from ideal to suboptimal scaling in finite statistics, and they reveal a bias–variance trade-off akin to static tomography. Numerically, adaptive strategies yield a constant-factor improvement over non-adaptive methods in realistic noise levels, with the advantage increasing as noise decreases, and detector tomography requirements are shown to significantly influence reconstruction fidelity in practical scenarios. The work highlights that while asymptotic benefits of adaptivity are limited by noise, adaptive measurements remain practically valuable in well-calibrated experiments and motivate further noise-aware designs and experimental validation.

Abstract

Assumption-free reconstruction of quantum states from measurements is essential for benchmarking and certifying quantum devices, but it remains difficult due to the extensive measurement statistics and experimental resources it demands. An approach to alleviating these demands is provided by adaptive measurement strategies, which can yield up to a quadratic improvement in reconstruction accuracy for pure states by dynamically optimizing measurement settings during data acquisition. A key open question is whether these asymptotic advantages remain in realistic experiments, where readout is inevitably noisy. In this work, we analyze the impact of readout noise on adaptive quantum state tomography with readout-error mitigation, focusing on the challenging regime of reconstructing pure states using mixed-state estimators. Using analytical arguments based on Fisher information optimization and extensive numerical simulations using Bayesian inference, we show that any nonzero readout noise eliminates the asymptotic quadratic scaling advantage of adaptive strategies. We numerically investigate the behavior for finite measurement statistics for single- and two-qubit systems with exact readout-error mitigation and find a gradual transition from ideal to sub-optimal scaling. We furthermore investigate realistic scenarios where detector tomography is performed with a limited number of state copies for calibration, showing that insufficient detector characterization leads to estimator bias and limited reconstruction accuracy. Although our result imposes an upper bound on the reconstruction accuracy that can be achieved with adaptive strategies, we nevertheless observe numerically a constant-factor gain in reconstruction accuracy, which becomes larger as the readout noise decreases. This indicates potential practical benefits in using adaptive measurement strategies in well-calibrated experiments.

Figures (12)

• Figure 1: Simulated single-qubit state reconstruction with and without readout-error mitigation for various levels of depolarizing noise $p$. Without readout-error mitigation (dashed lines), the reconstruction infidelity eventually saturates at a bias value set by the distance to the effectively measured noisy state. When exact readout-error mitigation is applied (solid lines) the mean infidelity curves keep decreasing but shift further to the right with increasing readout noise strength. Each curve is averaged over 2000 Haar-random pure states.
• Figure 2: Comparison of mean infidelity scaling between adaptive and non-adaptive Bayesian measurement strategies for single- and two-qubit state reconstruction. A power-law curve $N^a$ is fitted between $10^4$ and $10^5$ measurements, where the scaling exponent $a$ is displayed for (adaptive, non-adaptive) in the legend. The single-qubit non-adaptive approach uses static Pauli-6 measurements. Both adaptive and non-adaptive two-qubit strategies use factorized measurements, i.e. $\mathcal{M} = \mathcal{N}_1\otimes \mathcal{N}_2$. The non-adaptive approach uses static factorized Pauli-6 measurements, i.e. $\mathcal{M} =\mathcal{M}_\text{Pauli-6}\otimes \mathcal{M}_\text{Pauli-6}$. The single- and two-qubit curves are averaged over 1000 and 250 Haar-random states respectively.
• Figure 3: Single- and two-qubit mean infidelities with different amount of depolarizing readout noise. A power-law $N^a$ is fitted to all adaptive curves within the range of $10^4$ to $10^5$ measurements. The resulting scaling exponents $a$ are displayed in the legend. Left: Average over 1000 Haar-random single-qubit states. Right: Average over 250 Haar-random two-qubit states.
• Figure 4: Top: Relative reduction factor between adaptive and non-adaptive method. The zero-noise case continuously decreases while noisy cases level off towards a constant reduction factor. The stronger the readout noise is, the larger the relative improvement factor becomes. Bottom: Rolling power-law fit $I \propto N^{a}$ applied to the adaptive curves in Fig. \ref{['fig:infidelity_comparison']}, where $a$ denotes the scaling exponent. The rolling power-law fit is applied in the range $[x,5x]$, where $x$ is the measurement number. Only ranges that can be filled are plotted. The black dashed horizontal lines are for guidance, showing expected scaling for adaptive and non-adaptive curves. The asymptotic scaling decays faster for stronger readout noise.
• Figure 5: Distribution of the infidelities reached at a given number of measurements for single-qubit curves in Fig. \ref{['fig:infidelity_comparison']}. The noiseless adaptive cases separate from the remaining curves which stay relatively close to each other.