Learning Distributions over Quantum Measurement Outcomes

• URL: http://arxiv.org/abs/2209.03007v1
• Date: Wed, 7 Sep 2022 09:08:58 GMT
• Title: Learning Distributions over Quantum Measurement Outcomes
• Authors: Weiyuan Gong and Scott Aaronson
• Abstract summary: Shadow tomography for quantum states provides a sample efficient approach for predicting the properties of quantum systems. We develop an online shadow tomography procedure that solves this problem with high success probability.
• Score: 4.467248776406005
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Shadow tomography for quantum states provides a sample efficient approach for predicting the properties of quantum systems when the properties are restricted to expectation values of $2$-outcome POVMs. However, these shadow tomography procedures yield poor bounds if there are more than 2 outcomes per measurement. In this paper, we consider a general problem of learning properties from unknown quantum states: given an unknown $d$-dimensional quantum state $\rho$ and $M$ unknown quantum measurements $\mathcal{M}_1,...,\mathcal{M}_M$ with $K\geq 2$ outcomes, estimating the probability distribution for applying $\mathcal{M}_i$ on $\rho$ to within total variation distance $\epsilon$. Compared to the special case when $K=2$, we need to learn unknown distributions instead of values. We develop an online shadow tomography procedure that solves this problem with high success probability requiring $\tilde{O}(K\log^2M\log d/\epsilon^4)$ copies of $\rho$. We further prove an information-theoretic lower bound that at least $\Omega(\min\{d^2,K+\log M\}/\epsilon^2)$ copies of $\rho$ are required to solve this problem with high success probability. Our shadow tomography procedure requires sample complexity with only logarithmic dependence on $M$ and $d$ and is sample-optimal for the dependence on $K$.

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