A quantum process matrix reconstructed via compressive sensing. Instead of potentially 576 measurement configurations, a mere 18, selected at random, suffice for a high-fidelity estimate.

We have a new paper, titled “*Efficient measurement of quantum dynamics via compressive sensing*” in PRL. I already spent a significant amount of time with my co-authors on writing the paper and the press release so I’m not gonna invent the wheel a third time and just post some excerpts from the UQ press release as introduction:

At present it is extremely difficult to characterise quantum systems — the number of measurements required increases exponentially with the number of quantum parts. For example, an 8-qubit quantum computer would require over a billion measurements.

“Imagine that you’re building a car but you can’t test-drive it. This is the situation that quantum engineers are facing at the moment,” said UQ’s Dr Alessandro Fedrizzi, co-author of the study that was recently published in *Physical Review Letters*.

“We have now found a way to test quantum devices efficiently, which will help transform them from small-scale laboratory experiments to real-world applications.”

The team also include UQ collaborators Dr Marcelo de Almeida, Professor Andrew White and PhD student Matthew Broome, as well as researchers from Princeton University, the Massachusetts Institute of Technology (MIT), and SC Solutions, Inc. The researchers adapted techniques from “compressive sensing”, a hugely successful mathematical data compression method and for the first time, have applied it to experimental quantum research.

“Audio signals have natural patterns which can be compressed to vastly smaller size without a significant quality loss: this means we now store in a single CD what used to take hundreds. In the same way, compressive sensing now allows us to drastically simplify the measurement of quantum systems,” said Dr Alireza Shabani, the study’s main author from Princeton University.

“A common example for data compression is a Sudoku puzzle: only a few numbers will allow you to fill in the whole grid. Similarly, we can now estimate the behaviour of a quantum device from just a few key parameters,” said co-author Dr Robert Kosut from SC Solutions, Inc., who developed the algorithm with Dr Shabani, Dr Masoud Mohseni (MIT) and Professor Hershel Rabitz (Princeton University).

The researchers tested their compressive sensing algorithm on a photonic two-qubit quantum computer built at UQ, and demonstrated they could obtain high-fidelity estimates from as few as 18 measurements, compared to the 240 normally required.

The team expects its technique could be applied in a wide range of architectures including quantum-based computers, communication networks, metrology devices and even biotechnology.

To summarize, we have performed process tomography of a two-qubit quantum gate with just 18 measurement configurations out of the potential 576 (which we call an overcomplete set). The compressive sensing algorithm therefore offers a huge reduction in resources and time already for the really small scale lab demonstrations we’re working on at the moment.

It should be noted that, at the time of writing, there is one other (theory) proposal on quantum tomography with compressive sensing. The paper by D. Gross et al. explicitly treats quantum *state* tomography, which is not quite the same as process tomography. However, according to the authors, and in particular Steve Flammia, who happened to visit UQ recently, their algorithm can be extended to process tomography in a straightforward manner.

The main difference between the two methods is, in a nutshell, the following: Our method scales with *~o(s log d)*, where *s* indicates the *sparsity *of the process matrix in a chosen basis and *d* the dimension of the quantum system. The method by Gross et al. scales with ~*o(r d^2 log d)*, where *r* is the rank of the quantum state, or, in extension, the quantum process.

At a first glance, our method scales more favorably. This is indeed the case, but only when the process basis is known, because a process will only be maximally sparse in its eigenbasis. The algorithm is therefore best applied for certification of a device with a defined target process. The method by Flammia et al., in contrast, can be applied to black-box processes, because rank is basis-independent.

The two methods are therefore, as I like to think, very complementary, and should both be further investigated as there are still plenty of open questions. One promising, and maybe not immediately obvious feature we found during our tests was that the process estimates returned by the compressive sensing algorithm allowed us to improve our gates in practice. This is a key requirement for any efficient tomographic estimation, and can thus be seen as a successful litmus test for compressive sensing methods entering the field of quantum information processing.