Compressing measurements in quantum dynamic parameter estimation

We present methods that can provide an exponential savings in the resources required to perform dynamic parameter estimation using quantum systems. The key idea is to merge classical compressive sensing techniques with quantum control methods to significantly reduce the number of signal coefficients that are required for reconstruction of time-varying parameters with high fidelity. We show that incoherent measurement bases and, more generally, suitable random measurement matrices can be created by performing simple control sequences on the quantum system. Random measurement matrices satisfying the restricted isometry property can be used efficiently to reconstruct signals that are sparse in any basis. Because many physical processes are approximately sparse in some basis, these methods can benefit a variety of applications such as quantum sensing and magnetometry with nitrogen-vacancy centers.

Figure 1

Simulated (blue solid lines) and successful CS reconstructed (red dotted lines) magnetic fields (nT) (five events with $m=200$ and $\mathrm{MSQE}=7.6109\times {10}^{-12}$).

Figure 2

Simulated (blue solid lines) and unsuccessful CS reconstructed (red dotted lines) magnetic fields (nT) (five events with $m=200$ and $\mathrm{MSQE}=0.005$).

Figure 3

CS reconstruction of (frequency)-sparse magnetic field (nT) with eight random frequencies (reconstruction with $m=250$ and $\mathrm{MSQE}=0.005$).

Figure 4

CS reconstruction of time-sparse magnetic field produced by firing neuron: simulated (blue solid lines) and CS reconstructed (red dotted lines) magnetic fields (nT) (five events with $m=250$ random Walsh measurements and MSQE $=3.2184\times {10}^{-4}$).