Gradient-descent methods for fast quantum state tomography

TL;DR

This work tackles the exponential scalability of quantum state tomography by introducing gradient-descent–based QST (GD-QST) with three density-matrix parameterizations—Cholesky decomposition, Stiefel manifold, and projective normalization—that preserve positivity and normalization while enabling rank control. By employing mini-batch GD and these parameterizations, the method achieves fast, high-fidelity reconstructions for both discrete- and continuous-variable systems, including full-rank seven-qubit states in under three minutes on a laptop. Compared to constrained convex optimization, iMLE, and CGANs, GD-QST often delivers superior time efficiency and robustness to noise, with performance that depends on the chosen parameterization and target state rank. The work provides broad practical impact for characterizing noisy intermediate-scale quantum devices and offers open-source Python code to facilitate adoption and further extensions to quantum process tomography.

Abstract

Quantum state tomography (QST) is a widely employed technique for characterizing the state of a quantum system. However, it is plagued by two fundamental challenges: computational and experimental complexity grows exponentially with the number of qubits, rendering experimental implementation and data post-processing arduous even for moderately sized systems. Here, we introduce gradient-descent (GD) algorithms for the post-processing step of QST in discrete- and continuous-variable systems. To ensure physically valid state reconstruction at each iteration step of the algorithm, we use various density-matrix parameterizations: Cholesky decomposition, Stiefel manifold, and projective normalization. These parameterizations have the added benefit of enabling a rank-controlled ansatz, which simplifies reconstruction when there is prior information about the system. We benchmark the performance of our GD-QST techniques against state-of-the-art methods, including constrained convex optimization, conditional generative adversarial networks, and iterative maximum likelihood estimation. Our comparison focuses on time complexity, iteration counts, data requirements, state rank, and robustness against noise. We find that rank-controlled ansatzes in our stochastic mini-batch GD-QST algorithms effectively handle noisy and incomplete data sets, yielding significantly higher reconstruction fidelity than other methods. Simulations achieving full-rank seven-qubit QST in under three minutes on a standard laptop, with 18 GB of RAM and no dedicated GPU, highlight that GD-QST is computationally more efficient and outperforms other techniques in most scenarios, offering a promising avenue for characterizing noisy intermediate-scale quantum devices. Our Python code for GD-QST algorithms is publicly available at https://github.com/mstorresh/GD-QST.

Figures (10)

• Figure 1: Graphical illustration of our GD-QST methods. (a) Depiction of GD-QST employing CD (green), SM (yellow), and PN (blue), starting from an ansatz $\varrho_{\text{ans}}$. For CD and SM, GD updates occur within the space of physical density matrices; for PN, the GD updates can go outside the physical space (black arrows) and are then projected back into the physical space (dashed blue lines). All methods approach $\varrho_{\text{opt}}$, the optimal density matrix corresponding to the minimum loss. (b) Depiction of the behavior of the loss function in a physical space as a function of iteration count.
• Figure 2: Time required for GD-QST algorithms to achieve a state fidelity of $\mathcal{F} > 0.99$ for (a) full-rank states and (b) pure states, as a function of the number of qubits in the system. The total time in each scenario is averaged over 30 randomly generated full-rank and pure states. The legend indicates different methods: GD-QST with CD (teal circles), SM (orange squares), and PN (green upward triangles), and the CCO tool CVX (black downward triangles). In panel (a), the PN algorithm is excluded for systems with more than five qubits, and SM and CVX are excluded beyond six qubits, while in panel (b) CVX is omitted beyond six qubits, due to their failure to converge within a reasonable time frame.
• Figure 3: GD-QST performance for a five-qubit system. (a) Reconstruction fidelity obtained using the parameterizations CD (blue), SM (orange), PN (green), and CD-tri (red), as a function of the number of iterations. (b) Reconstruction fidelity as a function of cumulative time. The sold lines represent average fidelity values calculated over 30 full-rank random states with full rank-ansatz; shaded areas indicate respective standard deviation.
• Figure 4: Time complexity analysis with respect to rank-varying state and rank-varying ansatz for a five-qubit system. (a) Time complexity of GD-QST (with the ansatz having the same rank as the state) and CVX for reconstructing states of specific rank with $\mathcal{F} > 0.99$. (b) Time (in milliseconds) per iteration as a function of the rank of the ansatz. (c) Time complexity of GD-QST of five-qubit rank-1 states as a function of the ansatz rank. In all scenarios, the time complexity is averaged over 30 randomly chosen states for each data point. The CVX case is not presented in panels (b) and (c) since it does not support rank-controlled ansatzes. The CD-tri case is also omitted because it does not achieve high fidelity, either within the specified number of iterations or in reasonable time.
• Figure 5: Quantum state tomography of five-qubit (a) Hadamard and (b) GHZ states using reduced data sets of varying sizes. In both cases, the average fidelity (solid line) and the corresponding standard deviation (shaded area) are calculated by randomly sampling reduced data sets of given size from the full data set 15 times. For both cases, the rank of the ansatz for the GD-QST methods is set to one.