Recovering Quantum Gates from Few Average Gate Fidelities

Characterizing quantum processes is a key task in the development of quantum technologies, especially at the noisy intermediate scale of today’s devices. One method for characterizing processes is randomized benchmarking, which is robust against state preparation and measurement errors and can be used to benchmark Clifford gates. Compressed sensing techniques achieve full tomography of quantum channels essentially at optimal resource efficiency. In this Letter, we show that the favorable features of both approaches can be combined. For characterizing multiqubit unitary gates, we provide a rigorously guaranteed and practical reconstruction method that works with an essentially optimal number of average gate fidelities measured with respect to random Clifford unitaries. Moreover, for general unital quantum channels, we provide an explicit expansion into a unitary 2-design, allowing for a practical and guaranteed reconstruction also in that case. As a side result, we obtain a new statistical interpretation of the unitarity—a figure of merit characterizing the coherence of a process.

Figure 1

Reconstruction of a Haar-random 3-qubit channel using the optimization (3): The plots show the dependence of the observed average reconstruction error ${ϵ}_{\mathrm{rec}}≔\parallel {\mathcal{Z}}^{♯}-\mathcal{X}\parallel$ on the number of AGFs $m$ for different noise strengths $\eta ≔{\parallel \epsilon \parallel }_{{\ell }_2}$. The error bars denote the observed standard deviation. The averages are taken over 100 samples of random i.i.d. measurements and channels (nonuniform). The matlab code and data used to create these plots can be found on GitHub [76].