Boson sampling's Achilles heel is its fuel. It needs n identical single photons in the same clock cycle, and a heralded SPDC source delivers one only a few percent of the time — so the n-fold coincidence rate collapses as p_{\text{source}}^{\,n}. Twenty photons: essentially never. The escape, found by Hamilton and colleagues in 2017, is a change of fuel rather than of engine. The same nonlinear crystal that makes heralded pairs makes, when you simply don't herald, squeezed light — a state that pours out deterministically, every single pump pulse. Feed squeezed light straight into the mesh and count photons at the output: this is Gaussian boson sampling (GBS), the design behind the first photonic quantum-advantage claims — China's Jiuzhang and Xanadu's Borealis. To understand it we need this course's one excursion into continuous-variable quantum optics: field quadratures, the vacuum's noise floor, and what it means to squeeze below it.

Quadratures: the harmonic oscillator returns

A single optical mode is, mathematically, a quantum harmonic oscillator — the photon number is the oscillator's level. Its "position" and "momentum" are the field's quadratures,

the cosine and sine components of the oscillating field — what a radio engineer calls I and Q. Being conjugate, they obey an uncertainty relation \Delta X\,\Delta P \ge \tfrac12. The vacuum saturates it symmetrically: \Delta X^2 = \Delta P^2 = \tfrac12. This vacuum fuzz is not an instrument artefact — it is the shot noise that Module 5 met as the fundamental noise floor of optical detection: measure any quadrature of "nothing" and you get a Gaussian jitter of variance ½. Laser light (a coherent state) is just displaced vacuum, carrying the same circle of noise around a nonzero mean.

Squeezing: reshaping the vacuum

The uncertainty product is a floor on the product — not on either factor. A squeezed state trades the two off. Pumping a χ² crystal applies the squeezing operator S(r) and turns the noise circle into an ellipse:

quieter than the vacuum along one axis, correspondingly louder along the other. The squeezing strength is quoted in decibels below shot noise, -10\log_{10}\!\bigl(e^{-2r}\bigr) \approx 8.69\,r dB; today's best crystals reach about 15 dB (r \approx 1.7). Two facts make squeezed vacuum the perfect GBS fuel. First, it is deterministic: every pump pulse squeezes, no herald, no lottery. Second, expand it in photon number and only even terms appear,

because the crystal creates photons strictly in pairs — squeezed vacuum is the un-heralded face of SPDC, with mean photon number \sinh^2 r. That pair structure is about to leave a fingerprint on the mathematics.

GBS: sampling from hafnians

The Gaussian boson sampler squeezes several input modes of an m-mode mesh, lets the unitary interfere them, and counts photons at every output. The output probabilities are again given by a matrix function of the mesh unitary — but the permanent, which pairs inputs with outputs one-to-one, is replaced by the function that pairs output photons with each other, respecting their pairwise birth:

where A is a matrix built from U and the squeezing strengths, and the hafnian Haf sums over all perfect matchings — all ways of splitting the detected photons into pairs. (The permanent is recoverable as a special case, so the hafnian is the harder object; it is likewise #P-hard, with the best classical algorithms scaling roughly as 2^{n/2}.) The complexity story then runs parallel to Aaronson–Arkhipov: efficient classical GBS simulation would collapse the polynomial hierarchy, with analogous conjectures covering the noisy case. What changes is the economics: with deterministic fuel in every input port, experiments jumped from boson sampling's handful of photons to hundreds, essentially overnight.

Hence the record books. Jiuzhang (USTC, December 2020): 50 squeezed states into a 100-mode free-space interferometer, up to 76 photons detected, and a sampling task estimated at the time to need billions of years classically — the first quantum-advantage claim on light. Borealis (Xanadu, 2022): the same mathematics rebuilt as engineering — a single squeezed source firing into a loop of fibre delay lines, 216 time-bin modes interfered by three dynamically programmed beamsplitters, and quantum advantage claimed on a programmable, fibre-networked machine you could queue jobs to over the cloud. Between them sits the pattern this course keeps finding: the physics fits on one line; the leap is in the packaging.

Loss: the advantage-eating channel

Now the villain. Every photonic system loses photons, but for squeezed light loss is crueller than a dimmer switch. Transmission \eta acts as a beamsplitter that swaps a fraction 1-\eta of your state for vacuum — in the language of quantum channels, a Gaussian channel mixing in fresh noise. A quadrature variance V becomes

so the squeezed variance e^{-2r} is dragged back toward the vacuum's 1. The floor is brutal: even infinite squeezing behind transmission \eta can never show a variance below 1-\eta. At 50% transmission, no source in the universe looks better than 3 dB. Play with the slider — watch how loss crushes the squeezed curve toward vacuum while barely denting the noisy one:

For GBS machines this is existential. Squeezing is the quantum resource; as loss mounts, the output state slides toward a classically simulable thermal state, and classical "spoofing" algorithms exploit exactly this — several later analyses narrowed or contested the early advantage claims by simulating the lossy experiment rather than the ideal one. The lesson generalises far beyond sampling: in photonic quantum computing, loss is not one error among many but the error, the quantity every architecture is shaped around. The final lesson takes that as its founding constraint.

Worked example: a squeezing budget

Your crystal produces 10 dB of squeezing: e^{-2r} = 0.1. The route to the detector — coupling, propagation, detection efficiency — has total transmission \eta = 0.5. The observed variance is

Ten decibels of hard-won laboratory squeezing arrive as 2.6. Run the budget the other way and the engineering targets write themselves: to observe even 6 dB (V = 0.25) from a perfect source you need \eta > 0.75; state-of-the-art squeezing experiments fight for every percent of detection efficiency, and GBS machines live or die on end-to-end transmission in a way qubit machines — which can retry a lost photon — do not.

Where this goes next

GBS closed the argument that photons can do classically impossible things at scale — and underlined, twice, what stands between here and a useful machine: loss, and the absence of error correction in sampling architectures. The final lesson of this module assembles everything — dual-rail qubits, HOM interference, KLM-style heralded entanglement, and loss-as-erasure — into the architecture that industry has actually bet on: fusion-based quantum computing, where small photonic resource states are stitched into a fault-tolerant whole by Bell measurements, on silicon photonics stamped out in a semiconductor fab.