A Novel Oscillator Ising Machine Coupling Scheme for High-Quality Optimization

Abstract

Oscillator Ising Machines (OIMs) are networks of coupled nonlinear oscillators that solve the NP-hard Ising problem heuristically. Conventionally, the oscillators in an OIM are coupled using resistors. However, the phase-domain properties of such couplers are unsatisfactory; resistively-coupled OIMs do not realize the optimization performance predicted by simulations of idealized OIMs. This has been a major hurdle impeding the development of high quality analog OIMs on integrated circuits. In this paper, we present a novel coupling scheme, the sampling coupler, that addresses this issue theoretically and practically. Essentially, a sampling coupler injects a current that depends on the phase difference between interacting oscillators. We prove analytically that using sampling couplers leads to idealized OIMs, abstracting away the waveforms and innate phase sensitivities of the oscillators. We evaluate sampling-coupler OIMs (using simulation) on a practically-important digital wireless communication problem and show that the performance is near-optimal. Sampling couplers therefore open up a way to implement practically feasible, high-performance analog OIMs using virtually any oscillator.

A Sampling-Coupler Based OIMs Achieve \(\mathrm {F_{c,d}}(\cdot )\)

Here, we prove that the actual $\textrm{F}_{\textrm{c}}(\cdot )$ of the sampling coupler is in fact equal to a scaled version of the ideal $\mathrm {F_{c,d}}(\cdot )$. We start from the PPV equation ((7), repeated here):

$$\begin{aligned} \frac{1}{f_0} \frac{d}{dt}\varDelta \phi (t) = \boldsymbol{v}^T(f_0 t + \varDelta \phi (t)) \cdot \boldsymbol{b}(t). \end{aligned}$$

(9)

Here, $\varDelta \phi (t)$ is the phase change of the oscillator, $\boldsymbol{v}^T(\cdot )$ is its vector of 1-periodic PPVs, and $\boldsymbol{b}(t)$ is the vector of inputs applied to the oscillator.

Applying the above PPV model to the two oscillators coupled via sampling couplers, we get

$$\begin{aligned} \begin{aligned} \frac{1}{f_0} \frac{d}{dt}\varDelta \phi _i(t) &= v^T(f_0t + \varDelta \phi _i(t)) \cdot b_i(t), \\ \frac{1}{f_0} \frac{d}{dt}\varDelta \phi _j(t) &= v^T(f_0t + \varDelta \phi _j(t)) \cdot b_j(t), \end{aligned} \end{aligned}$$

(10)

where $b_i(t)$ and $b_j(t)$ represent the inputs into each oscillator from the other via the sampling couplers in Fig. 3. Note that vector PPVs and inputs (i.e., $\boldsymbol{v}(\cdot )$ and $\boldsymbol{b}(t)$ respectively) have been replaced by scalars; this is a simplification (for exposition) assuming a single scalar input, i.e., $\boldsymbol{b}$ has only one nonzero component.¹⁴

We now focus on $OSC_i$ and derive its actual $\textrm{F}_{\textrm{c}}(\cdot )$. The input $b_{i}(t)$, represented in Fig. 3 by $I_{src,i,j}$, has the form

$$\begin{aligned} b_{i}(t) = J_{i,j} \cdot \textrm{F}_{\textrm{c,d}}\left( \varDelta \phi _i(t)-\varDelta \phi _j(t)\right) \cdot w\big (\varDelta \phi _i(t)+f_0t\big ), \end{aligned}$$

(11)

where:

• $J_{i,j}$ is the Ising coupling coefficient (from (1)) between the $i^\text {th}$ and the $j^\text {th}$ oscillator.

• $\textrm{F}_{\textrm{c,d}}\left( \varDelta \phi _i(t)-\varDelta \phi _j(t)\right)$ is the value of the sample held by $DFF_{i,j}$ in Fig. 3, as established in Sect. 3. Note that $\varDelta \phi _i$ and $\varDelta \phi _j$ in (11) now change with time as the system evolves. However, the flip-flops in Fig. 3 hold the value at the last sampling instant until the next sample; this is not captured by $\textrm{F}_{\textrm{c,d}}\left( \varDelta \phi _i(t)-\varDelta \phi _j(t)\right)$.

• The $w(\cdot )$ term captures the sampling aspect of the flip-flop. Note that the flip-flop $DFF_{i,j}$ samples at transitions of $x_i(t)$, i.e., its sampling instant is timed using the phase $f_0t + \varDelta \phi _i(t)$ of $OSC_i$. Ideal sampling would be captured by a weighted delta function of this phase, i.e., $C w(f_0t + \varDelta \phi _i(t) + \theta )$, where $w(\cdot )$ is a unit impulse train with period 1, C is a weight, and $\theta$ is a constant phase offset, useful for adjusting the sampling instant within each cycle and/or to model clock-to-Q delay in the flip-flop.¹⁵ However, $w(\cdot )$ can in fact be almost any 1-periodic function for the scheme to work, as we show below; incorporating C into, and using a $\theta$-shifted version of, a given $w(\cdot )$ simplifies the expression to $w(f_0t + \varDelta \phi _i(t))$.

Substituting $b_{i}(t)$ in (10), we obtain

$$\begin{aligned} \begin{aligned} \varDelta \dot{\phi _i}(t) = f_0 \cdot v\big (\varDelta \phi _i(t)&+f_0t\big ) \cdot J_{i,j} \\ &\cdot \textrm{F}_{\textrm{c,d}}\left( \varDelta \phi _i(t)-\varDelta \phi _j(t)\right) \cdot w\big (\varDelta \phi _i(t)+f_0t\big ). \end{aligned} \end{aligned}$$

(12)

Now, we assume that the phases $\varDelta \phi _i(t)$ and $\varDelta \phi _j(t)$ vary ‘slowly’— this is a standard assumption for averaging or “Adlerization”; [2, 4]. With this assumption, the Adlerization of (12) is

(13)

where $\phi$ represents the nominal oscillator phase, $f_0t$. Simplifying the above leads to

$$\begin{aligned} \begin{aligned} \varDelta \dot{\phi _i}(t) &= f_0 \cdot J_{i,j} \cdot \textrm{F}_{\textrm{c,d}}\left( \varDelta \phi _i(t)-\varDelta \phi _j(t)\right) \cdot \\ &\qquad \qquad \quad \quad \ \int _{\phi =0}^{\phi =1} v\big (\varDelta \phi _i(t)+\phi \big ) \cdot w\big (\varDelta \phi _i(t)+\phi \big ) \cdot d\phi \\ &= f_0 \cdot J_{i,j} \cdot \textrm{F}_{\textrm{c,d}}\left( \varDelta \phi _i(t)-\varDelta \phi _j(t)\right) \int _{\psi =\varDelta \phi _i(t)}^{\psi =1+\varDelta \phi _i(t)} v(\psi ) \cdot w(\psi ) \cdot d\psi \\ &= f_0 \cdot J_{i,j} \cdot \textrm{F}_{\textrm{c,d}}\left( \varDelta \phi _i(t)-\varDelta \phi _j(t)\right) \int _{\psi =0}^{\psi =1} v(\psi ) \cdot w(\psi ) \cdot d\psi , \\ \end{aligned} \end{aligned}$$

(14)

where $\psi \triangleq \varDelta \phi _i(t)+\phi$. For the last step, we used the fact that the integral remains the same over any interval of length 1 since both $v(\cdot )$ and $w(\cdot )$ are 1-periodic.

It is convenient, though not necessary,¹⁶ to assume that

$$\begin{aligned} K_c \triangleq \int _{\phi =0}^{\phi =1} v(\phi ) \cdot w(\phi ) \cdot d\phi > 0. \end{aligned}$$

(15)

Using this definition of $K_c$ in (14), we get

$$\begin{aligned} \begin{aligned} \varDelta \dot{\phi _i}(t) &= f_0 \cdot J_{i,j} \cdot K_c~ \textrm{F}_{\textrm{c,d}}\left( \varDelta \phi _i(t)-\varDelta \phi _j(t)\right) . \\ \end{aligned} \end{aligned}$$

(16)

Comparing (16) to (6), we have shown that the actual $\textrm{F}_{\textrm{c}}(\cdot )$ from using sampling couplers is indeed equal to a scaled version of $\textrm{F}_{\textrm{c,d}}\left( \theta \right)$.

The above generalizes straightforwardly to the case of N coupled oscillators, resulting in

$$\begin{aligned} \begin{aligned} \forall i, \quad \varDelta \dot{\phi _i}(t) &= f_0 \sum _{j=1,j\ne i}^{j=N} J_{i,j} \cdot K_c~\textrm{F}_{\textrm{c,d}}\left( \varDelta \phi _i(t)-\varDelta \phi _j(t)\right) . \end{aligned} \end{aligned}$$

(17)