# Quantum tomography

An important task in quantum physics is to fully learn a quantum state or quantum process from measurements. This is called quantum state or quantum process tomography, respectively.

Most quantum state or processes of interest are compressible: They can be described by matrices of low rank. Hence, low-rank matrix reconstruction from compressed sensing can be used to reduce the number of measurements.

## Quantum states

Ongoing projects:

- Self-certification, see here
- Basis measurements

## Quantum processes

- The optimal scaling from quantum state tomography carries over to quantum process tomography:

Guaranteed recovery of quantum processes from few measurements - Measurements based on Clifford group operations in a randomized benchmarking setup:

Recovering quantum gates from few average gate fidelities

## Efficiently measuring quantum gates with imperfect devices

Quantum computers have the power to solve some problems in polynomial time for which no efficient classical algorithm is known. They also have the ability to simulate complex quantum systems, which has potential applications in quantum chemistry, condensed matter physics and high energy physics. A key obstacle, however, is that even the best quantum devices will have noticeable amounts of noise. In order to obtain accurate results, this noise must be characterized and controlled. Characterizing the noise is difficult for two reasons: First, the time evolution of even a few-qubit system is described by a very large number of parameters. Second, the tools used to characterize the noise are themselves subject to errors, called “state preparation and measurement” or “SPAM” errors. Previous work has succeeded in working around each of these issues in isolation, but never simultaneously. Here we present the first method for characterizing noise in quantum gates that does solve both of these problems at the same time. The method works by combining two different techniques, "randomized benchmarking" and "compressed sensing". The correctness of the method follows from remarkable mathematical properties of the Clifford group.

Ongoing projects:

- Measurements based on Clifford group operations in for natural measurements, see arXiv:1701.03135
- Self-certification, see here
- Efficient algorithms