Spectrum measurement of quantum channels and application to Hamiltonian parameter estimation

Abstract

Quantum channels describe the most general dynamics of open quantum systems. A quantum channel, as a linear map on vectorized quantum states, can be represented by a single matrix, whose spectrum is called the channel spectrum. Here we propose a general method to measure the channel spectrum and apply this method to Hamiltonian parameter estimation. We first demonstrate that the channel spectrum can be measured by tracking the probability of a specific outcome in repeated application of the same channel. Then we construct and analyze {a class of concatenated channels, with each one being a unitary channel followed by a weak-measurement channel induced by a Ramsey sequence of a probe qubit}. We show that the spectrum measurement of such concatenated channels can be utilized for estimating the parameters in the free Hamiltonians generating the unitary channels of the target system. As practical examples, we numerically demonstrate that a probe spin qubit can accurately sense nuclear spin clusters for nanoscale nuclear magnetic resonance.

Figures (2)

• Figure 1: Schematic illustration of quantum channel spectrum measurement. (a) The system evolves with $N$ repetitive quantum channels. Each channel has $r$ measurement outcomes, and we track the frequency $f_i^{(m)}$ of a particular outcome $i$ in the $m$th channel to measure the channel spectrum. (b) Schematic of $f_i^{(m)}$ (blue solid line) as a function of measurement cycle number $m$. A real eigenvalue of the channel contributes an exponential decay (dashed red line), while a pair of complex conjugate eigenvalues contribute a damped oscillation (dashed green line). (c) The channel spectrum can be accurately inferred from $\{f_i^{(m)}\}_{m=1}^N$ by the MP method.
• Figure 2: Applying quantum channel spectrum measurement to Hamiltonian parameter estimation. (a) Illustration of the quantum circuit for Hamiltonian parameter estimation. The target system evolves under repetitive quantum channels, with each channel $\hat{\Phi}$ concatenated by $\hat{\Phi}_B$ generated by a free evolution and $\hat{\Phi}_A$ induced by a probe qubit under RIM. (b) Frequency $f_1^{(m)}$ as a function of measurement number $m$ with different RIM evolution time $\tau_A$. The oscillation damping rate increases with $\tau_A$ (corresponding to increasing measurement strength). (c) Comparison of the ideal spectra of $\hat{\Phi}_B$ (blue diamonds), $\hat{\Phi}$ (red circles) and the estimated spectra of $\hat{\Phi}$ (blue stars). Under the perturbation of $\hat{\Phi}_A$, the eigenvalues of $\hat{\Phi}$ have almost the same phase angles as those of $\hat{\Phi}_B$ but reduced amplitudes. Parameters are $|\vb*{h}_1|/2\pi=1.20$ kHz, $|\vb*{h}_2|/2\pi=1.33$ kHz, $D/2\pi=105.34$ Hz, $\omega/2\pi=1$ kHz, $\tau_A=100\,\,\mu$s and $\tau_B=227.3\,\,\mu$s.