Protecting quantum information through quantum error correction (QEC) is a cornerstone of future fault-tolerant quantum computation. However, current QEC-protected logical qubits have only achieved coherence times about twice those of their best physical constituents. Here, we show that the primary barrier to higher QEC gains is ancilla-induced operational errors rather than intrinsic cavity coherence. To overcome this bottleneck, we introduce error-detectable universal control of bosonic modes, wherein ancilla relaxation events are detected and the corresponding trajectories discarded, thereby suppressing operational errors on logical qubits. For binomial codes, we demonstrate universal gates with fidelities exceeding $99.6\%$ and QEC gains of $8.33\times$ beyond break-even. Our results establish that gains beyond $10\times$ are achievable with state-of-the-art devices, establishing a path toward fault-tolerant bosonic quantum computing.
Protecting quantum information through quantum error correction (QEC) is a cornerstone of future fault-tolerant quantum computation. However, current QEC-protected logical qubits have only achieved coherence times about twice those of their best physical constituents. Here, we show that the primary barrier to higher QEC gains is ancilla-induced operational errors rather than intrinsic cavity coherence. To overcome this bottleneck, we introduce error-detectable universal control of bosonic modes, wherein ancilla relaxation events are detected and the corresponding trajectories discarded, thereby suppressing operational errors on logical qubits. For binomial codes, we demonstrate universal gates with fidelities exceeding 99.6% and QEC gains of 8.33× beyond break-even. Our results establish that gains beyond 10× are achievable with state-of-the-art devices, establishing a path toward fault-tolerant bosonic quantum computing.
Fault-tolerant quantum computation with bosonic qubits often necessitates the use of noisy discrete-variable ancillae. In this work, we establish a comprehensive and practical fault-tolerance framework for such a hybrid system and synthesize it with fault-tolerant protocols by combining bosonic quantum error correction (QEC) and advanced quantum control techniques. We introduce essential building blocks of error-corrected gadgets by leveraging ancilla-assisted bosonic operations using a generalized variant of path-independent quantum control. Using these building blocks, we construct a universal set of error-corrected gadgets that tolerate a single-photon loss and an arbitrary ancilla fault for four-legged cat qubits. Notably, our construction requires only dispersive coupling between bosonic modes and ancillae, as well as beam-splitter coupling between bosonic modes, both of which have been experimentally demonstrated with strong strengths and high accuracy. Moreover, each error-corrected bosonic qubit is comprised of only a single bosonic mode and a three-level ancilla, featuring the hardware efficiency of bosonic QEC in the full fault-tolerant setting. We numerically demonstrate the feasibility of our schemes using current experimental parameters in the circuit-QED platform. Finally, we present a hardware-efficient architecture for fault-tolerant quantum computing by concatenating the four-legged cat qubits with an outer qubit code utilizing only beam-splitter couplings. Our estimates suggest that the overall noise threshold can be reached using existing hardware. These developed fault-tolerant schemes extend beyond their applicability to four-legged cat qubits and can be adapted for other rotation-symmetrical codes, offering a promising avenue toward scalable and robust quantum computation with bosonic qubits.
Bosonic codes offer hardware-efficient approaches to logical qubit construction and hosted the first demonstration of beyond-break even logical quantum memory. However, such accomplishments were done for idling information, and realization of fault-tolerant logical operations remains a critical bottleneck for universal quantum computation in scaled systems. Error-transparent (ET) gates offer an avenue to resolve this issue, but experimental demonstrations have been limited to phase gates. Here, we introduce a framework based on dynamic encoding subspaces that enables simple linear drives to accomplish universal gates that are error semi-transparent (EsT) to oscillator photon loss. With an EsT logical gate set of {X, H, T}, we observe a five-fold reduction in infidelity conditioned on photon loss, demonstrate extended active-manipulation lifetimes with quantum error correction, and construct a composite EsT non-Clifford operation using a sequence of eight gates from the set. Our approach is compatible with methods for detectable ancilla errors, offering an approach to error-mitigated universal control of bosonic logical qubits with the standard quantum control toolkit.
Fault-tolerant quantum computation with bosonic qubits often necessitates the use of noisy discrete-variable ancillae. In this work, we establish a comprehensive and practical fault-tolerance framework for such a hybrid system and synthesize it with fault-tolerant protocols by combining bosonic quantum error correction (QEC) and advanced quantum control techniques. We introduce essential building blocks of error-corrected gadgets by leveraging ancilla-assisted bosonic operations using a generalized variant of path-independent quantum control. Using these building blocks, we construct a universal set of error-corrected gadgets that tolerate a single-photon loss and an arbitrary ancilla fault for four-legged cat qubits. Notably, our construction requires only dispersive coupling between bosonic modes and ancillae, as well as beam-splitter coupling between bosonic modes, both of which have been experimentally demonstrated with strong strengths and high accuracy. Moreover, each error-corrected bosonic qubit is comprised of only a single bosonic mode and a three-level ancilla, featuring the hardware efficiency of bosonic QEC in the full fault-tolerant setting. We numerically demonstrate the feasibility of our schemes using current experimental parameters in the circuit-QED platform. Finally, we present a hardware-efficient architecture for fault-tolerant quantum computing by concatenating the four-legged cat qubits with an outer qubit code utilizing only beam-splitter couplings. Our estimates suggest that the overall noise threshold can be reached using existing hardware. These developed fault-tolerant schemes extend beyond their applicability to four-legged cat qubits and can be adapted for other rotation-symmetrical codes, offering a promising avenue toward scalable and robust quantum computation with bosonic qubits. Published by the American Physical Society 2024
Quantum information is vulnerable to environmental noise and experimental imperfections, hindering the reliability of practical quantum information processors. Therefore, quantum error correction (QEC) that can protect quantum information against noise is vital for universal and scalable quantum computation. Among many different experimental platforms, superconducting quantum circuits and bosonic encodings in superconducting microwave modes are appealing for their unprecedented potential in QEC. During the last few years, bosonic QEC is demonstrated to reach the break-even point, i.e. the lifetime of a logical qubit is enhanced to exceed that of any individual components composing the experimental system. Beyond that, universal gate sets and fault-tolerant operations on the bosonic codes are also realized, pushing quantum information processing towards the QEC era. In this article, we review the recent progress of the bosonic codes, including the Gottesman-Kitaev-Preskill codes, cat codes, and binomial codes, and discuss the opportunities of bosonic codes in various quantum applications, ranging from fault-tolerant quantum computation to quantum metrology. We also summarize the challenges associated with the bosonic codes and provide an outlook for the potential research directions in the long terms.
Quantum computers promise to solve computational problems that are inaccessible to classical machines. The reliable execution of large-scale quantum algorithms requires robust operations in the presence of noise. Quantum error correction addresses this challenge by encoding quantum information redundantly across several physical qubits, such that errors can be detected and corrected during a computation. However, no single error-correcting code intrinsically supports an inherently robust, fault-tolerant implementation of all gates needed for universal quantum computing. Moreover, most modern approaches to both quantum error correction and the implementation of universal logical operations on encoded qubits rely on mid-circuit measurements and feed-forward control. These types of operations are technologically demanding, often slow, and prone to high error rates, and they limit the overall success probability of quantum algorithms. In this thesis, we develop fault-tolerant protocols and circuits for quantum error correction and universal logical computation, with an emphasis on implementations compatible with current hardware systems. We first focus on fault-tolerant quantum error correction on small codes. By comparing different protocols, we identify trade-offs between compact circuit constructions and their robustness to noise. We then assess their practical performance by comparing numerical simulations of noisy circuits with experimental data. Beyond the preservation of encoded information, we study how universal logic can be implemented in a fault-tolerant manner on encoded qubits. A central approach considered in this thesis is code switching, in which logical information is transferred between two distinct error-correcting codes that support complementary sets of inherently fault-tolerant logical gates. In combination, these codes give access to a complete universal fault-tolerant logical gate set. In this thesis, we determine conditions that allow us to construct fault-tolerant protocols for code switching. We adapt these protocols to the capabilities of current hardware, thereby enabling the first experimental realization of universal fault-tolerant logic by means of code switching. However, the overall performance of these schemes is limited by slow and noisy mid-circuit measurements. To address this bottleneck, we construct a suite of measurement-free fault-tolerant universal operations that do not require any measurements during the execution of an algorithm. In this framework, error correction and logical operations are implemented coherently as part of the quantum circuit, without mid-circuit measurements or in-sequence classical control. We introduce measurement-free code-switching schemes that can be scaled to arbitrary code distances through concatenation, and which thereby provide, in principle, arbitrarily strong error suppression. We further tailor our approach to the capabilities of current trapped-ion quantum processors. New circuit constructions support the first experimental implementation of a logical quantum algorithm without mid-circuit measurements, demonstrated through Grover's search on three encoded qubits. Together, the results of this thesis expand the toolbox for early fault-tolerant quantum computation. They show how protocols for quantum error correction and universal logical gates can be designed without mid-circuit measurements with realistic hardware constraints in mind, opening new pathways towards universal quantum logic.
Abstract 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 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.