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...
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...
Learning faithful representations of quantum states is crucial to fully characterizing the variety of many-body states created on quantum processors. While various tomographic methods, such as classical shadow and matrix product state (MPS) tomography have shown promise in characterizing a wide class of quantum states, they face unique limitations in detecting topologically ordered two-dimensional states. To address this problem, we implement and study a heuristic tomographic method that combines variational optimization on tensor networks with randomized measurement techniques. Using this approach, we demonstrate its ability to learn the ground state of the surface-code Hamiltonian as well as an experimentally realizable quantum spin liquid state. In particular, we perform numerical experiments using MPS ansätze and systematically investigate the sample complexity required to achieve high fidelities for systems with sizes of up to 48 qubits. In addition, we provide theoretical insights into the scaling of our learning algorithm by analyzing the statistical properties of maximum-likelihood estimation. Notably, our method is sample-efficient and experimentally friendly, only requiring snapshots of the quantum state measured randomly in the X or Z bases. Using this subset of measurements, our approach can effectively learn any real pure states represented by tensor networks, and we rigorously prove that random- X Z measurements are tomographically complete for such states.
The ability of modern quantum simulators--both digital and analogue--to generate large ensembles of single-shot projective "snapshots" has opened a data-rich avenue for the study of quantum many-body systems. Unsupervised machine learning analysis of such snapshots has gained traction, with numerous works reconstructing phase diagrams by learning and clustering low-dimensional representations of quantum states. Here, we forgo such representation learning in favour of distance learning: we infer the pairwise distances between quantum states--already sufficient for clustering--directly from snapshots. Specifically, we use a single neural discriminator to estimate Csiszar f-divergences--statistical distances between distributions--in an unsupervised manner. The resulting clusters reveal regimes with different dominant correlations, often coinciding with, but not limited to, conventionally defined phases of matter. Beyond phase-diagram exploration, we connect the infinitesimal limit of the inferred divergences to the Fisher information metric and analyse its finite-size scaling. This yields critical exponents of the discovered transitions and enables snapshot-based analysis of universality classes. We apply distance learning to a diverse set of systems characterised by conventional local order parameters (1D transverse-field and 2D classical Ising models), non-local topological order (extended toric code), and higher-order correlations (fermionic t-J model on a triangular lattice). In all cases, we correctly recover boundaries between distinct correlation regimes and, where applicable, quantitatively match established critical behaviour. Finally, we show that distances to suitably chosen reference snapshot distributions help identify the dominant correlations within the discovered clusters, positioning distance learning as a versatile information-geometric probe of quantum many-body physics.
Learning faithful representations of quantum states is crucial to fully characterizing the variety of many-body states created on quantum processors. While various tomographic methods, such as classical shadow and matrix product state (MPS) tomography have shown promise in characterizing a wide class of quantum states, they face unique limitations in detecting topologically ordered two-dimensional states. To address this problem, we implement and study a heuristic tomographic method that combines variational optimization on tensor networks with randomized measurement techniques. Using this approach, we demonstrate its ability to learn the ground state of the surface-code Hamiltonian as well as an experimentally realizable quantum spin liquid state. In particular, we perform numerical experiments using MPS ansätze and systematically investigate the sample complexity required to achieve high fidelities for systems with sizes of up to 48 qubits. In addition, we provide theoretical insights into the scaling of our learning algorithm by analyzing the statistical properties of maximum-likelihood estimation. Notably, our method is sample-efficient and experimentally friendly, only requiring snapshots of the quantum state measured randomly in the 𝑋 or 𝑍 bases. Using this subset of measurements, our approach can effectively learn any real pure states represented by tensor networks, and we rigorously prove that random-𝑋𝑍 measurements are tomographically complete for such states.
Quantum computers are devices that can solve certain problems faster than ordinary, classical computers. The fundamental units of quantum information are qubits, superpositions of two states, a "zero" state and a "one" state. There are various approaches to construct such two-level systems, among others, using superconducting circuits, trapped ions or photons. A common feature of these physical systems is that their coherence times are relatively short compared to the length of useful computations. Superconducting qubits, for instance, are currently the most advanced solid-state qubits, but they decohere after around 100 microseconds, and any information stored in these qubits is lost. On the other hand, useful quantum computations may require quantum information to survive on time scales that are many orders of magnitude longer, as their runtimes can reach several hours or even days. Topological quantum computing is an approach to construct qubits that survive for the entire duration of such a long computation. Topological quantum computing comes in two flavors. The condensed matter approach is to build error-resilient qubits using exotic quasiparticles in topological materials, most prominently Majorana zero modes in topological superconductors. Even though no such qubit has been built to date, the hope is that their coherence times may be significantly longer than the coherence times of currently available solid-state qubits, but are still expected to be too short for large-scale quantum computing. The quantum information approach is to combine many error-prone qubits to build more robust logical qubits using topological error-correcting codes, e.g., surface codes. Even though the first approach is hardware-based and the second approach is software-based, they are deeply related. With Majorana-based qubits, the main logical operations are Majorana fermion parity measurements. By replacing Majorana-based qubits with surface-code patches and parity measurements with lattice-surgery operations, schemes for quantum computation with Majorana-based qubits or with surface codes can be identical. In this thesis, we explore how to construct a large-scale topological fault-tolerant quantum computer that can perform useful quantum computations. Here, topological refers to the nature of the quantum error-correcting code, while the underlying hardware may be based on non-topological qubits, but could also be composed of Majorana-based qubits. We provide a complete picture of such a large-scale device, breaking down large quantum computations into logical qubits and logical operations, describing how these logical operations are performed on the level of physical qubits and physical gates, and finally discussing how these physical qubits can be pieced together in a Majorana-based system using topological superconducting nanowires.
We present a versatile and mathematically rigorous technique for bounding recovery thresholds in topological codes subject to noise. Our method captures the effect of applying an arbitrary (possibly non-Pauli) local noise channel to the code state of a broad class of two-dimensional codes, including surface codes, non-Abelian quantum doubles, and string-net codes. In each case, we prove that for noise strengths up to some explicit constant value, any initially encoded logical information can be recovered to high precision, and that the noise-corrupted state exhibits key hallmarks of mixed-state topological order: long-range entanglement and emergent higher-form symmetries. We also describe how these methods can be adapted to higher dimensions and correlated noise models.
Quantum error correction protects quantum information against decoherence, provided the noise strength remains below a critical threshold. This threshold marks the critical point for the decoding phase transition. Here, we connect this transition in the toric code to a topological phase transition in disordered Majorana fermions at high temperatures. A quantum memory in the error-correctable phase is captured by the presence of a Majorana zero mode, trapped in vortex defects associated with twisted boundary conditions. These results are established by expressing the coherent information, which measures the amount of recoverable quantum information in a given noisy code, in terms of a mixed-state topological order parameter of fermions. Our work hints at a broader connection of the robustness of quantum information in stabilizer codes and mixed-state topological phase transitions in symmetry-protected fermion matter.