



01874nas a2200181   4500008004100000245008500041210006900126260001400195300001100209490000600220520130000226100002301526700002301549700002001572700002201592700002101614856005701635       2024                        eng d00aPrecision Bounds on Continuous-Variable State Tomography Using Classical Shadows0 aPrecision Bounds on ContinuousVariable State Tomography Using Cl  c3/18/2024  a0103460 v53 a<p>Shadow tomography is a framework for constructing succinct descriptions of quantum states using randomized measurement bases, called &ldquo;classical shadows,&rdquo; with powerful methods to bound the estimators used. We recast existing experimental protocols for continuous-variable quantum state tomography in the classical-shadow framework, obtaining rigorous bounds on the number of independent measurements needed for estimating density matrices from these protocols. We analyze the efficiency of homodyne, heterodyne, photon-number-resolving, and photon-parity protocols. To reach a desired precision on the classical shadow of an&nbsp;N-photon density matrix with high probability, we show that homodyne detection requires order&nbsp;O(N4+1/3) measurements in the worst case, whereas photon-number-resolving and photon-parity detection require&nbsp;O&nbsp;(N4)measurements in the worst case (both up to logarithmic corrections). We benchmark these results against numerical simulation as well as experimental data from optical homodyne experiments. We find that numerical and experimental analyses of homodyne tomography match closely with our theoretical predictions. We extend our single-mode results to an efficient construction of multimode shadows based on local measurements.</p>1 aGandhari, Srilekha1 aAlbert, Victor, V.1 aGerrits, Thomas1 aTaylor, Jacob, M.1 aGullans, Michael  uhttps://link.aps.org/doi/10.1103/PRXQuantum.5.01034601776nas a2200181   4500008004100000245004200041210004100083260001400124520125200138653002701390653003101417100002301448700002301471700002001494700002201514700002101536856003701557       2022                        eng d00aContinuous-Variable Shadow Tomography0 aContinuousVariable Shadow Tomography  c11/9/20223 a<p>Shadow tomography is a framework for constructing succinct descriptions of quantum states, called classical shadows, with powerful methods to bound the estimators used. We recast existing experimental protocols for continuous-variable tomography in the classical-shadow framework, obtaining rigorous bounds on the sample complexity for estimating density matrices from these protocols. We analyze the efficiency of homodyne, heterodyne, photon number resolving (PNR), and photon-parity protocols. To reach a desired precision on the classical shadow of an N-photon density matrix with a high probability, we show that homodyne detection requires an order O(N5) measurements in the worst case, whereas PNR and photon-parity detection require O(N4) measurements in the worst case (both up to logarithmic corrections). We benchmark these results against numerical simulation as well as experimental data from optical homodyne experiments. We find that numerical and experimental homodyne tomography significantly outperforms our bounds, exhibiting a more typical scaling of the number of measurements that is close to linear in N. We extend our single-mode results to an efficient construction of multimode shadows based on local measurements.</p>10aFOS: Physical sciences10aQuantum Physics (quant-ph)1 aGandhari, Srilekha1 aAlbert, Victor, V.1 aGerrits, Thomas1 aTaylor, Jacob, M.1 aGullans, Michael  uhttps://arxiv.org/abs/2211.05149