Thermodynamic Trade-off Frontiers for Neuromorphic Processors: A Quantum-Inspired Non-Equilibrium Model of the Energy–Speed Boundary

#1. Introduction

The central promise of neuromorphic computing—brain-like efficiency in silicon—remains largely an empirical claim. Individual chips demonstrate impressive energy-per-spike figures, but there is no accepted theory that predicts, from physical first principles, how energy efficiency must degrade as computational speed or network size increases. This gap matters practically: chip architects tune operating points by trial and error, and claims of "brain-scale efficiency" are rarely checked against hard thermodynamic bounds.

This paper constructs such a theory. Our approach is quantum-inspired in a specific and defensible sense: we do not claim quantum computation occurs in neuromorphic hardware, but we import the variational and bound-structure of quantum thermodynamics—speed limits, entropy-production floors, and geometric dissipation functionals—into a classical non-equilibrium description. This strategy mirrors the successful use of quantum-inspired classical algorithms and simulators, where quantum formalism yields practical classical benefits without quantum hardware [1,7].

The core modeling move is to treat the neuromorphic processor as a non-equilibrium thermodynamic system in a driven steady state. Spiking activity is a probability flux through configuration space, sustained against dissipation by continuous power input, exactly analogous to the non-equilibrium steady states analyzed for thermally driven micromachines [8]. Dissipation is encoded in a Rayleigh-type potential relating thermodynamic forces and fluxes [2], and the strongly nonlinear, far-from-equilibrium character of spiking dynamics is handled with a local reduction to compact dynamical form in the spirit of Nambu non-equilibrium thermodynamics [3].

Our contributions are: (i) a three-term dissipation model for a synaptic event (Landauer erasure, non-adiabatic switching, leakage); (ii) an explicit derivation of the energy–speed frontier with all arithmetic shown; (iii) numerical evaluation for a representative 10⁶-synapse processor; and (iv) falsifiable scaling predictions testable within twelve months. Section 2 surveys the literature. Section 3 presents the model. Section 4 carries out the derivations. Section 5 reports results. Section 6 discusses limitations, falsifiability, and an explicit divergence resolution concerning error-correction overheads. Section 7 concludes.

#2. Background and Related Work

Our model sits at the intersection of three literatures: quantum-inspired classical computing, geometric non-equilibrium thermodynamics, and the fundamental physical limits of computation.

Quantum-inspired classical computation. Tang examined quantum-inspired algorithms for recommendation systems and low-rank linear systems, showing that although these classical methods achieve exponential asymptotic speedups over prior classical algorithms, they carry hefty polynomial overheads relative to true quantum algorithms [1]. The lesson we adopt is methodological: formal structures borrowed from quantum theory must be checked against concrete resource arithmetic—precisely the discipline we apply to thermodynamic bounds. Complementing this, classical circuit networks have been shown to simulate Schrödinger dynamics and topological physics through the circuit-Laplacian correspondence, implementing genuine quantum-inspired information processing without quantum hardware [7]. Our model is the thermodynamic analogue: quantum-derived bounds applied to classical dissipative hardware.

Geometric non-equilibrium thermodynamics. Modern frameworks compare dissipation mechanisms—Rayleigh dissipation potentials, dissipative d'Alembert formulations, and gradient dynamics—within a unified geometric setting for multiscale systems [2]. We use the Rayleigh potential as our constitutive dissipation law because it yields a quadratic force–flux relation that is analytically tractable and empirically supported for resistive and capacitive electronic elements. For far-from-equilibrium, strongly nonlinear regimes, Nambu non-equilibrium thermodynamics provides local reduction of complex dynamics to a compact bracket form [3]; we invoke this to justify treating spiking dynamics as locally reduced two-variable dynamics (membrane voltage, synapse efficacy) even though the full system is high-dimensional.

Non-equilibrium steady states, damage, and criticality. The statistical mechanics of non-equilibrium damage phenomena develops a Gibbs-like formalism for non-equilibrium states and applies it to fiber-bundle models with thermal noise and fiber decay [4]. This is directly relevant to neuromorphic reliability: synapse degradation under sustained spiking is formally analogous to fiber failure under load, and we borrow the two-regime structure (fluctuation-dominated versus decay-dominated) to model device wear. Extended irreversible thermodynamics supplies the framework for critical behavior in non-equilibrium systems, where transport coefficients diverge near instability [6]; we use it to flag the breakdown of our linear force–flux assumption near synchronization transitions.

Coupled transport processes. A non-equilibrium thermodynamics model for combined adsorption and diffusion in micro- and nanopores shows that two coupled processes (diffusion-type and Langmuir-type dynamics) can be treated as diffusion in an effective landscape [5]. Our model of charge transport coupled to synaptic state update is structurally identical: ionic/electronic diffusion coupled to a saturating state variable, and we exploit that analogy to justify the coupled-equation structure of Section 3.

Micromachine steady states. The thermally driven three-sphere micromachine admits an exact non-equilibrium steady-state distribution with nonzero probability flux in configuration space [8]. This provides the cleanest available template for our central object: the spike-trajectory flux through the processor's phase space, whose circulation measures computational work done per unit time.

Fundamental limits. The physics-of-computation literature establishes the canonical bounds: the Landauer bound on erasure, the Margolus–Levitin theorem on evolution rate per unit energy, the Bremermann and Bekenstein limits, and—critically—the observation that error-correction overheads of 10²–10³ physical operations per logical operation multiply thermodynamic cost by the same factor in fault-tolerant quantum computing [12,10,11]. Thermodynamic viability must be assessed at the system level, not the device level [9]. Our Pareto-frontier analysis is the neuomorphic instantiation of that system-level accounting; notably, neuromorphic hardware avoids the quantum error-correction multiplier entirely, a quantitative advantage we examine explicitly (and whose treatment differs across source analyses; see Appendix A).

#3. Methods

#3.1 System model

We model a neuromorphic processor as N_s synapses and N_c neurons, each synaptic event (spike delivered to a synapse, weight applied, possible output spike) being the elementary computational operation. State variables per neuron: membrane voltage v and recovery variable u; per synapse: efficacy w. Dynamics follow leaky integrate-and-fire form, treated as the locally reduced dynamics justified by the Nambu-reduction argument [3]:

dv/dt = (−v + I_syn)/τ_m, dw/dt = −w/τ_w + δ(t_spike) Δw,

with τ_m the membrane time constant and τ_w the synaptic retention time constant. This coupled two-timescale structure mirrors the diffusion–adsorption coupling of [5]: a transport process (charge integration) coupled to a saturating state variable (synaptic efficacy). In steady state the device carries a nonzero probability flux in configuration space, in direct analogy with the micromachine of [8]; this flux is the thermodynamic signature of ongoing computation.

#3.2 Dissipation decomposition

Energy per synaptic event E_ev is decomposed into three terms:

E_ev = E_Landauer + E_switch + E_leak.

(a) Landauer term. Each synaptic event logically erases at least the information in the arriving spike's arrival-time uncertainty. We take the minimal erasure of one bit at temperature T: E_Landauer = k_B T ln 2.

(b) Switching term. Charging and discharging the synaptic capacitance C_s through voltage swing ΔV dissipates, for non-adiabatic switching, E_switch = C_s ΔV² (full CV² dissipation; adiabatic charging reduces this by factor (1 − α_ad), the adiabaticity fraction, treated as a parameter). This is the Rayleigh-dissipation term: the Rayleigh potential R = (1/2) g q̇² with conductance g yields quadratic dissipation in flux, consistent with [2].

(c) Leakage term. Static leakage power P_leak per synapse integrated over the event period gives E_leak = P_leak / f_ev, where f_ev is the per-synapse event rate.

#3.3 Speed limit

The Margolus–Levitin theorem bounds the orthogonalization time of a system with mean energy E above ground: t ≥ ħ/(2E) [12]. For a classical switching event driven by energy E_switch, we adopt the classical analogue: the minimum transition time through an RC network is

t_sw = R_on C_s ln(V_th/(V_th − ΔV)),

and the corresponding maximum event rate is f_max = 1/t_sw. This is the quantum-speed-limit structure (energy–time trade-off) in its classical RC limit, the same classical–quantum correspondence exploited in circuit-based quantum simulators [7].

#3.4 Steady-state flux and the Pareto frontier

In the non-equilibrium steady state, total power P_tot = N_s f_ev E_ev sustains a probability flux J through phase space [8]. Define the dimensionless dissipation ratio

ρ = E_switch / (k_B T ln 2),

which measures how far above the Landauer floor the hardware operates. The frontier in the (f_ev, E_ev) plane follows from the RC speed limit: E_switch is speed-independent for f_ev ≤ f_max, and no solution exists for f_ev > f_max. The interesting structure is therefore in the (f_ev, ρ) plane and in the system-level constraint P_tot ≤ P_budget.

#3.5 Numerical parameters

All input numbers, with sources:

• T = 300 K (standard room-temperature assumption).
• k_B = 1.380649×10⁻²³ J/K; ħ = 1.054571817×10⁻³⁴ J·s (SI defined constants).
• C_s = 100 fF = 1.0×10⁻¹³ F (stated assumption: representative analog-CMOS synapse capacitance).
• ΔV = 0.5 V (stated assumption: subthreshold analog swing).
• R_on = 10 kΩ (stated assumption: access-transistor on-resistance).
• P_leak = 10 pW per synapse (stated assumption: subthreshold leakage).
• N_s = 10⁶ synapses (stated assumption: representative mid-scale chip).
• f_ev = 1 Hz per synapse baseline, swept to 10⁴ Hz (stated assumption).
• P_budget = 100 mW (stated assumption: embedded-power envelope).

#4. Analysis

#4.1 Landauer term

E_Landauer = k_B T ln 2 = (1.380649×10⁻²³)(300)(0.693147). Step 1: k_B T = 4.141947×10⁻²¹ J. Step 2: × ln 2 → E_Landauer = 2.87×10⁻²¹ J (0.693 k_B T).

#4.2 Switching term

E_switch = C_s ΔV² = (1.0×10⁻¹³)(0.25) = 2.5×10⁻¹⁴ J. ρ = 2.5×10⁻¹⁴ / 2.87×10⁻²¹ = 8.7×10⁶. The representative synapse operates ~8.7 million times above the Landauer floor (8.7×10⁶ k_B T ln 2). With α_ad = 0.9: E_switch^eff = 2.5×10⁻¹⁵ J, ρ = 8.7×10⁵.

#4.3 Switching time and speed limit

With V_th = 0.9 ΔV = 0.45 V: ln(0.45/0.05) = ln 9 = 2.19722. t_sw = (10⁴)(1.0×10⁻¹³)(2.19722) = 2.20×10⁻⁹ s ≈ 2.2 ns; f_max = 4.6×10⁸ events/s. Cross-check against the Margolus–Levitin analogue: t ≥ ħ/(2E_switch) = 1.0546×10⁻³⁴/(5.0×10⁻¹⁴) = 2.1×10⁻²¹ s—negligible compared to the RC limit. Neuromorphic speed is RC-limited, not quantum-limited, by ~10¹²; the quantum speed-limit formalism is structurally useful but numerically inert here, an honest negative finding.

#4.4 Leakage term and crossover

E_leak = P_leak/f_ev: 1.0×10⁻¹¹ J at 1 Hz; 1.0×10⁻¹³ J at 10² Hz; 1.0×10⁻¹⁵ J at 10⁴ Hz. The switching and leakage terms cross when P_leak/f_ev = C_s ΔV²: f_cross = 10⁻¹¹/2.5×10⁻¹⁴ = 400 Hz. Below 400 Hz per synapse, leakage dominates; above, switching dominates. At f_ev = 400 Hz: E_ev ≈ 5.0×10⁻¹⁴ J (the Landauer term is negligible at every operating point, by ρ ≈ 8.7×10⁶).

#4.5 System-level power and the frontier

P_tot = N_s f_ev E_ev. In the switching-dominated regime: P_tot = 10⁶ × (2.5×10⁻¹⁴ f_ev + 10⁻¹¹). At f_ev = 1 Hz: ≈10 µW; at 10² Hz: 12.5 µW; at 10³ Hz: 35 µW; at 10⁴ Hz: 260 µW. Power-budget check: setting P_tot = 0.1 W gives 2.5×10⁻¹⁴ f_ev = 10⁻⁷ − 10⁻¹¹ ≈ 9.999×10⁻⁸, so f_ev = 4.0×10⁶ Hz per synapse—i.e., 4.0×10¹² synaptic events/s chip-wide—comfortably below f_max = 4.6×10⁸ Hz. The binding constraint at this scale is power, not switching physics.

#4.6 Adiabatic improvement

With α_ad = 0.9, the crossover moves to f_cross = 10⁻¹¹/2.5×10⁻¹⁵ = 4000 Hz, and chip-wide power at 10⁴ Hz becomes 10⁶ × (2.5×10⁻¹¹ + 10⁻¹¹) = 35 µW—a 7.4× reduction (260/35 = 7.43).

#4.7 Redundancy overhead analogy

Following the system-level lesson of fault-tolerant quantum computing, where 10²–10³ physical operations per logical operation multiply thermodynamic cost [12,10], suppose neuromorphic robustness requires redundancy factor r = 100 (stated assumption: triple-modular-style voting with spare synapses). Chip-wide logical throughput at fixed power falls by r: at 100 mW, logical rate = 4.0×10¹⁰ events/s, and energy per logical event rises to 2.5×10⁻¹² J = 8.7×10⁸ k_B T ln 2. Results scale linearly in r; for r ∈ [10, 1000], logical energy ∈ [2.5×10⁻¹³, 2.5×10⁻¹¹] J.

#5. Results

All numbers are computed in Section 4 from stated inputs; none are empirical measurements.

1. Landauer floor: 2.87×10⁻²¹ J (0.693 k_B T) per synaptic event at 300 K.
2. Representative switching energy: 2.5×10⁻¹⁴ J (8.7×10⁶ k_B T ln 2), ρ = 8.7×10⁶; with 90% adiabatic recovery, 2.5×10⁻¹⁵ J (ρ = 8.7×10⁵).
3. RC speed limit: t_sw = 2.2 ns, f_max = 4.6×10⁸ events/s per synapse; the Margolus–Levitin analogue gives 2.1×10⁻²¹ s, confirming the RC limit binds by ~10¹².
4. Leakage–switching crossover: f_cross = 400 Hz per synapse (non-adiabatic); 4000 Hz with α_ad = 0.9.
5. Minimum energy per event: ≈5.0×10⁻¹⁴ J ≈ 1.74×10⁷ k_B T ln 2 at the crossover (non-adiabatic)—the model's predicted optimal operating point.
6. Chip-level power curve (N_s = 10⁶): 10 µW (1 Hz/synapse), 12.5 µW (10² Hz), 35 µW (10³ Hz), 260 µW (10⁴ Hz); with α_ad = 0.9, 35 µW at 10⁴ Hz (7.4× improvement).
7. Power-budget-limited throughput: at 100 mW, 4.0×10⁶ events/s per synapse (4.0×10¹² chip-wide), two orders of magnitude below the RC limit; power, not device physics, binds.
8. Projection (labeled): with redundancy r = 100, energy per logical event rises to 2.5×10⁻¹² J and logical throughput falls to 4.0×10¹⁰ events/s at 100 mW (range 2.5×10⁻¹³–2.5×10⁻¹¹ J for r ∈ [10, 1000]).
9. Flat fundamental trade-off: because E_switch is speed-independent below f_max, the fundamental energy–speed curve is flat from DC to ~4.6×10⁸ Hz; observed speed–efficiency slopes in neuromorphic hardware are engineering artifacts (charging, leakage, routing), not fundamental physics.

#6. Discussion

Limitations. The model's weakest link is the parameter set. C_s = 100 fF, R_on = 10 kΩ, ΔV = 0.5 V, and P_leak = 10 pW are stated assumptions representative of analog CMOS, not measurements of any specific chip; memristive or spintronic synapses would shift E_switch by orders of magnitude in either direction. The Rayleigh quadratic force–flux law [2] breaks down near criticality—extended irreversible thermodynamics predicts divergent transport coefficients near non-equilibrium instabilities [6]—so our frontier is invalid near synchronization transitions. The Nambu-style local reduction [3] is justified only away from strong-coupling regimes; correlated firing is outside the model. The single-bit Landauer erasure assumption is conservative in one direction (analog spike timing may erase less) and optimistic in another (reset, routing, and conversion are ignored).

Failure modes and falsifiability. The central falsifiable prediction is the leakage–switching crossover at f_cross = 400 Hz (non-adiabatic parameters): a chip measured to have rate-independent energy per event well below 400 Hz, or a crossover displaced by more than an order of magnitude from the value implied by its measured C_s, ΔV, and P_leak, falsifies the three-term decomposition as applied. If measured energy per spike falls below our E_switch floor, the capacitance assumption is wrong, not the framework. If energy per spike rises with firing rate beyond the RC-predicted step, additional dissipation channels must be added as further Rayleigh terms. The redundancy projection is falsified if system-level energy per logical operation shows no overhead factor relative to device-level energy per event.

Arguing against ourselves. A skeptic could say the model is elaborate bookkeeping around the trivial fact that CV² dominates: the Landauer term, the quantum speed limit, and the geometric formalism contribute nothing numerically (ρ ≈ 8.7×10⁶; quantum bound inert by 10¹²). This is partly fair—and it is itself a result, echoing the cautionary lesson of quantum-inspired algorithms whose formal elegance outran practical advantage [1]. Our defense is that the framework identifies which bound binds and where the frontier bends (the 400 Hz crossover, the power-limited regime at 4×10⁶ Hz/synapse), which bare CV² arithmetic does not. The damage-physics analogy [4] suggests a further open direction: synapse wear under sustained flux may impose a lifetime–rate trade-off we have not modeled. Open questions include: the correct dissipation potential for subthreshold adiabatic charging; whether critical slowing near synchronization alters the frontier's slope [6]; and whether coupled transport–state dynamics [5] predict additional cross-terms measurable as rate-dependent leakage.

On error-correction overheads. One source analysis applied a combined quantum-inspired algorithmic and error-correction multiplier (η ≈ 3.16×10⁴) directly to neuromorphic operations, yielding a 0.5 pJ optimum under a 0.5 W / 1 TOPS scenario. We adopt the opposite convention—neuromorphic hardware carries no QEC multiplier, only an optional classical redundancy factor r—because B and C converge on the structural argument that QEC overheads are specific to fault-tolerant quantum architectures [10,11,12]; the conflict is documented in Appendix A rather than silently resolved.

#7. Conclusion

We constructed a non-equilibrium thermodynamic model of neuromorphic computation that treats the processor as a driven dissipative system sustaining computational flux, decomposes energy per synaptic event into Landauer, switching, and leakage terms, and derives an explicit energy–speed frontier. For representative analog-CMOS parameters, the model predicts a minimum energy per synaptic event of ~5.0×10⁻¹⁴ J at a 400 Hz crossover, a power-limited chip-wide throughput of 4.0×10¹² events/s at 100 mW for 10⁶ synapses, and a 7.4× efficiency gain available from 90% adiabatic charging. The Landauer floor and quantum speed limits, while structurally informative, are numerically inert at current operating points—efficiency gains must come from adiabaticity and leakage reduction, not from approaching fundamental bounds. The framework's crossover and scaling predictions are testable on existing hardware within twelve months, and its system-level accounting, informed by the redundancy overheads known from fault-tolerant quantum computing, provides the honest boundary assessment that neuromorphic efficiency claims have lacked.

#References

[1] Quantum-inspired algorithms in practice. arXiv:1905.10415v3. https://arxiv.org/abs/1905.10415v3 [2] Comparison of some geometric frameworks for dissipative evolution in multiscale non-equilibrium thermodynamics. arXiv:2512.05168v2. https://arxiv.org/abs/2512.05168v2 [3] Reduction of Complex Dynamics in Far-from-equilibrium Systems: Nambu Non-equilibrium Thermodynamics. arXiv:2508.19455v3. https://arxiv.org/abs/2508.19455v3 [4] Non-equilibrium statistical mechanics of non-equilibrium damage phenomena. arXiv:0905.0292v3. https://arxiv.org/abs/0905.0292v3 [5] A non-equilibrium thermodynamics model for combined adsorption and diffusion processes in micro- and nanopores. arXiv:1204.3841v1. https://arxiv.org/abs/1204.3841v1 [6] Non-equilibrium critical behavior: An extended irreversible thermodynamics approach. arXiv:0908.2161v1. https://arxiv.org/abs/0908.2161v1 [7] Engineering topological states and quantum-inspired information processing using classical circuits. arXiv:2409.09919v2. https://arxiv.org/abs/2409.09919v2 [8] Non-equilibrium probability flux of a thermally driven micromachine. arXiv:1905.06796v2. https://arxiv.org/abs/1905.06796v2 [9] DOI 10.5281/zenodo.18036068. QNFO: Thermodynamic Viability and the Universality of Feynman Matter. [10] DOI 10.5281/zenodo.17955898. QNFO: Thermodynamic and Informational Bottlenecks of Scalable Fault-Tolerant Quantum Computation. [11] DOI 10.5281/zenodo.17937531. QNFO: Thermodynamic and Quantum Constraints on Scalable Quantum Computing. [12] DOI 10.5281/zenodo.22753039. QNFO: The Physics of Computation: Fundamental Limits and the Honest Boundaries of Post-Classical Computing.

#Appendix A. Divergence report

D1. Applicability of quantum error-correction overhead to neuromorphic hardware. Draft A applies a combined multiplier η = 10² × η_QEC ≈ 3.16×10⁴ (algorithmic prefactor from [1] times median QEC overhead from [10]) directly to neuromorphic logical operations, obtaining an optimal E_op of 5.0×10⁻¹³ J under a 0.5 W / 1 TOPS scenario. Drafts B and C apply no QEC multiplier to neuromorphic operations, arguing (B, §2; C, §2/§4.8) that the 10²–10³ overhead of [10,11,12] is specific to fault-tolerant quantum architectures, and that neuromorphic hardware avoids it entirely; C instead introduces an optional classical redundancy factor r as a labeled assumption. Underlying convention disagreement: whether "quantum-inspired" algorithms running on classical neuromorphic hardware inherit quantum error-correction costs. Resolution: the main text adopts the B/C convention (no QEC multiplier; optional classical redundancy r), because two independent drafts converge on the structural argument and A's application of a quantum-QEC factor to non-quantum hardware lacks a mechanism. A's 0.5 pJ / 1 TOPS scenario is retained as a documented alternative convention, not as the main result.

D2. Power budget and scale. Draft A assumes P_max = 0.5 W and v_target = 10¹² ops/s (chip-level TOPS framing). Draft C assumes P_budget = 100 mW with N_s = 10⁶ synapses and per-synapse event rates. Draft B uses per-channel and per-chip floor framing with N = 10⁶ synapses at 1% activity. Underlying disagreement: chip class (high-performance module vs. embedded envelope) and accounting unit (logical TOPS vs. synaptic events). Resolution: the main text adopts C's embedded-envelope framing (100 mW, 10⁶ synapses) because it yields a fully explicit, self-consistent parameter set; A's scenario is reported here for completeness. Under A's convention, the power-budget constraint gives E_op ≤ P/v_target = 0.5/10¹² = 5.0×10⁻¹³ J, with the Margolus–Levitin ceiling v_max = 2P/h = 1.51×10³³ ops/s non-binding—consistent with the main text's finding that quantum limits are inert.

D3. Optimal operating point. Draft A reports E_op = 5.0×10⁻¹³ J (0.5 pJ) as the optimum of a weighted cost functional. Draft C reports ≈5.0×10⁻¹⁴ J at the 400 Hz crossover. Draft B reports a practical floor of ~110 fJ/op (range 10 fJ–1 pJ) and a fundamental floor of 2.87×10⁻²¹ J. Underlying disagreement: different objective functions (weighted energy/speed cost vs. minimum of a three-term decomposition vs. leakage+I/O practical floor) and different parameter sets. Resolution:* the main text adopts C's three-term decomposition optimum, which follows from the explicitly stated parameter set; the spread across drafts (5×10⁻¹⁴–5×10⁻¹³ J) reflects parameter assumptions, not physical disagreement, and all drafts agree the value lies ≥10⁷ above the Landauer floor.

D4. Role of the Margolus–Levitin bound. All three drafts agree the quantum speed limit does not bind (A: ceiling 1.51×10³³ ops/s far above target; B: crossover at 58 fs; C: RC limit binds by ~10¹²). CONVERGENT; no conflict.

#Appendix B. Claim attribution