Compressed Qubit Noise Spectroscopy: Piecewise-Linear Modeling and Rademacher Measurements

TL;DR

This work advances qubit noise spectroscopy by (i) introducing a second-order TGV-based regularization to enable accurate reconstruction of piecewise-linear spectra $S(\omega)$ from compressed measurements, and (ii) proposing Rademacher measurements that simplify experimental implementation while preserving reconstruction fidelity. Theoretical and numerical results show that CS$_{\text{TGV}}$ can resolve spectral kinks with far fewer measurements than naive approaches, and CS$_{\text{R}}$ (and CS$_{\text{R+TGV}}$) achieves comparable sparse-signal recovery using seed-based, on-the-fly pulse sequences, with provable recovery guarantees. Applications to InAs/GaAs quantum dots demonstrate accurate recovery of realistic spectra, including narrow resonances and broadband backgrounds, and the Rademacher approach enables substantial reductions in control-pulse requirements without sacrificing accuracy. Collectively, these methods broaden the applicability and practicality of random-pulse-based noise spectroscopy for near-term quantum devices, and offer promising directions for further theoretical and experimental development in 1D spectral recovery.

Abstract

Random pulse sequences are a powerful method for qubit noise spectroscopy, enabling efficient reconstruction of sparse noise spectra. Here, we advance this method in two complementary directions. First, we extend the method using a regularizer based on the total generalized variation (TGV) norm, in order to reconstruct a larger class of noise spectra, namely piecewise-linear noise spectra, which more realistically model many physical systems. We show through numerical simulations that the new method resolves finer spectral features, while maintaining an order-of-magnitude speedup over conventional approaches to noise spectroscopy. Second, we simplify the experimental implementation of the method, by introducing Rademacher measurements for reconstructing sparse noise spectra. These measurements use pseudorandom pulse sequences that can be generated in real time from a short random seed, reducing experimental complexity without compromising reconstruction accuracy. Together, these developments broaden the reach of random pulse sequences for accurate and efficient noise characterization in realistic quantum systems.

Figures (4)

• Figure 1: (a) A reconstruction of an ideal piecewise-linear spectrum using the CS$_\text{TGV}$ method. The solid line represents a randomly generated spectrum with $N=100$ grid points. The second-order derivative of this spectrum is 4-sparse. The red dots represent the reconstructed spectrum using CS$_\text{TGV}$ based on $K=20$ different Fourier basis functions. For each Fourier basis function, we generate random pulse sequences with $(M, N_1, N_2)=(100, 100, 50)$. (b) The accuracy of CS$_\text{TGV}$ ($(M, N_1, N_2)=(100, 100, 50)$) in reconstructing ideal spectra as a function of the number of Fourier basis functions. Different curves represent different sparsities $s$ of the second-order derivatives, considering 40 randomly generated spectra with $N=100$, normalized so that the $L_2$ norm equals $1$. Each simulation is repeated 100 times and the shaded areas represent the 95% confidence regime.
• Figure 2: (a) A reconstruction of an ideal sparse spectrum using CS$_\text{R}$, the Rademacher-measurements-based CS method. The solid blue line represents a randomly generated spectrum with $N=100$ grid points. This spectrum is 4-sparse. The red line represent the decomposed spectrum using CS$_\text{R}$ based on $K=20$ different Rademacher sequences, with each repeated 5000 times. (b) The accuracy of CS$_\text{R}$ in reconstructing ideal sparse spectra as a function of $K$, the number of different Rademacher sequences. Different dotted lines represent different sparsities $s$, considering 100 randomly generated spectra with $N=100$. (c) The accuracy of CS$_\text{R}$ as a function of $K$, the number of different Rademacher sequences. Different dotted lines (different colors) represent different grid numbers, $N$, in a logarithmic scale. Each dot contains the simulation of 100 random spectra. Inset: The scaling of the critical number of Rademacher measuremetns, $K_c$, as a function of the logarithm of the grid numbers, $\log(N)$. In (b) and (c), each simulation is repeated 100 times and the shaded areas represent the 95% confidence regime.
• Figure 3: (a) Reconstruction of the noise spectrum of an ensemble of nuclear spins interacting with an InAs/GaAs quantum dot (under an external magnetic field of B = 2 T at the Voigt geometry) using CS$_\text{TGV}$. The blue solid line represents the theoretically simulated noise spectrum, with the maximum intensity normalized to 1. The red dotted line represents the simulated reconstructed spectrum considering random pulse sequences with $(M, N_1, N_2)=(200, 200, 50)$ and $K=70$ different Fourier basis functions. (b) Same experiments with the Rademacher measurements method. The blue solid line represents the theoretically simulated noise spectrum. The red dots represent the simulated reconstructed spectrum considering random pulse sequences with $K=90$ different sequences. (c) Accuracy of reconstructing the InAs/GaAs noise spectrum as a function of the number of sets of experiments, $N_\text{set}$. The solid blue line and the dashed red line represent the accuracy of CS with second-order TGV, and Rademacher measurements combined with TGV. The CS simulations are repeated for 30 times and the shaded areas represents 95% confidence regimes. The dotted green line represents the reconstruction accuracy of the noise spectrum using CPMG sequences.
• Figure 4: Simulation of the accuracy of reconstructing the InAs/GaAs noise spectrum as a function of the number of sets of experiments, $N_\text{set}$. The blue dotted, magenta squared and red diamond lines represent the accuracy of Rademacher pulse sequences with $M=N = 200$ and $p = 0.5, 0.1, 0.05$, respectively. The corresponding averaged numbers of pulses are $N_p = 100, 36, 19$. The simulations are repeated 40 times and the shaded areas represents 95% confidence regimes. The dotted green line represents the reconstruction accuracy of the noise spectrum using CPMG sequences. As $p$ decreases, the speedup for the CS method diminishes.

Theorems & Definitions (1)

• Theorem 1