Teleportation is a facet where quantum measurements can act as a powerful resource in quantum physics, as local measurements allow us to steer quantum information in a nonlocal way. While this has long been established for a single Bell pair, the teleportation of a many-qubit entangled state using nonmaximally entangled resources presents a fundamentally different challenge. Here, we investigate a tangible protocol for teleporting a long-range entangled surface-code state using elementary Bell measurements and its stability in the presence of coherent errors that weaken the Bell entanglement. We relate the underlying threshold problem to the physics of anyon condensation under weak measurements and map it to a variant of the Ashkin-Teller model of statistical mechanics with Nishimori-type disorder, which gives rise to a cascade of phase transitions. Tuning the angle of the local Bell measurements, we find a continuously varying threshold. Notably, the threshold moves to infinity for the X + Z angle along the self-dual line—indicating that infinitesimally weak entanglement is sufficient in teleporting a self-dual topological surface code. Our teleportation protocol, which can be readily implemented in dynamically configurable Rydberg-atom arrays, thereby gives guidance for a practical demonstration of the power of quantum measurements. Published by the American Physical Society 2024
Teleportation is a facet where quantum measurements can act as a powerful resource in quantum physics, as local measurements allow us to steer quantum information in a nonlocal way. While this has long been established for a single Bell pair, the teleportation of a many-qubit entangled state using nonmaximally entangled resources presents a fundamentally different challenge. Here, we investigate a tangible protocol for teleporting a long-range entangled surface-code state using elementary Bell measurements and its stability in the presence of coherent errors that weaken the Bell entanglement. We relate the underlying threshold problem to the physics of anyon condensation under weak measurements and map it to a variant of the Ashkin-Teller model of statistical mechanics with Nishimori-type disorder, which gives rise to a cascade of phase transitions. Tuning the angle of the local Bell measurements, we find a continuously varying threshold. Notably, the threshold moves to infinity for the X + Z angle along the self-dual line—indicating that infinitesimally weak entanglement is sufficient in teleporting a self-dual topological surface code. Our teleportation protocol, which can be readily implemented in dynamically configurable Rydberg-atom arrays, thereby gives guidance for a practical demonstration of the power of quantum measurements. Published by the American Physical Society 2024
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.
For the surface code, topological quantum order allows one to encode logical quantum information in a robust, long-range entangled many-body quantum state. However, if an observer probes this quantum state by performing measurements on the underlying qubits, thereby collecting an ensemble of highly correlated classical snapshots, two closely related questions arise: (i) do measurements decohere the topological order of the quantum state; and (ii) how much of the logical information can one learn from the snapshots? Here we address these questions for measurements in a uniform basis on all qubits. We find that for generic measurement angles, sufficiently far away from the Clifford X, Y, and Z directions (such as the X+Y+Z basis) the logical information is never lost in one of the following two ways: (i) for weak measurement, the topological order is absolutely robust; (ii) for projective measurement, the quantum state inevitably collapses, but the logical quantum information is faithfully transferred from the quantum system to the observer in the form of a tomographically complete classical shadow. At these generic measurement angles and in the projective-measurement limit, the measurement ensemble enforced by Born probabilities can be represented by a 2D tensor network that can be fermionized into a disordered, free-fermion network model in symmetry class DIII, which gives rise to a Majorana "metal" phase. When the measurement angle is biased towards the X or Z limits, a critical angle indicates the threshold of a learning transition beyond which the classical shadow no longer reveals full tomographic information (but reduces to a measurement of the logical X or Z state). This learning transition can be described in the language of the network model as a "metal to insulator" transition...
Entanglement, measurement, and classical communication together enable teleportation of quantum states between distant parties, in principle, with perfect fidelity. To what extent do correlations and entanglement of a many-body wave function transfer under teleportation protocols? We address this question for the case of an imperfectly teleported quantum critical wave function, focusing on the ground state of a critical Ising chain. We demonstrate that imperfections, e.g., in the entangling gate adopted for a given protocol, effectively manifest as weak measurements acting on the otherwise pristinely teleported critical state. Armed with this perspective, we leverage and further develop the theory of measurement-altered quantum criticality to quantify the resilience of critical-state teleportation. We identify classes of teleportation protocols for which imperfection (i) preserves both the universal long-range entanglement and correlations of the original quantum critical state, (ii) weakly modifies these quantities away from their universal values, and (iii) obliterates long-range entanglement altogether while preserving power-law correlations, albeit with a new set of exponents. We also show that mixed states describing the average over a series of sequential imperfect teleportation events retain pristine power-law correlations due to a “built-in” decoding algorithm, though their entanglement structure measured by the negativity depends on errors similarly to individual protocol runs. These results may allow one to design teleportation protocols that optimize against errors—highlighting a potential practical application of measurement-altered criticality. Published by the American Physical Society 2024
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.
Nishimori Threshold Estimation for Bayesian Inference and Zq Surface Code Decoding
In quantum error correction, the error threshold provides essential quantitative guidance for the ability to bring about fault-tolerance through decoding the effects of incoherent noise, weak measurement or inference. However, the numerical value of an error threshold is typically only accessible through large-scale numerical simulations of the underlying noise model. Here we introduce an analytical estimate of error thresholds falling into the Nishimori universality class via a Fourier--Walsh projection scheme that maps the critical point of the underlying disorder-free statistical-mechanics model to the Born-disordered Nishimori critical point. Using a minimal replica theory approach, this closed-form estimate is obtained from a projection of the exact replicated single-bond weight which we find to reproduce (within a percentage point) the known numerical thresholds of random-bond and random-plaquette Ising models / Z2 stabilizer codes in spatial dimensions d=2−5, and extends to Potts variables with q≤4. The main application of our projection scheme is to Zq surface codes, whose decoding problem maps to the disordered q-state clock model. For q≥5 the clean clock model has \textit{two} Berezinskii--Kosterlitz--Thouless transitions, which the projection maps to two Nishimori temperatures that bound an intermediate information-critical phase. The resulting threshold values not only accurately agree with recent decohered-Zq-toric-code numerics, but are found to satisfy the Gilbert--Varshamov self-dual entropy relation lnq≃Hq(T1∗)+Hq(T2∗), although no duality condition is imposed in the construction. Our approach thereby points to a deeper connection between the clean and Born-disordered models, while allowing for instant analytical estimates of error thresholds for a variety of stabilizer codes.
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≈0.013, 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 pc≈0.014>pth, a physically significant finite-size effect relevant to near-term devices. Causality enforces an irreducible quadratic infrastructure cost Wbulk∝N2, strictly satisfying the second law at all separations and defining a fundamental thermodynamic horizon Nmax≈78 beyond which positive net work extraction is impossible regardless of code distance or decoder quality.