The development of prototype quantum information processors has progressed to a stage where small instances of logical qubit systems perform better than the best of their physical constituents. Advancing toward fault-tolerant quantum computing will require an understanding of the underlying error mechanisms in logical primitives as they relate to the performance of quantum error correction. In this work we demonstrate the novel capability to characterize the physical error properties relevant to fault-tolerant operations via cycle error reconstruction. We illustrate this diagnostic capability for a transversal controlled-NOT (CNOT) gate, a prototypical component of quantum logical operations, in a 16-qubit register of a trapped-ion quantum computer. Our error characterization technique offers three key capabilities: (i) identifying context-dependent physical layer errors, enabling their mitigation, (ii) contextualizing component gates in the environment of logical operators, validating the performance differences in terms of characterized component-level physics, and (iii) providing a scalable method for predicting quantum error correction performance using pertinent error terms, differentiating correctable versus uncorrectable physical layer errors. The methods with which our results are obtained have scalable resource requirements that can be extended with moderate overhead to capture overall logical performance in increasingly large and complex systems.

Magic-state injection is a standard route to realize universal fault-tolerant quantum computation. Whereas the set of Clifford gates in combination with the non-Clifford T gate is a widely used universal gate set, extending the available set of non-Clifford primitives can reduce compilation overhead, provided that the additional primitives can be prepared fault-tolerantly with competitive resource costs and at sufficiently low logical noise rates. In this work, we introduce flag fault-tolerant protocols for preparing logical |√(T)⟩ magic states on the 3D tetrahedral color code and its smaller morphed variant. Our simulations under circuit-level noise verify fault tolerance, quantify acceptance and logical error rates, and we reconstruct the effective logical channels of the corresponding circuits for gate injection via logical process tomography. We find that access to √(T) reduces the average space-time cost of synthesizing Haar-random single-qubit unitaries by approximately 20-30

We introduce the random coupled-plaquette gauge model (RCPGM), which enables optimal accounting for Y-errors in decoding a surface code with noisy syndrome measurements. Using Parallel Tempering Monte Carlo simulations, we determine the code’s fundamental error thresholds. For phenomenological depolarizing data and bit-flip syndrome noise we determine a threshold of 6%, to be compared to the “uncoupled” random plaquette gauge model (RPGM) with a mere 4.3%. We then tackle the circuit-level noise scenario, where an approximate reduction technique allows us to exploit the RCPGM. Within the assumptions of the reduction technique, we find a threshold of 1.4%, to be compared to 0.7% when marginalizing Y-errors for the “uncoupled” RPGM. These results crucially enlarge the landscape of statistical mechanical mappings for quantum error correction. In particular they show further room for improvement of the surface code for fault-tolerant quantum computation and should be highly encouraging for practical decoder development.

Qubit shuttling promises to advance some quantum computing platforms to the qubit register sizes needed for effective quantum error correction (QEC), but also introduces additional errors whose impact must be evaluated. The established method to investigate the performance of QEC codes in a realistic scenario is to employ a standard noise model known as circuit-level noise, where all quantum operations are modeled as noisy. In the present work, we take this noise model and single out the effect of shuttling errors by introducing them as an additional so-called error location. This hardware abstraction is motivated by the SpinBus architecture and allows a systematic numerical investigation to map out the resulting two-dimensional parameter space. To this end, we take the Surface code and perform large scale simulations, most notably extracting the threshold across said two-dimensional parameter space. We study two scenarios for shuttling errors, depolarization on the one hand and dephasing on the other hand. For a purely dephasing shuttling error, we find a threshold of several percent, provided that all other operations have a high fidelity. The qubit overhead needed to reach a logical error rate of 10^-12 (known as the "teraquop" regime ) increases only moderately for shuttling error rates up to about 1 % per shuttling operation. The error rates at which practically useful, i.e. well below threshold error correction is predicted to be possible are comfortably higher than what is expected to be achievable for spin qubits. Our results thus show that it is reasonable to expect shuttling operations to fall below threshold already at surprisingly large error rates. With realistic efforts in the near term, this offers positive prospects for spin qubit based quantum processors as a viable avenue for scalable fault-tolerant error-corrected quantum computing.

Analyzing and developing new quantum error-correcting (QEC) schemes is one of the most prominent tasks in quantum computing research. In such efforts, introducing time dynamics explicitly in both analysis and design of error-correcting protocols constitutes an important cornerstone. In this work, we present a graphical formalism based on tensor networks to capture the logical action and error-correcting capabilities of any Clifford circuit with Pauli measurements. We showcase the functioning of the formalism on new Floquet codes derived from topological subsystem codes, which we call XYZ ruby codes. Based on the projective symmetries of the building blocks of the tensor network we develop a framework of Pauli flows. Pauli flows allow for a graphical understanding of all quantities entering an error-correction analysis of a circuit, including different types of QEC experiments, such as memory and stability experiments. We lay out how to derive a well-defined decoding problem from the tensor-network representation of a protocol and its Pauli flows alone, independent of any stabilizer code or fixed circuit. Importantly, this framework applies to all Clifford protocols and encompasses both measurement-based and circuit-based approaches to fault tolerance. We apply our method to our new family of dynamical codes, which are in the same topological phase as the 2+1-dimensional color code, making them a promising candidate for low-overhead logical gates. In contrast to its static counterpart, the dynamical protocol applies a Z_{3} automorphism to the logical Pauli group every three time steps. We highlight some of its topological properties and comment on the anyon physics behind a planar layout. Lastly, we benchmark the performance of the XYZ ruby code on a torus by performing both memory and stability experiments and find competitive circuit-level noise thresholds of approximately equal to 0.18%, comparable with other Floquet codes and 2+1-dimensional color codes.

Encoding information redundantly using quantum error-correcting (QEC) codes allows one to overcome the inherent sensitivity to noise in quantum computers to ultimately achieve large-scale quantum computation. The Steane QEC method involves preparing an auxiliary logical qubit of the same QEC code used for the data register. The data and auxiliary registers are then coupled with a logical CNOT gate, enabling a measurement of the auxiliary register to reveal the error syndrome. This study presents the implementation of multiple rounds of fault-tolerant Steane QEC on a trapped-ion quantum computer. Various QEC codes are employed, and the results are compared to a previous experimental approach utilizing flag qubits. Our experimental findings show improved logical fidelities for Steane QEC. This establishes experimental Steane QEC as a competitive paradigm for fault-tolerant quantum computing.

We map the decoding problem of the surface code under depolarizing and syndrome noise to a disordered spin model, which we call the random coupled-plaquette gauge model (RCPGM). By coupling X- and Z-syndrome volumes, this model allows us to optimally account for genuine Y-errors in the surface code in a setting with noisy measurements. Using Parallel Tempering Monte Carlo simulations, we determine the code's fundamental error threshold. Firstly, for the phenomenological noise setting we determine a threshold of 6% under uniform depolarizing and syndrome noise. This is a substantial improvement compared to results obtained via the previously known "uncoupled" random plaquette gauge model (RPGM) in the identical setting, where marginalizing Y-errors leads to a threshold of 4.3%. Secondly, we tackle the circuit-level noise scenario, where we use a reduction technique to find effective asymmetric depolarizing and syndrome noise rates to feed into the RCPGM mapping. Despite this reduction technique breaking up some of the correlations contained in the intricacies of circuit-level noise, we find an improvement exceeding that for the phenomenological case. We report a threshold of up to 1.4%, to be compared to 0.7% under the identical noise model when marginalizing the Y-errors and mapping to the anisotropic RPGM. These results enlarge the landscape of statistical mechanical mappings for quantum error correction. In particular they provide an underpinning for the broadly held belief that accounting for Y-errors is a major bottleneck in improving surface code decoders. This is highly encouraging for leading efficient practical decoder development, where heuristically accounting for Y-error correlations has seen recent developments such as belief-matching. This suggests that there is further room for improvement of the surface code for fault-tolerant quantum computation.

Topological color codes are widely acknowledged as promising candidates for fault-tolerant quantum computing. Neither a two-dimensional nor a three-dimensional topology, however, can provide a universal gate set $\{$H, T, CNOT$\}$, with the T-gate missing in the two-dimensional and the H-gate in the three-dimensional case. These complementary shortcomings of the isolated topologies may be overcome in a combined approach, by switching between a two- and a three-dimensional code while maintaining the logical state. In this work, we construct resource-optimized deterministic and non-deterministic code switching protocols for two- and three-dimensional distance-three color codes using fault-tolerant quantum circuits based on flag-qubits. Deterministic protocols allow for the fault-tolerant implementation of logical gates on an encoded quantum state, while non-deterministic protocols may be used for the fault-tolerant preparation of magic states. Taking the error rates of state-of-the-art trapped-ion quantum processors as a reference, we find a logical failure probability of $3\%$ for deterministic logical gates, which cannot be realized transversally in the respective code. By replacing the three-dimensional distance-three color code in the protocol for magic state preparation with the morphed code introduced in [1], we reduce the logical failure rates by two orders of magnitude, thus rendering it a viable method for magic state preparation on near-term quantum processors. Our results demonstrate that code switching enables the fault-tolerant and deterministic implementation of a universal gate set under realistic conditions, and thereby provide a practical avenue to advance universal, fault-tolerant quantum computing and enable quantum algorithms on first, error-corrected logical qubits.

We use the recently introduced lifted product to construct a family of quantum low density parity check codes (QLDPC codes). The codes we obtain can be viewed as stacks of surface codes that are interconnected, leading to the name lift-connected surface (LCS) codes. LCS codes offer a wide range of parameters-a particularly striking feature is that they show interesting properties that are favorable compared to the standard surface code. For example, already at moderate numbers of physical qubits in the order of tens, LCS codes of equal size have lower logical error rate or similarly, require fewer qubits for a fixed target logical error rate. We present and analyze the construction and provide numerical simulation results for the logical error rate under code capacity and phenomenological noise. These results show that LCS codes attain thresholds that are comparable to corresponding (non-connected) copies of surface codes, while the logical error rate can be orders of magnitude lower, even for representatives with the same parameters. This provides a code family showing the potential of modern product constructions at already small qubit numbers. Their amenability to 3D-local connectivity renders them particularly relevant for near-term implementations.

Neutral Atom Quantum Computing (NAQC) emerges as a promising hardware platform primarily due to its long coherence times and scalability. Additionally, NAQC offers computational advantages encompassing potential long-range connectivity, native multi-qubit gate support, and the ability to physically rearrange qubits with high fidelity. However, for the successful operation of a NAQC processor, one additionally requires new software tools to translate high-level algorithmic descriptions into a hardware executable representation, taking maximal advantage of the hardware capabilities. Realizing new software tools requires a close connection between tool developers and hardware experts to ensure that the corresponding software tools obey the corresponding physical constraints. This work aims to provide a basis to establish this connection by investigating the broad spectrum of capabilities intrinsic to the NAQC platform and its implications on the compilation process. To this end, we first review the physical background of NAQC and derive how it affects the overall compilation process by formulating suitable constraints and figures of merit. We then provide a summary of the compilation process and discuss currently available software tools in this overview. Finally, we present selected case studies and employ the discussed figures of merit to evaluate the different capabilities of NAQC and compare them between two hardware setups.

Quantum error correcting (QEC) stabilizer codes enable protection of quantum information against errors during storage and processing. Simulation of noisy QEC codes is used to identify the noise parameters necessary for advantageous operation of logical qubits in realistic quantum computing architectures. Typical quantum error correction techniques contain intermediate measurements and classical feedback that determine the actual noisy circuit sequence in an instance of performing the protocol. Dynamical subset sampling enables efficient simulation of such non-deterministic quantum error correcting protocols for any type of quantum circuit and incoherent noise of low strength. As an importance sampling technique, dynamical subset sampling allows one to effectively make use of computational resources to only sample the most relevant sequences of quantum circuits in order to estimate a protocol's logical failure rate with well-defined error bars. We demonstrate the capabilities of dynamical subset sampling with examples from fault-tolerant (FT) QEC. We show that, in a typical stabilizer simulation with incoherent Pauli noise of strength $p = 10^{-3}$, our method can reach a required sampling accuracy on the logical failure rate with two orders of magnitude fewer samples than direct Monte Carlo simulation. Furthermore, dynamical subset sampling naturally allows for efficient simulation of realistic multi-parameter noise models describing faulty quantum processors. It can be applied not only for QEC in the circuit model but any noisy quantum computing framework with incoherent fault operators including measurement-based quantum computation and quantum networks.

Fault-tolerant quantum error correction provides a strategy to protect information processed by a quantum computer against noise which would otherwise corrupt the data. A fault-tolerant universal quantum computer must implement a universal gate set on the logical level in order to perform arbitrary calculations to in principle unlimited precision. We characterize the recent demonstration of a fault-tolerant universal gate set in a trapped-ion quantum computer [Postler et al. Nature 605.7911 (2022)] and identify aspects to improve the design of experimental setups to reach an advantage of logical over physical qubit operation. We show that various criteria to assess the break-even point for fault-tolerant quantum operations are within reach for the ion trap quantum computing architecture under consideration. We analyze the influence of crosstalk in entangling gates for logical state preparation circuits. These circuits can be designed to respect fault tolerance for specific microscopic noise models. We find that an experimentally-informed depolarizing noise model captures the essential noise dynamics of the fault-tolerant experiment, and crosstalk is negligible in the currently accessible regime of physical error rates. For deterministic Pauli state preparation, we provide a fault-tolerant unitary logical qubit initialization circuit, which can be realized without in-sequence measurement and feed-forward of classical information. We show that non-deterministic state preparation schemes for logical Pauli and magic states perform with higher logical fidelity over their deterministic counterparts for the current and anticipated future regime of physical error rates. Our results offer guidance on improvements of physical qubit operations and validate the experimentally-informed noise model as a tool to predict logical failure rates in quantum computing architectures based on trapped ions.

Belief propagation (BP) is well-known as a low complexity decoding algorithm with a strong performance for important classes of quantum error correcting codes, e.g. notably for the quantum low-density parity check (LDPC) code class of ran-dom expander codes. However, it is also well-known that the performance of BP breaks down when facing topological codes such as the surface code, where naive BP fails entirely to reach a below-threshold regime, i.e. the regime where error cor-rection becomes useful. Previous works have shown, that this can be remedied by resorting to post-processing decoders out-side the framework of BP. In this work, we present a generalized belief propagation method with an outer re-initialization loop that successfully decodes surface codes, i.e. opposed to naive BP it recovers the sub-threshold regime known from de-coders tailored to the surface code and from statistical-mechanical mappings. We report a threshold of 17% under indepen-dent bit-and phase-flip data noise (to be compared to the ideal threshold of 20.6%) and a threshold value of 14% under de-polarizing data noise (compared to the ideal threshold of 18.9%), which are on par with thresholds achieved by non-BP post -processing methods.

Mapping the decoding of quantum error correcting (QEC) codes to classical disordered statistical mechanics models allows one to determine critical error thresholds of QEC codes under phenomenological noise models. Here, we extend this mapping to admit realistic, multi-parameter noise models of faulty QEC circuits, derive the associated strongly correlated classical spin models, and illustrate this approach for a quantum repetition code with faulty stabilizer readout circuits. We use Monte-Carlo simulations to study the resulting phase diagram and benchmark our results against a minimum-weight perfect matching decoder. The presented method provides an avenue to assess fundamental thresholds of QEC circuits, independent of specific decoding strategies, and can thereby help guiding the development of near-term QEC hardware.

Fault-tolerant quantum computation relies on scaling up quantum error correcting codes in order to suppress the error rate on the encoded quantum states. Topological codes, such as the surface code or color codes are leading candidates for practical scalable quantum error correction and require efficient and scalable decoders. In this work, we propose and study the efficiency of a decoder for 2D topological color codes on the 4.8.8 lattice, by building on the work of [1] for color codes on hexagonal lattices. The decoder is based on a rescaling approach, in which syndrome information on a part of the qubit lattice is processed locally, and then the lattice is rescaled iteratively to smaller sizes. We find a threshold of 6.0% for code capacity noise.

Quantum computers can be protected from noise by encoding the logical quantum information redundantly into multiple qubits using error-correcting codes1,2. When manipulating the logical quantum states, it is imperative that errors caused by imperfect operations do not spread uncontrollably through the quantum register. This requires that all operations on the quantum register obey a fault-tolerant circuit design3-5, which, in general, increases the complexity of the implementation. Here we demonstrate a fault-tolerant universal set of gates on two logical qubits in a trapped-ion quantum computer. In particular, we make use of the recently introduced paradigm of flag fault tolerance, where the absence or presence of dangerous errors is heralded by the use of auxiliary flag qubits6-10. We perform a logical two-qubit controlled-NOT gate between two instances of the seven-qubit colour code11,12, and fault-tolerantly prepare a logical magic state8,13. We then realize a fault-tolerant logical T gate by injecting the magic state by teleportation from one logical qubit onto the other14. We observe the hallmark feature of fault tolerance-a superior performance compared with a non-fault-tolerant implementation. In combination with recently demonstrated repeated quantum error-correction cycles15,16, these results provide a route towards error-corrected universal quantum computation.

Based on numerically-optimized real-device gates and parameters we study the performance of the phase-flip (repetition) code on a linear array of Gallium Arsenide (GaAs) quantum dots hosting singlet-triplet qubits. We first examine the expected performance of the code using simple error models of circuit-level and phenomenological noise, reporting, for example, a circuit-level depolarizing noise threshold of approximately 3%. We then perform density-matrix simulations using a maximum-likelihood and minimum-weight matching decoder to study the effect of real-device dephasing, read-out error, quasi-static as well as fast gate noise. Considering the trade-off between qubit read-out error and dephasing time (T2) over measurement time, we identify a sub-threshold region for the phase-flip code which lies within experimental reach.

Projected entangled-pair states (PEPS) provide a framework for the construction of models where a single tensor gives rise to both Hamiltonian and ground state wave function on the same footing. A key problem is to characterize the behavior which emerges in the system in terms of the properties of the tensor, and thus of the Hamiltonian. In this paper, we consider PEPS models with Z(2) on-site symmetry and study the occurrence of long-range order and spontaneous symmetry breaking. We show how long-range order is connected to a degeneracy in the spectrum of the PEPS transfer operator, and how the latter gives rise to spontaneous symmetry breaking under perturbations. We provide a succinct characterization of the symmetry-broken states in terms of the PEPS tensor, and find that using the symmetry-broken states we can derive a local entanglement Hamiltonian, thereby restoring locality of the entanglement Hamiltonian for all gapped phases.