Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code

TL;DR

This paper experimentally compares fault-tolerant entangling gates on two quantum error-correcting codes—the pieceable fault-tolerance implementation for the five-qubit code and the transversal CNOT gate for the color code—on two Quantinuum trapped-ion platforms with real-time classical decoding. It demonstrates that color-code-based FT gates can achieve higher state and logical fidelities than the five-qubit code under similar hardware noise, including a break-even-like milestone for a logical two-qubit gate. Simulations suggest that, at sufficiently low physical error rates, the color code could outperform unencoded gates, highlighting a viable near-term path toward scalable, fault-tolerant quantum computation on QCCD architectures. The study underscores the trade-offs between qubit overhead and circuit depth, and motivates exploring alternative codes and decoding strategies to reach robust logical quantum processing.

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.

Figures (15)

• Figure 1: An illustration of the (a) H1-2 and (b) H1-1 systems as they were configured for the five-qubit code and color code experiments. H1-2 has three zones capable of performing parallel gates, marked by the crossing laser beams, and the system can use up to twelve qubits. H1-1 has five parallel gate zones, and the system can use up to twenty qubits.The $^{171}\textrm{Yb}+$ qubit ions (red circles) are always paired with $^{138}\textrm{Ba}+$ ions (white circles) for sympathetic cooling. Note that the ion crystals are not drawn to scale as the gating zones are $750 \mu$m apart and the four-ion crystals are $8 \mu$m long.
• Figure 2: The stabilizer generators of the joint $[[10,2,3]]$ code (a) before and (b) after the application of the first part of the pieceable fault-tolerant CNOT gate. Before the application of any physical-level entangling gates, the stabilizer generators are just two sets of the rotated five-qubit code stabilizers. The first subscript identifies which code the stabilizer generator belongs to, either $f$ for the five-qubit code or $t$ for the $[[10,2,3]]$ code, with the second subscript labeling the individual generator. In a), the superscripts denote which logical qubit the generators belong to. After applying physical-level entangling gates (the first two-thirds of the round-robin gate) the two sets of stabilizer generators become mixed to form new higher-weight stabilizer generators. The higher weight stabilizers are measured in the intermediate QEC cycle to maintain FT. The higher weight stabilizer generators can be derived from the original stabilizers by considering the action of the circuitry shown in Fig. \ref{['main_char']}. For example, $\mathcal{S}_{t,6}=R_{2/3}U\mathcal{S}_{f,2}^2U^{\dagger}R_{2/3}^{\dagger}$, where $R_{2/3}$ denotes the first two-thirds (6 physical CNOTs) of the round-robin gate.
• Figure 3: The quantum circuit layout for the different CNOT constructions. The circuit elements are colored differently to indicate that some of the elements are constant in all of the constructions listed in Table \ref{['FidelityTable_v1']}, while some of them vary. All of the circuit elements colored gray are constant in each construction: Encoding, the first local Clifford rotation to prepare the input state to the gate, the first part of logical gate, the second part of logical gate, and the second local Clifford rotation which determines the measurement basis. The circuit elements that are colored green vary in our different constructions: fault-tolerant initialization, the two-qubit QEC cycle, and the measure-out scheme.
• Figure 4: Comparison of logical fidelity vs two-qubit gate count for five different five-qubit code experiments. Note that the ordering here is different than that in Table \ref{['FidelityTable_v1']}, as denoted by the vertical line labels. Note that we also separate the experiments by the input state that was used, with red squares denoting the $\overline{X}$ basis states, blue triangles denoting $\overline{Z}$ basis states, and green stars denoting mixtures of $\overline{X}$ and $\overline{Z}$ basis states which ideally produce Bell pairs when acted upon by a CNOT gate.
• Figure 5: The circuit used for testing the different instantiations of the color code's transversal CNOT gate. Using the same notation as with the five-qubit code circuit diagram, the grey boxes denote parts of the circuit that are used in all instantiations, and the green boxes denote additional circuitry used to increase the amount of FT.