Multi-Value Probabilistic Computing with Skyrmions

• MPC is a computing paradigm where a single spintronic element represents a multi-state categorical probability distribution, bypassing conventional binary approaches.
• It uses current-controlled skyrmion diffusion among physical pinning sites to enact softmax-like functions and scalable probabilistic operations.
• The approach enables invertible logic and hardware-native probability sampling, potentially overcoming the limitations of traditional binary p-bit systems.

Multi-Value Probabilistic Computing (MPC) denotes a probabilistic-computing paradigm in which a single physical element represents a discrete random variable with more than two states, rather than a binary stochastic bit. In the recent spintronic realization reported in "Multi-value Probabilistic Computing with current-controlled Skyrmion Diffusion" (Winkler et al., 27 Aug 2025), MPC is implemented by a single thermally diffusing magnetic skyrmion moving among a discrete set of pinning sites, with the long-time occupation of those sites directly defining a categorical distribution. The state probabilities are reweighted by current-generated spin-orbit torques, which enables softmax-like computation and invertible logic without constructing a network of probabilistic devices. The acronym is not uniform across adjacent literatures: recent arXiv papers also use MPC for Magnetic Probabilistic Computing (Wang et al., 23 Jul 2025) and for cryptographic multi-party computation (Humphries et al., 2021, Gold et al., 15 May 2025). In the multi-value sense, however, MPC refers specifically to hardware-native representation and manipulation of multi-state probability distributions.

1. Terminology and conceptual scope

The term “multi-value” is used here in the strict sense of a discrete state space with cardinality $K>2$. The central object is therefore a categorical distribution

$$\pi = (\pi_1,\pi_2,\dots,\pi_K), \qquad \sum_{i=1}^K \pi_i = 1,$$

rather than a Bernoulli random variable. In the skyrmion implementation, the $K$ values are physical pinning sites, and the probabilities are the normalized dwell times of a single diffusing skyrmion among those sites (Winkler et al., 27 Aug 2025).

Acronym disambiguation is necessary because recent work uses the same three letters for distinct concepts.

Acronym usage	Meaning	Representative paper
MPC	Multi-value Probabilistic Computing	(Winkler et al., 27 Aug 2025)
MPC	Magnetic Probabilistic Computing	(Wang et al., 23 Jul 2025)
MPC	multi-party computation	(Humphries et al., 2021, Gold et al., 15 May 2025)

Within probabilistic hardware, the multi-value interpretation differs from two neighboring paradigms. First, it differs from binary p-bit hardware, where the primitive fluctuates between two states and more complex distributions are synthesized from networks of coupled stochastic units. Second, it differs from continuous-valued stochastic analog samplers, such as domain-wall-based Gaussian generators, where the device output is an analog random variable with tunable mean and variance rather than a directly realized finite-state categorical distribution (Daniel et al., 2023, Wang et al., 23 Jul 2025).

The skyrmion-based formulation presents MPC as a response to specific limitations of binary probabilistic computing: binary elements represent only two states and therefore require large, often densely interconnected networks to represent more complex distributions or logic constraints. The reported consequences include scalability limits, device mismatch and calibration issues, interconnect complexity, energy inefficiency, and sampling inefficiency when invalid states are present but only energetically suppressed (Winkler et al., 27 Aug 2025).

2. Statistical representation and control law

The core abstraction of MPC in the skyrmion work is a finite-state thermodynamic sampler. A skyrmion at finite temperature moves in an effective energy landscape $V(x)$, and its stationary distribution is assumed to follow a Boltzmann form

$$\pi(x) = \frac{e^{-V(x)}}{Z(V)}, \qquad Z(V) = \int e^{-V(x)}\,dx.$$

After coarse-graining continuous motion into $K$ metastable pinning sites, the discrete occupation law becomes

$\pi_i = \frac{e^{-E_i}}{\sum_j e^{-E_j}},$

with $E_i$ the energy of state $i$ (Winkler et al., 27 Aug 2025).

This discrete Boltzmann law is the mathematical basis of multi-value probabilistic operation. The physical state space is not imposed by digital encoding; it emerges from a pinning-dominated diffusion regime in which disorder, thickness fluctuations, grain boundaries, local variations of anisotropy or exchange, and geometric confinement create metastable sites. The stochastic dynamics between those sites are modeled with Markov State Modeling (MSM), using a $K\times K$ row-stochastic transition matrix $$\pi = (\pi_1,\pi_2,\dots,\pi_K), \qquad \sum_{i=1}^K \pi_i = 1,$$0 with lag time

$$\pi = (\pi_1,\pi_2,\dots,\pi_K), \qquad \sum_{i=1}^K \pi_i = 1,$$1

chosen as one camera frame (Winkler et al., 27 Aug 2025).

External control is implemented by voltage-driven current and spin-orbit torque (SOT). In the reported operating regime, the skyrmion Hall angle is taken to be negligible, Joule heating is neglected, and SOT is approximated as a conservative force,

$$\pi = (\pi_1,\pi_2,\dots,\pi_K), \qquad \sum_{i=1}^K \pi_i = 1,$$2

Under this approximation, the energy of pinning site $$\pi = (\pi_1,\pi_2,\dots,\pi_K), \qquad \sum_{i=1}^K \pi_i = 1,$$3 is modeled as

$$\pi = (\pi_1,\pi_2,\dots,\pi_K), \qquad \sum_{i=1}^K \pi_i = 1,$$4

so that the stationary distribution becomes

$$\pi = (\pi_1,\pi_2,\dots,\pi_K), \qquad \sum_{i=1}^K \pi_i = 1,$$5

A single global voltage therefore reweights all states simultaneously and traces a one-parameter family of distributions through the probability simplex (Winkler et al., 27 Aug 2025).

Experimentally, the state probabilities are extracted from occupancies,

$$\pi = (\pi_1,\pi_2,\dots,\pi_K), \qquad \sum_{i=1}^K \pi_i = 1,$$6

where $$\pi = (\pi_1,\pi_2,\dots,\pi_K), \qquad \sum_{i=1}^K \pi_i = 1,$$7 is the number of frames assigned to site $$\pi = (\pi_1,\pi_2,\dots,\pi_K), \qquad \sum_{i=1}^K \pi_i = 1,$$8. The inverse relation

$$\pi = (\pi_1,\pi_2,\dots,\pi_K), \qquad \sum_{i=1}^K \pi_i = 1,$$9

provides an energy estimate up to an additive gauge term $K$0. Fitting is performed with

$K$1

and loss

$K$2

The practical significance is that a compact linear energy model is sufficient to parameterize the observed multi-valued probabilistic response (Winkler et al., 27 Aug 2025).

3. Skyrmion-diffusion hardware realization

The realized MPC device is a triangular thin-film element with electrical contacts at edges or corners. The material stack is

$K$3

with thicknesses in nm. The CoFeB layer supports skyrmions stabilized by interfacial DMI and dipolar effects, and the observed skyrmions are of order one to a few micrometers (Winkler et al., 27 Aug 2025).

Two prototype devices are central to the experimental program.

Prototype	States	Function
First device	$K$4	MPC modeling and softmax discussion
Second device	$K$5	Invertible OR logic

State-resolved dynamics are observed with magneto-optical Kerr effect (MOKE) microscopy at frame rate $K$6 and exposure $K$7. Long-time measurements last at least $K$8 min per voltage, and some datasets extend to a minimum of $K$9 h at $V(x)$0. Temperature control spans $V(x)$1 with stability $V(x)$2; the two main devices are operated at $V(x)$3 and $V(x)$4, respectively. Skyrmions are nucleated by an in-plane field pulse under a stabilizing out-of-plane bias field, and an oscillatory out-of-plane field is used to enhance diffusion (Winkler et al., 27 Aug 2025).

For the six-state prototype, cumulative occurrence maps reveal $V(x)$5 pinning sites. When the applied voltage is swept between

$V(x)$6

the occurrence distribution shifts visibly across the device, showing direct control of a six-state categorical distribution. At the largest voltage, the current density at half-width is estimated to remain below

$V(x)$7

The extracted occupancies agree well with the fitted Boltzmann model. The reported qualitative trend is that the center pinning site has a small slope $V(x)$8, so its occupation changes weakly with $V(x)$9, whereas outer pinning sites have larger positive or negative slopes depending on their location relative to current flow (Winkler et al., 27 Aug 2025).

The paper describes this as MPC realized by a single skyrmion rather than by a network of stochastic devices. That distinction matters because the computational state space is the physical state space itself: pinning sites are not auxiliary hidden states but the directly sampled values of the probabilistic variable.

4. Softmax computation and invertible logic

A principal computational consequence of the Boltzmann occupation law is its direct equivalence to softmax. For logits $$\pi(x) = \frac{e^{-V(x)}}{Z(V)}, \qquad Z(V) = \int e^{-V(x)}\,dx.$$0,

$$\pi(x) = \frac{e^{-V(x)}}{Z(V)}, \qquad Z(V) = \int e^{-V(x)}\,dx.$$1

while skyrmion occupations obey

$$\pi(x) = \frac{e^{-V(x)}}{Z(V)}, \qquad Z(V) = \int e^{-V(x)}\,dx.$$2

Identifying

$$\pi(x) = \frac{e^{-V(x)}}{Z(V)}, \qquad Z(V) = \int e^{-V(x)}\,dx.$$3

yields

$$\pi(x) = \frac{e^{-V(x)}}{Z(V)}, \qquad Z(V) = \int e^{-V(x)}\,dx.$$4

The paper therefore interprets the device as a physical softmax engine when negative pre-activations are encoded as pinning-site energies (Winkler et al., 27 Aug 2025).

The experimental six-state device establishes the MPC mechanism, but the classifier demonstration is performed in Monte Carlo simulation rather than full hardware inference. The example uses the Iris dataset with architecture 4 input, 64 dense + ReLU, 32 dense + ReLU, 3 dense outputs, and softmax at the output. In the diffusion-based implementation, a circular sample of radius $$\pi(x) = \frac{e^{-V(x)}}{Z(V)}, \qquad Z(V) = \int e^{-V(x)}\,dx.$$5 is divided into three regions, each assigned an energy equal to the negative pre-activation, and particle motion is simulated with a Metropolis-Hastings procedure. Output probabilities are estimated from dwell times over

$$\pi(x) = \frac{e^{-V(x)}}{Z(V)}, \qquad Z(V) = \int e^{-V(x)}\,dx.$$6

time steps. The approximation is reported to be good especially when the pre-activations are close; discrepancies increase when one class logit is much larger than the others, because finite-time sampling estimates very small probabilities poorly unless simulation time is increased (Winkler et al., 27 Aug 2025).

The second major demonstration is invertible logic. Instead of encoding a Boolean gate in a network Hamiltonian over many p-bits, the OR truth table is mapped directly onto four pinning sites: $$\pi(x) = \frac{e^{-V(x)}}{Z(V)}, \qquad Z(V) = \int e^{-V(x)}\,dx.$$7 These are the valid assignments of

$$\pi(x) = \frac{e^{-V(x)}}{Z(V)}, \qquad Z(V) = \int e^{-V(x)}\,dx.$$8

The four-state prototype is clustered to match those rows exactly, so invalid logical states are absent by construction. Operation is evaluated for unclamped, $$\pi(x) = \frac{e^{-V(x)}}{Z(V)}, \qquad Z(V) = \int e^{-V(x)}\,dx.$$9-clamped, and $K$0-clamped targets by minimizing

$K$1

The experimentally reported values are:

• $K$2-clamped: $K$3 at $K$4,
• $K$5-clamped: $K$6 at $K$7,
• unclamped: $K$8 at $K$9.

The paper emphasizes that the $\pi_i = \frac{e^{-E_i}}{\sum_j e^{-E_j}},$0-clamped and unclamped cases achieve KLD below $\pi_i = \frac{e^{-E_i}}{\sum_j e^{-E_j}},$1, while the $\pi_i = \frac{e^{-E_i}}{\sum_j e^{-E_j}},$2-clamped case is limited by the intentionally small voltage range. The architectural implication is a form of invertible logic in which valid truth-table entries are hardware states, rather than low-energy corners of a much larger binary configuration space (Winkler et al., 27 Aug 2025).

5. Relation to binary p-bits and other probabilistic spintronics

MPC is best understood against the background of binary probabilistic hardware. The integrated on-chip p-bit core reported in "Experimental demonstration of an integrated on-chip p-bit core utilizing stochastic Magnetic Tunnel Junctions and 2D-MoS$\pi_i = \frac{e^{-E_i}}{\sum_j e^{-E_j}},$3 FETs" (Daniel et al., 2023) is a foundational binary probabilistic device: a stochastic MTJ fluctuates between parallel and antiparallel states, and a transistor-plus-inverter circuit converts those fluctuations into a voltage-controllable binary stochastic output. That architecture is not multi-value in the one-device-one-symbol sense; its microscopic state variable remains binary. The paper is nevertheless relevant because it identifies circuit-level constraints that also matter for any extension toward MPC: resistance matching matters strongly, excessively large TMR is not always beneficial, slow telegraphic MTJs are undesirable, and very high-gain inverters can worsen p-bit behavior by creating output plateaus (Daniel et al., 2023).

The skyrmion approach differs from such p-bits at the representational level. Binary p-bits represent a Bernoulli variable and generally require networks to emulate multi-valued or constrained distributions. By contrast, the skyrmion device directly realizes a $\pi_i = \frac{e^{-E_i}}{\sum_j e^{-E_j}},$4-state distribution in a single physical object, so the multi-valued variable is native rather than encoded across several coupled binary elements (Winkler et al., 27 Aug 2025).

A second neighboring paradigm is the domain-wall-based "Magnetic Probabilistic Computing" platform for Bayesian hardware (Wang et al., 23 Jul 2025). There, the device output is not a categorical distribution over discrete symbolic states. Instead, thermally fluctuating domain-wall dynamics, VCMA, and TMR implement a tunable Gaussian-distributed analog output whose mean is controlled by domain-wall position and whose standard deviation is controlled by anisotropy tuning. The paper therefore describes a physical Gaussian sampler or stochastic analog weight source for Bayesian neural networks rather than a multi-valued finite-state logic architecture. Its direct relevance to MPC is indirect: it demonstrates that spintronic probabilistic hardware can move beyond simple Bernoulli p-bits, but its computational model is continuous and distributional, not discrete and categorical (Wang et al., 23 Jul 2025).

Taken together, these works delimit three distinct hardware regimes. Binary p-bits furnish compact stochastic bits and network-level programmability. Domain-wall Gaussian samplers furnish tunable continuous distributions for uncertainty-aware inference. Skyrmion MPC furnishes a native discrete categorical variable in a single device. The last of these is the most literal realization of “multi-value probabilistic computing.”

6. Architectural implications, limitations, and open directions

The skyrmion MPC paper explicitly proposes several routes to richer hardware. The present experiment uses one skyrmion and one global control parameter, but the authors note that multiple skyrmions would produce combinatorial growth in the number of states, and that additional current paths could generalize the energy law to

$\pi_i = \frac{e^{-E_i}}{\sum_j e^{-E_j}},$5

They also discuss top electrodes centered above pinning sites as a route toward locally addressable control approximating

$\pi_i = \frac{e^{-E_i}}{\sum_j e^{-E_j}},$6

Engineered pinning landscapes, MTJ readout above pinning sites, and broader links to attention mechanisms, Bayesian computing, probabilistic inference, and optimization are all presented as natural extensions (Winkler et al., 27 Aug 2025).

The current realization also has clear limitations. Probability estimates require long sampling windows ranging from tens of minutes to hours. The present pinning sites arise largely from uncontrolled inhomogeneities, so practical hardware would require reproducible pinning energies, predictable site locations, and controlled energy barriers. State-resolved readout in the reported experiments is MOKE-based rather than integrated on-chip electrical detection. The demonstrated control uses only one scalar voltage $\pi_i = \frac{e^{-E_i}}{\sum_j e^{-E_j}},$7, which restricts the reachable family of distributions. The model further assumes negligible skyrmion Hall angle, conservative SOT, negligible Joule heating, and linear energy shifts with voltage; these approximations may not hold in other geometries or drive regimes. The paper also estimates tracking failures on the order of $\pi_i = \frac{e^{-E_i}}{\sum_j e^{-E_j}},$8, and systematic reproducibility across devices is not established (Winkler et al., 27 Aug 2025).

These limitations do not negate the central result. They delimit its current status as a proof of concept in which the computational primitive is already present, while speed, controllability, readout integration, and fabrication determinism remain open engineering problems. A plausible implication is that future MPC hardware will be defined less by emulating multi-valued probability with large binary networks and more by physically embedding the desired state space into the energy landscape of a compact spintronic system. In that sense, the distinctive contribution of current-controlled skyrmion diffusion is not merely another probabilistic device, but a change in the level at which probabilistic state spaces are instantiated in hardware (Winkler et al., 27 Aug 2025).