Reconstructing quantum states is hard. Full tomography of an n-qubit quantum system requires a number of measurements, computing power, and memory that grows exponentially with n.
There are a variety of more efficient (e.g. polynomial scaling) methods for reconstructing classes of quantum states under simplifying assumptions such as sparsity or short-ranged entanglement.
Quantum shadow tomography is a new efficient technique applicable to arbitrary quantum states.
The key idea is that in most situations we don't need to reconstruct the full density matrix of the quantum state; what we are interested in is expectation values of certain observables of that state. By performing measurements of the state in a set of randomly-chosen bases, one can predict O(M) low weight observables using only O(log M) measurements.
This is important because many important potential applications of quantum computers including quantum chemistry scale poorly with the problem size, e.g. requiring O(n^8) observables to be measured with high precision to obtain molecular energies. Existing methods for more efficiently obtaining these observables (e.g. by grouping commuting observables) are challenging to scale, since finding the optimal grouping is an NP-hard problem.
Last year an efficient protocol for quantum shadow tomography was proposed.
A PRL paper published last week has now experimentally demonstrated quantum shadow tomography using a 4-qubit system.
Related work using spatial modes of photons and re-analysis of existing experimental data.
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