We present a universal scheme of pulsed operations suitable for the IBM oscillator-stabilized flux qubit comprising the controlled-σ z (cphase) gate, single-qubit preparations and measurements. Based on numerical simulations, we argue that the error rates for these operations can be as low as about 0.5% and that noise is highly biased, with phase errors being stronger than all other types of errors by a factor of nearly 10 3 . In contrast, the design of a controlled-σ x (cnot) gate for this system with an error rate of less than about 1.2% seems extremely challenging. We propose a special encoding that exploits the noise bias allowing us to implement a logical cnot gate where phase errors and all other types of errors have nearly balanced rates of about 0.4%. Our results illustrate how the design of an encoding scheme can be adjusted and optimized according to the available physical operations and the particular noise characteristics of experimental devices.
We present a universal scheme of pulsed operations suitable for the IBM oscillator-stabilized flux qubit comprising the controlled-σ z (cphase) gate, single-qubit preparations and measurements. Based on numerical simulations, we argue that the error rates for these operations can be as low as about 0.5% and that noise is highly biased, with phase errors being stronger than all other types of errors by a factor of nearly 10 3 . In contrast, the design of a controlled-σ x (cnot) gate for this system with an error rate of less than about 1.2% seems extremely challenging. We propose a special encoding that exploits the noise bias allowing us to implement a logical cnot gate where phase errors and all other types of errors have nearly balanced rates of about 0.4%. Our results illustrate how the design of an encoding scheme can be adjusted and optimized according to the available physical operations and the particular noise characteristics of experimental devices.
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.
We formulate a scheme for fault-tolerant quantum computation that works effectively against highly biased noise, where dephasing is far stronger than all other types of noise. In our scheme, the fundamental operations performed by the quantum computer are single-qubit preparations, single-qubit measurements, and conditional-phase (CPHASE) gates, where the noise in the CPHASE gates is biased. We show that the accuracy threshold for quantum computation can be improved by exploiting this noise asymmetry; e.g., if dephasing dominates all other types of noise in the CPHASE gates by four orders of magnitude, we find a rigorous lower bound on the accuracy threshold higher by a factor of 5 than for the case of unbiased noise.
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 full potential of quantum computers will be attainable when large devices can employ quantum error correction and attain fault-tolerance. Great strides have been made, with experimental demonstrations of pivotal components for quantum error correction on a variety of platforms, substantiating that the foundations for fault-tolerant logic are realisable. This thesis explores the kinds of noise we expect to find at the logical level while employing schemes for fault-tolerance in this emerging period and how we may optimise such schemes. We study how a noise characteristic called bias is transformed in the implementation of a fault-tolerant gadget for the T-gate. We show that noise bias is amplified at the logical level due to error correction, which is furthermore affected by the noise-transforming properties of the injection gadget, which separately induces logical phase bias. We experimentally characterise the noise on a trapped-ion quantum computer. We target a logical CNOT with this method such that we are able to: understand and mitigate control layer imperfections, account for the distinction in noise characteristics between individual CNOT and transversal CNOTs, and make predictions on logical level performance if these transversal CNOTs are used as a logical CNOT for the Steane code. This is achieved by leveraging information about correlated noise across the device from the physical characterisation of gates, demonstrating that such diagnostic information can be learned scalably. Finally, noting the transversal gate capabilities of recent architectures, we optimise fault-tolerant circuits for preparing magic states in that setting. We describe an algorithm that recompiles and synthesises such circuits for minimal T-depth and low CNOT depth, while maintaining the total qubit count. We apply this algorithm to fault-tolerant circuits for |CCZ>, |CS> and |T> and show how we can reduce overheads for preparing magic states.
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.
Noise-biased qubits have bit-flip errors that are exponentially suppressed relative to phase-flip errors, and offer a promising route toward fault-tolerant quantum computing. However, this bias can be compromised during gate operations with non-biased control qubits. To address this limitation, we propose a "0-n" Fock qubit architecture that maintains the noise bias by encoding information in the ground state and n-th excited state of a nonlinear multi-level system, such as a transmon. This encoding is achieved via a dissipative stabilization that acts as decay and gain for lower and upper intermediate levels, respectively. We first analytically demonstrate that bit-flip errors are exponentially suppressed with the number of levels. Then, we present a practical implementation using a multi-mode lossy filter to achieve the frequency-selective dissipation. Finally, we numerically demonstrate that bit-flip probabilities approaching 10−8 are achievable for controlled-X gates on cat qubits using the 0-n qubit as an ancilla with n≥9 (i.e. ten or more levels), for realistic experimental parameters. Building on this, we simulate syndrome extraction in a repetition code, achieving logical error rates in the megaquop regime with only a distance of d=9.