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arXiv:2211.12880

Faster Stochastic First-Order Method for Maximum-Likelihood Quantum State Tomography

Tsai, Chung-En, Hao–Chung Cheng, Yen-Huan Li

Published Nov 23, 2022• 1 citation
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Abstract

In maximum-likelihood quantum state tomography, both the sample size and dimension grow exponentially with the number of qubits. It is therefore desirable to develop a stochastic first-order method, just like stochastic gradient descent for modern machine learning, to compute the maximum-likelihood estimate. To this end, we propose an algorithm called stochastic mirror descent with the Burg entropy. Its expected optimization error vanishes at a $O ( \sqrt{ ( 1 / t ) d \log t } )$ rate, where $d$ and $t$ denote the dimension and number of iterations, respectively. Its per-iteration time complexity is $O ( d^3 )$, independent of the sample size. To the best of our knowledge, this is currently the computationally fastest stochastic first-order method for maximum-likelihood quantum state tomography.

DOI
10.48550/arxiv.2211.12880

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In maximum-likelihood quantum state tomography, both the sample size and dimension grow exponentially with the number of qubits. It is therefore desirable to develop a stochastic first-order method, just like stochastic gradient descent for modern machine learning, to compute the maximum-likelihood estimate. To this end, we propose an algorithm called stochastic mirror descent with the Burg entropy. Its expected optimization error vanishes at a O((1/t)dlog⁡t)O ( \sqrt{ ( 1 / t ) d \log t } )O((1/t)dlogt​) rate, where ddd and ttt denote the dimension and number of iterations, respectively. Its per-iteration time complexity is O(d3)O ( d^3 )O(d3), independent of the sample size. To the best of our knowledge, this is currently the computationally fastest stochastic first-order method for maximum-likelihood quantum state tomography.
Quantum state tomography (QST), the task of estimating an unknown quantum state given measurement outcomes, is essential to building reliable quantum computing devices. Whereas computing the maximum-likelihood (ML) estimate corresponds to solving a finite-sum convex optimization problem, the objective function is not smooth nor Lipschitz, so most existing convex optimization methods lack sample complexity guarantees; moreover, both the sample size and dimension grow exponentially with the number of qubits in a QST experiment, so a desired algorithm should be highly scalable with respect to the dimension and sample size, just like stochastic gradient descent. In this paper, we propose a stochastic first-order algorithm that computes an ε\varepsilonε-approximate ML estimate in O((Dlog⁡D)/ε2)O( ( D \log D ) / \varepsilon ^ 2 )O((DlogD)/ε2) iterations with O(D3)O( D^3 )O(D3) per-iteration time complexity, where DDD denotes the dimension of the unknown quantum state and ε\varepsilonε denotes the optimization error. Our algorithm is an extension of Soft-Bayes to the quantum setup.
We propose, to the best of our knowledge, the first online algorithm for maximum-likelihood quantum state tomography. Suppose the quantum state to be estimated corresponds to a D D D-by-D D D density matrix. The per-iteration computational complexity of the algorithm is O(D3) O ( D ^ 3 ) O(D3), independent of the data size. The expected numerical error of the algorithm is O((1/T)Dlog⁡D)O(\sqrt{ ( 1 / T ) D \log D })O((1/T)DlogD​), where TTT denotes the number of iterations. The algorithm can be viewed as a quantum extension of Soft-Bayes, a recent algorithm for online portfolio selection (Orseau et al. Soft-Bayes: Prod for mixtures of experts with log-loss. \textit{Int. Conf. Algorithmic Learning Theory}. 2017).
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We propose an iterative algorithm that computes the maximum-likelihood estimate in quantum state tomography. The optimization error of the algorithm converges to zero at an O((1/k)log⁡D)O ( ( 1 / k ) \log D )O((1/k)logD) rate, where kkk denotes the number of iterations and DDD denotes the dimension of the quantum state. The per-iteration computational complexity of the algorithm is O(D3+ND2)O ( D ^ 3 + N D ^2 )O(D3+ND2), where NNN denotes the number of measurement outcomes. The algorithm can be considered as a parameter-free correction of the RρRR ρRRρR method [A. I. Lvovsky. Iterative maximum-likelihood reconstruction in quantum homodyne tomography. \textit{J. Opt. B: Quantum Semiclass. Opt.} 2004] [G. Molina-Terriza et al. Triggered qutrits for quantum communication protocols. \textit{Phys. Rev. Lett.} 2004.].
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 github.com/mstorresh/GD-QST .
We initiate the study of online quantum state tomography (QST), where the matrix representation of an unknown quantum state is reconstructed by sequentially performing a batch of measurements and updating the state estimate using only the measurement statistics from the current round. Motivated by recent advances in non-convex optimization algorithms for solving low-rank QST, we propose non-convex mini-batch stochastic gradient descent (SGD) algorithms to tackle online QST, which leverage the low-rank structure of the unknown quantum state and are well-suited for practical applications. Our main technical contribution is a rigorous convergence analysis of these algorithms. With proper initialization, we demonstrate that the SGD algorithms for online low-rank QST achieve linear convergence both in expectation and with high probability. Our algorithms achieve nearly optimal sample complexity while remaining highly memory-efficient. In particular, their time complexities are better than the state-of-the-art non-convex QST algorithms, in terms of the rank and the logarithm of the dimension of the unknown quantum state.