Predicting features of complex, large-scale quantum systems is essential to the characterization and engineering of quantum architectures. We present an efficient approach for predicting a large number of linear features using classical shadows obtained from very few quantum measurements. This approach is guaranteed to accurately predict @math linear functions with bounded Hilbert-Schmidt norm from only @math measurement repetitions. This sampling rate is completely independent of the system size and saturates fundamental lower bounds from information theory. We support our theoretical findings with numerical experiments over a wide range of problem sizes (2 to 162 qubits). These highlight advantages compared to existing machine learning approaches.
Shadow tomography aims at simultaneously estimating the probability associated with @math 2-outcome measurements up to accuaracy @math : @math , where each @math is a positive semidefinite matrix whose with operator norm at most one @cite_15 @cite_5 @cite_47 . This may be viewed as a generalization of direct fidelity estimation. The best existing result is due to Aaronson @cite_47 who showed that copies of the unknown state The scaling symbol @math suppresses logarithmic expressions in other problem-specific parameters. suffice to achieve this task. In a nutshell, his protocol is based on gently measuring the 2-outcome measurements one-by-one and subsequently (partially) reverting the perturbative effects a measurement exerts on quantum states. This task is achieved by explicit quantum circuits of exponential size that act on all copies of the unknown state simultaneously. This rather intricate procedure bypasses the no-go result advertised in Theorem and results in a sampling rate that is independent of the measurement in question -- only their cardinality @math matters.
@cite_5 @cite_15 @cite_47
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We introduce the problem of *shadow tomography*: given an unknown D-dimensional quantum mixed state ρ, as well as known two-outcome measurements E1,…,EM, estimate the probability that Ei accepts ρ, to within additive error e, for each of the M measurements. How many copies of ρ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only O( e−5·log4 M·logD) copies. This means, for example, that we can learn the behavior of an arbitrary n-qubit state, on *all* accepting rejecting circuits of some fixed polynomial size, by measuring only nO( 1) copies of the state. This resolves an open problem of the author, which arose from his work on private-key quantum money schemes, but which also has applications to quantum copy-protected software, quantum advice, and quantum one-way communication. Recently, building on this work, have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time. Traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n . But using ideas from computational learning theory, we show that one can do exponentially better in a statistical setting. In particular, to predict the outcomes of most measurements drawn from an arbitrary probability distribution, one needs only a number of sample measurements that grows linearly with n . This theorem has the conceptual implication that quantum states, despite being exponentially long vectors, are nevertheless ‘reasonable’ in a learning theory sense. The theorem also has two applications to quantum computing: first, a new simulation of quantum one-way communication protocols and second, the use of trusted classical advice to verify untrusted quantum advice. We construct a practically implementable classical processing for the Bennett-Brassard 1984 (BB84) protocol and the six-state protocol that fully utilizes the accurate channel estimation method, which is also known as the quantum tomography. Our proposed processing yields at least as high a key rate as the standard processing by Shor and Preskill. We show two examples of quantum channels over which the key rate of our proposed processing is strictly higher than the standard processing. In the second example, the BB84 protocol with our proposed processing yields a positive key rate even though the so-called error rate is higher than the 25 limit. Direct quantum state tomography—deducing the state of a system from measurements—is mostly unfeasible due to the exponential scaling of measurement number with system size. The authors present two new schemes, which scale linearly in this respect, and can be applied to a wide range of quantum states.