Bosonic Quantum Error Correction Achieves Gains Beyond Break-Even with New Control

Scientists are tackling a critical challenge in quantum computing: extending the coherence of logical qubits protected by quantum error correction (QEC). Weizhou Cai, Zi-Jie Chen, and Ming Li, alongside Qing-Xuan Jie, Xu-Bo Zou, Guang-Can Guo, and et al from the CAS Key Laboratory of Quantum Information at the University of Science and Technology of China, demonstrate that limitations in current QEC gains stem primarily from ancilla-induced operational errors, rather than inherent qubit limitations. Their research introduces an error-detectable universal control method for bosonic modes, effectively filtering out trajectories affected by ancilla relaxation and suppressing errors on logical qubits. This approach achieves universal gate fidelities exceeding 99.9% and QEC gains beyond break-even, suggesting that gains exceeding 100 are now within reach using existing technology and paving the way for practical, fault-tolerant bosonic quantum computation.

By employing higher energy levels for redundancy, the team could detect ancilla relaxation events, allowing them to discard trajectories where these errors occurred and thus mitigate their impact on logical qubit fidelity. This error-detectable universal control relies on extending a numerical optimization technique, Gradient Ascent Pulse Engineering, to a two-photon drive on the ancilla, generating an effective Hamiltonian for precise control. Numerical simulations confirm the efficacy of this approach, demonstrating a universal gate set for binomial codes with process fidelities surpassing 99.5%. Analysis of dominant errors in these cycles revealed a crossover behaviour, indicating that increasing ancilla lifetime improves quantum error correction gains up to a critical point, beyond which further improvements yield diminishing returns. This work establishes a clear pathway towards building practically useful, fault-tolerant quantum computers based on bosonic modes, addressing a long-standing challenge in the field.

Ancilla Error Suppression via Bosonic Mode Control

Current QEC-protected logical qubits typically exhibit coherence times only marginally exceeding those of their physical counterparts, hindering substantial improvements in stability. The team engineered a system employing the {|g⟩,|f⟩} states as the qubit, utilising the intermediate state |e⟩ for error detection. This approach enables the detection of dominant ancilla errors, specifically longitudinal relaxation, by leveraging higher energy levels for redundancy. Consequently, experimental trajectories exhibiting ancilla errors are discarded, mitigating their contribution to the overall operational error εop.

Universal control was realised by extending Gradient Ascent Pulse Engineering (GRAPE) to a two-photon drive on the ancilla, generating an effective Hamiltonian described by a complex equation incorporating drive quadratures and Pauli operators. Experiments employed a binomial code with specific parameters: χe/2π = 1MHz, χf/2π = 2MHz, Ec/2π = 400MHz, κe = 1/40μs, κf = 1/20μs, and κ = 1/2ms. The researchers numerically investigated the gate set on a binomial logical qubit, characterising gate performance using the Pauli transfer matrix. Figures 2(a) and 2(b) illustrate the deviations from ideal values for logical T and Hadamard gates, respectively, both with and without error-detection and post-selection (PS) of the ED ancilla.

The optimised waveform for the logical Hadamard gate, with a duration of 2μs, is presented in Figure 2(c). Furthermore, the ED universal control was extended to multi-qubit gates, validating a logical controlled-phase (CZ) gate between two binomial-encoded logical qubits. Figure 2(d) displays the differences between simulated and ideal Pauli transfer matrices for the CZ gate, again comparing performance with and without PS. Table I summarises the process fidelities for ED control, demonstrating significant improvements achieved through post-selection, with success probabilities exceeding 0.89 for various logical operations, and achieving εop < 0.4% for all logical operations.

Ancilla errors limit bosonic QEC gains significantly

Experiments revealed that the dominant barrier to higher QEC gains is not intrinsic cavity coherence, but rather errors stemming from the ancilla qubit itself. This work establishes a pathway towards fault-tolerant bosonic quantum computing with substantial improvements in logical qubit coherence and fidelity. The team measured universal gate fidelities exceeding 99.6% using binomial codes, a crucial step towards reliable quantum computation. Data shows that this error-detectable control significantly reduces the impact of imperfect ancilla qubits on the overall QEC performance. Numerical simulations confirm that gains beyond 10× are achievable with current state-of-the-art devices, highlighting the potential for near-term advancements.

Researchers established a conceptual framework involving a coupled bosonic mode and an ancilla transmon qubit, enabling universal control for encoding, decoding, and error correction. For binomial codes, the study focused on correcting single-photon-loss errors, utilising the code space span{|0L⟩= (|0⟩+ |4⟩)/ √ 2, |1L⟩= |2⟩}. Measurements confirm that the achievable QEC operation error, previously limited to approximately 0.05, can be substantially reduced through this error-detectable approach. The team’s analysis indicates that improving ancilla lifetime enhances QEC gains up to a critical point, beyond which further improvements yield diminishing returns. The breakthrough delivers a fundamental understanding of the error budget in bosonic QEC, demonstrating that suppressing ancilla errors is more effective than solely improving cavity coherence. Results demonstrate that the maximum achievable QEC gain is governed by the relationship Gbreak ∼ α 2√αWεop, where εop represents the ancilla-induced operation errors.

Ancilla Errors Limit Quantum Error Correction performance

This innovative approach yielded universal gates with fidelities exceeding 99.6% and QEC gains surpassing the break-even point by a factor of 5.6. Numerical results indicate that gains exceeding a factor of ten are attainable with existing device parameters, suggesting a viable pathway towards practical fault-tolerant bosonic computing. The authors acknowledge that residual first-order cavity loss during the correction step currently limits further improvements, and propose replacing displacement-based operations with two-photon drives to suppress these errors. Future research will focus on exploring the impact of dephasing and other uncorrectable errors as the dominant limitations are overcome, potentially leading to the development of early fault-tolerant quantum technologies based on error-detectable universal control.