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Mitigating photon loss in linear optical quantum circuits

James Mills, Rawad Mezher

Abstract

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 n photons, an m-mode interferometer, and m single photon detectors. Central to these techniques is the construction recycled probabilities. 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.

Mitigating photon loss in linear optical quantum circuits

Abstract

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 n photons, an m-mode interferometer, and m single photon detectors. Central to these techniques is the construction recycled probabilities. 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.
Paper Structure (45 sections, 28 theorems, 258 equations, 8 figures)

This paper contains 45 sections, 28 theorems, 258 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Overview of main results
  3. Preliminaries
  4. Method
  5. Recycled probabilities
  6. Construction of recycled probabilities from lossy outputs
  7. Decomposition of recycled probabilities into $n$--photon output bit string probabilities
  8. A classical simulation algorithm for recycled probabilities
  9. Bounding the bias error
  10. Generating the loss-mitigated outputs
  11. Linear solving
  12. Extrapolation
  13. Normalising the mitigated distribution
  14. Numerical simulations
  15. Properties of the mitigated probabilities in the absence of statistical error
  16. ...and 30 more sections

Key Result

Theorem 1

Consider $N_{tot}$ samples generated from a DVLOQC circuit with $n$ photons and $m$ modes. Let $U \in \mathsf{U}(m)$ be the unitary implemented by the circuit , $\eta \in [0,1]$ the uniform probability a photon is lost in any mode. Assume we are in the no-collision regime, $m \in \Omega(n^2)$, where samples where $k>0$ photons were lost, runs in time $N_{rec} + \mathsf{poly}(m,n,k)$, and that outp

Figures (8)

  • Figure 1: A schematic illustrating the main steps of the recycling mitigation protocol. (a) An input state of $n$ photons is introduced to a lossy $m$-mode linear optical interferometer implementing a unitary transformation. The classical measurement outcomes of each of the output modes is denoted by a classical $m-$bit string. (b) The classical data set generated by repeatedly sampling from the quantum circuit is then used as input for classical postprocessing. This classical postprocessing consists of three stages. First, the data set is used to generate lossy probability estimators. These estimators are then used to construct recycled probability estimators, which are in turn then used to generate mitigated values.
  • Figure 2: The results of numerical simulations where $D_k$ for $k\in \{1,2\}$ is plotted for $30$ random unitaries with $m=20$ for (a) $n=3$, (b) $n=4$, (c) $n=5$, (d) $n=6$, and (e) $n=7$. From these results, as with aaronson_bosonsampling_2014 where it was shown that $D_0$ is lower bounded by a constant, it appears this also holds for $D_k$ when $k \in O(1)$.
  • Figure 3: A numerical performance comparison of the different methods of recycling mitigation and postselection for random unitary circuits with $m=20$ modes and $n=4$ photons. (a) For a uniform loss parameter of $\eta=0.8$, the KL divergence from the ideal output distribution for linear extrapolation, exponential extrapolation, linear solving and linear solving with dependency versions of recycling mitigation and postselection is plotted against total sample number. (b) For a total number of samples of $N_{tot}=1 \times 10^5$ and a uniform loss parameter in the range $\eta \in [0.5,0.9]$, the KL divergence from the ideal output distribution of linear extrapolation, exponential extrapolation, linear solving and linear solving with dependency versions of recycling mitigation and postselection is plotted against total sample number.
  • Figure 4: The mean absolute value of the distance of the interference terms from the uniform probability were computed for 20 randomly selected unitaries. For all unitaries, and for $k=1$ and $k=2$, the magnitude of the computed values were observed to be exponentially small in terms of $m$ and $n$ (being of the order of ${m \choose n}^{-1}$). This indicates that it may be possible to derive tighter analytical bounds than those stated in thm. \ref{['Haarbound']} and thm. \ref{['nonHaarbound']}.
  • Figure 5: The numerical simulations that produced these plots involved exact computation of the mitigated probabilities and so only the bias error is present. The magnitude of the average error per output bit string is compared with that of the uniform probability. For all chosen parameters the average error per bit string is below the uniform probability. (a) The average error per bit string plotted for a range of modes $[16,30]$ for a fixed number of input photons $n=5$. (b) The average error per bit string plotted for a range of input photons $[3,9]$ for a fixed number of modes photons $m=20$.
  • ...and 3 more figures

Theorems & Definitions (54)

  • Theorem 1
  • Corollary 2
  • Conjecture 3
  • Conjecture 4
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • Theorem 8
  • Lemma 9
  • Theorem 10
  • ...and 44 more