Efficient Convex Optimization for Bosonic State Tomography

TL;DR

The paper tackles the challenge of reconstructing bosonic quantum states from dense phase-space measurements in high-dimensional spaces by formulating quantum state tomography as a convex optimization problem with ρ ≽ 0 and Tr(ρ) = 1. It introduces three efficiency-enhancing components—Efficient Displacement Operator Computation (EDOC), Hilbert Space Truncation (HST), and Stochastic Convex Optimization (SO)—together with a sample-based maximum-likelihood estimation (MLE) approach for flying modes. The authors demonstrate that EDOC speeds up displacement evaluations, HST controls dimensionality with provable truncation bounds, and stochastic solvers (PGD/SPGD with ASSG-r) enable scalable reconstruction for up to nine modes while maintaining high fidelity even under noise. Compared to prior methods, the framework offers improved stability, scalability, and practical runtimes, enabling reliable continuous-variable quantum state tomography with potential impact on CV quantum error correction and quantum interconnects.

Abstract

Quantum states encoded in electromagnetic fields, also known as bosonic states, have been widely applied in quantum sensing, quantum communication, and quantum error correction. Accurate characterization is therefore essential yet difficult when states cannot be reconstructed with sparse Pauli measurements. Tomography must work with dense measurement bases, high-dimensional Hilbert spaces, and often sample-based data. However, existing convex optimization-based techniques are not efficient enough and scale poorly when extended to large and multi-mode systems. In this work, we explore convex optimization as an effective framework to address problems in bosonic state tomography, introducing three techniques to enhance efficiency and scalability: efficient displacement operator computation, Hilbert space truncation, and stochastic convex optimization, which mitigate common limitations of existing approaches. Then we propose a sample-based, convex maximum-likelihood estimation (MLE) method specifically designed for flying mode tomography. Numerical simulations of flying four-mode and nine-mode problems demonstrate the accuracy and practicality of our methods. This method provides practical tools for reliable bosonic mode quantum state reconstruction in high-dimensional and multi-mode systems.

Figures (6)

• Figure 1: Comparison of time cost and speed acceleration between the proposed method EDOC (green line with circle marker) and the Padé approximation method (blue line with square marker) across different dimensions $N$. The time cost is shown as a line plot on the left y-axis, while acceleration ratio is represented as a bar chart on the right y-axis. The acceleration ratio indicates the ratio of the time cost of the Padé approximation method to that of the EDOC.
• Figure 2: Single-flying‐mode quantum‐state reconstruction with histogram method. Figures show the performance of three algorithms as a function of Hilbert‐space dimension $N$ and phase‐space limit $\alpha_{\mathrm{max}}$. Blue lines are for EDOC+HST+SCS, orange dotted lines are for baseline convex optimization method and green dashed lines are for MLE method. (a) Reconstruction fidelity versus $N$. (b) Computational runtime versus $N$ (logarithmic scale). (c) Reconstruction fidelity versus $\alpha_{\mathrm{max}}$. (d) Computational runtime versus $\alpha_{\mathrm{max}}$ (logarithmic scale).
• Figure 3: Two-stationary‐mode state reconstruction from Wigner functions. (a) Computational runtime versus single mode dimension $N$ (total dimension is $N^2$). Blue lines with triangle markers represent EDOC+HST+PGD, while orange with square markers and green lines with circle markers correspond to EDOC+HST+SCS and the baseline convex optimization method, respectively. (b) Fidelity versus single mode dimension $N$ (total dimension is $N^2$). Line colors correspond to those in panel (a). (c) Fidelity versus the number of data points with $N=6$. The batch size is fixed at 3000, and the number of data points is varied by changing the number of batches used in the optimization. Error bars indicate the standard deviation of fidelity over different random data initializations.
• Figure 4: Results of flying mode tomography under different setups. (a) Final fidelity within 1000 training steps for various single-mode dimensions $N$ (four modes in total); blue: baseline stochastic convex optimization, red: ASSG-r. (b) Characteristic decay time of infidelity from panel (a), defined as the training time for the infidelity to drop to $1/e$ of its initial value. (c) Reconstruction fidelity versus training time for systems with 1–9 modes (dimension 2 per mode). Color code: gray (1), black (2), dark purple (3), light green (4), red (5), orange (6), blue (7), dark green (8), light purple (9). (d) Final fidelity within 1000 steps as a function of the number of modes, corresponding to panel (c). (e) Reconstruction fidelity versus training time for different noise photon numbers (purple: 0, red: 1, orange: 2, green: 3, blue: 4). (f) Final fidelity within 1000 steps as a function of noise photon numbers, corresponding to panel (e).
• Figure 5: Fidelities versus training time for different mini-batch size $B$ and inner iteration number $T$. (a) Fidelities versus training time for different mini-batch size $B$, $B= 10,25,50,100,200$ for line with color blue, purple, red, orange and green. (b) Fidelities versus training time for different inner iteration number $T$, $T=16,32,64,128,256,512$ for line with color black, purple, red, orange, green and blue.