15,838 research outputs found
Sample-Optimal Tomography of Quantum States
© 1963-2012 IEEE. It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error \epsilon in trace distance required O(dr^{2}/\epsilon ^{2}) copies for a d -dimensional density matrix of rank r. Here, we give a theoretical measurement scheme (POVM) that requires O (dr/ \delta) \ln ~(d/\delta) copies to estimate \rho to error \delta in infidelity, and a matching lower bound up to logarithmic factors. This implies O((dr / \epsilon ^{2}) \ln ~(d/\epsilon)) copies suffice to achieve error \epsilon in trace distance. We also prove that for independent (product) measurements, \Omega (dr^{2}/\delta ^{2}) / \ln (1/\delta) copies are necessary in order to achieve error \delta in infidelity. For fixed d , our measurement can be implemented on a quantum computer in time polynomial in n
Learning Distributions over Quantum Measurement Outcomes
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 -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 -dimensional quantum state
and unknown quantum measurements with
outcomes, estimating the probability distribution for applying
on to within total variation distance .
Compared to the special case when , 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 copies of . We further prove an information-theoretic
lower bound that at least copies of
are required to solve this problem with high success probability. Our
shadow tomography procedure requires sample complexity with only logarithmic
dependence on and and is sample-optimal for the dependence on .Comment: 25 page
Learning quantum states and unitaries of bounded gate complexity
While quantum state tomography is notoriously hard, most states hold little
interest to practically-minded tomographers. Given that states and unitaries
appearing in Nature are of bounded gate complexity, it is natural to ask if
efficient learning becomes possible. In this work, we prove that to learn a
state generated by a quantum circuit with two-qubit gates to a small trace
distance, a sample complexity scaling linearly in is necessary and
sufficient. We also prove that the optimal query complexity to learn a unitary
generated by gates to a small average-case error scales linearly in .
While sample-efficient learning can be achieved, we show that under reasonable
cryptographic conjectures, the computational complexity for learning states and
unitaries of gate complexity must scale exponentially in . We illustrate
how these results establish fundamental limitations on the expressivity of
quantum machine learning models and provide new perspectives on no-free-lunch
theorems in unitary learning. Together, our results answer how the complexity
of learning quantum states and unitaries relate to the complexity of creating
these states and unitaries.Comment: 8 pages, 1 figure, 1 table + 56-page appendi
Deterministic realization of collective measurements via photonic quantum walks
Collective measurements on identically prepared quantum systems can extract
more information than local measurements, thereby enhancing
information-processing efficiency. Although this nonclassical phenomenon has
been known for two decades, it has remained a challenging task to demonstrate
the advantage of collective measurements in experiments. Here we introduce a
general recipe for performing deterministic collective measurements on two
identically prepared qubits based on quantum walks. Using photonic quantum
walks, we realize experimentally an optimized collective measurement with
fidelity 0.9946 without post selection. As an application, we achieve the
highest tomographic efficiency in qubit state tomography to date. Our work
offers an effective recipe for beating the precision limit of local
measurements in quantum state tomography and metrology. In addition, our study
opens an avenue for harvesting the power of collective measurements in quantum
information processing and for exploring the intriguing physics behind this
power.Comment: Close to the published versio
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