Case Study: Fast Gates or Good Gates — Choosing a Platform for a Workload

Executive Summary

Superconducting qubits run gates in 300 ns with $7\times10^{-3}$ error. Trapped ions run them in 200 μs with $8\times10^{-4}$ error — 600× slower, 9× more accurate, and with all-to-all connectivity.

Which is better depends entirely on the workload, and the answer flips between the three workloads examined here. This case study does the comparison quantitatively for a variational chemistry calculation, a deep algorithmic circuit, and an error-correction experiment, and extracts the rule that determines which platform wins.

Skills applied

• Comparing platforms on total circuit fidelity, not per-gate error (§27.9).
• Accounting for routing overhead differences (§27.7).
• Computing wall-clock time including shot counts.
• Matching platform characteristics to workload structure.

The two devices

Workload 1: Variational chemistry (VQE, 12 qubits)

Ansatz needs 180 two-qubit gates as written, with the interaction pattern requiring near-all-to-all connectivity.

Superconducting. Transpilation inserts SWAPs: 180 → 428 two-qubit gates (2.4× routing overhead).

$$F = (1 - 0.007)^{428} \times (1-0.013)^{12} = 0.050 \times 0.855 = \mathbf{0.043}$$

Trapped ion. No routing needed: 180 gates.

$$F = (1 - 0.0008)^{180} \times (1-0.0003)^{12} = 0.866 \times 0.996 = \mathbf{0.862}$$

20× better fidelity on the ion trap — from 9× better gates and 2.4× fewer of them.

Now wall-clock, since VQE needs many evaluations. At $10^{8}$ shots total (Chapter 19's budget):

The ion trap gives a usable answer per shot and cannot take enough shots. The superconducting device takes shots quickly and each is noise.

Verdict: neither. This is the honest conclusion, and it illustrates why measurement-heavy variational algorithms are hard on every platform — the two failure modes are different but equally fatal.

Workload 2: A deep algorithmic circuit (QPE, 14 qubits, 2,400 two-qubit gates)

Structured, mostly local interactions; routing overhead only 1.3×.

Superconducting. 3,120 gates:

$$F = (0.993)^{3120} \approx 3\times10^{-10}$$

Also check coherence: 3,120 × 300 ns = 936 μs against $T_2 = 100$ μs — nine times the coherence time. Doubly impossible.

Trapped ion. 2,400 gates:

$$F = (0.9992)^{2400} \approx 0.147$$

Coherence: 2,400 × 200 μs = 480 ms against $T_2 = 10$ s — comfortably inside, using 5% of the budget.

Verdict: trapped ion, decisively. 14.7% fidelity is marginal but a real signal; $3\times10^{-10}$ is not. And critically, the ion trap's long coherence means gate error, not decoherence, is the binding constraint — leaving a clear improvement path.

The general point. For deep circuits, slow gates are nearly free if coherence scales with them. What matters is the ratio $T_2 / t_{\text{gate}}$ — the number of gates that fit in a coherence time.

The ion trap affords 150× more gates per coherence time. Gate speed alone is misleading; this ratio is the meaningful figure.

Workload 3: Surface-code error correction

Requires many physical qubits, 2D nearest-neighbour connectivity, fast repeated syndrome extraction, and mid-circuit measurement with reset.

Superconducting. 127 qubits supports a distance-5 patch. Syndrome round: ~1 μs. Thousands of rounds run in milliseconds. Native 2D layout matches the code exactly.

Trapped ion. 32 qubits supports at most a distance-3 patch. Syndrome round: ~1 ms (1,000× slower). All-to-all connectivity is wasted — the surface code only needs nearest-neighbour. And shuttling for larger systems adds latency to every round.

Verdict: superconducting, decisively. Error correction needs many cheap fast qubits repeating a local operation, which is exactly the superconducting profile. This is why below-threshold demonstrations have come from superconducting and neutral-atom platforms.

The rule

The summarizing question: is your bottleneck fidelity-per-operation or operations-per-second? Ions win the first, superconductors the second, and almost every real comparison reduces to which one binds.

Discussion Questions

1. VQE failed on both platforms for opposite reasons. What does that suggest about the algorithm rather than the hardware?
2. The ratio $T_2/t_{\text{gate}}$ favoured ions 150×. Why is this a better comparison than gate speed or coherence separately?
3. All-to-all connectivity was an advantage in workload 1 and irrelevant in workload 3. Explain.
4. Which platform would you choose for a company betting on fault tolerance by 2035? Defend it.

Your Turn: Extensions

• Compute $T_2/t_{2q}$ for neutral atoms and photonics; add them to the comparison.
• Take a circuit you care about, transpile it for a heavy-hex map, and compute both fidelities.
• Estimate the shot throughput for each platform and find the crossover for a variational workload.
• Work out the largest surface-code distance each platform's qubit count supports.

Key Takeaways

• Compare total circuit fidelity on the transpiled circuit; per-gate error rates omit routing overhead, which can be 2–3×.
• The ratio $T_2/t_{\text{gate}}$ — gates per coherence time — is the meaningful depth figure, and trapped ions lead it by ~150×.
• Slow gates are nearly free for deep circuits and fatal for shot-hungry variational ones, where throughput dominates.
• Error correction wants many fast local qubits, favouring superconducting and neutral-atom platforms regardless of per-gate fidelity.
• The deciding question is whether your bottleneck is fidelity-per-operation or operations-per-second.