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arXiv:2207.08800

Quantum tomography using state-preparation unitaries

Joran van Apeldoorn, Arjan Cornelissen, András Gilyén, Giacomo Nannicini

Published Jul 18, 2022• 6 citations
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Abstract

We describe algorithms to obtain an approximate classical description of a $d$-dimensional quantum state when given access to a unitary (and its inverse) that prepares it. For pure states we characterize the query complexity for $\ell_q$-norm error up to logarithmic factors. As a special case, we show that it takes $\widetildeΘ(d/\varepsilon)$ applications of the unitaries to obtain an $\varepsilon$-$\ell_2$-approximation of the state. For mixed states we consider a similar model, where the unitary prepares a purification of the state. In this model we give an efficient algorithm for obtaining Schatten $q$-norm estimates of a rank-$r$ mixed state, giving query upper bounds that are close to optimal. In particular, we show that a trace-norm ($q=1$) estimate can be obtained with $\widetilde{\mathcal{O}}(dr/\varepsilon)$ queries. This improves (assuming our stronger input model) the $\varepsilon$-dependence over the algorithm of Haah et al.\ (2017) that uses a joint measurement on $\widetilde{\mathcal{O}}(dr/\varepsilon^2)$ copies of the state. To our knowledge, the most sample-efficient results for pure-state tomography come from setting the rank to $1$ in generic mixed-state tomography algorithms, which can be computationally demanding. We describe sample-optimal algorithms for pure states that are easy and fast to implement. Along the way we show that an $\ell_\infty$-norm estimate of a normalized vector induces a (slightly worse) $\ell_q$-norm estimate for that vector, without losing a dimension-dependent factor in the precision. We also develop an unbiased and symmetric version of phase estimation, where the probability distribution of the estimate is centered around the true value. Finally, we give an efficient method for estimating multiple expectation values, improving over the recent result by Huggins et al.\ (2021) when the measurement operators do not fully overlap.

DOI
10.48550/arxiv.2207.08800

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We study the complexity of two closely related learning problems, one quantum and one classical. In the quantum setting, we consider agnostic tomography for the natural class of product mixed states. Given NNN copies of an nnn-qubit state ρρρ, the goal is to output a nearly optimal product mixed state approximation in trace distance. While recent work has focused on pure-state ansatz (e.g., product or stabilizer states), no polynomial-time guarantees were previously known for mixed-state ansatz. In the classical setting, we study robust learning of binary product distributions: given samples from an unknown distribution on 0,1n{0,1}^n0,1n, the goal is to output a nearly optimal product approximation. Our main contributions are as follows. (1) We give a semi-agnostic tomography algorithm for product mixed states with polynomial sample and computational complexity achieving error O(optlog⁡(1/opt))O(\mathrm{opt}\log(1/\mathrm{opt}))O(optlog(1/opt)), where opt\mathrm{opt}opt is the trace distance to the best product approximation. This is the first efficient algorithm with any nontrivial agnostic guarantee for mixed-state ansatz, using only single-qubit, single-copy measurements. We also prove a Quantum Statistical Query lower bound showing near-optimality, and an unconditional lower bound demonstrating that adaptivity is necessary under single-qubit measurements. (2) We give a semi-agnostic algorithm for robustly learning binary product distributions with matching guarantees and establish a Statistical Query lower bound, essentially resolving the efficient robust learnability of this class and improving on prior work since Diakonikolas et al. (2016).
We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a d-dimensional qudit, we aim to output a classical description of a unitary that is ε\varepsilonε-close to the unknown unitary in diamond norm. We design an algorithm achieving error ε\varepsilonε using O( d2/ε)O\left(\mathrm{~d}^{2} / \varepsilon\right)O( d2/ε) applications of the unknown channel and only one qudit. This improves over prior results, which use O( d3/ε2)O\left(\mathrm{~d}^{3} / \varepsilon^{2}\right)O( d3/ε2) [via standard process tomography] or O( d2.5/ε)O\left(\mathrm{~d}^{2.5} / \varepsilon\right)O( d2.5/ε) [Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to “bootstrap” an algorithm that can produce constant-error estimates to one that can produce ε\varepsilonε-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires Ω(d2/ε)\Omega\left(\mathrm{d}^{2} / \varepsilon\right)Ω(d2/ε) applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity.