[Paper Review] Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code

C. Ryan-Anderson, N. C. Brown+35 more|arXiv (Cornell University)|Aug 3, 2022

Quantum Computing Algorithms and Architecture54 citations

TL;DR

The paper experimentally compares fault-tolerant entangling gate implementations on two quantum error correction codes—the [[5,1,3]] code with pieceable fault tolerance and the [[7,1,3]] color code—on trapped-ion devices, evaluating performance via state fidelities and process bounds, including real-time decoding.

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.

Motivation & Objective

• Assess practical fault-tolerant (FT) entangling gate implementations on two QEC codes: the five-qubit code and the color code.
• Evaluate how different FT primitives (flagging, pieceable FT, FT SPAM, FT measure-out, QEC cycles) affect logical state fidelity.
• Compare logical gate performance to physical-level performance under realistic noise.
• Demonstrate real-time decoding integrated with the quantum processor.
• Provide simulations to understand code performance at lower physical error rates.

Proposed method

• Implement a FT logical CNOT between two logical qubits encoded in the [[5,1,3]] code using pieceable FT with intermediate QEC and flagging.
• Implement a transversal logical CNOT between two logical qubits encoded in the [[7,1,3]] color code.
• Embed runtime decoding via a classical co-processor using look-up table decoders decoupled from QASM via WebAssembly.
• Characterize circuits with varied FT primitives (initialization, QEC cycles, flagging, measure-out) and measure state fidelities in X, Z, and Bell bases to bound process fidelity.
• Use two different Quantinuum trapped-ion systems (H1-2 for five-qubit code; H1-1 for color code) with integrated SPAM, QEC, and decoding."],
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Experimental results

Research questions

• RQ1 What fidelity gains or penalties arise when using FT primitives (flagging, pieceable FT, FT SPAM) in the five-qubit code versus a color code implementation?
• RQ2 How do different fault-tolerant encodings and CNOT constructions affect logical state fidelity and the inferred process fidelity?
• RQ3 What are the comparative impacts of including intermediate QEC cycles and decode-on-the-fly on logical gate performance?
• RQ4 How do logical fidelities compare to physical fidelities under similar noise environments and what do simulations suggest about break-even behavior at lower error rates?
• RQ5 How does real-time decoding integrated with the quantum processor influence overall FT gate performance?

Key findings

• Color code experiments with the [[7,1,3]] code yield higher fidelities than the five-qubit code under the tested conditions.
• FT SPAM in the color code improves eigenstate preparation fidelities to near 0.9994–0.9996 for X and Z bases.
• Logical CNOT with color code achieves average fidelity bounds around [0.9957, 0.9963], which exceed the corresponding physical-level bounds [0.9850, 0.9903].
• Five-qubit code experiments show that higher FT circuit complexity does not always improve fidelity under the examined noise environment; CNOT1f (non-FT SPAM) provides the highest fidelity among tested five-qubit code sequences.
• SPAM-only and FT-SPAM results indicate logical SPAM can approach or surpass physical SPAM in some setups, but overall logical fidelity is still bounded by practical noise and gate counts.
• Simulations indicate potential performance improvements at lower physical error rates, illustrating the long-term promise of these QEC codes.