Markus Müller

Trapped ions are one of the most promising quantum-information-processing platforms, yet conventional entangling gates mediated by collective motion remain slow and difficult to scale. Exciting trapped ions to high-lying electronic Rydberg states provides a promising route to overcome these limitations by enabling strong, long-range dipole–dipole interactions that support much faster multi-qubit operations. Here, we introduce the first scheme for implementing a native controlled–controlled-Z gate with microwave-dressed Rydberg ions by optimizing a single-pulse protocol that accounts for the finite Rydberg-state lifetime. Under realistic conditions, the resulting gate achieves fidelities above 97 % with execution times of about 2 μ s at cryogenic temperatures, making it more than 8 μ s faster than standard decompositions into one- and two-qubit gates. To explore the potential of trapped Rydberg ions for fault-tolerant (FT) quantum error correction (QEC), and to illustrate the utility of three-qubit Rydberg-ion gates in this context, we develop and analyze a proposal for FT, measurement-free QEC using the nine-qubit Bacon–Shor code. Our simulations confirm that QEC can be performed in a fully FT manner on a linear Rydberg-ion chain despite its limited qubit connectivity. These results establish native multi-qubit Rydberg-ion gates as a valuable resource for fast, high-fidelity quantum computing and highlight their potential for FT QEC.

Quantum error correction is needed for quantum computers to be capable of executing algorithms using hundreds of logical qubits in a fault-tolerant manner. Recent experiments have progressed towards this by demonstrating sufficiently low error rates for state preservation of a single logical qubit. However, quantum computation algorithms also require that these logical qubits can be entangled and that gate operations can be performed on them. Lattice surgery is a technique that offers a practical approach for implementing such gates, particularly in planar quantum processor layouts. Here we demonstrate lattice surgery between two distance-three repetition-code qubits by splitting a single distance-three surface-code qubit. Using a quantum circuit that is fault-tolerant for bit-flip errors, we achieve an improvement in the value of the decoded Z Z logical two-qubit observable compared with a similar non-encoded circuit. We therefore demonstrate the functional building blocks needed for lattice-surgery operations on larger-distance codes based on superconducting circuits.

The ability to perform quantum error correction (QEC) and robust gate operations on encoded qubits opens the door to demonstrations of quantum algorithms. Contemporary QEC schemes typically require mid-circuit measurements with feed-forward control, which are challenging for qubit control, often slow, and susceptible to relatively high error rates. In this work, we propose and experimentally demonstrate a universal toolbox of fault-tolerant logical operations on error-detecting codes without mid-circuit measurements on a trapped-ion quantum processor. We present modular logical state teleportation between two four-qubit error-detecting codes without measurements during algorithm execution. Moreover, we realize a fault-tolerant universal gate set on an eight-qubit error-detecting code hosting three logical qubits, based on state injection, which can be executed by coherent gate operations only. We apply this toolbox to experimentally realize Grover’s quantum search algorithm fault-tolerantly on three logical qubits encoded in eight physical qubits, with the implementation displaying clear identification of the desired solution states. Our work demonstrates the practical feasibility and provides first steps into the largely unexplored direction of measurement-free quantum computation.

High-fidelity decoding of quantum error correction codes relies on an accurate experimental model of the physical errors occurring in the device. Because error probabilities can depend on the context of the applied operations, the error model is ideally calibrated using the same circuit as is used for the error correction experiment. Here, we present an experimental approach guided by an analytical formula to characterize the probability of independent errors using correlations in the syndrome data generated by executing the error correction circuit. Using the method on a distance-three surface code, we analyze error channels that flip an arbitrary number of syndrome elements, including Pauli Y ̂ errors, hook errors, multiqubit errors, and leakage, in addition to standard Pauli X ̂ and Z ̂ errors. We use the method to find the optimal weights for a minimum-weight perfect matching decoder without relying on a theoretical error model. Additionally, we investigate whether improved knowledge of the Pauli Y ̂ error channel, based on correlating the X- and Z-type error syndromes, can be exploited to enhance matching decoding. Furthermore, we find correlated errors that flip many syndrome elements over up to eight cycles, potentially caused by leakage of the data qubits out of the computational subspace. The presented method provides the tools for accurately calibrating a broad family of decoders, beyond the minimum-weight perfect matching decoder, without relying on prior knowledge of the error model.

Stabilizer quantum error correction (QEC) codes, in particular topological surface codes, are prime candidates to enable practical quantum computing. While it is widely believed that strictly fault-tolerant protocols can only be implemented using single- and two-qubit gates, several quantum computing platforms, including trapped ions, neutral atoms, and superconducting qubits, support native multi-qubit operations. In this Letter, we show that stabilizer measurement circuits for unrotated surface codes can be fault tolerant using single auxiliary qubits and gates. These gates enable lower-depth circuits with fewer fault locations and potentially shorter QEC cycle times. We find that in an optimistic parameter regime where fidelities of three-qubit gates are the same as those of two-qubit gates, the logical error rate can be up to one order of magnitude lower and the threshold significantly higher, increasing from ≈ 0.63 % to ≈ 0.83 % . Our results, applicable to a wide range of platforms, motivate further investigation into multi-qubit gates for fault-tolerant QEC as they can offer substantial time and physical qubit resource advantages to reach a given target logical error rate.

In this paper, we derive optimized measurement-free protocols for quantum error correction and the implementation of a universal gate set optimized for an error model that is noise biased. The noise bias is adapted for neutral-atom platforms, where two- and multi-qubit gates are realized with Rydberg interactions and are thus expected to be the dominant source of noise. Careful design of the gates allows us to further reduce the noise model to Pauli- Z errors. In addition, the presented circuits are robust to arbitrary single-qubit gate errors, and we demonstrate that the break-even point can be significantly improved compared to fully fault-tolerant measurement-free schemes. The obtained logical qubits with their suppressed error rates on logical gate operations can then be used as building blocks in a first step of error correction in order to push the effective error rates below the threshold of a fully fault-tolerant and scalable quantum error-correction scheme.

In loop quantum gravity (LQG), states of the gravitational field are represented by labeled graphs called spin networks. Their dynamics can be described by a Hamiltonian constraint, which acts on the spin network states, modifying both spins and graphs. Fixed-graph approximations of the dynamics have been extensively studied, but its full graph-changing action so far remains elusive. The latter, alongside the solutions of its constraint, are arguably the missing features in canonical LQG to access phenomenology in all its richness. Here, we discuss a recently developed numerical tool that, for the first time, implements graph-changing dynamics via the Hamiltonian constraint. We explain how it is used to find new solutions to that constraint and to show that some quantum geometric observables behave differently than in the graph-preserving truncation. We also point out that these new numerical methods can find applications in other domains.

Quantum error correcting (QEC) codes protect quantum information against environmental noise. Computational errors caused by the environment change the quantum state within the qubit subspace, whereas quantum erasures correspond to the loss of qubits at known positions. Correcting either type of error involves different correction mechanisms, which makes studying the interplay between erasure and computational errors particularly challenging. In this work, we propose a framework based on the coherent information (CI) of the mixed-state density operator associated to noisy QEC codes, for treating both types of errors together. We show how to rigorously derive different families of statistical mechanics mappings for generic stabilizer QEC codes in the presence of both types of errors. We observe that the erasure errors enter as a classical average over fully depolarizing channels. Further, we show that computing the CI for erasure errors only can be done efficiently upon sampling over erasure configurations. We then test our approach on the two-dimensional toric and color codes and compute optimal thresholds for erasure errors only, finding a 50 % threshold for both codes. This strengthens the notion that both codes share the same optimal thresholds. When considering both computational and erasure errors, the CI of small-size codes yields thresholds in very accurate agreement with established results that have been obtained in the thermodynamic limit. Next, we perform a similar analysis for a low-density parity-check (LDPC) code—the lift-connected surface code. We find a 50 % threshold under erasure errors alone and, for the first time, derive the exact statistical mechanics mappings in the presence of both computational and erasure errors. We thereby further establish the CI as a practical tool for studying optimal thresholds for code classes beyond topological codes under realistic noise, and as a means for uncovering new relations between QEC codes and statistical physics models.

A major hurdle for building a large-scale quantum computer is increasing the number of qubits while maintaining connectivity between them. In trapped-ion devices, this connectivity can be achieved by moving subregisters consisting of a few ions across the processor. Here, we focus on an architecture, which we refer to as the quantum spring array (QSA), that is based on a rectangular two-dimensional lattice of linear strings of ions. Connectivity between adjacent ion strings can be controlled by adjusting their separation. This requires control of trapping potentials along two directions, one along the axis of the ion string and one radial to it. In this work, we investigate key elements of the QSA architecture along both directions: We show that the coupling rate between neighboring lattice sites increases with the number of ions per site and the motion of the coupled system can be resilient to electrical noise, both being key requisites for fast and high-fidelity quantum gate operations. The coherence of the coupling is assessed and an entangling gate between qubits stored in radially separated trapping regions is demonstrated. Moreover, we demonstrate control over radio-frequency signals to adjust the radial separation, and thus the coupling rate, between strings. We further present constructions for the implementation of parallelized, transversal gate operations, and map the QSA architecture to code primitives for fault-tolerant quantum error correction, providing a step towards a quantum processor architecture that is optimized for large-scale operation.

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.