Finn Eckstein of the University of Cologne and The Hong Kong University of Science and Technology (Guangzhou), along with colleagues, found that information is either preserved through robust topological order with weak measurement or faithfully transferred to an observer as a “tomographically complete classical shadow” during projective measurement. This occurs for measurement angles sufficiently far from the standard Clifford X, Y, and Z directions. The work addresses fundamental questions of whether measurements destroy topological order and how much information can be extracted from the resulting data.

Surface Code Topology Protects Logical Quantum Information

Specifically, for weak measurement, the topological order is absolutely robust. However, even when employing projective measurements, which inevitably collapse the quantum state, the logical information is faithfully transferred from the quantum system to the observer in the form of a tomographically complete classical shadow. The researchers demonstrate that the resulting measurement ensemble, dictated by Born probabilities, can be elegantly represented as a two-dimensional tensor network.

This network, in turn, can be transformed into a disordered, free-fermion network model belonging to symmetry class DIII, resulting in a unique “Majorana metal” phase. This mathematical mapping provides a deeper understanding of how information is encoded and transferred during the measurement process and offers a new lens through which to view the interplay between quantum and classical information.

The diagnostic for determining whether information is retained relies on the correlation between a reference qubit and the observer, with the quantum degrees of freedom within the surface code itself being discarded during the calculation. This approach allows for a focused assessment of information transfer, independent of the internal dynamics of the code. The coherent information, a key metric in this analysis, quantifies the amount of quantum information remaining within the code after measurement.

A negative value would indicate a loss of both quantum and classical information, but the researchers found this was not the case under the conditions studied. “Therefore the phase diagram depends essentially only on the ensemble of the reference states P(s), $ρ_{R} (s)$ ,” the paper states, highlighting the importance of the initial state of the reference qubit in determining the outcome.

Measurement Basis Impacts Topological Order Preservation

Measurements performed on a topological surface code do not necessarily destroy the encoded quantum information, even when those measurements force a collapse of the quantum state. This outcome challenges the conventional understanding of measurement as a disruptive process in quantum systems, as the surface code’s topological order remains robust under specific conditions.

The research details two distinct scenarios. Weak measurements maintain the topological order entirely, while projective measurements transfer the logical quantum information to an observer. Analysis of the measurement ensemble, dictated by Born probabilities, reveals a surprising mathematical structure. The extent of topological order preservation is also linked to a critical point that shifts as the measurement basis changes.

Around specific angles, a maximal infinite threshold is found, indicating a substantial region where the code remains stable against measurement. This threshold is particularly pronounced when adding a Y-component to the measurement basis, causing a critical point to transition into a line with volume-law entanglement entropy scaling in its 1D boundary state. This metric is an efficient order parameter for diagnosing measurement-induced phase transitions and topological quantum memory.

This finding suggests that the phase diagram of the system depends primarily on the ensemble of reference states, rather than the specific internal dynamics of the qubits. The work demonstrates that the projective-measurement limit allows for a non-interacting fermion model, applicable for arbitrary measurement angles.

This simplification facilitates analysis of the information transfer process and provides a deeper understanding of the underlying physics. The team’s findings suggest that the surface code’s ability to maintain logical information, even during measurement, is not merely a consequence of robust encoding but also a result of the specific mathematical properties of the measurement basis and the resulting classical shadow.

Weak Measurements Maintain Robust Topological Order

This outcome challenges the expectation that measurement invariably leads to decoherence, suggesting a more detailed relationship between observation and quantum state disturbance. This threshold is particularly notable, suggesting a considerable area where the surface code can reliably maintain logical information even under measurement.

The team’s analysis extends beyond qubit measurements to include syndrome measurements, revealing that even these can trigger a collapse of the topological order and enable the observer to learn the logical information, provided the measurement strength exceeds a certain threshold. This learnability threshold, described as a “metal to insulator” transition in the context of network models, indicates a critical point beyond which the surface code’s protective properties break down. For generic measurement angles, the surface code exhibits robustness against even strong, non-projective measurements.

Critical Angle Defines Learning Transition Threshold

Measurements at angles deviating from standard Clifford directions reveal a critical threshold governing information transfer in surface codes, with the measurement ensemble describable as a two-dimensional tensor network. The emergence of this phase indicates a fundamental connection between measurement, topology and the underlying quantum state.

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. The coherent information, a measure of the logical qubit’s integrity, exhibits a distinct behavior around this threshold, starting at log 2 in the unperturbed surface code phase and decreasing to zero in the projective-measurement limit.

Researchers observed a crossing point in the coherent information at a critical measurement strength, denoted as $t_{c}$ , for a measurement angle of 0.1π, close to the X point in the XY plane. Boundary entanglement entropy, calculated along a one-dimensional chain of qubits attached to the surface code, shows a peak approaching the transition point with log(L) scaling, a characteristic of (1+1)D quantum critical points.

In the “thermodynamic limit” where the code distance approaches infinity, the coherent information converges to a step function, maintaining one bit of information as long as the measurement strength remains below the threshold. Determining the precise threshold requires finite-size scaling, allowing researchers to pinpoint the location of the learning phase transition. This precise determination is essential for understanding the limits of quantum information extraction and the robustness of topological codes.

The topological phase exhibits spontaneous symmetry breaking of underlying 1-form symmetries, while the dephased phase displays a strong-to-weak symmetry breaking of the same symmetries. This symmetry breaking behavior provides further insight into the nature of the transition and the underlying physics governing the system. The boundary entanglement entropy serves as another diagnostic tool to detect and characterize the critical theory of the learning transition, offering a complementary perspective to the coherent information analysis.

Syndrome Measurements Enable Logical Information Learning

Syndrome measurements, traditionally employed to detect and correct errors, can also enable an observer to learn the logical information encoded within a topological surface code, a finding that challenges conventional understanding of measurement’s role in quantum systems. This learning occurs even as the topological order of the quantum state undergoes a transition, demonstrating a surprising resilience in the face of probing. The work reveals that the observer doesn’t necessarily destroy the information, but rather transforms it into a form they can access.

The transition from preserving topological order to enabling information learning is analogous to a “metal to insulator” shift within a network model, with a defined threshold governing the process. Below this threshold, the surface code remains absolutely robust to measurement, retaining its encoded information. However, exceeding this threshold, defined by measurement strength, allows the observer to extract logical information, even if it means collapsing the original quantum state.

This collapse doesn’t equate to information loss; instead, the logical information is faithfully transferred as a “tomographically complete classical shadow,” a representation containing all the necessary data to reconstruct the original quantum state. This phenomenon is distinct from scenarios involving incoherent noise, where information is irrevocably lost to the environment. In the case of coherent errors, the logical information isn’t entirely erased but is partially reconstructed by the observer through syndrome measurements.

This challenges the typical assumption that the syndrome space, designed to reveal noise, remains entirely orthogonal to the logical space, preserving the quantum state. This readout, however, comes at the cost of collapsing the quantum code, losing the complementary information. The findings extend beyond simple qubit measurements, applying equally to syndrome measurements, traditionally used for quantum error correction. The researchers note that the surface code’s ability to maintain logical information, even during measurement, is a significant step toward realizing fault-tolerant quantum computation.

👉 More information
🗞 Learning transitions of topological surface codes
✍️ Finn Eckstein, Bo Han, Simon Trebst and Guo-Yi Zhu
🧠 DOI: http://link.aps.org/doi/10.1103/tv5x-scmc

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