A new paradigm for thermodynamically efficient, computation-universal information processing using hidden momentum degrees of freedom.
Continuous-time Markov chains (CTMCs) — the standard tool of stochastic thermodynamics — forbid many basic computations. A bit flip maps 0→1 and 1→0, but CTMC input-output matrices must have positive determinants. A bit flip has determinant −1.
Once distributions overlap at x=0, a memoryless (Markov) process cannot selectively route particles to opposite sides. Information is irrecoverably lost.
Expand state space to include momentum p. A particle's current momentum distinguishes its future trajectory even when positions overlap. This makes the dynamics a continuous-time hidden Markov chain (CTHMC).
During computation, the system is briefly decoupled from the heat bath and a quadratic well Vcomp = kx²/2 is applied. Particles undergo half-period harmonic oscillation: x(τ) = −x(0). The swap costs zero net work.
The protocol has three steps — decouple, oscillate for τ = π√(μ/k), recouple. The work at t=0 is exactly cancelled by the work at t=τ because the potential is even and the position flips sign.
The key symmetry: Vstore is even in x, so the energy injection at t=0 is exactly recovered at t=τ. Net thermodynamic work ⟨W⟩ = 0. This is forbidden by CTMC but allowed by CTHMC.
Generalizing to three particles implements the Fredkin (controlled swap) gate — a reversible, computation-universal operation. The x-particle acts as a control bit; if x>0, the y and z particles undergo a coordinate-rotation swap.
The same half-period oscillation trick, now in a rotated y'–z' basis, implements the swap only when x>0. Fredkin gates are Turing complete — any computation is reachable. Implementation: zero average work in the ideal case.
The second paper (Phys. Rev. Applied, 2023) implements momentum computing in a superconducting Josephson junction circuit, achieving sub-Landauer operation at GHz speeds. Compare with conventional approaches:
L = 1 Landauer = kBT ln2 ≈ 4×10⁻²¹ J at room temperature, ≈ 10⁻²³ J at 1 K. The Josephson junction implementation achieves minimum ⟨W⟩ = 0.43 Landauers — roughly 10,000× more efficient than the best current CMOS — operating at nanosecond timescales.
| Property | CTMC | CTHMC |
|---|---|---|
| State space | Position only | Position + momentum |
| Bit flip | × Forbidden | ✓ Zero work |
| Bit swap | × Forbidden | ✓ Sub-Landauer |
| Fredkin gate | × Forbidden | ✓ Universal |
| Computation speed | W ∝ 1/τ (slow = cheap) | Optimal τ* (fast + cheap) |
| Fidelity vs. cost | Trade-off | Aligned: both improve together |
| Input-output det. | Must be positive | No restriction |
| Physical substrate | Overdamped Langevin | Underdamped (e.g., Josephson junctions, NEMS) |
When a system is underdamped — weakly coupled to its environment — its momentum persists long enough to carry information. This is a hidden state: not directly observable as a memory bit, but physically real and causally potent.
In the Josephson junction device, ring-down times are O(10³) oscillations. The computation completes in under 15 ns — far faster than the thermalization timescale τR. During this window, momentum is a reliable hidden memory register.
Momentum computing works because computation can be faster than thermalization. When a particle's momentum persists, position space alone is an incomplete description — the system lives in full phase space. Exploiting this hidden structure enables bit operations that are at once fast, accurate, and thermodynamically free.