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arXiv:2408.12968 · Japanese Journal of Applied Physics

Single-qubit rotations on a binomial code without ancillary qubits

Yuki Tanaka, Yuichiro Mori, Yuta Shingu, Aiko Yamaguchi, Tsuyoshi Yamamoto, Yuichiro Matsuzaki

Published Nov 22, 2024• 0 citations
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Abstract

Abstract Great attention has been paid to binomial codes utilizing bosonic systems as logical qubits with error correction capabilities. However, implementing single-qubit rotation operations on binomial codes has proven challenging, requiring an ancillary qubit in previous approaches. Here, we propose a method for performing logical qubit rotation on binomial codes without requiring an ancillary qubit. Specifically, we explain how to implement X-axis rotations by simultaneously applying two-frequency parametric drives to resonators with nonlinearity. Furthermore, we show that Z-axis rotations could be realized with the detuning. Due to the reduction of the need for the ancillary qubit for the logical qubit rotation, our proposed approach is advantageous for quantum computation in the noisy intermediate-scale quantum era, where the number of qubits is limited.

DOI
10.35848/1347-4065/ad95f4

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Abstract Great attention has been paid to binomial codes utilizing bosonic systems as logical qubits with error correction capabilities. However, implementing single-qubit rotation operations on binomial codes has proven challenging, requiring an ancillary qubit in previous approaches. Here, we propose a method for performing logical qubit rotation on binomial codes without requiring an ancillary qubit. Specifically, we explain how to implement X-axis rotations by simultaneously applying two-frequency parametric drives to resonators with nonlinearity. Furthermore, we show that Z-axis rotations could be realized with the detuning. Due to the reduction of the need for the ancillary qubit for the logical qubit rotation, our proposed approach is advantageous for quantum computation in the noisy intermediate-scale quantum era, where the number of qubits is limited.
We introduce a class of bosonic quantum error-correcting codes, termed extended binomial codes, which generalize the structure of one-mode binomial codes by incorporating ideas from high-rate qubit stabilizer codes. These codes are constructed in close analogy to [[n,k,d]][[n,k,d]][[n,k,d]] qubit codes, where the parameter nnn corresponds to the total excitation budget rather than the number of physical qubits. Our construction achieves a significant reduction in average excitation per mode while preserving error-correcting capabilities, offering improved compatibility with hardware constraints in the strong-dispersive regime. We demonstrate that extended binomial codes not only reduce the mean excitation required for encoding but also simplify syndrome extraction and logical gate implementation, particularly the logical X‾\overline{X}X operation. These advantages suggest that extended binomial codes offer a scalable and resource-efficient approach for bosonic quantum error correction.
High-fidelity two-logical-qubit gates are essential for realizing fault-tolerant quantum computation with bosonic codes, yet experimentally reported fidelities have rarely exceeded 90\%. Here, we propose a geometric phase engineering approach for implementing controlled-phase gates for binomially encoded logical qubits. This method leverages the structural simplicity of geometric drives to reduce the numerical optimization dimensionality while fully incorporating system nonlinearities, enabling fast and high-fidelity logical operations. As an example, we experimentally demonstrate a process fidelity of 97.4$\pm$0.8\% for a controlled-Z gate between two binomial codes, surpassing all previously reported two-logical-qubit gates in bosonic codes. This work demonstrates that geometric phase engineering provides an effective and experimentally feasible route to fast, high-fidelity logical operations in bosonic quantum processors.
Bosonic codes offer hardware-efficient approaches to quantum error correction, with the best encodings offering effective protection of idle quantum information against loss and dephasing - particularly rotation-symmetric codes, which include the cat and binomial code families. However, rotation-symmetric codes are only naturally endowed with a single logical Pauli gate, while other logical gates require the use of non-linear operations, obstructing the utility of these codes for realizing quantum algorithms. Here, we balance error protection with controllability by introducing bosonic cyclic codes: a generalization of rotation-symmetric codes that enable the measured tradeoff of error protection properties for fault-tolerant logical phase gates. Through our general construction, we find that sacrificing the detectability of a single photon loss relative to a rotation-symmetric code can yield a number of logical phase gates commensurate with the original rotation symmetry order of the code, all achievable via passive Gaussian rotations. Giving the corresponding generalizations of cat and binomial codes - which we dub cyclic cat and Vandermonde codes, respectively - we further find that many of the desirable properties of these codes transfer to the bosonic cyclic code setting. We go on to discuss the larger SU(2)SU(2)SU(2) symmetry and rotation gates of the codes, which yield additional stabilizers and logical Pauli gates, as well as new non-Clifford gates for the smallest `kitten' binomial code, and provide a new error detection protocol. Finally, we introduce a general paradigm for converting higher-order stabilizers to logical gates, as in our generalization of rotation-symmetric codes, and apply it to several multimode bosonic codes.
Encoding a qubit in the continuous degrees of freedom of a quantum system, such as bosonic modes, is a powerful alternative to modern quantum error correction (QEC). Among the most prominent bosonic QEC codes, binomial codes provide protection against loss and dephasing errors by encoding logical states in finite superpositions of Fock states with binomially weighted coefficients. While much attention has been given to their error-correcting capabilities and integration into fault-tolerant architectures, efficient methods for generating arbitrary binomial codewords remain scarce. In this work, we propose a scheme for generating these codewords by exploiting nonlinear multiphoton interactions between a continuous-variable bosonic mode (oscillator) and a two-level system (spin/qubit). Our proposed scheme assumes the ability to prepare the oscillator in an arbitrary Fock state and the qubit in an arbitrary superposition of its basis states and access to arbitrarily high multiphoton interactions. To enhance the experimental feasibility of our scheme, we further demonstrate how to reduce the required order parameter of multiphoton interactions by a factor of two for a special class of code states.
Rotation symmetric bosonic codes are an attractive encoding for qubits into oscillator degrees of freedom, particularly in superconducting qubit experiments. While these codes can tolerate considerable loss and dephasing, they will need to be combined with higher level codes to achieve large-scale devices. We investigate concatenating these codes with the planar code in a measurement-based scheme for fault-tolerant quantum computation. We focus on binomial codes as the base level encoding, and estimate break-even points for such encodings under loss for various types of measurement protocol. These codes are more resistant to photon loss errors, but require both higher mean photon numbers and higher phase resolution for gate operations and measurements. We find that it is necessary to implement adaptive phase measurements, maximum likelihood quantum state inference, and weighted minimum weight decoding to obtain good performance for a planar code using binomial code qubits.
Bosonic error correction provides a hardware-efficient route to protected qubits for accurate quantum information processing. A critical component for error correction of rotation-symmetric bosonic (RSB) codes, like the well-known cat codes, is the two-mode controlled-rotation (CROT) operation. The CROT gate is useful because of its code-agnostic generality and its error-propagation properties. While the CROT gate can in principle be composed from existing bosonic physical primitives, the desirable properties are typically lost when composed as a sequence of imperfect primitive operations. Past work has focused on direct implementations that rely on features of specific codes. Here, we present a direct route to CROT within the circuit-QED architecture that retains its code-agnostic and error-propagation characteristics. By driving a transmon simultaneously coupled to two microwave cavities, we controllably enhance the effective nonlinearities between the microwave cavities and thus engineer the necessary two-mode interaction underlying the CROT gate. Using only a single drive frequency, we can achieve an on-off ratio sufficient for gate implementation, while minimizing mode distortions. We provide simple analytical formulas that permit the identification of potential working regimes, further refined by exact numerics, and illustrate the efficacy of our CROT approach in an error-correction example involving different RSB codes within the same circuit. Our CROT gate adds to the arsenal of direct two-mode gates available for general bosonic information processing.