Quantum error correction recently reached logical error rates below the physical threshold on superconducting processors, making topologically ordered states experimentally accessible. A whole line of research has grown around this work: neighbouring papers explore quantum computers that sweep away their own errors, quantum sensors that beat their own noise, and iceberglike codes where errors make the computer smarter. This paper sits off to the side of that current: it asks not about computation but about thermodynamics, namely whether topological order can be used for the nonlocal transfer of useful work at finite temperature. The question had remained open because quantum energy teleportation works only near absolute zero and carries bare energy, not ergotropy.
The protocol is split into five stages. Alice holds a charged battery and a surface code on a rectangular lattice of size N×L, where L is the code distance and N is the separation between the sides. She spends energy ΔE to encode a logical qubit via interaction with a boundary string, then measures 2L boundary stabilizers and sends the classical syndrome record to Bob. Bob applies a minimum-weight perfect-matching decoder and, depending on the outcome, conditionally charges his battery. The simulation was run in the Stim simulator with circuit-level noise (depolarization on all Clifford gates and bit-flip measurement errors) and the PyMatching decoder; 8000 trajectories were collected for each point. Active syndrome monitoring keeps the measurement rate far above the error rate, which exponentially suppresses logical errors below threshold.
Below the topological threshold p_th ≈ 0.013, the probability of successful decoding P_succ approaches unity exponentially with growing code distance L, so the extracted ergotropy E_B = (2P_succ − 1)ΔE approaches ΔE. For a finite code with L = 7, the net work changes sign at a thermodynamic critical error rate p_c ≈ 0.014, which is noticeably above the topological threshold; this is a finite-size effect that matters for today's devices. Causality requires the number of syndrome measurement rounds to grow in proportion to N, so the cost of maintaining the channel W_bulk grows as N² (the numerical exponent is 2.05 ± 0.04). The intersection of the nearly constant ergotropy with the cost parabola yields a thermodynamic horizon N_max ≈ 78 cells, beyond which the net work becomes negative regardless of the quality of the code or decoder. In addition, an information threshold was found: if Bob receives less than roughly 62% of the boundary syndromes, the decoding graph loses connectivity and the ergotropy drops to zero.
The work follows the same broad line as its neighbours in quantum error correction but differs in its object of study: instead of computational reliability, it measures thermodynamic benefit. This is the first quantitative check that topological order can serve as a channel for the nonlocal transfer of ergotropy at finite temperature, and not only for protecting qubits. The difference from quantum energy teleportation is fundamental: there, bare energy is transferred, and it need not be useful work; here, it is ergotropy itself that is transmitted. The quadratic infrastructure cost turns out to be a fundamental consequence of causality rather than an engineering imperfection, which places it on a par with Landauer's erasure principle.
A natural next step is ergotropy distribution networks, in which several demons share a single topological backbone to route quantum energy. An open question is whether a more efficient decoder could shift the horizon N_max without eliminating the quadratic penalty. Similar horizons are worth looking for in other topologically ordered phases, such as the toric code or non-Abelian phases. A broader prospect: the thermodynamic cost of maintaining quantum correlations across space may turn out to be a fundamental quantity comparable to the Landauer bound.
The results bear directly on distributed quantum computing and quantum networks, where useful work must be transferred between nodes. They also provide a thermodynamic estimate of the cost of quantum error correction, which matters for designing future quantum batteries and energy interfaces.
Test the protocol on a real superconducting processor with a surface code at error rates achievable today, p ≈ 0.003. Investigate how decoder complexity affects the thermodynamic budget and whether the horizon can be optimized without violating causality.
In quantum thermodynamics, the question of how information resources are converted into useful work in spatially separated systems remains open. This work links it to the problem of the passive instability of two-dimensional topological order at finite temperature and shows that active error correction is necessary for stable ergotropy transfer. This sheds light on the fundamental limits of distributed quantum thermodynamic machines.
The thermodynamic horizon N_max ≈ 78 cells corresponds to roughly 78×7 = 546 physical qubits in the channel, which already fits on modern superconducting chips. The critical temperature T_c ≈ 0.937 J/k_B, at a typical coupling constant J/h ~ 5–10 GHz, amounts to tens of millikelvins, that is, the operating range of an ordinary dilution refrigerator.
The idea of a nonlocal demon that extracts work from information echoes Stanisław Lem's story 'The Fourteenth Voyage' from The Star Diaries of Ijon Tichy, in which the hero encounters a being that violates the second law of thermodynamics. In this protocol the second law is strictly obeyed, thanks to the quadratic cost of maintaining the channel.