Unitary k -designs are distributions of unitary gates that match the Haar distribution up to its k th statistical moment. They are a crucial resource for randomized quantum protocols. However, their implementation on encoded logical qubits is nontrivial due to the need for magic gates, which can require a large resource overhead. In this work, we propose an efficient approach to generate unitary designs for encoded qubits in surface codes by applying local unitary rotations (“coherent errors”) on the physical qubits followed by syndrome measurement and error correction. We prove that, under some conditions on the coherent errors (notably including all single-qubit unitaries) and on the error-correcting code, this process induces a unitary transformation of the logical subspace. We numerically show that the ensemble of logical unitaries (indexed by the random syndrome outcomes) converges to a unitary design in the thermodynamic limit, provided that the density or strength of coherent errors is above a finite threshold. This “unitary design” phase transition coincides with the code’s coherent error threshold under optimal decoding. Furthermore, we propose a classical algorithm to simulate the protocol based on a “staircase” implementation of the surface code encoder and decoder circuits. This enables a mapping to a ( 1 + 1 ) -dimensional monitored circuit, where we observe an entanglement phase transition (and thus a classical complexity phase transition of the decoding algorithm) coinciding with the aforementioned unitary design phase transition. Our results provide a practical way to realize unitary designs on encoded qubits, with applications including quantum state tomography and benchmarking in error-correcting codes.
Unitary k -designs are distributions of unitary gates that match the Haar distribution up to its k th statistical moment. They are a crucial resource for randomized quantum protocols. However, their implementation on encoded logical qubits is nontrivial due to the need for magic gates, which can require a large resource overhead. In this work, we propose an efficient approach to generate unitary designs for encoded qubits in surface codes by applying local unitary rotations (“coherent errors”) on the physical qubits followed by syndrome measurement and error correction. We prove that, under some conditions on the coherent errors (notably including all single-qubit unitaries) and on the error-correcting code, this process induces a unitary transformation of the logical subspace. We numerically show that the ensemble of logical unitaries (indexed by the random syndrome outcomes) converges to a unitary design in the thermodynamic limit, provided that the density or strength of coherent errors is above a finite threshold. This “unitary design” phase transition coincides with the code’s coherent error threshold under optimal decoding. Furthermore, we propose a classical algorithm to simulate the protocol based on a “staircase” implementation of the surface code encoder and decoder circuits. This enables a mapping to a ( 1 + 1 ) -dimensional monitored circuit, where we observe an entanglement phase transition (and thus a classical complexity phase transition of the decoding algorithm) coinciding with the aforementioned unitary design phase transition. Our results provide a practical way to realize unitary designs on encoded qubits, with applications including quantum state tomography and benchmarking in error-correcting codes.
We investigate the stability of logical information in quantum stabilizer codes subject to coherent unitary errors. Beginning with a logical state, we apply a random unitary error channel and subsequently measure stabilizer checks, resulting in a syndrome-dependent post-measurement state. By examining both this \emph{syndrome state} and the associated syndrome distribution, we identify a phase transition in the behavior of the logical information. In the Clifford/stabilizer setting, the change of logical stabilizer structure in a syndrome branch determines the state-level MAP Pauli-frame return probability for a chosen logical-basis input. We separately use quantum coherent information to diagnose recoverability of arbitrary encoded inputs at the channel level. Below a critical error threshold pc, the syndrome branches remain compatible with the original logical sector, enabling a high state-return probability. Above pc, the data are consistent with effective scrambling of the encoded subspace relative to the stabilizer checks, where the syndrome-resolved logical maps approach a global-Clifford-induced stabilizer instrument. This common post-threshold picture organizes both the toric-code and finite-rate random-stabilizer-code results within the same scrambling mechanism. The syndrome distribution supplies complementary local and global structural diagnostics whose scaling depends on the code family. We refer to this phenomenon as a \emph{coherent error induced phase transition}. To illustrate this transition, we study two classes of quantum error-correcting codes, the toric code and finite-rate random stabilizer codes, thereby shedding light on the design and performance limits of quantum error correction under coherent errors.
The surface code, one of the leading candidates for quantum error correction, is known to protect encoded quantum information against stochastic, i.e., incoherent errors. The protection against coherent errors, such as from unwanted gate rotations, is however understood only for special cases, such as rotations about the X or Z axes. Here we consider generic single-qubit coherent errors in the surface code, i.e., rotations by angle α about an axis that can be chosen arbitrarily. We develop a statistical mechanical mapping for such errors and perform entanglement analysis in transfer matrix space to numerically establish the existence of an error-correcting phase, which we chart in a subspace of rotation axes to estimate the corresponding maximum-likelihood thresholds α th . The classical statistical mechanics model we derive is a random-bond Ising model with complex couplings and four-spin interactions (i.e., a complex-coupled Ashkin-Teller model). The error-correcting phase, α < α th , where the logical error rate decreases exponentially with code distance, is shown to correspond in transfer matrix space to a gapped one-dimensional quantum Hamiltonian exhibiting spontaneous breaking of a Z 2 symmetry. Our numerical results rest on two key ingredients: (i) we show that the state evolution under the transfer matrix, a nonunitary ( 1 + 1 ) -dimensional quantum circuit, can be efficiently numerically simulated using matrix product states; and (ii) based on this approach, we also develop an algorithm to (approximately) sample syndromes based on their Born probability. The α th values we find show that the maximum-likelihood thresholds for coherent errors are larger than those for the corresponding incoherent errors (from the Pauli twirl), and significantly exceed the values found using minimum weight perfect matching.
The maximum likelihood (ML) decoder in the two-dimensional surface code with generic unitary errors is governed by a statistical mechanics model with complex weights, which can be simulated via (1+1)D transfer matrix contraction. Information loss with an increasing error rate manifests as a ferromagnetic-to-paramagnetic transition in the contraction dynamics. In this work, we establish entanglement as a separate obstruction to decoding; it can undergo a transition from area- to volume-law scaling in the transfer matrix contraction with increasing unitary error rate. In particular, the volume-law entanglement can coexist with ferromagnetic order, giving rise to a phase in which the encoded information is retained yet is effectively undecodable. We numerically simulate the ML decoding in the surface code subject to both single- and two-qubit Pauli-X rotations and obtain a phase diagram that contains a ferromagnetic area-law, a paramagnetic volume-law, and a potential ferromagnetic volume-law phase. We further show that, starting from the paramagnetic volume-law phase, tilting the single-qubit rotation away from the X-axis couples the stat-mech models for X and Z errors and can lead to a ferromagnetic volume-law phase in which, although Z errors remain in principle correctable, the encoded classical information is hard to recover. To perform numerical simulations, we develop an algorithm for syndrome sampling based on an isometric tensor network representation of the surface code.
Quantum error correction (QEC) enables fault-tolerant quantum computation but requires operating quantum hardware at physical error rates below code-dependent thresholds, which remains challenging for current devices. We introduce syndrome resampling, a general method that increases QEC thresholds of any decoder and suppresses logical errors without additional hardware, decoding modifications, or code-specific assumptions beyond syndrome statistics. The method exploits the fact that syndromes with low probability are likely to lead to logical failure, therefore biasing syndrome averages towards most likely syndromes effectively increases logical fidelities. We establish a direct connection between the Rényi coherent information (RCI) and powers of the syndrome probability distribution, showing that resampling syndromes according to these powers combined with maximum likelihood decoding (MLD) realizes a family of optimal thresholds associated with phase transitions in the RCI. Numerical simulations of surface codes demonstrate that syndrome resampling substantially increases thresholds for both optimal and suboptimal decoders and reduces logical error rates by up to four orders of magnitude in experimentally relevant regimes. We further show that syndrome resampling can be effectively implemented from finite data and combined with decoding-based post-selection to achieve additional gains. Finally, applying the method to existing experimental QEC data yields up to two orders of magnitude reduction in logical error rates without requiring additional measurements. Our results provide a practical and decoder-agnostic route to improved logical fidelities in near-term QEC experiments.
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.
A quantum error correcting code protects encoded logical information against errors. Transversal gates are a naturally fault-tolerant way to manipulate logical qubits but cannot be universal themselves. Protocols such as magic state distillation are needed to achieve universality via measurements and postselection. A phase is a region of parameter space with smoothly varying large-scale statistical properties except at its boundaries. Here, we find a phase of continuously tunable logical unitaries for the surface code implemented by transversal operations and decoding that is robust against dephasing errors. The logical unitaries in this phase have an infidelity that is exponentially suppressed in the code distance compared to their rotation angles. We exploit this to design a simple fault-tolerant protocol for continuous-angle logical rotations. This lowers the overhead for applications requiring many small-angle rotations such as quantum simulation.