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arXiv:2605.14924

Nonlocal Topological Maxwell Demon Teleporting Ergotropy via Surface-Code Quantum Error Correction

M. Y. Abd-Rabbou, Cong-Feng Qiao

quant-phSubmitted May 14, 2026• Announced May 15, 2026• 0 citations• Revised Jun 24, 2026 (v4)
v1 · May 14, 2026v2 · May 15, 2026v3 · May 30, 2026v4 · Jun 24, 2026
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Abstract

Surface-code quantum error correction has recently achieved logical error rates below the physical threshold on superconducting processors, establishing topologically ordered states as experimentally accessible resources. Whether these resources can support thermodynamic operations beyond fault-tolerant computation remains open. We introduce a nonlocal Maxwell demon protocol that transfers ergotropy between spatially separated quantum batteries using only local operations and classical communication over a shared surface code. Alice expends ergotropy to encode a logical qubit and transmits a classical syndrome record to Bob, who decodes via minimum-weight perfect matching and conditionally charges his battery, with no direct energy exchange across the channel. Active syndrome monitoring exponentially suppresses logical errors below the topological threshold $p_{\rm th} \approx 0.013$, converting physical qubits directly into recoverable ergotropy. For finite-size codes at distance $L = 7$, net extracted work changes sign at a thermodynamic critical error rate $p_c \approx 0.014 > p_{\rm th}$, a physically significant finite-size effect relevant to near-term devices. Causality enforces an irreducible quadratic infrastructure cost $W_{\rm bulk} \propto N^2$, strictly satisfying the second law at all separations and defining a fundamental thermodynamic horizon $N_{\rm max} \approx 78$ beyond which positive net work extraction is impossible regardless of code distance or decoder quality.

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Surface-code quantum error correction has recently achieved logical error rates below the physical threshold on superconducting processors, establishing topologically ordered states as experimentally accessible resources. Whether these resources can support thermodynamic operations beyond fault-tolerant computation remains open. We introduce a nonlocal Maxwell demon protocol that transfers ergotropy between spatially separated quantum batteries using only local operations and classical communication over a shared surface code. Alice expends ergotropy to encode a logical qubit and transmits a classical syndrome record to Bob, who decodes via minimum-weight perfect matching and conditionally charges his battery, with no direct energy exchange across the channel. Active syndrome monitoring exponentially suppresses logical errors below the topological threshold pth≈0.013p_{\rm th} \approx 0.013pth​≈0.013, converting physical qubits directly into recoverable ergotropy. For finite-size codes at distance L=7L = 7L=7, net extracted work changes sign at a thermodynamic critical error rate pc≈0.014>pthp_c \approx 0.014 > p_{\rm th}pc​≈0.014>pth​, a physically significant finite-size effect relevant to near-term devices. Causality enforces an irreducible quadratic infrastructure cost Wbulk∝N2W_{\rm bulk} \propto N^2Wbulk​∝N2, strictly satisfying the second law at all separations and defining a fundamental thermodynamic horizon Nmax≈78N_{\rm max} \approx 78Nmax​≈78 beyond which positive net work extraction is impossible regardless of code distance or decoder quality.
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Fault-tolerant logical operations are fundamental for scalable quantum computation. Here, we report the experimental realization of lattice-surgery operations between a pair of distance-three surface-code logical qubits on a planar superconducting processor. During repeated syndrome extraction cycles, the logical qubits exhibit per-cycle error rates of 0.0365(2)0.0365(2)0.0365(2) and 0.0282(1)0.0282(1)0.0282(1), respectively, after leakage events are rejected. By leveraging joint initialization and lattice splitting, we deterministically prepare a logical Bell state, confirming genuine bipartite entanglement via the error-corrected logical state fidelity. We further execute a two-qubit Deutsch-Jozsa algorithm at the logical level to demonstrate algorithmic utility in a fault-tolerant framework. Finally, to achieve universal control, we implement magic-state injection and gate teleportation to realize continuous non-Clifford rotations about the logical XXX axis. For the logical RX(π/4)R_{X}(π/4)RX​(π/4) gate, we achieve a logical gate fidelity of 0.943−9+100.943_{-9}^{+10}0.943−9+10​ conditioned on the absence of detected errors. These results establish lattice surgery as a practical and versatile paradigm for logical computation in near-term surface-code architectures, representing a critical milestone toward scalable fault-tolerant quantum advantage in superconducting circuits.
FINDING: Surface code logical error rate scales exponentially with code distance (lattice size), enabling practical quantum error correction via topological order. | MATH: Logical error rate PL∝(Λ)−dP_L \propto (\Lambda)^{-d}PL​∝(Λ)−d, where ddd = code distance (linear lattice dimension), Λ=Pth/Pphys\Lambda = P_{th}/P_{phys}Λ=Pth​/Pphys​ (error suppression factor per round). Threshold Pth≈0.01P_{th} \approx 0.01Pth​≈0.01 (1% per qubit per cycle). Physical-to-logical overhead scales as O(d2)O(d^2)O(d2) qubits per logical qubit. | CONNECTION: The surface code is a **Z₂ toric code** — a lattice gauge theory on a square lattice with **crystallographic symmetry (p4m plane group)**. The logical operators are **non-contractible loops** on the torus (genus-1 surface), whose weight scales as ddd — directly tied to the **topological winding number** and the **fundamental group π1(T2)=Z×Z\pi_1(T^2) = \mathbb{Z} \times \mathbb{Z}π1​(T2)=Z×Z**. The threshold ~1% is not a golden ratio, but the **error suppression per distance** Λ\LambdaΛ often approaches values Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
For planar architectures, surface code-based quantum error correction is one of the most promising approaches to fault-tolerant quantum computation. This is partially due to the variety of fault-tolerant logical protocols that can be implemented in two dimensions using local operations. One such protocol is the lattice surgery-based logical state teleportation, which transfers a logical quantum state from an initial location on a quantum chip to a target location through a linking region of qubits. This protocol serves as a basis for higher-level routines, such as the entangling cnot gate or magic state injection. In this work, we investigate the correctability phase diagram of this protocol for distinct error rates inside the surface code patches and within the linking region. We adopt techniques from statistical physics to describe the numerically observed crossover regime between correctable and uncorrectable quantum error correction phases, where the correctability depends on the separation between the initial and target locations. We find that inside the crossover regime the correctability-threshold lines decay as a power law with increasing separation, which we explain accurately using a finite-size scaling analysis. Our results indicate that the logical state teleportation protocol can tolerate much higher noise rates in the linking region compared to the bulk of the surface code patches, provided the separation between the positions is relatively small.
The topological surface code is a leading candidate for harnessing long-range entanglement to protect logical quantum information against errors, and teleportation of logical states is desirable for robust quantum information processing. Nevertheless, scaling up the surface code in quantum teleportation poses a formidable challenge to experiment. Here on a superconducting quantum processor with 125 qubits, we demonstrate the robust teleportation of topological rotated surface code prepared by a linear-depth unitary circuit, with code distances up to 7. We obtain the teleportation phase diagram by tuning the local entangling gates uniformly across a finite threshold. Furthermore, we show that the entangling threshold can be boosted by coherent qubit rotations that inject magic resources beyond the Clifford regime, restoring the duality symmetry of the topological phase, which serves as a guiding principle to minimize the entanglement resource. Our results shed light on simulating and leveraging topological quantum matter on quantum devices, and pave the way to the ultimate goal of distributed fault tolerant quantum computation.