[Paper Review] Benchmarking logical three-qubit quantum Fourier transform encoded in the Steane code on a trapped-ion quantum computer

Karl Mayer, Ciarán Ryan-Anderson+21 more|arXiv (Cornell University)|Apr 12, 2024

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper benchmarks a logical three-qubit quantum Fourier transform (QFT) encoded in the [[7,1,3]] Steane code on the Quantinuum H2-1 trapped-ion quantum computer, using transversal two-qubit gates and non-fault-tolerant $T$-gate teleportation. Despite high-fidelity logical CNOT (0.9980(8)) and $T$ gates (0.990(1)), the full QFT achieves only 0.78(1) and 0.66(1) average output state fidelity, indicating that current logical circuits do not yet outperform physical ones due to $T$-gate errors.

ABSTRACT

We implement logically encoded three-qubit circuits for the quantum Fourier transform (QFT), using the [[7,1,3]] Steane code, and benchmark the circuits on the Quantinuum H2-1 trapped-ion quantum computer. The circuits require multiple logical two-qubit gates, which are implemented transversally, as well as logical non-Clifford single-qubit rotations, which are performed by non-fault-tolerant state preparation followed by a teleportation gadget. First, we benchmark individual logical components using randomized benchmarking for the logical two-qubit gate, and a Ramsey-type experiment for the logical $T$ gate. We then implement the full QFT circuit, using two different methods for performing a logical control-$T$, and benchmark the circuits by applying it to each basis state in a set of bases that is sufficient to lower bound the process fidelity. We compare the logical QFT benchmark results to predictions based on the logical component benchmarks.

Motivation & Objective

• To evaluate whether logical quantum circuits encoded in the Steane code can outperform unencoded circuits on current noisy hardware.
• To benchmark individual logical components—specifically transversal CNOT and non-Clifford $T$ gates—using randomized benchmarking and teleportation-based fidelity estimation.
• To implement and benchmark a full logical three-qubit QFT using two different methods for logical control-$T$ gates.
• To assess whether component-level error rates account for system-level circuit errors in logical quantum circuits.
• To model logical QFT performance using component benchmarks and identify gaps in error accounting.

Proposed method

• The logical QFT is implemented using the [[7,1,3]] Steane code, encoding one logical qubit in seven physical qubits with additional ancillas for magic state distillation.
• Transversal two-qubit logical CNOT gates are benchmarked via two-qubit randomized benchmarking (RB), yielding average fidelities of 0.9991(2) on H1-1 and 0.9980(8) on H2-1.
• Non-Clifford logical $T$ gates are implemented via non-fault-tolerant state preparation followed by gate teleportation, with fidelity estimated using a Ramsey-type protocol.
• The full logical QFT is implemented using two distinct methods for realizing logical control-$T$ gates, both based on ancilla-assisted and teleportation-based $T$-gate protocols.
• Process fidelity is lower-bounded by applying the QFT to all basis states in computational and Fourier bases, with and without post-selection on syndrome information from teleportation gadgets.
• A custom software framework, Simple Logical Representation (SLR), is used to design and compile the logical circuits for execution on the H2-1 processor.

Figure 4: Logical two-qubit RB decay curves for the devices H1-1 and H2-1. Ten random circuits per sequence length were chosen, and all circuits were run with 100 shots and submitted in a random order. The curves are fit to Eq. ( 3 ), and the estimated average fidelities per logical CNOT are listed

Experimental results

Research questions

• RQ1 Can logical quantum circuits encoded in the Steane code achieve lower error rates than unencoded circuits on current trapped-ion hardware?
• RQ2 To what extent do component-level benchmarks of logical CNOT and $T$ gates predict the performance of a full logical QFT circuit?
• RQ3 How do errors in non-Clifford $T$ gates limit the overall fidelity of logical quantum circuits, even when two-qubit gates are highly accurate?
• RQ4 Does post-selection on syndrome information from teleportation gadgets improve logical circuit fidelity, and by how much?
• RQ5 What is the gap between predicted and measured logical circuit error rates based on component benchmarks, and what does it imply for fault-tolerant scaling?

Key findings

• The logical CNOT gate achieves an average fidelity of 0.9980(8) on the H2-1 processor, approaching the performance of physical two-qubit gates.
• The logical $T$ gate exhibits significantly lower fidelity at 0.990(1), which is the primary source of error in the logical QFT circuit.
• The full logical QFT achieves average output state fidelities of 0.78(1) and 0.66(1) in the computational and Fourier bases, respectively, without syndrome post-selection.
• With post-selection on syndrome information from teleportation gadgets, average output state fidelities improve to 0.89(1) and 0.77(2), indicating error mitigation potential.
• The measured logical QFT fidelity remains below that of an unencoded QFT circuit, indicating that current logical circuits do not yet outperform physical ones.
• Component-level benchmarks explain part but not all of the logical circuit error, highlighting a significant gap that must be closed for large-scale logical quantum computation.

Figure 5: Logical $T$ gate benchmarking decay curves for the device H2-1 and the two methods for the teleportation gadget shown in Fig. 1 . Solid lines are fit to the raw data and dashed lines to the data including post-selection on the syndromes obtained from the logical measurement in the teleport