# Quantum tomography using state-preparation unitaries

## 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 ℓq-norm error up to logarithmic factors. As a special case, we show that it takes Θ(equation presented)(d/ε) applications of the unitaries to obtain an ε-ℓ2-approximation of the state. For mixed states we consider a similar model, where the unitary prepares a purification of the state. We characterize the query complexity for obtaining Schatten q-norm estimates of a rank-r mixed state, up to polylogarithmic factors. In particular, we show that a trace-norm (q = 1) estimate can be obtained with Θ(equation presented)(dr/ε) queries. This improves (assuming our stronger input model) the ε-dependence over the works of O'Donnell and Wright (STOC 2016) and Haah et al. (IEEE Trans. Inf. Theory, 63.9, 2017), that use a joint measurement on Õ(dr/ε2) 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 require a large amount of computing resources. We describe sample-optimal algorithms for pure states that are simple and fast to implement. Along the way we show that an ℓ∞-norm estimate of a normalized vector induces a (slightly worse) ℓqnorm 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. (arXiv:2111.09283) when the measurement operators do not fully overlap. More specifically, we show that for E1, ..., Em normalized measurement operators, all expectation values Tr(Ejρ) can be efficiently learned up to error ε with Õ(√∣∣-j E2j∣∣/ε) applications of a state-preparation unitary for a purification of ρ.