All-Photonic Artificial Neural Network Processor Via Non-linear Optics

Stefan Krastanov | MIT ⟶ UMass Amherst
work by Basani, Heuck, Englund, Krastanov

Room-Temperature Optical Quantum Computing

Multimode cavity in a nonlinear optical medium¹
  1. Krastanov et al.
    Room-temperature photonic logical qubits via second-order nonlinearities
Multimode cavity¹
... and its spectrum
\[ \hat{H} = \hat{a}^\dagger \hat{b}\hat{b} + p(t)\hat{b}^\dagger\hat{c} + H.c. \]
  1. Krastanov et al.
    Room-temperature photonic logical qubits via second-order nonlinearities
\[ \hat{H} = \htmlClass{bc0}{\hat{a}^\dagger} \htmlClass{bc1}{\hat{b}\hat{b}} + \htmlClass{bc4}{p(t)} \htmlClass{bc1}{\hat{b}^\dagger} \htmlClass{bc2}{\hat{c}} + H.c. \]
  1. Krastanov et al.
    Room-temperature photonic logical qubits via second-order nonlinearities
\[ \hat{H} = \htmlClass{bc0}{\hat{a}^\dagger} \htmlClass{bc1}{\hat{b}\hat{b}} + \htmlClass{bc4}{p(t)} \htmlClass{bc1}{\hat{b}^\dagger} \htmlClass{bc2}{\hat{c}} + H.c. \]

A rather "poor" Hamiltonian...

  1. Krastanov et al.
    Room-temperature photonic logical qubits via second-order nonlinearities

Can be twisted into something useful:

Can be twisted into something useful:
Full error correction

\[\begin{aligned} |\htmlClass{bc0}{0}\htmlClass{bc1}{4}\htmlClass{bc2}{1}\rangle \rightarrow & |\htmlClass{bc0}{0}\htmlClass{bc1}{4}\htmlClass{bc2}{1}\rangle \\ |\htmlClass{bc0}{0}\htmlClass{bc1}{3}\htmlClass{bc2}{1}\rangle \rightarrow & |\htmlClass{bc0}{0}\htmlClass{bc1}{4}\htmlClass{bc2}{0}\rangle \\ \end{aligned}\]

Can be twisted into something useful:
Full error correction

\[\begin{aligned} |\htmlClass{bc0}{0}\htmlClass{bc1}{4}\htmlClass{bc2}{1}\rangle \rightarrow & |\htmlClass{bc0}{0}\htmlClass{bc1}{4}\htmlClass{bc2}{1}\rangle \\ |\htmlClass{bc0}{0}\htmlClass{bc1}{3}\htmlClass{bc2}{1}\rangle \rightarrow & |\htmlClass{bc0}{0}\htmlClass{bc1}{4}\htmlClass{bc2}{0}\rangle \\ |\htmlClass{bc0}{0}\htmlClass{bc1}{2}\htmlClass{bc2}{1}\rangle \rightarrow & |\htmlClass{bc0}{0}\htmlClass{bc1}{2}\htmlClass{bc2}{1}\rangle \\ |\htmlClass{bc0}{0}\htmlClass{bc1}{1}\htmlClass{bc2}{1}\rangle \rightarrow & |\htmlClass{bc0}{0}\htmlClass{bc1}{2}\htmlClass{bc2}{0}\rangle \\ |\htmlClass{bc0}{0}\htmlClass{bc1}{0}\htmlClass{bc2}{1}\rangle \rightarrow & |\htmlClass{bc0}{0}\htmlClass{bc1}{0}\htmlClass{bc2}{1}\rangle \\ \end{aligned}\]

Can be twisted into something useful:
Full error correction

\[\begin{aligned} |\htmlClass{bc0}{0}\htmlClass{bc1}{4}\htmlClass{bc2}{1}\rangle \rightarrow & |\htmlClass{bc0}{0}\htmlClass{bc1}{4}\htmlClass{bc2}{1}\rangle \\ |\htmlClass{bc0}{0}\htmlClass{bc1}{3}\htmlClass{bc2}{1}\rangle \rightarrow & |\htmlClass{bc0}{0}\htmlClass{bc1}{4}\htmlClass{bc2}{0}\rangle \\ |\htmlClass{bc0}{0}\htmlClass{bc1}{2}\htmlClass{bc2}{1}\rangle \rightarrow & |\htmlClass{bc0}{0}\htmlClass{bc1}{2}\htmlClass{bc2}{1}\rangle \\ |\htmlClass{bc0}{0}\htmlClass{bc1}{1}\htmlClass{bc2}{1}\rangle \rightarrow & |\htmlClass{bc0}{0}\htmlClass{bc1}{2}\htmlClass{bc2}{0}\rangle \\ |\htmlClass{bc0}{0}\htmlClass{bc1}{0}\htmlClass{bc2}{1}\rangle \rightarrow & |\htmlClass{bc0}{0}\htmlClass{bc1}{0}\htmlClass{bc2}{1}\rangle \\ \end{aligned}\]
Coherent feedback-free error correcting circuit for a \(\left\{|22\rangle,\frac{|40\rangle+|04\rangle}{\sqrt{2}}\right\}\) code.

Is this a realistic device?

\(\times 10^{10}\) of magnitude improvement in figures of merit in just 10 years...

All-photonic Linear Algebra Accelerator Via Non-linear Optics

All-to-all neuron interactions in a single on-chip component

Reversible computation

Interesting performance-power relationship

The "Computational Core"

A single multimode resonator.
A single multimode resonator...
and its spectrum

System dynamics

\(\frac{\mathrm{d} A_{i}}{\mathrm{d}t} = \left( -\frac{\Gamma}{2} + i\chi|P_{1}|^{2} \right) A_{i} \\ - \chi \left[ \sum_{j\gt i}^{N} \left( P_{1}P_{j}^{*} \right) A_{j} - \sum_{j\lt i}^{i - 1} \left( P_{1}^{*}P_{j} \right) A_{j} \right]\)

Where \(A_\dots\) stands for "neural amplitudes"

and \(P_\dots\) stands for "pump amplitudes".

\(P_1\gg P_i\) is a convenient simplification, but unimportant

Simplifying Assumptions and Hardware alternatives

We assumed strong pumps to avoid neural Kerr

  • ... but we could have forgone linearity
  • ... maybe also assume in-situ training
  • ... or switch to \(\chi^{(2)}\) media
    • ... but that would require orders of magnitude difference in pump and neuron frequencies

Effective dynamics - matrix multiplication

\(\vec{A}(t = \Delta t) = e^{\Delta t \textbf{P}} \vec{A}(t = 0)\)
\[ \textbf{P} = \begin{pmatrix} -\Gamma/2 + i\chi|P_{1}|^{2} & P_{1}P_{2}^{*}\chi & P_{1}P_{3}^{*}\chi & & & \dots & P_{1}P_{N}^{*}\chi \\ -P_{1}^{*}P_{2}\chi & -\Gamma/2 + i\chi|P_{1}|^{2} & P_{1}P_{2}^{*}\chi & & & \dots & P_{1}P_{N-1}^{*}\chi \\ -P_{1}^{*}P_{3}\chi & -P_{1}^{*}P_{2}\chi & -\Gamma/2 + i\chi|P_{1}|^{2} & \ddots & & \dots & P_{1}P_{N-2}^{*}\chi \\ \vdots & \ddots & \ddots & \ddots & & & \vdots\\ \vdots & & \ddots & \ddots & & & \vdots \\ & & & & & & \\ -P_{1}^{*}P_{N-1}\chi & \dots & & -P_{1}^{*}P_{2}\chi & & -\Gamma/2 + i\chi|P_{1}|^{2} & P_{1}P_{2}^{*}\chi\\ -P_{1}^{*}P_{N}\chi & \dots & & -P_{1}^{*}P_{3}\chi & & -P_{1}^{*}P_{2}\chi & -\Gamma/2 + i\chi|P_{1}|^{2} \end{pmatrix} \]

Expressivity

Expressivity (or maybe just "unitarity") decreases in the presence of loss

Linear classification example

Activation Function

sigmoid

nonlinearity

Fully optical implementation

capable of working with negative (or complex) numbers

Architecture

System dynamics

\(\frac{\partial E_{\mathrm{n}}}{\partial z} + \frac{\eta}{c} \frac{\partial E_{\mathrm{n}}}{\partial t} = -\kappa E_{\mathrm{sub}}^{2} - \alpha E_{\mathrm{n}}\)
\(\frac{\partial E_{\mathrm{sub}}}{\partial z} + \frac{\eta}{c} \frac{\partial E_{\mathrm{sub}}}{\partial t} = \kappa E_{\mathrm{n}} E_{\mathrm{sub}}^{*} - \alpha E_{\mathrm{sub}}\)

Interactive supplementary materials at pluto.krastanov.org.

Effective activation function via a sub-harmonic pump.

(Nonlinear) Amplification

Activation function via a second-harmonic pump.

A Case Study

Digit recognition with MNIST

Increase at first thanks to higher expressivity, followed by a decrease, followed by an absorption cliff.

Main Takeaways

All-to-all neuron interactions in a single on-chip component.

Self-funded dev program for room-temperature quantum computing.

Consider a postdoc at UMass Amherst:

Design of optical/mechanical/spin devices with Sandia, Mitre, and MIT.

Creating new modeling tools for the entire community.