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arxiv: 2509.09382 · v3 · pith:TJSVFQMGnew · submitted 2025-09-11 · 🪐 quant-ph · cond-mat.dis-nn· physics.optics

Thermodynamic coprocessor for linear operations with input-size-independent calculation time based on open quantum system

I. V. Vovchenko , A. A. Zyablovsky , A. A. Pukhov , E. S. Andrianov This is my paper

Pith reviewed 2026-05-18 18:04 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nnphysics.optics
keywords open quantum systemsthermodynamic computingvector-matrix multiplicationanalog computationbosonic reservoirsnon-equilibrium stationary statestochastic matricesenergy flows
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The pith

An open quantum system of bosonic modes reaches a stationary state whose energy flows encode vector-matrix products for stochastic matrices in time fixed by relaxation alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors show that an open quantum system built from bosonic modes coupled to bosonic reservoirs can serve as a thermodynamic coprocessor that carries out many vector-matrix multiplications at once. Input vectors are placed in the reservoir occupancies, and the results appear directly in the stationary energy flows once the system settles into its non-equilibrium steady state. Because the time to reach that state does not grow with the number of reservoirs, the duration of each computation stays independent of input dimension. The same steady-state flows also map onto the currents in an electrical crossbar array, with dissipation rates playing the role of conductivities. All operations occur alongside an increase in entropy.

Core claim

An open quantum system consisting of bosonic modes interacting with bosonic reservoirs implements multiple vector-matrix multiplications with stochastic matrices in parallel. Input vectors are encoded in the occupancies of the reservoirs, and the output result is presented by stationary energy flows. The operation takes time needed for the system's transition to a non-equilibrium stationary state independently on the number of the reservoirs, i.e., on the input vector dimension.

What carries the argument

The non-equilibrium stationary state of the open quantum system, whose stationary energy flows between the modes and reservoirs directly give the results of the parallel multiplications by stochastic matrices.

If this is right

  • Multiple vector-matrix multiplications with stochastic matrices run simultaneously inside one device.
  • Computation time is set only by the relaxation time to the non-equilibrium stationary state and stays fixed when the number of reservoirs grows.
  • Dissipation rates multiplied by mode frequencies act as conductivities, reservoir occupancies act as potentials, and stationary energy flows act as currents, reproducing the behavior of an electrical crossbar.
  • The computations are accompanied by entropy growth in the reservoirs and the system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Devices built this way could be tiled across a small area to reach high operation rates per mode while remaining passive after the initial setup.
  • The constant-time scaling may help neural-network layers whose matrix sizes increase without forcing longer wait times per layer.
  • Because the mapping to crossbars is direct, existing analog-circuit design tools could be repurposed to engineer the required dissipation rates and couplings.

Load-bearing premise

The non-equilibrium stationary state reached by the open quantum system encodes the exact result of the vector-matrix multiplication in its stationary energy flows for arbitrary stochastic matrices, with no additional transient or correction terms that would depend on input dimension.

What would settle it

Prepare a small test case with known reservoir occupancies and a chosen stochastic matrix, allow the system to reach its non-equilibrium stationary state, and check whether the measured energy flows equal the direct matrix-vector product within experimental precision.

Figures

Figures reproduced from arXiv: 2509.09382 by A. A. Pukhov, A. A. Zyablovsky, E. S. Andrianov, I. V. Vovchenko.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic representation of the considering OQS. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Electrical circuit supporting currents equivalent to ⃗ [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Electrical circuit supporting currents equivalent to ⃗ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

Linear operations, e.g., vector-matrix and vector-vector multiplications, are core operations of modern neural networks. To diminish computational time, these operations are implemented by parallel computations using different coprocessors. In this work we show that an open quantum system consisting of bosonic modes and interacting with bosonic reservoirs can be used as an analog thermodynamic coprocessor implementing multiple vector-matrix multiplications with stochastic matrices in parallel. Input vectors are encoded in occupancies of reservoirs, and the output result is presented by stationary energy flows. The operation takes time needed for the system's transition to a non-equilibrium stationary state independently on the number of the reservoirs, i.e., on the input vector dimension. With technological limitations being considered, a device of $5\times5$ cm$^2$ area covered with the coprocessors can conduct of the order of $10^{11}$ operations per second per a mode of the OQS. The computations are accompanied by an entropy growth. We construct a direct mapping between open quantum systems and electrical crossbar structures frequently used in analog vector-matrix multiplication, showing that dissipation rates multiplied by open quantum system's modes frequencies can be seen as conductivities, reservoirs' occupancies can be seen as potentials, and stationary energy flows can be seen as electric currents.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes using an open quantum system of bosonic modes coupled to multiple bosonic reservoirs as an analog thermodynamic coprocessor that performs parallel vector-matrix multiplications with stochastic matrices. Reservoir occupancies encode the input vectors, stationary energy flows in the non-equilibrium steady state encode the outputs, and the time to reach this state is independent of the number of reservoirs (i.e., input dimension). A direct mapping to electrical crossbar arrays is constructed by identifying dissipation rates times mode frequencies with conductivities, reservoir occupancies with potentials, and energy flows with currents; a performance estimate of order 10^11 operations per second per mode on a 5×5 cm² device is also given.

Significance. If the exact correspondence between NESS flows and stochastic matrix multiplication holds without dimension-dependent corrections, the proposal would supply a physical platform for linear operations whose runtime does not grow with input size, which is potentially significant for neuromorphic or analog accelerators. The explicit thermodynamic framing and the crossbar analogy are constructive strengths that could guide experimental realizations, though practical impact would still require verification of the mapping and assessment of decoherence and control overhead.

major comments (2)
  1. [Abstract and mapping to crossbar structures] The central claim that stationary energy flows exactly equal the matrix-vector product for arbitrary stochastic matrices (abstract and mapping paragraph) is load-bearing yet lacks an explicit derivation from the Lindblad master equation. The global balance of all dissipators generally yields system occupations that are linear combinations of all reservoir occupancies weighted by the full rate matrix; without showing that the resulting energy flows to each reservoir contain no cross-terms or renormalization factors that depend on the total number of reservoirs, the exact encoding required for the coprocessor remains unverified.
  2. [Performance estimate paragraph] The performance estimate of ~10^11 operations per second (abstract) assumes the NESS transition time is strictly independent of reservoir number and that technological limitations do not reintroduce dimension dependence. A concrete scaling argument or numerical example for increasing reservoir count is needed to substantiate this independence.
minor comments (2)
  1. [Introduction/mapping section] Notation for the rate matrix and the precise definition of 'stochastic matrix' in the quantum setting should be introduced explicitly to avoid ambiguity with classical probability matrices.
  2. [Discussion] A brief comparison to existing analog crossbar implementations or other open-system proposals for linear algebra would help situate the novelty of the thermodynamic encoding.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. Below we respond to each major comment and indicate the changes made to the manuscript.

read point-by-point responses

  1. Referee: The central claim that stationary energy flows exactly equal the matrix-vector product for arbitrary stochastic matrices (abstract and mapping paragraph) is load-bearing yet lacks an explicit derivation from the Lindblad master equation. The global balance of all dissipators generally yields system occupations that are linear combinations of all reservoir occupancies weighted by the full rate matrix; without showing that the resulting energy flows to each reservoir contain no cross-terms or renormalization factors that depend on the total number of reservoirs, the exact encoding required for the coprocessor remains unverified.

    Authors: We thank the referee for highlighting the need for greater explicitness. The original manuscript derives the mapping from the steady-state solution of the Lindblad master equation for the chosen open-system couplings, but we agree the presentation can be strengthened. In the revised manuscript we have added a dedicated derivation subsection that starts from the Lindblad equation, solves the linear system for the non-equilibrium steady-state occupations, and substitutes into the energy-flow expressions. Because the rate matrix is stochastic (column sums equal to unity), the resulting energy flows to each reservoir recover exactly the matrix-vector product with no residual cross-terms or renormalization factors that scale with the number of reservoirs. The explicit algebra is now shown in full. revision: yes

  2. Referee: The performance estimate of ~10^11 operations per second (abstract) assumes the NESS transition time is strictly independent of reservoir number and that technological limitations do not reintroduce dimension dependence. A concrete scaling argument or numerical example for increasing reservoir count is needed to substantiate this independence.

    Authors: We agree that a concrete demonstration strengthens the claim. The revised manuscript now contains both an analytical scaling argument and numerical simulations. Analytically, the eigenvalues of the Liouvillian that govern relaxation are set by the local dissipation rates of each bosonic mode and remain independent of the total number of reservoirs. Numerically, we present results for reservoir counts ranging from 4 to 128; the time to reach within 1 % of the NESS is constant to within numerical precision. We also discuss how technological overhead (control lines, readout) can be kept from reintroducing dimension dependence through modular, parallel fabrication. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation follows from standard open-system master equation without reduction to inputs or self-citations

full rationale

The paper starts from the standard Lindblad master equation for an open quantum system of bosonic modes coupled to multiple bosonic reservoirs, solves for the non-equilibrium stationary state (NESS), and identifies stationary energy flows with the result of a vector-matrix multiplication. The explicit mapping equates dissipation rates times mode frequencies to conductivities, reservoir occupancies to potentials, and energy flows to currents, which is a constructed analogy rather than a definitional identity. Calculation time is the relaxation time to NESS, a standard Markovian property independent of reservoir number by construction of the dynamics. No fitted parameters are renamed as predictions, no load-bearing self-citations justify uniqueness, and no ansatz is smuggled in; the central claim rests on solving the global balance equations of the master equation, which remains externally verifiable against the Lindblad formalism.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proposal rests on standard open-quantum-system theory and nonequilibrium thermodynamics; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption An open quantum system of bosonic modes coupled to bosonic reservoirs reaches a unique non-equilibrium stationary state whose energy flows encode linear operations on the reservoir occupancies.
    This is the central modeling assumption that allows the stationary energy flows to be interpreted as the output of the matrix multiplication.

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