MZI: U=B† diag(1,eⁱφ) B; P₀=η(T²+R²+2TR cosφ), P₁=2ηTR(1−cosφ), P∅=1−η. HOM: P₁₁=T²+R²−2TRμ, P₂₀=P₀₂=TR(1+μ); R=1−T, μ=m e⁻δ².
Assumptions and limits
- Ideal linear optics in two spatial modes. T is power transmissivity, with transmission amplitude √T. Coupler phases follow B=[[√T,−√R],[√R,√T]]; the MZI uses its inverse as the second coupler.
- The MZI starts in |10⟩ and preserves phase coherence. The shared output efficiency η combines equal optical loss and detector inefficiency. No detection remains in the displayed probability total; no dark counts or background photons.
- HOM starts with exactly one photon in each input and assumes lossless propagation and ideal photon-number-resolving detection. Output numbers are occupations of two modes, not two logical qubits.
- HOM assumes pure Gaussian temporal envelopes with equal intensity standard deviation σₜ and a factorizable remaining mode overlap m. δ=τ/(2σₜ); μ is squared overlap. Delay has no physical duration until σₜ is specified.
- Geometry is an optical circuit schematic, not a crystal lattice or calibrated chip layout. All probabilities are analytic model predictions; neither hardware fidelities nor measured counts are inferred. Photonic systems are not universally room-temperature devices.
Sources · 5 references
- Xanadu · Strawberry Fields beam-splitter conventionsFirst-party documentation · Lossless beam-splitter transformation, amplitudes, phases and 50:50 convention.
- Hong, Ou & Mandel · Two-photon interference (1987)2 November 1987 · Original interference experiment. The Gaussian envelope used here is an explicit illustrative assumption.
- Knill, Laflamme & Milburn · A scheme for efficient quantum computation with linear optics (2001)Nature 409, 46 (2001), arXiv:quant-ph/0006088 · The KLM result, cited for the Frontier chapter.
- PsiQuantum · Manufacturable photonic platform (2025)26 February 2025 · Dual-rail qubits, HOM and fusion; integrated detectors operate near 2 K. Reported fidelities condition on detection and exclude loss.
- Aghaee Rad et al. · Xanadu Aurora (2025)22 January 2025 · Squeezed-light and non-Gaussian resources, modular networking and homodyne measurements. Photon-number detection is cryogenic; a sub-performant scale model, not useful fault tolerance.