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Thermodynamic characterization of message-passing: Framework and initial study

Kalana G. Abeywardena, Stark C. Draper

arXiv:2610.36348v1cs.ITcond-mat.stat-mechmath.IT

TL;DR

Message-passing decoders motivate an algorithm-level thermodynamic analysis because detailed physical dynamics are difficult to characterize and existing energy bounds can require such information. The paper applies mismatch cost to a decoder modeled as a main program with message-update subroutines, finding that cost depends on decoding dynamics, input statistics, and prior choice. Its Hamming-code study over the BEC also exposes limits of direct computation and the need to extend beyond this illustrative setting.

  • Problem

    Detailed physical dynamics needed for exact entropy-production characterization are generally unavailable, while existing bounds can require substantial system information.

    Source: §I · pp. 2–3

  • Method

    The paper applies mismatch cost to a message-passing decoder modeled as a main program with message-update subroutines, treating memory as an external subsystem.

    Source: F. Memory as an external subsystem AM · p. 22, §V · p. 34

  • Results

    Mismatch cost depends on decoding dynamics, input statistics, and prior choice; for ϵ < 0.3, the cumulative step-wise cost corresponds to 0.224 aJ of energy dissipation.

    Source: B. Mismatch costs of message-passing… · pp. 26–27, §V · p. 34

  • Takeaways & Limitations

    The framework provides a basis for studying thermodynamic costs across decoder inputs and hardware implementations to inform energy-efficient decoder design.

    Source: §V · pp. 34–35

    The Hamming code and BEC limit trajectory diversity and practical-channel relevance, while exponentially growing state spaces make direct computation for practical LDPC codes infeasible.

    Source: §V · p. 34

Abstract

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We study the thermodynamic efficiency of message-passing decoders using the mismatch cost framework. This framework quantifies the excess entropy generated when a system operates away from the thermodynamically optimal initial distribution. It provides a computable lower bound on irrecoverable energy loss. The decoder is modeled as a main decoding program with local message updates implemented as subroutines. Memory is treated as an external copying subsystem whose thermodynamic cost is excluded. Studying the $[7,4]$ Hamming code transmitted over a binary erasure channel, we show how the channel-induced initialization statistics and the step-wise evolution of the algorithmic state distribution during decoding determine the total mismatch cost. We further show that different thermodynamically optimal state distributions, representing different underlying physical implementations of the decoder, can produce different thermodynamic costs for identical decoding dynamics. Conditioning the input distribution on correctable and non-correctable erasure patterns further reveals distinct thermodynamic signatures. These results demonstrate that thermodynamic cost is jointly governed by decoding dynamics, input statistics, and hardware characteristics. This points to opportunities for thermodynamically-informed decoder design to realize energy-efficient physical implementations.

I. INTRODUCTION

The introduction extends mismatch-cost analysis to message-passing decoders, addressing the gap between hardware-level energy models and the thermodynamic cost of algorithmic updates. It presents a decoder model and computational approach for examining how input statistics and prior choices shape that cost.

I. INTRODUCTION

  • Mismatch cost offers a computable lower bound on entropy production from observable state dynamics, addressing the lack of algorithm-level thermodynamic analysis for message-passing decoders.The introduction situates this gap alongside prior hardware- and graph-level energy models that do not capture the message-update sequence itself.Source: §I · pp. 3–4

A. Our contributions

  • The framework models decoding as a main program with modular variable- and check-node update subroutines.It focuses on algorithmic states rather than detailed circuit-level dynamics.Source: A. Our contributions · pp. 4–5
  • The proposed block decomposition addresses transition-structure scale: even the Hamming-code main program has over 4.2 million states, making direct computation intractable.The method decomposes the transition matrix into smaller blocks.Source: A. Our contributions · p. 5
  • Different prior distributions, representing different physical implementations, can assign different thermodynamic costs to the same computation.The paper considers mismatch cost under uniform and non-uniform priors.Source: A. Our contributions · p. 5
  • Conditioning on classes of BEC erasure patterns reveals input-dependent thermodynamic signatures, including in small codes.The paper studies correctable and non-correctable patterns as distinct input classes.Source: A. Our contributions · p. 5

II. PRELIMINARIES

This section introduces message-passing decoding and a thermodynamic framework for describing computation as stochastic evolution of a system’s state distribution. It distinguishes entropy production from the Landauer cost and motivates mismatch cost as a lower bound when microscopic dynamics are unavailable.

A. Message-passing decoding on Tanner Graphs

  • Message-passing decoders iteratively pass updates between variable and check nodes on a Tanner graph, with simple local computations suited to thermodynamic analysis.In low-density codes, each node processes a small set of incoming messages from adjacent nodes.Source: A. Message-passing decoding on Tanner… · pp. 6–7

B. Thermodynamic cost of computation in a general physical system

  • A finite-state system evolves from an initial to a final distribution through stochastic dynamics, exchanging heat with one or more reservoirs.The transition matrix maps the initial distribution to the final distribution; reservoir heat flow contributes to entropy flow.Source: B. Thermodynamic cost of computation… · pp. 7–8
  • Expected entropy production combines reservoir entropy flow with the system’s Shannon-entropy change, distinguishing irreversible dissipation from the Landauer cost of logical state changes.Entropy production represents irreversibly dissipated entropy; the Landauer cost captures the reversible contribution associated with the transformation.Source: B. Thermodynamic cost of computation… · pp. 8–9
  • Exact entropy-production evaluation requires generally unavailable microscopic physical dynamics, motivating a lower bound based on mismatch cost.The physical model can be interpreted through coarse-grained logical states and transitions implemented by circuit switching and charge movement.Source: B. Thermodynamic cost of computation… · pp. 8–9

C. Mismatch cost as a lower bound on entropy production

The mismatch-cost framework compares actual initialization with the entropy-minimizing prior for fixed dynamics and gives a lower bound on entropy production. Its algorithmic extension tracks mismatch across repeated execution steps, where state distributions typically depart from the prior.

C. Mismatch cost as a lower bound on entropy production

  • Mismatch cost is the reduction in KL divergence between actual and prior distributions under the transition map, measuring the penalty for initialization away from the prior.The prior minimizes entropy production for the given implementation and is determined by its transition structure and state-dependent entropy flows.Source: C. Mismatch cost as a lower bound on… · pp. 10–11
  • Entropy production decomposes into residual entropy production and mismatch cost, which is nonnegative and provides a lower bound on total entropy production.When residual entropy production is zero, mismatch cost fully accounts for entropy production and the bound is tight.Source: C. Mismatch cost as a lower bound on… · p. 11
  • Once the prior and transition map are specified, mismatch cost can be evaluated without microscopic implementation details, isolating dissipation due to initialization mismatch.Source: C. Mismatch cost as a lower bound on… · p. 11

D. Mismatch cost of algorithmic computation

  • The algorithmic extension models computation as repeated applications of one transition matrix to distributions over coarse-grained algorithmic states, including states revisited through control flow.A periodic machine represents this repeated evolution; its state distribution reaches a steady state, with a minimum step count for periodic closure.Source: D. Mismatch cost of algorithmic… · p. 12
  • Even when initialized at the prior, repeated algorithmic updates typically move the state distribution away from it, generating mismatch cost across execution steps.The total algorithmic mismatch cost sums the step-wise costs.Source: D. Mismatch cost of algorithmic… · p. 13

III. PROBLEM SETUP

The decoder is modeled as a main program that coordinates check-node and variable-node update subroutines, with its registers and stepwise execution defining algorithmic state. Its input and carried-forward internal state determine initialization, while memory reads and writes are treated as copying.

III. PROBLEM SETUP

  • The decoder comprises a main program AD, check-node and variable-node subroutines AC and AV, and external memory AM for channel observations and Tanner-graph messages.AD reads and writes messages around subroutine updates; AM holds n channel observations and |E| edge messages.Source: §III · p. 13, Figure 1 · p. 14
  • The framework models each device with RASP registers and call-by-value subroutine inputs, representing deterministic computation as stepwise state transitions.The registers may hold symbols beyond binary values.Source: §III · p. 14

A. State space design for main program AD

  • AD's registers jointly define its state, and each computational step induces a transition over the resulting state space.The program counter tracks instructions as the decoder processes nodes and iterations until convergence or the maximum iteration count.Source: A. State space design for main program… · pp. 15–16

B. Initial distribution and mismatch cost of main program AD

  • The input state is the channel output, whose distribution is determined by the message prior, encoding map, and channel.The message components are modeled as i.i.d. biased Bernoulli variables, with Pr(ui = 0) = α.Source: B. Initial distribution and mismatch… · p. 17
  • Internal non-input variables retain the previous execution's terminal-state marginal, while special variables reset to a fixed state for each run.Input reinitialization is excluded from the algorithm's thermodynamic cost because the channel supplies those inputs externally.Source: B. Initial distribution and mismatch… · pp. 17–18

C. State space design for the subroutines AC and AV

E. Total mismatch cost of the message-passing decoder

The decoder's total mismatch cost accounts for AD and its message-update subroutines. Without early termination, the subroutine contribution grows linearly with the maximum number of iterations.

E. Total mismatch cost of the message-passing decoder

  • The total mismatch cost accounts for interactions among AD, AV, and AC, while AM is treated as an external subsystem rather than included in the analyzed program.Memory reads and writes are idealized as copying, and the thermodynamic cost of data movement is left for future work.Source: E. Total mismatch cost of the… · p. 22, F. Memory as an external subsystem AM · p. 22
  • The subroutine mismatch contribution scales linearly with ℓmax when the decoder runs all iterations without early termination.The per-iteration factors n and m account for the number of variable-node and check-node updates.Source: E. Total mismatch cost of the… · p. 22

G. Correctable vs. non-correctable erasure patterns and their initial distributions

The input distribution can be conditioned on correctable or non-correctable erasure patterns while keeping the decoder hardware and its transition structure fixed. These conditional distributions allow mismatch costs to be evaluated separately for each error category.

G. Correctable vs. non-correctable erasure patterns and their initial distributions

  • Conditioning on correctable versus non-correctable patterns changes the input distribution but leaves the state space, transition matrix, prior, and decoder hardware unchanged.The same main program is run on each restricted input ensemble.Source: G. Correctable vs. non-correctable… · pp. 22–23
  • For the [7, 4] Hamming code over a BEC, all weight-2 erasure patterns are correctable under sum-product decoding, while patterns with more than four erasures may be non-correctable.The latter may belong to a stopping set.Source: G. Correctable vs. non-correctable… · p. 23
  • The conditional input distributions initialize AD's input variables and, together with its prior, permit separate mismatch-cost calculations for each error category.The categories are correctable and non-correctable erasure patterns.Source: G. Correctable vs. non-correctable… · p. 23

IV. EXPERIMENTAL EVALUATIONS

The evaluations use a [7, 4] Hamming code to examine mismatch costs under different priors and erasure-pattern classes. The reported comparisons link the cost to input statistics, implementation-dependent priors, and state-transition structure.

IV. EXPERIMENTAL EVALUATIONS

  • A non-uniform prior induced by state-dependent entropy flows shifts the message-prior and erasure-probability combinations that minimize mismatch cost relative to the uniform-prior case.Under the uniform prior, state-dependent entropy flows are approximately equal across module states.Source: §IV · p. 23
  • Correctable and non-correctable error patterns have distinct mismatch costs associated with differences in their state-transition structures.The comparison conditions the input distribution by error category.Source: §IV · p. 24

A. Experimental setup

The study uses the Hamming code over a BEC as a tractable setting for examining message-passing decoder state evolution and mismatch cost. It enumerates reachable states and transitions, while noting that state-space growth limits scaling.

A. Experimental setup

  • BEC simulations vary erasure probability ǫ from 0 to 1; the code uses 19 memory cells for seven channel outputs and 12 edge messages.The code has dmax = 4, and ℓmax = 3 suffices for convergence.Source: A. Experimental setup · p. 24
  • The [7, 4] Hamming code has about 4.2 million reachable main-program states, compared with a Cartesian state space of roughly 7.3 billion.The state-space growth with block length motivates using a small code for these initial explorations.Source: A. Experimental setup · p. 24
  • The transition matrix is built by enumerating initial combinations, simulating each to termination, and normalizing reachable transitions; calculations are blockwise with κ = 310.The reachable state space makes memory management necessary.Source: A. Experimental setup · p. 24
  • The small-code analysis is an initial exploration: exponential state-space growth makes trajectory-level cost analysis increasingly difficult as block length increases.Message-passing failures on stopping sets include the listed weight-three non-correctable patterns.Source: A. Experimental setup · p. 25
  • The main decoder and check- and variable-node update subroutines are modeled as interacting machines, while messages and channel observations reside in external memory.The setup assumes a single thermal reservoir at 300 K and uniform thermodynamically optimal priors for the programs.Source: A. Experimental setup · p. 25

B. Mismatch costs of message-passing decoding over the BEC

The main decoder’s mismatch cost depends on channel erasure probability and step-wise state evolution, while message prior α has little effect under the uniform prior. The full decoder’s cost includes its main program and update subroutines.

B. Mismatch costs of message-passing decoding over the BEC

C. Mismatch cost under a non-uniform prior distribution

A state-dependent entropy-flow functional induces a non-uniform prior over decoder states, changing how input parameters affect mismatch cost despite fixed decoding dynamics. Under this prior, the minimizing α shifts and the maximum associated dissipation rises.

C. Mismatch cost under a non-uniform prior distribution

  • The constructed state-dependent entropy-flow functional favors low-flow states and suppresses states with larger entropy flows, inducing the non-uniform prior.The functional represents hardware-dependent effects, without specifying a particular physical implementation.Source: C. Mismatch cost under a non-uniform… · pp. 29–30, Figure 7 · p. 30
  • Under the non-uniform prior, the minimizing α shifts from 0.5 to 0.9, and maximum associated dissipation rises from approximately 0.223 aJ to 0.31 aJ.For the same initial distribution, mismatch cost is consistently larger under the non-uniform prior than under the uniform prior.Source: C. Mismatch cost under a non-uniform… · p. 30, Figure 8 · p. 31
  • With the non-uniform prior, mismatch cost becomes more sensitive to α because input parameters affect which decoder states are preferentially visited.The dependence on ǫ remains qualitatively similar to the uniform-prior case.Source: C. Mismatch cost under a non-uniform… · pp. 30–31
  • The results suggest aligning the prior with computation’s initialization statistics by engineering hardware to favor low-entropy-flow states.This conclusion holds for the analyzed fixed decoding dynamics and initial distribution.Source: C. Mismatch cost under a non-uniform… · p. 31

D. Mismatch costs of correctable and non-correctable erasure patterns

Conditioning on correctable or non-correctable BEC erasure patterns reveals different mismatch-cost behavior under the same uniform prior. In the correctable ensemble, cost decreases at higher erasure probability; in the non-correctable ensemble, it is nearly constant.

D. Mismatch costs of correctable and non-correctable erasure patterns

  • Correctable-pattern mismatch cost has a nontrivial dependence on ǫ: early trajectory collapses create spikes at low ǫ, while higher ǫ produces fewer and smaller spikes.Weight-three erasure patterns have probability 4×10^-5 at ǫ = 0.1 and α = 0.5, versus 3×10^-3 for weight-one patterns.Source: D. Mismatch costs of correctable and… · p. 32
  • For correctable patterns, energy dissipation decreases from 0.215 aJ to 0.186 aJ as the analysis moves from low to high ǫ.At low ǫ, weight-one and weight-two patterns dominate and are corrected after one iteration; at high ǫ, weight-three patterns dominate and evolve more gradually.Source: D. Mismatch costs of correctable and… · pp. 32–33
  • For non-correctable patterns, mismatch cost remains almost constant across ǫ because decoding changes the state distribution only weakly.As ǫ approaches 1, unchanged higher-weight patterns increasingly dominate, reducing iteration-to-iteration variation.Source: D. Mismatch costs of correctable and… · pp. 33–34
  • Conditioning on erasure-pattern subsets exposes different thermodynamic costs under the same uniform prior, showing that input statistics matter alongside decoder dynamics.The figure distinguishes single-, double-, and triple-erasure probabilities by color.Source: D. Mismatch costs of correctable and… · pp. 32, 34, Figure 9 · p. 32, Figure 10 · p. 33

V. CONCLUSION AND FUTURE WORK

The mismatch-cost framework gives a computable lower bound on irreversibly lost energy in message-passing decoding, with costs shaped by decoding dynamics, input statistics, and prior selection. The study uses the Hamming code over the BEC, while identifying limits of this illustrative setting and directions for scaling the analysis.

V. CONCLUSION AND FUTURE WORK

  • The Hamming-code BEC example shows that decoding dynamics, input statistics, and prior selection all shape the lower bound on irreversibly lost energy.The decoder is modeled as a main program with two subroutines.Source: §V · p. 34
  • The short block length and BEC limit trajectory diversity and practical-channel relevance, while longer-code mismatch-cost computation faces an exponentially growing state space.Even block-wise sub-matrix methods require storing all trajectories to construct the full transition structure.Source: §V · p. 34
  • Future work proposes representative sampling across error regimes and finite-precision representations to estimate costs for longer codes and more realistic channels.The authors also aim to extend the framework to binary symmetric and additive Gaussian channels.Source: §V · pp. 34–35
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