Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code (2208.01863v1)

Abstract: We compare two different implementations of fault-tolerant entangling gates on logical qubits. In one instance, a twelve-qubit trapped-ion quantum computer is used to implement a non-transversal logical CNOT gate between two five qubit codes. The operation is evaluated with varying degrees of fault tolerance, which are provided by including quantum error correction circuit primitives known as flagging and pieceable fault tolerance. In the second instance, a twenty-qubit trapped-ion quantum computer is used to implement a transversal logical CNOT gate on two [[7,1,3]] color codes. The two codes were implemented on different but similar devices, and in both instances, all of the quantum error correction primitives, including the determination of corrections via decoding, are implemented during runtime using a classical compute environment that is tightly integrated with the quantum processor. For different combinations of the primitives, logical state fidelity measurements are made after applying the gate to different input states, providing bounds on the process fidelity. We find the highest fidelity operations with the color code, with the fault-tolerant SPAM operation achieving fidelities of 0.99939(15) and 0.99959(13) when preparing eigenstates of the logical X and Z operators, which is higher than the average physical qubit SPAM fidelities of 0.9968(2) and 0.9970(1) for the physical X and Z bases, respectively. When combined with a logical transversal CNOT gate, we find the color code to perform the sequence--state preparation, CNOT, measure out--with an average fidelity bounded by [0.9957,0.9963]. The logical fidelity bounds are higher than the analogous physical-level fidelity bounds, which we find to be [0.9850,0.9903], reflecting multiple physical noise sources such as SPAM errors for two qubits, several single-qubit gates, a two-qubit gate and some amount of memory error.

• The paper demonstrates that a transversal CNOT gate with the color code achieves logical fidelities up to 0.996, outperforming the five-qubit code under current noise conditions.
• It employs flagging and pieceable fault tolerance in the five-qubit code while contrasting these with the naturally fault-tolerant design of the color code.
• The study reveals practical trade-offs in quantum error correction, suggesting that reduced physical error rates may enhance the competitiveness of the five-qubit approach.

Implementing Fault-tolerant Entangling Gates: A Comparative Study of Quantum Error Correction Codes

The paper examines the implementation of fault-tolerant entangling gates using two prominent quantum error correction (QEC) codes: the five-qubit code and the color code. The research is conducted on quantum computers designed to operate with trapped-ion systems, providing a framework for comparing the practical effectiveness and efficiency of these codes in a controlled setting.

Overview of Implementations

The implementation focuses on logical qubits utilizing two distinct entangling gate strategies. The first uses a twelve-qubit trapped-ion system to execute a non-transversal logical CNOT gate between two logical qubits encoded with the $[[5,1,3]]$ five-qubit code. The operation implements varying degrees of fault tolerance, including methods known as flagging and pieceable fault tolerance, which collectively improve error suppression and management.

Conversely, the second implementation utilizes a twenty-qubit system for a transversal logical CNOT gate on two logical qubits using the $[[7,1,3]]$ color code. Transversal gates in the color code naturally exhibit fault tolerance, reducing the operational overhead required to suppress errors.

Experimental Results and Fidelity Measurements

The paper meticulously presents data on logical state fidelity after applying entangling gates, measured for different input states to bound the process fidelity. The color code implementation achieves significantly higher fidelity, with state preparation and measurement (SPAM) fidelity reaching as high as $0.99959(13)$, surpassing the physical qubit SPAM fidelity. When combined with a logical transversal CNOT gate, the color code achieves average fidelity bounds of $[0.9957,0.9963]$ for the logical operation sequence. This marks a milestone where operations at the logical level, utilizing fault-tolerance mechanisms, are executed with fidelity surpassing that of physical-level operations.

On the other hand, the five-qubit code demonstrates limitations under current noise conditions, as more FT circuitry did not correlate with improved performance. Simulations indicate the five-qubit code might only outperform its counterparts under significantly reduced physical error rates — a key consideration as advancements in quantum hardware continue.

Implications and Future Directions

Practically, this research underlines the advantages of using the color code in real-time quantum processing, particularly noting its compatibility with a more straightforward FT logic gate structure and higher achieved fidelities under similar experimental conditions. Theoretical implications point to the nuanced trade-offs inherent in QEC selection, such as between qubit number and computational depth, further informed by system error profiles and gate transversality.

Simulations suggest that should physical error rates decrease notably, codes like the five-qubit code could become more competitive. Future research could expand on optimizing FT entangling gates for various QEC codes under different noise regimes and hardware specifications, potentially exploring further transversal gate implementations that offer promising performance without excessive resource overhead.

In conclusion, this paper makes substantive contributions to the understanding of implementing and comparing fault-tolerant entangling gates in quantum computing, providing a detailed exploration of the practical applications of QEC codes under current and projected quantum system capabilities.