Experimental demonstration of Quantum Overlapping Tomography

Abstract

Quantum tomography is one of the major challenges of large-scale quantum information research due to the exponential time complexity. In this work, we develop and apply a Bayesian state estimation method to experimentally demonstrate quantum overlapping tomography [Phys. Rev. Lett. \textbf{124}, 100401 (2020)], a scheme intent on characterizing critical information of a many-body quantum system in logarithmic time complexity. By comparing the measurement results of full state tomography and overlapping tomography, we show that overlapping tomography gives accurate information of the system with much fewer state measurements than full state tomography.

Figures (4)

• Figure 1: (a) 2-qubit quantum overlapping tomography of a large-scale system. The whole system is divided into two groups, red and blue, in different strategies. For each dividing strategy, the two groups are measured on a different basis. (b) QOT dividing strategy for the $n=4, k=2$ case.
• Figure 2: Schematic of experimental set-up for generating 4-photon entanglement, with the detectors labeled by order. An ultrafast pulsed laser with the center wavelength of 390 nm, pulse duration of $\sim$100 fs, and repetition rate of 80 MHz was deployed to pump two sets of interference-based beam-like SPDC entanglement sources PhysRevLett.121.250505PhysRevLett.75.4337. The polarization-entangled photon pairs go through a post-selection interference to generate a 4-photon GHZ state, which is then measured by 8 single-photon detectors. To reduce the loss of two-fold fidelity caused by time-space correlation PhysRevA.64.063815PhysRevLett.117.210502, we applied narrow-band filters with $\lambda_{FWHM}$=3 nm and $\lambda_{FWHM}$=10 nm to the signal and idler photons respectively. The center wavelengths of both signal and idler photons are 780 nm. BBO: Barium Borate; PBS: polarization beam-splitter; HWP: half-wave plate; QWP: quarter-wave plate.
• Figure 3: Real part of density matrix of 2-qubit subsystem (a) ${\psi}_{2,3}^F$ obtained by 4-qubit full state tomography (FST) (b) ${\psi}_{2,3}^O$ obtained by overlapping tomography (QOT). (c) 2-qubit state fidelities of reconstructed ${\psi}_{i_1,i_2}$ and reference state $(\psi_{GHZ}^R)_{i_1,i_2}$ and (d) Von Neumann Entropy of reconstructed ${\psi}_{i_1,i_2}$. (e) Normalized fidelity $F(\theta_1,\theta_2)$ between 4-qubit state ${\psi'}^F$ reconstructed via 4-qubit FST and reference state ${\psi'}^R(\theta_1,\theta_2)$, $F$ reached peak at $\theta_1=175^\circ, \theta_2=-27^\circ$. (f) Normalized fidelity $F_{0,2}(\theta_1,\theta_2)$ between 2-qubit subsystem ${\psi'}_{0,2}$ reconstructed via overlapping tomography and reduced reference state ${\psi'}_{0,2}^R(\theta_1,\theta_2)$. $F_{0,2}$ reached peak at $\theta_1=183^\circ, \theta_2=-21^\circ$. For (c-d), QOT and FST results are estimated by Bayesian Estimation based on equal number of measurements. Error bars show 95% confidence interval obtained by Bayesian analysis based on Markov Chain Monte Carlo method, as 95% MCMC sample points (after pre-heating) lie within the intervals.
• Figure 4: 2-qubit state (a) fidelity with subsystem of 6-photon GHZ state $({\psi}_{{GHZ}_6}^R)_{i_1,i_2}$ and (b) Von Neumann Entropy of reconstructed density matrix ${\psi}_{i_1,i_2}$ obtained by 6-qubit QOT, error bars show 95% confidence interval obtained by Bayesian analysis based on MCMC method, as 95% sample points lie within the intervals.