# Techniques: Mitigating photon loss in linear optical quantum circuits

## What the paper contributes (one paragraph)

The paper introduces recycling mitigation, a family of classical postprocessing techniques that mitigate the effect of photon loss on the outputs of discrete variable linear optical quantum circuits (an input of n photons, an m-mode interferometer, and m single photon detectors). Lossy output statistics that postselection would discard are reused to build recycled probabilities, which amplify the signal of the ideal lossless probabilities. Two postprocessing routes then turn recycled probabilities into mitigated output probabilities or expectation values: linear solving, which uses a single loss order and comes with a proven additive-error bound, and exponential extrapolation, which fits the decay of recycled probabilities across several loss orders and performs better numerically. Both are shown, analytically and numerically, to outperform postselection up to large sample sizes above a loss threshold. The paper also gives a classical simulation algorithm for high-loss recycled probabilities and argues that zero-noise extrapolation cannot beat postselection here.

## Techniques offered

- **recycling mitigation (photon-loss error mitigation via recycled probabilities)** - classical postprocessing that reuses lossy output statistics to construct recycled probabilities and produce loss-mitigated output probabilities or expectation values for a linear optical circuit, outperforming postselection.
  - guarantee: statistical bound; biased estimators whose combined bias and statistical error fall below the postselection statistical error up to a quantified sample size, above a loss threshold lower bounded by a constant, with no extra sampling overhead relative to postselection.
  - quote: "We refer to these techniques collectively as \emph{recycling mitigation}, as they all involve the use of lossy output statistics that otherwise would be discarded."

- **linear solving recycling mitigation** - error mitigation that substitutes each recycled probability's interference term with its uniform-distribution expectation and solves for the ideal probabilities at a single loss order k, with an explicit additive-error guarantee.
  - guarantee: statistical bound; outputs with probability 1 minus delta an estimate p_mit(s) of the ideal probability p_id(s) with |p_mit(s) - p_id(s)| at most f(n,m,k,eta,N_tot), in runtime N_rec + poly(m,n,k).
  - quote: "There exists a classical algorithm (linear solving recycling mitigation) that uses a subset  of  $$N_{rec}= {n \choose k} (1-\eta)^{n-k} \eta^k N_{tot}   \in   O \Big(\sqrt{\frac{n}{k(n-k)}}\;\frac{n^n}{k^k (n-k)^{\,n-k}}\eta^k (1-\eta)^{n-k}N_{tot} \Big), $$ samples  where $k>0$ photons were lost, runs in time $N_{rec} + \mathsf{poly}(m,n,k)$, and that outputs with probability $1-\delta$ an approximation $p_{mit}(s)$ of the exact lossless circuit probability of bit string $s$, $p_{id}(s)$ such that"

- **exponential extrapolation recycling mitigation** - error mitigation that uses recycled probabilities at multiple loss orders k and fits the decay of their signal towards the uniform distribution to an exponential model to extract the ideal probabilities, with lower bias error than linear solving.
  - guarantee: statistical bound (conjectured bias and statistical bounds plus a worst-case theorem conditional on those conjectures); under Conjectures on bias and statistical error it outperforms postselection up to a larger additive precision than linear solving.
  - quote: "We present another recycling mitigation technique, exponential extrapolation, which outperforms the linear solving technique in numerical simulation experiments."

- **classical simulation of high-loss recycled probabilities** - a classical algorithm that exactly computes recycled probabilities in the high photon loss regime, delimiting which loss orders are classically easy and which retain potential quantum hardness.
  - guarantee: static proof of efficient classical computability; for k = n - r with r constant the recycled-probability sum is computed exactly in time O(2^{r-1} r (n choose n-r)^2), which is polynomial in n.
  - quote: "Let $k=n-r$, there is a classical algorithm running in time $O(2^{r-1}r\big({n \choose n-r}\big)^2)$  which exactly computes $\sum_{s_i^{n-k} \in \mathcal{L}(s^n_l)}p(s^{n-k}_i)$."

## Where it could apply

- Target primitive(s): sampling (boson sampling and DVLOQC sampling), expectation-value estimation, VQE (variational quantum eigensolvers), and quantum circuit Born machine training.
- Target application group(s): error_correction (loss mitigation), quantum_ml (QCBM and photonic quantum machine learning), platforms_tooling, chemistry_materials (variational eigensolvers), optimization (graph problems with DVLOQC).
- Code family / hardware assumptions: photonic; discrete variable linear optical quantum computing with single photon sources, an m-mode interferometer, and single photon detectors; uniform loss model; no-collision regime m in Omega(n^2) with threshold detectors (assumption stated as relaxable).

## Caveats

The techniques produce biased estimators, so a residual bias error persists that does not vanish with more samples, and the advantage over postselection holds only up to a finite sample size (of order (m choose n)^2) and only above a loss threshold. The sample cost to construct recycled probabilities at order k scales exponentially with system size, a limitation common to purely classical postprocessing mitigation. The strongest exponential-extrapolation performance guarantee rests on conjectures (Conjectures on bias and statistical error) that are supported numerically but not proved. The paper also presents a no-go result rather than a suppliable technique for zero-noise extrapolation: it gives evidence that ZNE and Richardson extrapolation cannot outperform postselection for photon loss in the DVLOQC setting, so ZNE is not offered as a usable photon-loss mitigation method here.