Shruti Puri, Lucas St-Jean, Jonathan A. Gross, Alexander Grimm, Nicholas Frattini, Pavithran Sridharan Iyer, Anirudh Krishna, Steven Touzard, Liang Jiang, Alexandre Blais, Steven T. Flammia, S. M. Girvin
The code capacity threshold for error correction using biased-noise qubits is known to be higher than with qubits without such structured noise. However, realistic circuit-level noise severely restricts these improvements. This is because gate operations, such as a controlled-NOT (CX) gate, which do not commute with the dominant error, unbias the noise channel. Here, we overcome the challenge of implementing a bias-preserving CX gate using biased-noise stabilized cat qubits in driven nonlinear oscillators. This continuous-variable gate relies on nontrivial phase space topology of the cat states. Furthermore, by following a scheme for concatenated error correction, we show that the availability of bias-preserving CX gates with moderately sized cats improves a rigorous lower bound on the fault-tolerant threshold by a factor of two and decreases the overhead in logical Clifford operations by a factor of five. Our results open a path toward high-threshold, low-overhead, fault-tolerant codes tailored to biased-noise cat qubits.
The code capacity threshold for error correction using biased-noise qubits is known to be higher than with qubits without such structured noise. However, realistic circuit-level noise severely restricts these improvements. This is because gate operations, such as a controlled-NOT (CX) gate, which do not commute with the dominant error, unbias the noise channel. Here, we overcome the challenge of implementing a bias-preserving CX gate using biased-noise stabilized cat qubits in driven nonlinear oscillators. This continuous-variable gate relies on nontrivial phase space topology of the cat states. Furthermore, by following a scheme for concatenated error correction, we show that the availability of bias-preserving CX gates with moderately sized cats improves a rigorous lower bound on the fault-tolerant threshold by a factor of two and decreases the overhead in logical Clifford operations by a factor of five. Our results open a path toward high-threshold, low-overhead, fault-tolerant codes tailored to biased-noise cat qubits.
Qubits with strongly biased noise, in which phase-flip errors are orders of magnitude more frequent than bit-flips, arise both naturally, as in electron and nuclear spins, and by engineering, as in stabilized cat qubits. This noise structure holds the promise of reducing the daunting hardware overhead of fault-tolerant quantum computing, but exploiting it requires physical operations that do not convert frequent phase-flips into rare bit-flips. In this review, we analyze the most prominent fault-tolerant protocols for biased-noise qubits, organized according to the available set of such bias-preserving operations. When this set is restricted to the CZ gate together with preparation and measurement in the X basis, we show that the complexity of the required syndrome extraction gadgets essentially cancels the benefit of the noise bias: at experimentally relevant error rates, one may as well ignore the bias and rely on standard error correction designed for depolarizing noise. The situation changes drastically when a bias-preserving CX gate is available: the hierarchy of errors can then be reflected in the structure of the code, with frequent phase-flips corrected by a dedicated high-threshold code and rare bit-flips by concatenation with a high-rate code. The same hierarchy also enables hardware-efficient preparation of magic states. Finally, as a bias-preserving CX is forbidden in naturally biased platforms and challenging in engineered ones, we present a measurement-based architecture in which a high-fidelity quantum non-demolition readout of multi-qubit Pauli Z operators takes its place, extending these overhead reductions to a much broader range of physical platforms.
The code capacity threshold for error correction using biased-noise qubits is known to be higher than with qubits without such structured noise. However, realistic circuit-level noise severely restricts these improvements. This is because gate operations, such as a controlled-NOT (CX) gate, which do not commute with the dominant error, unbias the noise channel. Here, we overcome the challenge of implementing a bias-preserving CX gate using biased-noise stabilized cat qubits in driven nonlinear oscillators. This continuous-variable gate relies on nontrivial phase space topology of the cat states. Furthermore, by following a scheme for concatenated error correction, we show that the availability of bias-preserving CX gates with moderately sized cats improves a rigorous lower bound on the fault-tolerant threshold by a factor of two and decreases the overhead in logical Clifford operations by a factor of five. Our results open a path toward high-threshold, low-overhead, fault-tolerant codes tailored to biased-noise cat qubits.
Stabilized cat codes can provide a biased noise channel with a set of bias-preserving (BP) gates, which can significantly reduce the resource overhead for fault-tolerant quantum computing. All existing schemes of BP gates, however, require adiabatic quantum evolution, with performance limited by excitation loss and nonadiabatic errors during the adiabatic gates. In this paper, we apply a derivative-based leakage-suppression technique to overcome nonadiabatic errors, so that we can implement fast BP gates on Kerr-cat qubits with improved gate fidelity while maintaining high noise bias. When applied to concatenated quantum error correction, the fast BP gates not only can improve the logical error rate but also can reduce resource overhead, which enables more efficient implementation of fault-tolerant quantum computing.
Stabilized cat qubits that possess biased noise channel with bit-flip errors exponentially smaller than phase-flip errors. Together with a set of bias-preserving (BP) gates, cat qubits are a promising candidate for realizing hardware efficient quantum error correction and fault-tolerant quantum computing. Compared to dissipatively stabilized cat qubits, the Kerr cat qubits can in principle support faster gate operations with higher gate fidelity, benefiting from the large energy gap that protects the code space. However, the leakage of the Kerr cats can increase the minor type of errors and compromise the noise bias. Both the fast implementation of gates and the interaction with environment can lead to such detrimental leakage if no sophisticated controls are applied. In this work, we introduce new fine-control techniques to overcome the above obstacles for Kerr cat qubits. To suppress the gate leakage, we use the derivative-based transition suppression technique to design derivative-based controls for the Kerr BP gates. We show that the fine-controlled gates can simultaneously have high gate fidelity and high noise bias and when applied to concatenated quantum error correction, can not only improve the logical error rate but also reduce resource overhead. To suppress the environment-induced leakage, we introduce colored single-photon dissipation, which can continuously cool the Kerr cats and suppress the minor errors while not enhancing the major errors.
Stabilized cat codes can provide a biased noise channel with a set of bias-preserving (BP) gates, which can significantly reduce the resource overhead for fault-tolerant quantum computing. All existing schemes of BP gates, however, require adiabatic quantum evolution, with performance limited by excitation loss and non-adiabatic errors during the adiabatic gates. In this work, we apply a derivative-based leakage suppression technique to overcome non-adiabatic errors, so that we can implement fast BP gates on Kerr-cat qubits with improved gate fidelity while maintaining high noise bias. When applied to concatenated quantum error correction, the fast BP gates can not only improve the logical error rate but also reduce resource overhead, which enables more efficient implementation of fault-tolerant quantum computing.
Fault-tolerant quantum computation with depolarization error often requires demanding error threshold and resource overhead. If the operations can maintain high noise bias---dominated by dephasing error with small bit-flip error---we can achieve hardware-efficient fault-tolerant quantum computation with a more favorable error threshold. Distinct from two-level physical systems, multilevel systems (such as harmonic oscillators) can achieve a desirable set of bias-preserving quantum operations while using continuous engineered dissipation or Hamiltonian protection to stabilize to the encoding subspace. For example, cat codes stabilized with driven-dissipation or Kerr nonlinearity can possess a set of bias-preserving gates while continuously correcting bosonic dephasing error. However, cat codes are not compatible with continuous quantum error correction against excitation loss error, because it is challenging to continuously monitor the parity to correct photon loss errors. In this work, we generalize the bias-preserving operations to pair-cat codes, which can be regarded as a multimode generalization of cat codes, to be compatible with continuous quantum error correction against both bosonic loss and dephasing errors. Our results open the door towards hardware-efficient robust quantum information processing with both bias-preserving operations and continuous quantum error correction simultaneously correcting bosonic loss and dephasing errors.