#4549 P-Bits in Mathematica

#4549 P-Bits in Mathematica

#4549 What can be done with probabilistic bits in Mathematica?

In Mathematica, probabilistic bits (p-bits)—the foundational units of probabilistic computing that fluctuate continuously between 0 and 1—can be modeled, simulated, and visualized using a blend of built-in statistical functions, matrix mathematics, and discrete-time simulation loops.

Because Mathematica natively handles symbolic mathematics, arbitrary-precision floats, and hardware-accelerated matrix operations, it is a highly capable environment for prototyping p-circuits, designing Ising machines, or exploring stochastic computing architectures before moving them to physical hardware (like FPGAs or specialized silicon).

Here is a breakdown of what you can do with p-bits in Mathematica, along with conceptual approaches and functional implementations.

1. Simulating the Physics of a Single p-Bit

A standard behavioral model for a hardware p-bit (such as one utilizing stochastic magnetic tunnel junctions or RRAM) relies on a thermal fluctuation governed by a sigmoid function. The probability P of the bit outputting a +1 (or 1) given an input bias (or local activation field) I_i at a given "inverse temperature" beta is modeled as:

P(m_i = 1) = sig(2beta I_i) = 1 / {1 + e^{-2beta I_i}}

In Mathematica, you can define a functional p-bit update step that evaluates a random choice based on this probability:

(* Define the activation profile of a single p-bit *)
PBitUpdate[I_List, beta_] := 
  Map[If[RandomReal[] < 1/(1 + Exp[-2 * beta * #]), 1, -1] &, I]

2. Modeling Coupled p-Circuits (Ising Machines)

When you connect multiple p-bits together, their interactions are defined by a weight matrix W and a bias vector B. The effective input to the i-th p-bit at time t+1 becomes:

I_i(t+1) = \sum_{j} W_{ij} m_j(t) + B_i

Mathematica’s optimized vectorization makes simulating these massive systems highly efficient. You can write a discrete-time Markov Chain / Gibbs Sampling engine to update an entire system of coupled p-bits:

(* Complete System Sync/Async Update Step *)
StepPCircuit[m_List, W_List, B_List, beta_] := Module[
  {inputs},
  inputs = W . m + B;
  (* Returns the new state array containing 1s and -1s *)
  Map[If[RandomReal[] < 1/(1 + Exp[-2*beta*#]), 1, -1] &, inputs]
]

3. Invertible Logic & Boolean Gates

One of the most powerful paradigms of p-bits is invertible logic (running logic gates backward). For example, by mapping the truth table of an AND gate to an energy landscape via a quadratic unconstrained binary optimization (QUBO) penalty matrix, a network of 3 p-bits (A, B, and the output C) can solve the problem in any direction.

In Mathematica, you can:

• Define the connectivity matrix W for an invertible gate (e.g., Full Adder, AND gate).

• Use NestList to run the p-circuit simulation for thousands of cycles.

• Clamp specific bits using a mask matrix to observe inverse operations (e.g., setting the output C = 1 to force the system to fluctuate only between the valid inputs A=1, B=1).

4. Combinatorial Optimization & Simulated Annealing

Because p-bits naturally explore energy landscapes via intrinsic randomness, they excel at optimization problems like the Traveling Salesperson Problem (TSP), graph coloring, or Max-Cut.

In Mathematica, you can map these optimization problems to an Ising Spin Glass formulation:

• Use LinearFractionalTransform or Rescale to sweep the beta parameter over time, effectively executing a hardware-accurate Simulated Annealing schedule.

• Track the system's current energy state over time using a simple dot product: E = -1/2 m^T W m - m^T B.

5. Statistical Analysis & Visualizing State Spaces

Once you run a simulation of a p-circuit yielding millions of samples, you can use Mathematica's robust statistics suite to evaluate performance:

• Tally and EmpiricalDistribution: Instantly extract the steady-state probability of all configurations to see if the ground states dominate.

• BarChart or Histogram: Plot the probability distribution of the states to visually confirm that the highest peaks correspond to the correct solutions of your system's objective function.

• DiscreteMarkovProcess: For smaller networks, you can construct an exact transition matrix analytically to compute the stationary distribution without running a stochastic simulation, finding exact analytical bounds for your network configuration.

Example: Visualizing a 3 p-bit State Space

If you've collected a massive matrix of timeline states from a 3-bit simulation, you can easily visualize how often the system visits correct solutions:

(* Generating mock sample data of 3 bits over 1000 clock cycles *)
mockSamples = Table[RandomChoice[{-1, 1}, 3], {1000}];

(* Count occurrences of each unique 3-bit configuration *)
counts = Tally[mockSamples];

(* Plotting the state distribution *)
BarChart[counts[[All, 2]], 
 ChartLabels -> (Row /@ counts[[All, 1]]), 
 FrameLabel -> {"State Configuration", "Frequency"},
 PlotLabel -> "Steady-State Statistics of a 3 p-Bit Network"]

6. Pre-Hardware Synthesis Verification

If you are planning to deploy a stochastic computing architecture to physical hardware (such as an FPGA cluster like the Xilinx Artix-7):

• You can use Mathematica to model the effects of reduced bit-width fixed-point arithmetic or pseudo-random number generator (PRNG) quality (like LFSR vs. Cellular Automata) on the p-bit's activation function.

• By wrapping your activation probabilities in Around[value, delta], you can simulate how hardware thermal noise or variance in stochastic components impacts the convergence rate of your network.

From <https://gemini.google.com/app/>  Google Gemini (3.5 Flash)