We introduce a classification of quantum phases of mixed states based on the ability to preserve and fault tolerantly transfer information. We formulate phase equivalence between sets of states that encode logical quantum information and are connected by shallow local channel circuits assisted by local measurements and global classical communication of measurement outcomes, where all quantum operations, including measurements, must be robust to noise. We show that the existence of a recovery threshold for local noise is a property of an entire phase. We prove that any two sets of states with nonzero thresholds that are related by finite-depth local channel circuits belong to the same fault tolerant phase. Our approach leads to a classification of subsystem codes without favoring any state of the gauge subsystem, and we demonstrate phase equivalence between 2D and 3D color codes using dimensional jumps within the 3D gauge color code. Lastly, we study how syndrome structure affects fault tolerance in circuits with measurements. We show that qudit stabilizer codes with extended syndromes permit single-shot state preparation, whereas a broad class of shallow circuits measuring point-like syndromes, including state preparation for 2D topological codes and non-Abelian topological orders with solvable anyon theories, is not fault tolerant.
We introduce a classification of quantum phases of mixed states based on the ability to preserve and fault tolerantly transfer information. We formulate phase equivalence between sets of states that encode logical quantum information and are connected by shallow local channel circuits assisted by local measurements and global classical communication of measurement outcomes, where all quantum operations, including measurements, must be robust to noise. We show that the existence of a recovery threshold for local noise is a property of an entire phase. We prove that any two sets of states with nonzero thresholds that are related by finite-depth local channel circuits belong to the same fault tolerant phase. Our approach leads to a classification of subsystem codes without favoring any state of the gauge subsystem, and we demonstrate phase equivalence between 2D and 3D color codes using dimensional jumps within the 3D gauge color code. Lastly, we study how syndrome structure affects fault tolerance in circuits with measurements. We show that qudit stabilizer codes with extended syndromes permit single-shot state preparation, whereas a broad class of shallow circuits measuring point-like syndromes, including state preparation for 2D topological codes and non-Abelian topological orders with solvable anyon theories, is not fault tolerant.
Quantum computation promises to revolutionize information processing by exploiting superposition and entanglement, enabling applications such as quantum simulation, combinatorial optimization, and secure quantum communication. Beyond individual processors, the quantum internet aims to connect quantum nodes through entanglement distribution, enabling distributed computation. However, quantum states are fragile: environmental interactions, imperfect gate operations, and measurement errors can quickly degrade encoded information. Reliable quantum computation therefore requires carefully designed quantum error-correcting codes and fault-tolerant protocols. Unlike classical redundancy schemes, quantum error correction must preserve delicate phase relationships and relies on indirect error detection through syndrome extraction with ancillary qubits, enabling error identification without collapsing encoded states. Topological codes, in particular, emerged early as the preferred choice for many architectures due to their planar structure, local interactions, and natural support for fault-tolerant logical operations. Color codes, for instance, allow transversal implementation of all Clifford gates, whereas non-Clifford operations require magic-state injection. More recent constructions, including quantum low-density parity-check codes, offer higher encoding rates and reduced resource overhead, but require long-range qubit connectivity and more complex protocols. Despite experimental progress, several challenges remain. Many platforms exhibit biased noise, motivating codes that exploit channel asymmetry. Real-time decoding remains a critical bottleneck, as fault-tolerant computation requires low-latency processing of rapidly generated syndrome data. Fault-tolerant logical state initialization also remains resource-intensive and scales poorly with code distance. Addressing these challenges, this thesis develops a unified framework spanning theory and experiment. It provides precise performance analysis of quantum error-correcting codes in the low physical error-rate regime, introduces codes tailored to biased noise, develops fast low-complexity decoding algorithms for real-time operation, and presents modular and efficient protocols for fault-tolerant state preparation of CSS codes. The latter is demonstrated experimentally through the preparation of a logical zero state of the Golay code on the Quantinuum H2 trapped-ion quantum computer.
Reliable execution of large-scale quantum algorithms requires robust underlying operations, which is addressed by quantum error correction (QEC). Most modern QEC protocols rely on measurements and feed-forward operations, which are experimentally demanding and often prone to high error rates. Additionally, no single-error-correcting code intrinsically supports the full set of logical operations required for universal quantum computing. In this work, we present a complete toolbox for fault-tolerant universal quantum computing without measurements during algorithm execution by combining the strategies of code switching and concatenation. We develop fault-tolerant, measurement-free protocols to transfer encoded information between 2D and 3D color codes that offer complementary and, in combination, universal sets of robust logical gates. Moreover, we extend the scheme to higher-distance codes by concatenating the 2D color code and integrating code switching for operations lacking a natively fault-tolerant implementation. Our measurement-free approach thereby provides a practical and scalable pathway for universal quantum computing on state-of-the-art quantum processors.
Noise is one of the central obstacles to building useful quantum computers, and quantum error correction (QEC) provides the framework for protecting quantum information against it. Unlike classical error correction, QEC must preserve fragile quantum states without copying them, measuring them directly, or destroying the information they encode. Driven by rapid progress in both theory and experiment, this challenge has grown into one of the most active areas of quantum information science. This tutorial gives a guided introduction to modern QEC, developing the core concepts of codes, syndromes, stabilizers, decoding, and fault tolerance before connecting them to major code families and current research directions. We cover both established constructions and newer developments, including topological and subsystem codes, bosonic and qudit codes, dynamical codes, and quantum low-density parity-check (qLDPC) codes. The emphasis is on building operational understanding: explaining not only what the main objects are, but how they are used in code design, error diagnosis, decoding, and fault-tolerant computation. The tutorial is intended for newcomers seeking a first path through QEC, as well as researchers looking for a coherent reference for the concepts, code families, and tools that arise in current work.
For closed quantum systems, topological orders are understood through the equivalence classes of ground states of gapped local Hamiltonians. The generalization of this conceptual paradigm to open quantum systems, however, remains elusive, often relying on operational definitions without fundamental principles. Here, we fill this gap by proposing an approach based on three axioms: (i) local recoverability, (ii) absence of long-range correlations, and (iii) spatial uniformity. States that satisfy these axioms are fixed points; requiring the axioms only after coarse-graining promotes each fixed point to an equivalence class, i.e., a phase, presenting the first step toward the axiomatic classification of mixed-state phases of matter: the . From these axioms, a rich set of topological data naturally emerges; importantly, these data are under relaxation of axioms. For example, each topological mixed state supports locally indistinguishable classical and/or quantum logical memories with distinct responses to topological operations. These data label distinct mixed-state phases, allowing one to distinguish them. We further uncover a hierarchy of secret-sharing constraints: In non-Abelian phases, reliable recovery—even of information that looks purely classical—demands a specific coordination among spatial subregions, a requirement different across non-Abelian classes. This originates from non-Abelian fusion rules that can stay robust under decoherence. Finally, we performed large-scale numerical simulations to corroborate stability: Weakly decohered fixed points respect the axioms once coarse-grained. Surprisingly, the axiom violations at different length scales exhibit an exact crossing at the critical point, enabling its identification without any rescaling or data-collapse ambiguities. These results lay the foundation for a systematic classification of topological states in open quantum systems.
We prove a fault-tolerance theorem for quantum computation against adversarial noise. For every quantum circuit on Nˉ logical qudits of depth Tˉ, we construct a fault-tolerant circuit on N=poly(Nˉ) physical qudits of depth Tˉ⋅Nˉo(1), which is robust against an adversary who may arbitrarily choose and corrupt an almost-linear number N1−o(1) of physical qudits at each time step. This robustness significantly improves upon prior fault-tolerance theorems, which assumed corruptions were either local and stochastic, or else only act on a polynomially vanishing fraction of qudits.
Our fault-tolerance scheme addresses a key bottleneck towards constructing quantum PCPs via the circuit-to-Hamiltonian mapping of Anshu, Breuckmann, and Nguyen (STOC'24). More fundamentally, our result demonstrates that fault-tolerant quantum computation remains possible under noise models that are global, worst-case, and non-Markovian over the full duration of the computation, directly countering concerns that correlated noise could fundamentally undermine quantum fault tolerance.
Our construction is based on a new family of subsystem product codes we develop, which have large dimension and distance along with low-weight parity-checks, and which support transversal non-Clifford gates. We show how to perform single-shot fault-tolerant error correction on these codes using a Floquet-like procedure based on the local testability of classical tensor codes. We then obtain a universal fault-tolerance scheme using repeated code switching in a hypercubic qudit architecture. Finally, we recursively compose our scheme with itself to reduce an initially exponential qudit dimension down to a constant.
Color codes are a promising topological code for fault-tolerant quantum computing. Insufficient research on color codes has delayed their practical application. In this work, we address several key issues to facilitate practical fault-tolerant quantum computing based on color codes. First, by introducing decoding graphs with error-rate-related weights, we obtain the threshold of 0.47 % of the 6.6.6 triangular color code under the standard circuit-level noise model, narrowing the gap to that of the surface code. Second, our work first investigates the circuit-level decoding of color code lattice surgery, then gives an efficient decoding algorithm, which is crucial to perform logical operations in a quantum computer with two-dimensional architectures. Last, a state injection protocol of the triangular color code is proposed, reducing the output magic state error rate in one round of 15 to 1 distillation by two orders of magnitude compared to a previous rough protocol. We also prove that our protocol offers the lowest logical error rates for state injection among all possible codes. Published by the American Physical Society 2024