How asymptotically efficient is quantum state tomography of a flat qubit?
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Quantum state tomography—deducing quantum states from measured data—is the gold standard for verification and benchmarking of quantum devices. It has been realized in systems with few components, but for larger systems it becomes unfeasible because the number of measurements and the amount of computation required to process them grows exponentially in the system size. Here, we present two tomography schemes that scale much more favourably than direct tomography with system size. One of them requires unitary operations on a constant number of subsystems, whereas the other requires only local measurements together with more elaborate post-processing. Both rely only on a linear number of experimental operations and post-processing that is polynomial in the system size. These schemes can be applied to a wide range of quantum states, in particular those that are well approximated by matrix product states. The accuracy of the reconstructed states can be rigorously certified without any a priori assumptions.
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let and . From the normalization condition we obtain
which we can satisfy if we set
and
since
for all . We now have the state
Since we can ommit the overall phase factor and rename
to we get the desired result.
Now why did we use instead of . If we used we would get every state twice.
which we can satisfy if we set
and
since
for all . We now have the state
Since we can ommit the overall phase factor and rename
to we get the desired result.
Now why did we use instead of . If we used we would get every state twice.
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