Compressed sensing for quantum state tomography in three papers

Short answer: Full quantum state tomography needs a number of measurement settings that grows very quickly with system size. Compressed sensing reconstructs a state from far fewer settings when the state is close to pure. A 2010 theory paper proposed the method, a 2017 photonic study tested it on complete experimental data, and a 2017 trapped-ion paper applied it to a seven-qubit state using 127 settings.

What problem does compressed tomography address?

The 2017 Nature Communications abstract says well-controlled quantum devices of increasing size face a roadblock: the effort of tomography, the reconstruction of states and processes, scales unfavourably, so that state-of-the-art systems can no longer be characterised. Quantum compressed sensing is described as a way to mitigate this by reconstructing states from incomplete data.

What did the 2010 theory paper establish?

The 2010 Physical Review Letters paper establishes methods for state tomography based on compressed sensing, specialised for states that are fairly pure. The authors say the methods can reconstruct an unknown density matrix of dimension d and rank r using on the order of r d log^2 d measurement settings, compared with standard methods that require on the order of d^2 settings. They list features that make the methods suited to experiments: they need only simple Pauli measurements, use fast convex optimisation, are stable against noise, and can be applied to states that are only approximately low rank. They add that the acquired data can be used to certify that the state is close to pure, so no prior assumption is needed. This is a theoretical result with performance claims by the authors.

What did the photonic analysis find?

The 2017 paper in Quantum Science and Technology analyses compressed sensing tomography in a regime where tomographically complete data are available with reliable statistics from a multi-mode photonic experiment. Because the full data set is known with high statistical significance, the authors could systematically explore how reconstruction quality depends on the number of measurement settings, chosen at random from the complete set, and on different model assumptions. They present and test a complete prescription for efficient compressed sensing, using model selection and cross validation to account for experimental imperfections and finite counting statistics. They conclude that compressed sensing is an effective tool for state tomography, specifically suited to photonic systems.

What did the seven-qubit experiment do?

The 2017 Nature Communications paper presents an experimental implementation of compressed tomography of a seven-qubit system, a topological colour code prepared in a trapped-ion architecture. The authors work in a regime they describe as highly incomplete, with 127 Pauli basis measurement settings, and highly noisy, with 100 repetitions each. Compressed sensing was originally advocated for states with few non-zero eigenvalues. The authors argue that low-rank estimates are appropriate in general, because statistical noise enables reliable reconstruction of only the leading eigenvectors, and that the remaining eigenvectors behave consistently with a random-matrix model that carries no information about the true state.

How to read this

The three papers address a theory result, a photonic data analysis and a trapped-ion experiment, so they are complementary. Compressed sensing relies on the state being close to low rank, which is an assumption and not a given for every device. This post is based on abstracts and visible introductions, and full texts were not checked. It makes no claim about products.

FAQ

How many settings does standard tomography need?

On the order of d squared for dimension d, per the 2010 abstract.

How many did compressed sensing need?

On the order of r d log^2 d in the 2010 theory, and 127 settings in the seven-qubit experiment.

Which state was reconstructed in the experiment?

A seven-qubit topological colour code in trapped ions.

Does it work for any state?

It is aimed at states that are fairly pure or approximately low rank, according to the 2010 authors.