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The pump-probe experimental setup in fig. S2 was used to carry out the multilevel, time-resolved, and scalar multiplication measurements. The setup was then extended to that in fig. S3 to carry out the matrix-vector multiplications within balanced splitters with two GST memory cells. In both setups, the optical signals were confined to the photonic circuit, that is, Write, Erase, read-out, and multiplications were all realized within the integrated chip. To avoid interference, two different C + L CW tunable laser sources were used; wavelengths of 1598 nm (TSL-550, Santec) and 1590 nm (N7711A, Keysight) were chosen for the probe and pump, respectively. The pump pulses—as well as the multiplicand pulse Pin—were subsequently generated with an electro-optical modulation (Lucent Technologies, 2623NA), which was controlled by a 100-MHz electrical pulse generator (AFG 3102C, Tektronix). The pulse was further power amplified by a low-noise erbium-doped fiber amplifier (AEDFA-CL-23, Amonics). Both the pump pulses and the probe were coupled into the photonic device using integrated grating couplers with transmission peak at 1598 nm and coupling efficiency of ~20%. The counter-propagating scheme was used to ease the separation of the signals, and tunable optical filters (OTF-320, Santec) were introduced to the optical lines to further suppress noise resulting from reflections. At the probe output of the device, the CW signal was divided into two beams using a 90/10 beam splitter to measure the time-resolved and the long-term transmission with a 200-kHz low-noise photoreceiver (NewFocus, 2011) and a 125 MHz photodetector (NewFocus, 1811), respectively. At the other output, the transmitted pulses were monitored using a 1-GHz photodetector (NewFocus, 1611).

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