Thermodynamic Computing

Someone has built the hardware. This page sets out what it does, what it does not do, and which questions it leaves open — including for the material on this site.

What exists now
It is not quantum computing, and they say so
The precise relationship: PSCs are channels with the coherence discarded
The gate set, and what it can reach
Relation to the ZX-calculus
Where this leaves the work on this site
Further reading

What exists now

Extropic has fabricated chips whose computational primitive is a probabilistic bit: a circuit whose thermal noise is the signal rather than the enemy. A pbit emits a random telegraph signal whose bias is set by a control voltage, at energies in the attojoule range. Their X0 test chip carries pbits, pdits (d-state), and continuous Gaussian primitives; their Z1 design hardwires a degree-16 interaction graph and runs chromatic Gibbs sweeps in silicon.

On top of that they have defined Parametrized Stochastic Circuits — a gate-based intermediate representation — and released torx, a JAX library that builds, executes and differentiates them.

This is a serious piece of work with hardware behind it, and anyone thinking about computation at finite temperature should know it exists.

It is not quantum computing, and they say so

Their paper devotes an appendix to the distinction, which is worth repeating because the circuit diagrams look similar.

	Quantum circuit	Stochastic circuit
Objects	unitary matrices	column-stochastic matrices
Invertible?	always	no — reset has rank 1
State	complex amplitudes	a probability distribution
Interference	yes — amplitudes cancel	none
Readout	collapses the state	a sample; leaves it unchanged

The last row is a genuine capability, and it runs in their favour: you can measure at any point in a stochastic circuit without disturbing the rest of the computation. No quantum circuit can do that.

The absence of interference is the sharper limitation. Quantum speedups that depend on amplitudes cancelling — Shor’s period finding, for instance — have no counterpart here. Probabilistic hardware is not a cheaper quantum computer; it is a different machine that samples distributions efficiently.

The precise relationship: PSCs are channels with the coherence discarded

The contrast above is the useful practical statement, but the two models are not merely analogous. Stochastic circuits sit inside the theory of quantum channels.

A CPTP map (completely positive, trace preserving) describes an open quantum system’s evolution: Φ(ρ) = Σₖ Kₖ ρ Kₖ† with Σₖ Kₖ†Kₖ = I. Restrict attention to states diagonal in the computational basis, ρ = diag(p). A channel taking diagonal states to diagonal states acts on p by a column-stochastic matrix, and every column-stochastic matrix arises this way.

Stochastic maps are exactly the CPTP maps that preserve the diagonal subalgebra. A PSC is what remains of a quantum channel once coherence is thrown away.

This is standard — it is what is meant by calling classical probability the commutative case of quantum theory. The gate-level version is the part worth writing out.

PSC gate	Quantum channel	Kraus operators
`PNOT(p)`	bit-flip channel	√(1−p)·I, √p·X
`PReset(p)`	amplitude damping	[[1,0],[0,√(1−p)]], [[0,√p],[0,0]]
`PSWAP(p)`	partial swap	√(1−p)·I⊗I, √p·SWAP
`PCNOT(p)`	probabilistic CNOT	√(1−p)·I⊗I, √p·CNOT
`PCopy(p)`	no faithful lift — see below	—
`PIsing(θ)`	Davies generator / thermal Lindbladian	exp(t·ℒ)
`POU(γ,D,t)`	quantum Ornstein–Uhlenbeck	Gaussian channel

Two rows are worth dwelling on.

PReset is amplitude damping. Their paper singles it out as the exception in the library, being rank 1 and non-invertible. In the quantum setting it is the most familiar channel there is — spontaneous emission, a qubit relaxing to |0⟩. The gate carrying irreversibility is the same operation that makes open quantum systems open.

PCopy has no faithful quantum lift. Copying a classical bit is unproblematic; copying an unknown quantum state is forbidden by no-cloning. The classical gate lifts only to the CNOT-with-ancilla construction, which copies in the computational basis and entangles on superpositions. This is where Markov categories and their quantum counterpart part company, and the divergence is already formalised: Parzygnat (2020) defines quantum Markov categories and identifies what is lost — the canonical copy map Δ: X → X ⊗ X.

The first two rows were checked numerically rather than assumed: applying the channel to a diagonal state and reading off the diagonal reproduces the stochastic gate to 1.1 × 10⁻¹⁶ across the parameter range.

The gate set, and what it can reach

Their elementary gates are given as explicit matrices: PNOT, PSWAP, PCNOT, PJUMP, PCopy, PDEMUX, PReset, PditShift, plus generator-derived kernels like PIsing(J, h₁, h₂, β, Δt).

They list as open work “a general theory of synthesizing a target kernel from a fixed gate set” — the stochastic analogue of universality for quantum gate sets.

One fragment of that question is settled by classical mathematics. Several of the gates are convex combinations of the identity with a permutation, and by Birkhoff–von Neumann such combinations are doubly stochastic. Doubly stochastic matrices are closed under both matrix and Kronecker product, so:

A circuit built only from permutation-mixture gates can never leave the uniform distribution, and therefore can never reach a non-uniform steady state.

We checked this numerically: drift from uniform is exactly zero for such circuits, and 0.61 once PReset is included. The useful work in their library is done by reset, ancillas, and the generator-derived gates — not by the permutation mixtures.

Worth stating plainly: this is Birkhoff (1946) applied to their gate list, not a new theorem. It is a necessary condition on one fragment. The harder half — whether reset plus permutation mixtures generate every stochastic kernel, and at what depth — is genuinely open, and the Solovay–Kitaev machinery does not transfer, because its error contraction needs group commutators UVU⁻¹V⁻¹ and stochastic matrices have no inverses.

Relation to the ZX-calculus

Both are graphical calculi for composing processes, but they differ in the way that matters.

ZX has rewrite rules and a completeness theorem. Every true equation between the processes it describes is derivable from its rules. That is what makes it a calculus rather than a notation, and it is what allows circuit optimisers to rewrite mechanically and know they have missed nothing.

PSCs have composition but no rewrite rules. Their four operations — series, parallel, prepare an ancilla, discard a wire — build circuits, but there is no stated rule set for transforming one circuit into an equal one. The question “when are two PSCs the same?” cannot yet be posed in their framework, which is precisely why the synthesis problem is open.

The gap is not that the mathematics is missing. Completeness for mixed-state processes is settled: the doubled ZX-calculus is proved sound and complete for quantum channels (Carette, Jeandel, Perdrix & Vilmart, 2021), building on Selinger’s CP construction. And the classical structure PSCs have — stochastic maps with copying and discarding — is that of a Markov category (Fritz; Cho and Jacobs).

So the calculi exist on both sides. What is absent is the connection: their four operations are a Markov category’s structure in all but name, and their paper does not cite that literature.

Where this leaves the work on this site

Three observations.

On temperature as a parameter. Treating β as a real quantity to be tuned rather than a limit to be taken is common ground — their PIsing gate carries β explicitly and their demonstrations run at β = 1.5. It is not a distinguishing idea. Extending β into the complex plane formally contains both the thermal (β real) and unitary (β imaginary) cases, but that is Wick rotation, which is textbook, and a parameterisation whose two limits are two known theories is a framing rather than a result. It would need to predict something neither limit does.

On instruction sets. Extropic built a gate-based intermediate representation because they have a substrate whose native operations are those gates. Their opcodes are matrices you can multiply; that is why a synthesis question can even be asked about them. An instruction set without a substrate is a naming convention, and should be described as one.

On scope. Their demonstrations are random walks on graphs, discrete diffusion, max-cut, jump diffusion and Ising rings. Molecular electronic structure is not an energy-based model over binary variables, and choosing which electrons need exact treatment is not a sampling problem — so the chemistry question is orthogonal to this hardware rather than addressed by it.