Probabilistic Bits (P-Bits) in Polymer Electronics

• Probabilistic bits are stochastic, tunably biased binary elements that exploit thermal fluctuations and chemical dynamics for probabilistic computing.
• Polymer-based p-bits utilize electropolymerized memristive devices and voltage divider circuits to convert analog resistance noise into digital signals.
• Their tunable entropy and flexible design enable scalable, energy-efficient hardware for applications in neural networks and optimization.

Probabilistic bits (p-bits) are stochastic, tunably biased binary hardware elements that continuously fluctuate between two logic states with probabilities dictated by their inputs. Unlike classical deterministic bits or quantum bits (qubits), p-bits are fundamentally classical, energy-efficient, and can be implemented using diverse hardware—including CMOS, magnetic tunnel junctions (MTJs), ferroelectric transistors, and, as recently demonstrated, even organic polymers. P-bits serve as the core resource for probabilistic or “thermodynamic” computing, opening new routes for hardware-accelerated optimizers, invertible logic, and massively parallel statistical processors.

1. Physical Basis and Stochastic Modeling of Polymer-Based p-bits

Polymer-based p-bits are realized in electropolymerized memristive devices wherein the active medium is a dithieno[3,2‐b:2′,3′‐d]pyrrole (DTP) backbone polymer grafted with pendant triphenylamine (TPA) groups. Stochastic binary outputs are generated by leveraging the intrinsic, thermally driven resistance fluctuations of the polymer film. These fluctuations arise from molecular-level relaxation dynamics associated with the pendant group reorientation and redox-driven conformation changes.

The analog resistance noise is transduced into a digital output by a voltage divider/comparator circuit: the fluctuating voltage at the divider midpoint is compared with a reference threshold. The digital output probability is governed by:

$$P_{1i} = \frac{1}{1 + \exp{[-k \cdot (V_{\text{offset},i} - V_0)]}}$$

where $V_{\text{offset},i}$ is the comparator bias offset, $k$ is the steepness (control parameter related to device/circuit noise and sharpness), and $V_0$ is the calibration offset. This logistic transfer is mathematically characteristic of stochastic binary neurons and underpins the tunable probabilistic output.

2. Device-Level Stochasticity, Dynamic Entropy, and Statistical Properties

The stochastic nature of the polymer p-bit derives from multiple coupled microscopic processes:

• Conformational/Redox dynamics of TPA groups introduce rapidly fluctuating local potentials, modulating the accessible conduction pathways.
• Dielectric analysis shows strong, low-temperature β-relaxations (activation energy ~0.44 eV), consistent with TPA reorientations and local hopping phenomena.
• Morphological flexibility of the polymer backbone further introduces variations in the percolation paths supporting charge transport.

Extensive statistical characterization is performed by pulsed I–V measurements: repeated DC bias pulses yield stochastic current distributions, binned to extract probability densities $P(I|V)$ for each bias. Discrete Shannon entropy is calculated as:

$H_\text{disc} = -\sum P(I|V) \log_2 P(I|V)$

The entropy quantifies the degree of randomness; maxima in $H_\text{disc}$ correspond to bias regimes where the device is most stochastic (i.e., where the output samples are nearly equiprobable). The effective number of sampled states is

$N_\text{eff} = 2^{H_\text{disc}}$

maximized when both conduction pathways are equally likely.

3. Structure–Function Relationship: Electronic Energetics and Bifurcated Percolation

The correspondence between device-level stochasticity and polymer structure is elucidated through energy-resolved electrochemical impedance spectroscopy (ER–EIS) and density functional theory (DFT):

• ER–EIS reveals that the density of states exhibits sharp edges at the highest occupied and lowest unoccupied molecular orbitals (HOMO/LUMO), minimizing mid-gap trap states, and aligns the HOMO/LUMO with the ITO substrate work function (HOMO ~–4.67 eV, LUMO ~–1.88 eV).
• DFT calculations indicate that, in the neutral polymer, both HOMO/LUMO are localized on the DTP backbone. Upon oxidation (e.g., radical cation/dication states), TPA units become active as frontier orbitals hybridize, producing near-degeneracy and spatially distinct (bifurcated) paths for charge transport.

This bifurcation means that, under critical bias, current can percolate via either pathway (DTP-dominated or TPA-dominated), and stochastic switching between these is enhanced by thermal and redox dynamics. The sharp increase in device entropy at these bias points directly links the chemical composition and microstructure to macroscopic p-bit probabilistic response.

4. Circuit Implementation and Practical Transfer Characteristics

The practical realization consists of a polymer-based memristive element, a voltage divider, and a comparator. Binary stochastic output is sampled at the comparator output as the reference (offset) voltage is varied:

Bias Offset ($V_{\text{offset}}$)	$P_\text{out}=1$ (fraction)	$H_\text{disc}$ (bits, approx.)
Low	$\sim 0$	$\sim 0$
Mid (optimal stochasticity)	$\sim 0.5$	$>0.8$ (peak)
High	$\sim 1$	$\sim 0$

The sigmoidal transfer confirmed experimentally matches the form $$P_{1i} = [1 + \exp{(-k(V_{\text{offset},i}-V_0))}]^{-1}$$. The entropic maximum (i.e., maximal stochasticity) coincides with the regime where percolation between DTP and TPA pathways becomes equiprobable.

5. Applications and Thermodynamic Computing Perspective

Polymer p-bits serve as compact, energy-efficient binary stochastic units with analog-tunable transfer curves, directly suitable for thermodynamic and probabilistic computing architectures. Key technical implications are:

• Chemically tunable entropy source: By adjusting the polymer composition and pendant group chemistry, the degree of stochasticity can be engineered at the molecular level.
• Probabilistic neural/inference hardware: The experimentally realized logistic transfer and high entropy regime make these devices directly compatible with Ising-type hardware, stochastic Boltzmann machines, and neural networks.
• Flexible/printed electronics compatibility: The solution-processable polymer nature lends itself to scalable, unconventional hardware manufacturing and integration with conventional as well as soft-matter platforms.

6. Comparison to Other p-bit Implementations and Research Outlook

Polymer-based p-bits complement inorganic implementations such as MTJs and ferroelectric FETs:

• Distinctive features: Soft-matter, multi-state relaxation, electronic bifurcation, and dynamic disorder lead to unique noise and entropy characteristics.
• Scalability and integration: Chemically tunable, printable, and compatible with flexible/stretchable form factors.
• Fundamental understanding: The direct relation between relaxation dynamics, percolation bifurcation (confirmed via DFT and ER–EIS), and output statistics enables true bottom-up design of stochastic hardware.

Open research directions include engineering the molecular relaxation spectrum for desired entropy characteristics, designing networks of coupled polymer p-bits, and embedding such units into hybrid probabilistic–deterministic systems for domain-specific thermodynamic or inference workloads.

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Polymer-based p-bits establish organic semiconductors as a versatile thermodynamic computing substrate, where the intimate link between chemical relaxation dynamics and probabilistic digital outputs enables a new class of energy-efficient, entropy-engineered stochastic hardware for advanced probabilistic and inference applications (Foulger et al., 22 Sep 2025).