TENSORIZED INTEGRATED COHERENT ISING MACHINE

Examples of the present technology provide “tensorized” integrated coherent Ising machines that improve scalability by leveraging a tensorized optical coupling matrix comprising layers of multi-wavelength photonic tensor-train (TT) cores cascaded together via passive optical cross-connects. A multi-wavelength photonic TT core may comprise a Mach Zehnder interferometer (MZI) mesh (i.e., a lattice/array of interconnected MZIs) that modulates the phase and/or amplitude of optical signals. Tensorized integrated CIMs of the present technology can achieve further scalability optimizations by implementing bistable Ising nodes via one or more multi-wavelength Ising node collections. A multi-wavelength Ising node collection may comprise a bistable Ising nodes implemented on a common MZI, where each bistable Ising node of the multi-wavelength Ising node collection is associated with a separate wavelength of light.

BACKGROUND

The Ising model has roots in solid-state physics as a model of ferromagnets. The Ising model maps a broad class of combinatorial optimization problems for which no efficient, accurate classical algorithm exists. The Ising model may be described by a Hamiltonian function (H) as follows where Jijare the elements of a coupling matrix between spins σiand σj.

The Ising model can be used to formulate NP-hard combinatorial optimization problems (i.e., problems such as the “traveling salesman” problem where the number of possible solutions increases exponentially with the number of components of a system) with only polynomial overhead. For this reason, machines/apparatuses that can efficiently and effectively solve the Ising model can be very valuable.

DETAILED DESCRIPTION

Ising machines are apparatuses built/designed to find the absolute or approximate ground states of the Ising model. Because NP-hard combinatorial optimization problems (referred to herein as NP-hard problems) can be reformulated using the Ising model, these Ising machines can be used to solve (i.e., find approximate solutions to) NP-hard problems.

Conventional Ising machines generally leverage quantum annealing to find the absolute or approximate ground states of the Ising model based on quantum fluctuations. These conventional Ising machines can be difficult to scale-up due to restrictions in Ising node interconnectivity characteristic of quantum annealing-based systems. Due to scalability limitations, these conventional Ising machines can struggle to solve NP-hard problems of increasing (i.e., higher) complexity without prohibitive footprint and cost increases.

More recently, coherent Ising machines (CIMs) have been introduced that find the absolute or approximate ground states of the Ising model by processing coherent optical signals (e.g., optical signals having the same/similar frequency and waveform). CIMs have the potential for greater scalability than quantum annealing-based Ising machines as they lack the Ising node interconnectivity restrictions characteristic of the quantum annealing-based systems.

Among various CIMs, integrated CIMs (i.e., integrated photonic circuits that implement CIMs) attract much interest due to their high energy efficiency, compact footprint, high speed, and low fabrication cost. For example, PCT/US2015/048952 (which is incorporated herein by reference in its entirety) proposes a CIM architecture comprising: (1) bistable Ising nodes implemented using tunable optical resonators/amplifiers with self-feedback; (2) a coupling matrix; and (3) waveguides connecting the bistable Ising nodes and coupling matrix to form an Ising machine feedback loop.

While an integrated CIM is generally more scalable than conventional/quantum annealing-based Ising machines, further improvements to scalability of the integrated CIM would still be highly desirable.

Against this backdrop, examples of the present technology provide “tensorized” integrated CIMs that improve scalability by leveraging a tensorized optical coupling matrix. The tensorized optical coupling matrix may comprise layers of multi-wavelength photonic tensor-train (TT) cores cascaded together via passive optical cross-connects. A multi-wavelength photonic TT core may comprise a Mach-Zehnder interferometer (MZI) mesh (i.e., a lattice/array of interconnected MZIs) that modulates the phase and/or amplitude of optical signals of multiple wavelengths.

The tensorized optical coupling matrix can implement a tensor-train (TT) decomposition algorithm that efficiently compresses over-parameterized coupling matrices (especially low-rank sparse coupling matrices) used for solving the Ising model. Furthermore, by cascading the multi-wavelength photonic TT cores via passive optical cross-connects, further reductions in hardware (e.g., fewer MZIs) and footprint can be realized. Accordingly, a tensorized integrated CIM of the present technology can be scaled-up to solve complex NP-hard problems with less hardware and a smaller footprint than existing integrated CIMs. Relatedly, the tensorized integrated CIM may be less expensive to fabricate, consume less power, and require less control complexity than existing integrated CIMs.

As described in greater detail below, tensorized integrated CIMs can achieve further scalability optimizations through intelligent/particularized design of bistable Ising nodes. For example, bistable Ising nodes of a tensorized integrated CIM can be implemented via one or more multi-wavelength Ising node collections. A multi-wavelength Ising node collection may comprise a plurality of bistable Ising nodes, where each bistable Ising node of the multi-wavelength Ising node collection is associated with a separate wavelength of light. A single multi-wavelength Ising node collection-to-tensorized optical coupling matrix waveguide can optically connect an output from the multi-wavelength Ising node collection to an input multi-wavelength photonic TT core of the tensorized optical coupling matrix. Relatedly, a single tensorized optical coupling matrix-to-multi-wavelength Ising node collection waveguide can optically connect an output multi-wavelength photonic TT core of the tensorized optical coupling matrix to an input of the multi-wavelength Ising node collection. Accordingly, the multi-wavelength Ising node collection-to-tensorized optical coupling matrix waveguide and the tensorized optical coupling matrix-to-multi-wavelength Ising node collection waveguide can optically connect the multi-wavelength Ising node collection and the tensorized optical coupling matrix to form an Ising machine feedback loop.

Multi-wavelength Ising node collections can reduce hardware and footprint required to implement a tensorized integrated CIM of the presently disclosed technology. That is, tensorized integrated CIMs that leverage a multi-wavelength Ising node collection architecture can realize greater scalability than potential alternative tensorized integrated CIM designs.

For example, bistable Ising nodes in existing integrated CIMs are not grouped into multi-wavelength Ising node collections. Instead, each bistable Ising node is associated with its own bistable Ising node-to-coupling matrix waveguide, and its own coupling matrix-to-bistable Ising node waveguide. For this reason, implementing a tensorized integrated CIM using this existing bistable Ising node arrangement would generally require a multiplexer to multiplex optical signals received from bistable Ising nodes onto a waveguide associated with an input multi-wavelength photonic TT core of the tensorized optical coupling matrix. Relatedly, a demultiplexer would be required to demultiplex signals received from an output multi-wavelength photonic TT core of the tensorized optical coupling matrix onto waveguides going to the bistable Ising nodes. Depending on the size of the tensorized integrated CIM (e.g., the number of constituent multi-wavelength photonic TT cores per layer), the number and/or size of these multiplexers and demultiplexers may also increase. Also, for a tensorized integrated CIM comprising N bistable Ising nodes, this simplistic bistable Ising node arrangement would require N bistable Ising node-to-multiplexer waveguides and N demultiplexer-to-bistable Ising node waveguides.

However, by leveraging multi-wavelength Ising node collections instead of the simplistic/conventional arrangement of bistable Ising nodes, tensorized integrated CIMs of the presently disclosed technology can: (a) reduce/eliminate the above-described multiplexer(s) and demultiplexer(s); (b) reduce the number of waveguides required to connect the bistable Ising nodes and the tensorized coupling matrix; and (c) improve on-chip arrangement/consolidation of bistable Ising node hardware. Accordingly, tensorized integrated CIMs that implement multi-wavelength Ising node collections can realize greater compactness (i.e., smaller footprints) with less hardware than potential alternative (and more simplistic) tensorized integrated CIM designs. Through these footprint and hardware reductions, larger (and more powerful systems) can be implemented on smaller chips.

Examples of the present technology will be described in greater detail below in conjunction with the following FIGS.

FIG.1depicts an example integrated CIM100. Integrated CIM100may represent a generic integrated CIM upon which examples of the presently disclosed technology improve.

As depicted, each bistable Ising node of bistable Ising nodes110can implement a nonlinear input-output relationship vi=T (ui) from its input uito its output vi. The function T may be a threshold-like function.

As depicted, coupling matrix130applies analog weights to outputs of bistable Ising nodes110. For example, coupling matrix130can compute weighted sums of the outputs of bistable Ising nodes110in the manner below:

Here, the outputs from bistable Ising nodes110vjare multiplied by weights wijto produce signals u′i. Coupling matrix-to-bistable Ising node waveguides125connect the i-th output from coupling matrix130to the input of the i-th bistable Ising node of bistable Ising nodes110(e.g., ui=u′i) in a recurrent feedback loop.

Through the above-described Ising machine feedback loop, integrated CIM100can find the absolute or approximate ground states of the Ising model. Because NP-hard problems can be reformulated using the Ising model, integrated CIM100can be used to solve such NP-hard problems.

Before describing hardware implementations for bistable Ising nodes110in greater detail in conjunction withFIG.2, it may be helpful to understand some of the advantages integrated CIM100provides over conventional quantum annealing-based Ising machines. For example, integrated CIM100includes all-to-all connectivity between bistable Ising nodes110and coupling matrix130. That is, N outputs from bistable Ising nodes110are connected to N inputs of coupling matrix130via bistable Ising node-to-coupling matrix waveguides120. Relatedly, N outputs from coupling matrix130are connected to N inputs bistable Ising nodes110via coupling matrix-to-bistable Ising node waveguides125. Due to this all-to-all connectivity, integrated CIM100can outperform conventional quantum annealing-based Ising machines based on integrated CIM100's associated higher connectivity and reconfigurability. For instance, the computational capacity of an Ising machine may be determined by the number of Ising nodes, but also by the number of possible connections between Ising nodes, corresponding to the number of independent non-zero weights wijin Eq. 2. As alluded to above, integrated CIMs such as integrated CIM100process coherent optical signals. Consequently, ui, vi, wij, etc. are complex-valued. While traditional applications may only use real-valued wijweights, integrated CIM100can use a phase-sensitive nonlinearity that acts on the real-quadrature of ui. This facilitates processing of the spin variable stored in the specific quadrature of the super-mode of the anti-correlated internal states of the two resonators210(a) and210(b).

FIG.2depicts an example bistable Ising node200. Bistable Ising node200may represent a bistable Ising node which is not implemented in a multi-wavelength bistable Ising node collection. Bistable Ising node200may be an example of a bistable Ising node of bistable Ising nodes110fromFIG.1.

As depicted, bistable Ising node200comprises two optical resonators (i.e., optical resonators210(a) and210(b)) implemented on the arms of a Mach-Zehnder interferometer (MZI) (i.e., first arm waveguide214and second arm waveguide215). In some examples, optical resonators210(a) and210(b) may comprise identical optical ring resonators with dispersive optical non-linearity.

The MZI on which optical resonators210(a) and210(b) are implemented includes: (a) a first arm waveguide214; (b) a second arm waveguide215; (c) an input-side optical coupler218; and (d) an output-side optical coupler219. As depicted, input-side optical coupler218receives a first input from a coupling matrix-to-bistable Ising node waveguide220and a second input from a feedback loop waveguide216. Input-side optical coupler218then outputs a first output to first arm waveguide214and a second output to second arm waveguide215. Output-side optical coupler219receives a first input from first arm waveguide214and a second input from second arm waveguide215. Output-side optical coupler219then outputs a first output to a bistable Ising node-to-coupling matrix waveguide222and a second output to feedback loop waveguide216. Input-side optical coupler218and output-side optical coupler219may be various types of optical couplers, including 50-50 splitters, multi-mode interferometers (MMIs), direction couplers, etc.

As depicted, bistable Ising node200also comprises a feedback loop waveguide216and a bias pump230. An output of bias pump230may be optically coupled to feedback loop waveguide216. As will be described in greater detail below, bias pump230and optical resonators210(a) and210(b) can be configured to store two anti-correlated states associated with bistable Ising node200. That is, through its feedback loop and MZI structure, bistable Ising node200can selectively modify the circulating power of symmetric super-modes of optical resonators210(a) and210(b) such that optical resonators210(a) and210(b) assume two anti-correlated states.

In various examples, bistable Ising node200may also include bias field couplers250and252. Bias field couplers250and252can inject bias fields for the spin variable (e.g., one of the four non-relativistic coordinates of an electron) and facilitate monitoring of current spin amplitude.

As examples of the presently disclosed technology are designed in appreciation of, asymmetry of the bistability of a single optical resonator can be circumvented by encoding spin states in two optical resonators (e.g., optical resonators210(a) and210(b)) configured such that their internal states are anti-correlated. Accordingly, bistable Ising node220can exhibit a pitchfork bifurcation as drive from bias pump230increases power for a bias field above a threshold level (the phase of the bias field may be weakly related to the length of feedback loop waveguide216). This may be analogous to the bifurcation occurring in a degenerate optical parametric oscillator (DOPO). In this sense, bistable Ising node200can emulate certain traditional CIMs based on DOPOs but with an additional advantage that the input field of bias pump230is of the same wavelength as the signal field of bias pump230. Furthermore, bistable Ising node200does not depend on any specific type of optical non-linearity and could also be adaptive to devices with non-linear loss.

As alluded to above, a strong bias field entering a first MZI input (e.g., a first input of input-side optical coupler218) can lead to tunable, phase-sensitive gain for the transmission of a small signal from a second MZI input (e.g., a second input of input-side optical coupler218) to an MZI output (e.g., an output from input-side optical coupler218). A maximum gain can be determined by detuning bias (and signal) drive from a common optical resonator resonance frequency. There may exist a threshold detuning beyond which optical resonators210(a) and210(b) individually become bistable over a certain input power range. To implement two states to encode spin, coherent feedback can be used to couple resonator modes for optical resonators210(a) and210(b). For example, using an appropriately chosen bias feedback phase, two metastable states can be made unstable such that optical resonators210(a) and210(b) can only assume anti-correlated internal states.

As alluded to above, bistable Ising node200may be considered a tunable amplifier with feedback (TAFB) that features a pitchfork bifurcation at a particular threshold bias field power. Above this threshold bias field power, the above-mentioned anti-correlated states may exist. Amplitude of bias pump230can be used as a parameter to drive bias field power to and/or through the threshold bias field power. The spin variable may be encoded in a specific quadrature that is a function of pump amplitude.

FIGS.3A-3Cdepict an example tensorized integrated CIM300, in accordance with various examples of the present technology.

As depicted, tensorized integrated CIM300comprises: (a) multi-wavelength Ising node collections (e.g., multi-wavelength Ising node collections312,314, and316); (b) a tensorized optical coupling matrix330; (c) multi-wavelength Ising node collection-to-tensorized optical coupling matrix waveguides320optically connecting outputs from the multi-wavelength Ising node collections to inputs of tensorized optical coupling matrix330; and (d) tensorized optical coupling matrix-to-multi-wavelength Ising node collection waveguides325optically connecting outputs from tensorized optical coupling matrix330to inputs of the multi-wavelength Ising node collections. Accordingly, multi-wavelength Ising node collection-to-tensorized optical coupling matrix waveguides320and tensorized optical coupling matrix-to-multi-wavelength Ising node collection waveguides325optically connect the multi-wavelength Ising node collections and tensorized optical coupling matrix330to form an Ising machine feedback loop. As depicted, tensorized integrated CIM300may also include a laser360(e.g., a comb laser) that introduces up to g wavelengths of light into tensorized integrated CIM300.

Example multi-wavelength Ising node collections of the present technology will be described in greater detail in conjunction withFIG.5. Similarly, an example tensorized optical coupling matrix will be described in greater detail in conjunction withFIGS.6A-6D. Notwithstanding, it may be useful to describe the composition and operation of tensorized integrated CIM300at a high level before proceeding to those figures.

As alluded to above, examples of the present technology provide “tensorized” integrated CIMs (e.g., tensorized integrated CIM300) that improve scalability by leveraging a tensorized optical coupling matrix (e.g., tensorized optical coupling matrix330) comprising layers of multi-wavelength photonic tensor-train (TT) cores (e.g., a first layer comprising multi-wavelength photonic TT cores332(a) and332(b), a second layer comprising multi-wavelength photonic TT cores334(a) and334(b), a third layer comprising multi-wavelength photonic TT cores336(a) and336(b), a fourth layer comprising multi-wavelength photonic TT cores338(a) and338(b), etc.) cascaded together via passive optical cross-connects (e.g., passive optical cross-connects335). A multi-wavelength photonic TT core (e.g., multi-wavelength photonic TT core332(a)) may comprise a Mach Zehnder interferometer (MZI) mesh (i.e., a lattice/array of interconnected MZIs-see e.g.,FIG.7for more details) that modulates the phase and/or amplitude of optical signals.

The tensorized optical coupling matrix/cascaded multi-wavelength photonic TT cores can implement a tensor-train (TT) decomposition algorithm that efficiently compresses over-parameterized coupling matrices (especially low-rank sparse coupling matrices) used for solving the Ising model. Furthermore, by cascading the multi-wavelength photonic TT cores via the passive cross-connects, further reductions in hardware (e.g., fewer MZIs) and footprint can be realized. Accordingly, a tensorized integrated CIM of the present technology (e.g., tensorized integrated CIM300) can be scaled-up to solve complex NP-hard problems with less hardware and a smaller footprint than existing integrated CIMs. Relatedly, the tensorized integrated CIM may be less expensive to fabricate, consume less power, and require less control complexity than existing integrated CIMs.

As alluded to above, tensorized integrated CIMs can achieve further scalability optimizations through intelligent/particularized design of bistable Ising nodes. For example, bistable Ising nodes of a tensorized integrated CIM can be implemented via one or more multi-wavelength Ising node collections (e.g., multi-wavelength Ising node collections312,314,316, etc.). A multi-wavelength Ising node collection (e.g., multi-wavelength Ising node collections312) may comprise multiple bistable Ising nodes, where each bistable Ising node of the multi-wavelength Ising node collection is associated with a separate wavelength of light (see e.g.,FIG.5for more details). A single multi-wavelength Ising node collection-to-tensorized optical coupling matrix waveguide (see e.g., an individual multi-wavelength Ising node collection-to-tensorized optical coupling matrix waveguide of the multi-wavelength Ising node collection-to-tensorized optical coupling matrix waveguides320) can optically connect output from the multi-wavelength Ising node collection to an input multi-wavelength photonic TT core (e.g., multi-wavelength photonic TT core332(a)) of the tensorized optical coupling matrix. Relatedly, a single tensorized optical coupling matrix-to-multi-wavelength Ising node collection waveguide (see e.g., an individual tensorized optical coupling matrix-to-multi-wavelength Ising node collection waveguide of the tensorized optical coupling matrix-to-multi-wavelength Ising node collection waveguides325) can optically connect an output multi-wavelength photonic TT core (e.g., multi-wavelength photonic TT core338(a)) of the tensorized optical coupling matrix to an input of the multi-wavelength Ising node collection. Accordingly, the multi-wavelength Ising node collection-to-tensorized optical coupling matrix waveguide and the tensorized optical coupling matrix-to-multi-wavelength Ising node collection waveguide can optically connect the multi-wavelength Ising node collection and the tensorized optical coupling matrix to form an Ising machine feedback loop.

As alluded to above, multi-wavelength Ising node collections can reduce hardware and footprint required to implement a tensorized integrated CIM of the presently disclosed technology. That is, tensorized integrated CIMs that leverage a multi-wavelength Ising node collection architecture can realize greater scalability than potential alternative tensorized integrated CIM designs.

For illustration,FIGS.4A-4Ddepict an example alternative tensorized integrated CIM400that does not leverage multi-wavelength Ising node collections.

As depicted, tensorized integrated CIM400comprises: (a) bistable Ising nodes (e.g., bistable Ising nodes411-416), where each bistable Ising node of the bistable Ising nodes is associated with a separate wavelength of light; (b) a tensorized optical coupling matrix430comprising layers of multi-wavelength photonic tensor-train (TT) cores (e.g., a first layer comprising multi-wavelength photonic TT cores432(a) and432(b), a second layer comprising multi-wavelength photonic TT cores434(a) and434(b), a third layer comprising multi-wavelength photonic TT cores436(a) and436(b), a fourth layer comprising multi-wavelength photonic TT cores438(a) and438(b), etc.) cascaded together via passive optical cross-connects (e.g., passive optical cross-connects435); (c) a multiplexer470that multiplexes optical signals received from bistable Ising node-to-multiplexer waveguides440onto input waveguides of tensorized coupling matrix430; (d) a demultiplexer480that demultiplexes signals received from output waveguides of tensorized coupling matrix430onto demultiplexer-to-bistable Ising node waveguides445; (e) bistable Ising node-to-multiplexer waveguides440optically connecting outputs from the bistable Ising nodes (e.g., outputs from bistable Ising nodes411-416) to multiplexer470; and (f) demultiplexer-to-bistable Ising node waveguides445optically connecting outputs from demultiplexer480to the bistable Ising nodes. As depicted, each bistable Ising node of tensorized integrated CIM400may be implemented on its own MZI, and have its own separate bistable Ising node-to-multiplexer waveguide and demultiplexer-to-bistable Ising node waveguide. Accordingly, each bistable Ising node of tensorized integrated CIM400may be implemented in a similar manner to bistable Ising node200ofFIG.2.

As depicted, tensorized integrated CIM400includes some of the same/similar features as tensorized integrated CIM300(e.g., a tensorized optical coupling matrix430that contains the same/similar structures as tensorized optical coupling matrix330, a laser460that introduces up to g wavelengths into the system, etc.)—with a few key differences. For example, lacking the multi-wavelength Ising node collection architecture of tensorized integrated CIM300, tensorized integrated CIM400includes a larger number of waveguides between its Ising nodes and its tensorized coupling matrix. For instance, if both tensorized integrated CIM300and tensorized integrated CIM400comprise N Ising nodes, and each multi-wavelength Ising node collection of tensorized integrated CIM300includes g Ising nodes, tensorized integrated CIM400may include 2 g times more waveguides between its Ising nodes and tensorized optical coupling matrix. Lacking the multi-wavelength Ising node collection architecture of tensorized integrated CIM300, tensorized integrated CIM400also includes multiplexer470(which may comprise one or more individual multiplexers) and demultiplexer480(which may comprise one or more individual demultiplexers).

As alluded to above, bistable Ising nodes in existing integrated CIMs are not grouped into multi-wavelength Ising node collections. Instead, each bistable Ising node is associated with its own MZI, its own bistable Ising node-to-coupling matrix waveguide, and its own coupling matrix-to-bistable Ising node waveguide (see e.g., bistable Ising node200ofFIG.2). For this reason, implementing a tensorized integrated CIM using this existing bistable Ising node arrangement—of which tensorized integrated CIM400is an example—would generally require a multiplexer to multiplex optical signals received from the bistable Ising nodes onto waveguides associated with input multi-wavelength photonic TT cores of the tensorized optical coupling matrix. Relatedly, a demultiplexer would be required to demultiplex signals received from output multi-wavelength photonic TT cores of the tensorized optical coupling matrix onto waveguides going to the bistable Ising nodes. Depending on the size of the tensorized integrated CIM (e.g., the number of constituent multi-wavelength photonic TT cores per layer), the number and/or size of these multiplexers and demultiplexers may also increase. Also (and as alluded to above), the tensorized integrated CIM implementing this simple/existing bistable Ising node arrangement (e.g., tensorized integrated CIM400) may require many more waveguides between bistable Ising nodes and tensorized optical coupling matrix than a tensorized integrated CIM (e.g., tensorized integrated CIM300) that leverages multi-wavelength Ising node collections.

Accordingly, by leveraging multi-wavelength Ising node collections instead of the simplistic/conventional arrangement of bistable Ising nodes, tensorized integrated CIMs of the presently disclosed technology (e.g., tensorized integrated CIM300) can: (a) reduce/eliminate the above-described multiplexer(s) and demultiplexer(s); (b) reduce the number of waveguides required to connect the bistable Ising nodes and the tensorized coupling matrix; and (c) improve on-chip arrangement/consolidation of bistable Ising node hardware. Accordingly, tensorized integrated CIMs that implement multi-wavelength Ising node collections can realize greater compactness (i.e., smaller footprints) with less hardware than potential alternative (and more simplistic) tensorized integrated CIM designs. Through these footprint and hardware reductions, larger (and more powerful systems) can be implemented on smaller chips.

FIG.5depicts an example multi-wavelength Ising node collection500, in accordance with various examples of the presently disclosed technology. Multi-wavelength Ising node collection500may be an example of a multi-wavelength Ising node collection of tensorized integrated CIM300ofFIG.3.

As depicted, multi-wavelength Ising node collection500comprises g bistable Ising nodes implemented on a Mach Zehnder interferometer (MZI), where each bistable Ising node of the multi-wavelength Ising node collection is associated with a separate wavelength of light. A given bistable Ising node may comprise two identical optical resonators configured to modulate light of the given bistable Ising node's associated wavelength. For example, a first bistable Ising node of multi-wavelength Ising node collection500may comprise optical resonators510(1)(a) and510(1)(b). Optical resonators510(1)(a) and510(1)(b) may be configured to modulate light of a first wavelength. Together with a bias pump530(which may be a multi-wavelength bias pump associated with g wavelengths), optical resonators510(1)(a) and510(1)(b) may also be configured to store two bistable/anti-correlated states. Relatedly, a g-th bistable Ising node of multi-wavelength Ising node collection500may comprise optical resonators510(g) (a) and510(g) (b). Optical resonators510(g) (a) and510(g) (b) may be configured to modulate light of a g-th wavelength. Together with bias pump530, optical resonators510(g) (a) and510(g) (b) may also be configured to store two bistable/anti-correlated states.

As alluded to above (and as depicted), the g bistable Ising nodes of multi-wavelength Ising node collection500are implemented on a common MZI.

The MZI on which the g bistable Ising nodes are implemented includes: (a) a first arm waveguide514; (b) a second arm waveguide515; (c) an input-side optical coupler518; and (d) an output-side optical coupler519. As depicted, input-side optical coupler518receives a first input from a tensorized optical coupling matrix-to-multi-wavelength Ising node collection waveguide520and a second input from a feedback loop waveguide516. Input-side optical coupler518then outputs a first output to first arm waveguide514and a second output to second arm waveguide515. Output-side optical coupler519receives a first input from first arm waveguide514and a second input from second arm waveguide515. Output-side optical coupler519then outputs a first output to a multi-wavelength Ising node collection-to-tensorized optical coupling matrix waveguide522and a second output to feedback loop waveguide516. Input-side optical coupler518and output-side optical coupler519may be various types of optical couplers, including 50-50 splitters, multi-mode interferometers (MMIs), direction couplers, etc.

Through its feedback loop/MZI structure, multi-wavelength Ising node collection500can selectively modify the circulating power of symmetric super-modes of the optical resonators implemented on the arms of the MZI such that the optical resonators assume two anti-correlated states. A coupler inside feedback loop waveguide516can be used to add bias signals from bias pump530(as alluded to above, these bias signals may be associated with the g wavelengths).

In various examples, multi-wavelength Ising node collection500may also include bias field couplers (not depicted) implemented on each of tensorized optical coupling matrix-to-multi-wavelength Ising node collection waveguide520and multi-wavelength Ising node collection-to-tensorized optical coupling matrix waveguide522respectively. Such bias field couplers can inject bias fields for the spin variable and facilitate monitoring of current spin amplitude.

As examples of the presently disclosed technology are designed in appreciation of, asymmetry of the bistability of a single optical resonator can be circumvented by encoding spin states in two optical resonators (e.g., optical resonators510(1)(a) and510(1)(b)) configured such that their internal states are anti-correlated. Accordingly, a given bistable Ising node of multi-wavelength Ising node collection500can exhibit a pitchfork bifurcation as drive from multi-wavelength bias pump530increases power for a bias field above a threshold level. This may be analogous to the bifurcation occurring in a degenerate optical parametric oscillator. In this sense, the given bistable Ising node can emulate certain traditional CIMs based on degenerate optical parametric oscillators (DOPOs) but with an additional advantage that the input field of bias pump530is of the same wavelength as the signal field of bias pump530. Furthermore, the given bistable Ising node does not depend on any specific type of optical nonlinearity and could also be adaptive to devices with nonlinear loss.

As alluded to above, a strong bias field entering a first MZI input (e.g., a first input of input-side optical coupler518) can lead to tunable, phase-sensitive gain for the transmission of a small signal from a second MZI input (e.g., a second input of input-side optical coupler518) to a second MZI output (e.g., an output from input-side optical coupler418). A maximum gain can be determined by detuning of the bias (and signal) drive from a common optical resonator resonance frequency. For example, there may exist a threshold detuning beyond which the optical resonators510(1)(a) and510(1)(b) individually become bistable over a certain input power range. To implement two states to encode spin, coherent feedback can be used to couple resonator modes for optical resonators510(1)(a) and510(1)(b). For example, using a first appropriately chosen bias feedback phase (driven by bias pump530), two metastable states can be made unstable such that optical resonators510(1)(a) and510(1)(b) can only assume anti-correlated internal states. Optical resonators510(2)(a) and510(2)(b) can be configured such that a second bias feedback phase (driven by bias pump530) causes optical resonators510(2)(a) and510(2)(b) to assume anti-correlated states. Relatedly, optical resonators510(g) (a) and510(g) (b) can be configured such that a third bias feedback phase (driven by bias pump530) causes optical resonators510(g) (a) and510(g) (b) to assume anti-correlated states, and so on.

Here it should be understood that multi-wavelength Ising node collection500is just one example implementation of a multi-wavelength Ising node collection. For instance, another multi-wavelength Ising node collection implementation may comprise one bus waveguide and multiple microring resonators with internal reflection. Another multi-wavelength Ising node collection implementation may comprise multiple directly-coupled dual microring resonators.

FIGS.6A-6Ddepict an example tensorized optical coupling matrix600, in accordance with various examples of the presently disclosed technology. It should be understood that tensorized optical coupling matrix600merely depicts an example, and that other implementations may include tensorized optical coupling matrices of different configurations that embody the same/similar principles.

Namely, examples of the present technology provide “tensorized” integrated CIMs that improve scalability by leveraging a tensorized optical coupling matrix (e.g., tensorized optical coupling matrix600) comprising layers of multi-wavelength photonic tensor-train (TT) cores (e.g., photonic TT core layer4, photonic TT core layer3, photonic TT core layer2, and photonic TT core layer1) cascaded together via passive optical cross-connects. A multi-wavelength photonic TT core may comprise a Mach Zehnder interferometer (MZI) mesh (i.e., a lattice/array of interconnected MZIs-see e.g.,FIG.7for more details) that modulates the phase and/or amplitude of optical signals.

The tensorized optical coupling matrix/cascaded multi-wavelength photonic TT cores can implement a tensor-train (TT) decomposition algorithm that efficiently compresses over-parameterized coupling matrices (especially low-rank sparse coupling matrices) used for solving the Ising model. Furthermore, by cascading the multi-wavelength photonic TT cores via the passive cross-connects, further reductions in hardware (e.g., fewer MZIs) and footprint can be realized. Accordingly, a tensorized integrated CIM of the present technology can be scaled-up to solve complex NP-hard problems with less hardware and a smaller footprint than existing integrated CIMs. Relatedly, the tensorized integrated CIM may be less expensive to fabricate, consume less power, and require less control complexity than existing integrated CIMs.

Referring now to the TT decomposition algorithm, as examples of the present technology are designed in appreciation of, a weight matrix W ∈ RM×Ncan be represented in a TT format. The matrix dimensions M and N may be assumed to be factored as

where d is defined as the number of factors of M and N. μ and v may be natural bijections from indices (i,j) of W to indices [μ1(i), v1(j) . . . , μd(i), vd(j)] of a second order-2d weight tensor W. Then, W(i, j)=W(μ1(i), v1(j) . . . , μd(i), vd(j)). TT decomposition can be interpreted as singular value decomposition (SVD) of multi-dimensional matrices. As seen inFIG.9, TT decomposition can express the weight tensor W as a series of tensor products as depicted in Eq. 3 below:

Here, a four-way tensor Gk∈ RRk−1×Mk×NK×RKis the TT-core and the total number of tensor cores is d. The vector (R0, R1, . . . . Rd) is the TT rank, and R0=Rd=1. In this way, the total number of parameters can be reduced from M×N into the summation of the parameters of the TT-cores,

Referring now to example tensorized optical coupling matrix600ofFIG.6, tensorized optical coupling matrix600may be configured to receive inputs from 1024 bistable Ising nodes associated with 32 separate wavelengths of light. Here, 1024×1024 is factorized as 8×4×4×8×8×4×4×8. As depicted, the total number of photonic TT core layers is four (i.e., d=4), and the total number of wavelengths is 32 (i.e., g=32). Accordingly, in various examples, multi-wavelength Ising node collections coupled to tensorized optical coupling matrix600may each comprise 32 bistable Ising nodes associated with the 32 separate wavelengths.

Here, the TT-rank can be set as R0=R2=R4=1 and R1=R3=2.

As a result, tensorized optical coupling matrix600comprises four photonic TT core layers-photonic TT core layer4, photonic TT core layer3, photonic TT core layer2, and photonic TT core layer1—with the dimensions of G4∈ R1×8×4×2, G3∈ R2×4×8×1, G2∈ R1×4×8×2and G1∈ R2×4×8×1respectively. Passive optical cross-connects between photonic TT core layer4and photonic TT core layer3can implement index switching between the space domain of the intermediate signals. The passive optical cross-connects between photonic TT core layer2and photonic TT core layer1can implement similar index switching. As depicted, the passive optical cross-connects between photonic TT core layer3and photonic TT core layer2may implement a different functionality. In particular, they may comprise wavelength-space cross-connects that effectively switch indices between the space and wavelength domains of the intermediate signals. The wavelength-space cross connects between photonic TT core layer3and photonic TT core layer2can be realized by using wavelength division multiplexing (WDM) transponders. In some examples, the wavelength-space cross-connects may be implemented using O/E/O conversions and passive electrical cross-connects.

As depicted, each multi-wavelength photonic TT-core of tensorized optical coupling matrix600may comprise an 8×8 MZI mesh. Accordingly (and as depicted), tensorized optical coupling matrix600comprises sixteen 8×8 MZI meshes, along with passive optical cross-connects between multi-wavelength photonic TT cores (i.e., the 8×8 MZI meshes) of different photonic TT core layers-leading to sixteen 8×8 MZI meshes,448MZIs, and 32 cascaded stages of MZIs in total. Input optical signals received by tensorized optical coupling matrix600from bistable Ising nodes may first be modulated by using thirty-two 32-wavelength wavelength division multiplexing (WDM) microring modulator arrays. The signals may then be multiplied by a multi-wavelength photonic TT core of each photonic TT-core layer, before tensorized optical coupling matrix600outputs the optical signals back to the bistable Ising nodes.

As depicted, a 32-wavelength comb laser can provide a light source for the 32-wavelength optical signals of the system.

FIG.7depicts an example Mach Zehnder interferometer (MZI) mesh700, in accordance with various examples of the presently disclosed technology.

As alluded to above, a multi-wavelength photonic TT core of the presently disclosed technology may comprise an MZI mesh (i.e., a lattice/array of interconnected MZIs). Accordingly, MZI mesh700may represent a multi-wavelength photonic TT core of the presently disclosed technology.

As depicted, MZI mesh700can implement an N×N unitary matrix represented by a “rectangular” MZI mesh with 2×2 MZIs as the building blocks. That is, each constituent element (e.g., U1,1) of MZI mesh700may comprise a 2×2 MZI.

FIG.8depicts an example MZI mesh element800, in accordance with various examples of the presently disclosed technology. MZI mesh element800may comprise one of the constituent MZI mesh elements Ui,jof MZI mesh700.

As depicted, MZI mesh element800may comprise a 2×2 MZI. The 2×2 MZI may comprise: (a) a first optical coupler810and a second optical coupler812; and (2) a first phase shifter820and a second phase shifter822. Optical couplers810and812may be various types of optical couplers, including 50-50 splitters, multi-mode interferometers (MMIs), direction couplers, etc.

As alluded to above,FIG.9depicts an example of weight matrix TT decomposition for parameter compression, in accordance with various examples of the presently disclosed technology.

As used herein, the term “optical connection” (and its variants “operatively connected,” “operatively connecting,” etc.) may refer to a direct or indirect connection between two components that allows an optical signal to pass from one component to another.