Quantum error correction plays a critical role in enabling fault-tolerant quantum computing by protecting fragile quantum information from noise. While general-purpose quantum error correction codes are designed to address a wide range of noise types, they often require substantial resources, making them impractical for near-term quantum devices. Approximate quantum error correction provides an alternative by tailoring codes to specific noise environments, reducing resource demands while still maintaining noise-robustness. Dynamical codes, including Floquet codes, introduce a dynamic approach to quantum error correction, employing time-dependent operations to stabilize logical qubits. In this work, we combine the flexibility of dynamical codes with the versatility of approximate quantum error correction to offer a promising avenue for addressing dominant noise in quantum systems. We construct several approximate dynamical codes using the recently developed strategic code framework. As a special case, we recover the approximate static codes widely studied in the existing literature. By analyzing these approximate dynamical codes through semidefinite programming, we establish the uniqueness and robustness of the optimal encoding, decoding, and check measurements. We also develop a temporal Petz recovery map suited to approximate dynamical codes.
Quantum error correction plays a critical role in enabling fault-tolerant quantum computing by protecting fragile quantum information from noise. While general-purpose quantum error correction codes are designed to address a wide range of noise types, they often require substantial resources, making them impractical for near-term quantum devices. Approximate quantum error correction provides an alternative by tailoring codes to specific noise environments, reducing resource demands while still maintaining noise-robustness. Dynamical codes, including Floquet codes, introduce a dynamic approach to quantum error correction, employing time-dependent operations to stabilize logical qubits. In this work, we combine the flexibility of dynamical codes with the versatility of approximate quantum error correction to offer a promising avenue for addressing dominant noise in quantum systems. We construct several approximate dynamical codes using the recently developed strategic code framework. As a special case, we recover the approximate static codes widely studied in the existing literature. By analyzing these approximate dynamical codes through semidefinite programming, we establish the uniqueness and robustness of the optimal encoding, decoding, and check measurements. We also develop a temporal Petz recovery map suited to approximate dynamical codes.
Quantum error correcting (QEC) plays a crucial role in protecting quantum information against decoherence and enabling scalable, reliable quantum computing. One of the most realistic and ubiquitous sources of noise affecting quantum hardware today is generalized amplitude-damping (GAD) noise. Conventional, deterministic QEC codes struggle to correct for GAD noise because of their inherent structure, leading to fidelity losses that scale linearly with the damping strength. In this work, we introduce the framework of probabilistic approximate quantum error correction (PAQEC), that combines the flexibility of approximate QEC with the potential of post-selected recovery, enabling high-fidelity, resource-efficient error correction. We construct a five-qubit permutation-invariant code that, under probabilistic recovery, achieves a fidelity loss quadratic in the damping strength, thus outperforming existing QEC codes. Formulating PAQEC as an optimization problem, we present a numerical technique based on Charnes-Cooper and semidefinite programming to identify the optimal recovery map for any PAQEC code. Our results establish PAQEC as a powerful tool for developing resource-efficient, high-fidelity quantum codes tailored to realistic noise, with promising implications for near-term quantum devices and future fault-tolerant architectures.
A central goal in quantum error correction is to reduce the overhead of fault-tolerant quantum computing by increasing noise thresholds and reducing the number of physical qubits required to sustain a logical qubit. We introduce a potential path towards this goal based on a family of dynamically generated quantum error correcting codes that we call "hyperbolic Floquet codes.'' These codes are defined by a specific sequence of non-commuting two-body measurements arranged periodically in time that stabilize a topological code on a hyperbolic manifold with negative curvature. We focus on a family of lattices for n qubits that, according to our prescription that defines the code, provably achieve a finite encoding rate ( 1 / 8 + 2 / n ) while still requiring only two-body measurements. Similar to hyperbolic surface codes, the distance of the code at each time-step scales at most logarithmically in n . The family of lattices we choose indicates that this scaling is achievable in practice. We develop and benchmark an efficient matching-based decoder that provides evidence of a threshold near 0.1% in a phenomenological noise model and 0.25% in an entangling measurements noise model. Utilizing weight-two check operators and a qubit connectivity of 3, one of our hyperbolic Floquet codes uses 400 physical qubits to encode 52 logical qubits with a code distance of 8, i.e., it is a [ [ 400 , 52 , 8 ] ] code. At small error rates, comparable logical error suppression to this code requires 5x as many physical qubits (1924) when using the honeycomb Floquet code with the same noise model and decoder.
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.
Quantum error correction (QEC) protects quantum systems against inevitable noises and control inaccuracies, providing a pathway toward fault-tolerant (FT) quantum computation. Stabilizer codes, including surface code and color code, have long been the focus of research and have seen significant experimental progress in recent years. Recently proposed time-dynamical QEC, including Floquet codes and generalized time-dynamical code implementations, opens up new opportunities for FT quantum computation. By employing a periodic schedule of low-weight parity checks, Floquet codes can generate additional dynamical logical qubits, offering enhanced error correction capabilities and, potentially, higher code performance. Here, we experimentally implement the Floquet-Bacon-Shor code on a superconducting quantum processor. We encode a dynamical logical qubit within a 3×3 lattice of data qubits alongside a conventional static logical qubit. We demonstrate FT encoding and measurement of the two-qubit logical states and stabilize these states using repeated error detection. We showcase universal single-qubit logical gates on the dynamical qubit. Furthermore, by implementing a logical controlled-not (cnot) gate, we entangle the dynamical and static logical qubits, generating an error-detected logical Bell state with a fidelity of 75.9%. Our results highlight the potential of Floquet codes for resource-efficient FT quantum computation.
Quantum error-correcting code (QECC) is the central ingredient in fault-tolerant quantum information processing. An emerging paradigm of dynamical QECC shows that one can robustly encode logical quantum information both temporally and spatially in a more resource-efficient manner than traditional QECCs. Nevertheless, an overarching theory of how dynamical QECCs achieve fault-tolerance is lacking. In this work, we bridge this gap by proposing a unified spatio-temporal QECC framework called the ``strategic code'' built around an ``interrogator'' device which sequentially measures and evolves the spatial QECC in an adaptive manner based on the ``quantum combs'' formalism, a generalization of the channel-state duality. The strategic code covers all existing dynamical and static QECC, as well as all physically plausible QECCs to be discovered in the future, including those that involve adaptivity in its operational dynamics. Within this framework, we show an algebraic and an information-theoretic necessary and sufficient error-correction conditions for a strategic code, which consider spatially and temporally correlated errors. These conditions include the analogous known static QECC conditions as a special case. Lastly, we also propose an optimization-theoretic approach to obtain an approximate strategic code adapting to a correlated error.
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.