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Does the optical idea survive the arithmetic?

Four checks that came up again and again while gating proposals for log-domain neurons and photonic coprocessors. Each one takes a minute and has ended more than one design: precision caps how often light can be recirculated, materials trade nonlinearity for loss, logarithmic arithmetic loses at inference precisions, and fixed analog aggregation collapses to a single matrix.

Background: The log-arithmetic gates and Analog chips can do nonlinear ML.

1 · Recirculation · how many passes through one optical circuit?

Loss is not the binding constraint — precision is. Each pass spends SNR; once the datapath drops below its usable bit depth (about 4 bits in the log-arithmetic experiments), quality falls off a cliff. ENOB = (SNR_dB − 1.76) / 6.02.

starting ENOB (bits)floor ENOB (bits)Where quality collapses.η — transmission per pass

SNR budget12.0 dB

passive · thermal-limited26

passive · shot-limited53

amplified loop16

Thermal-limited detection loses 0.458 dB per pass; shot-limited loses half that (0.229dB). Amplifying doesn't rescue it — ASE from n cascaded ideal amplifiers degrades OSNR as 1/n, the same order of headroom. And passes through one operator are recurrent depth with a shared weight matrix — the weakest use of a memory budget in the log-arithmetic experiments.

2 · Materials · does a different nonlinear material escape the tradeoff?

FoM = n₂ / (β_TPA· λ): below 1, two-photon absorption eats the signal before a π phase shift accumulates. The column to read is n₂ / loss — across ~600× in n₂, it stays within about an order of magnitude. Material choice slides you along the loss–nonlinearity tradeoff; it doesn't lift you off it.

material	n₂ (m²/W)	FoM	loss (dB/cm)	n₂ / loss
Silicon	4.50e-18	0.36	2	2.25e-18
Silicon nitride	2.50e-19	161.29	0.1	2.50e-18
Chalcogenide As₂S₃	3.00e-18	32.26	0.5	6.00e-18
Silica	2.70e-20	174.19	0.001	2.70e-17
AlGaAs (below half-gap)	1.60e-17	103.23	1	1.60e-17
Yours	1.00e-18	6.45	0.5	2.00e-18

n₂ (m²/W)β_TPA (m/W)linear loss (dB/cm)λ (nm)

The structural requirement isn't more variety. |E|², χ⁽²⁾ and Kerr are all polynomial, and a stack of polynomial nonlinearities computes a bigger polynomial — not a universal approximator (Leshno et al. 1993). The vocabulary needs at least one saturating element: saturable absorption, or a bistable phase-change cell.

3 · Logarithmic arithmetic · does LNS pay at the precisions inference runs at?

LNS turns a multiply into an add — and an add into a log-sum-exp: a compare/subtract, a table lookup and another add. A MAC is both. Per-op energies are Horowitz's ISSCC 2014 45 nm table; the ratios are what carry across nodes.

LUT cost (× 8 KB SRAM read)1.0 = 5 pJ. 0.1 is the optimistic dedicated small table.

precision	linear MAC (pJ)	LNS MAC (pJ)	LNS / linear
int8	0.23	0.59	2.57×	worse
fp16	1.50	1.70	1.13×	worse
fp32	4.60	3.20	0.70×	better

bytes per weight2 = bf16.

move one weight (HBM)40.00 pJ

free arithmetic saves3.4%

Decode is memory-bound: on the calibrated H100 constants, one weight enables one MAC (1.42 pJ) but costs 40.00 pJ to move. Making arithmetic entirely free saves 3.4%; LNS captures at most a fraction of that. The industry adopted the log idea where it pays — the floating-point exponent and the shared block scale of MXFP and NVFP4.

4 · Pre-ADC aggregation · K optical experts, one ADC

Broadcast x once, fan out to K linear operators, aggregate with fixed analog weights g, digitise once. It is exactly one matrix: Σ g_i W_i x = (Σ g_i W_i) x. The check below builds the bank in your browser and compares.

K — expertsd — dimension

‖bank − W_eff·x‖< 1e-12

‖bank output‖93.1

bank weights409,600

collapsed weights4,096

Identical up to floating-point rounding: the bank costs 100× the hardware for the same function. A bank is only worth it if the gates depend on x or the operators are nonlinear — and each escape has a price. Input-dependent routing must measure something first (the conversion the aggregation existed to avoid) or reprogram analog weights every token; nonlinearity between broadcast and aggregation hits the cascadability wall. Amortised conversions or input-dependent routing: pick one.

Run it locally (Python)

pip install numpy
  python recirculation_budget.py
  python material_fom.py
  python energy_lns.py
  python preadc_trilemma.py

recirculation_budget.pypass budget from the precision floor
material_fom.pynonlinear figure of merit
energy_lns.pyLNS vs. linear MAC energy
preadc_trilemma.pyaggregation collapse · requires numpy

Does the optical idea survive the arithmetic?

1 · Recirculation · how many passes through one optical circuit?

2 · Materials · does a different nonlinear material escape the tradeoff?

3 · Logarithmic arithmetic · does LNS pay at the precisions inference runs at?

4 · Pre-ADC aggregation · K optical experts, one ADC

Run it locally (Python)

Leadership

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