Quantum State Tomography via Compressed Sensing

We establish methods for quantum state tomography based on compressed sensing. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. In particular, they are able to reconstruct an unknown density matrix of dimension $d$ and rank $r$ using $O(rd{log}^{2}d)$ measurement settings, compared to standard methods that require $d^2$ settings. Our methods have several features that make them amenable to experimental implementation: they require only simple Pauli measurements, use fast convex optimization, are stable against noise, and can be applied to states that are only approximately low rank. The acquired data can be used to certify that the state is indeed close to pure, so no a priori assumptions are needed.

Figure 1

Average fidelity and trace distance vs (scaled) number of measurement settings $m$ for random states of $n=8$ qubits, so $d=2^n$. As discussed in the text, the sampled states had rank $r=3$, depolarizing noise of 5% and Gaussian statistical noise with $\sigma =0.1/d$. Both the random Pauli and hybrid approaches are shown.