Mitigating photon loss in linear optical quantum circuits

Photon loss rates set an effective upper limit on the size of computations that can be run on current linear optical quantum devices. We present a family of techniques designed to mitigate the effects of photon loss on both output probabilities and expectation values derived from noisy linear optical circuits composed of an input of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> photons, an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> --mode interferometer, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> single photon detectors. Central to these techniques is the construction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext class="MJX-tex-mathit" mathvariant="italic">recycled probabilities</mml:mtext> </mml:mrow> </mml:math> . Recycled probabilities are constructed from output statistics affected by loss, and are designed to amplify the signal of the ideal (lossless) probabilities. Classical postprocessing techniques then take recycled probabilities as input and output a set of loss-mitigated probabilities, or expectation values. Our postprocessing methods result in biased estimators of the lossless probabilities. Nevertheless, we provide both analytical and numerical evidence that these methods can be applied, up to large sample sizes, to produce output probabilities with lower combined bias and statistical errors than the statistical errors of the output probabilities obtained from postselection. Therefore, these methods can outperform postselection - currently the standard method of coping with photon loss when sampling from discrete variable linear optical quantum circuits. In contrast, we provide evidence that the popular zero-noise extrapolation technique cannot improve on the performance of postselection for any photon loss rate.