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Theory of Hidden Subgroup Sampling confirmed for tested instances on a trapped‑ion quantum computer

November 6, 2026arXiv: 2611.06716v1

A recent experiment on the IonQ Forte trapped‑ion device shows that the four sampling strategies for the non‑abelian hidden subgroup problem perform exactly as predicted by theory.

Weak Fourier sampling gives no clue about the hidden subgroup, while strong sampling in an adapted basis succeeds when the p‑to‑q ratio is favourable; strong sampling of adjacent pairs works for smaller q values. Forgetful abelian sampling only resolves the orbit of the hidden element, not the element itself, so its success probability never exceeds (p + q – 1)/(p q). These results confirm the mathematical analysis of Moore, Rockmore, Russell and Schulman (MRRS) on q‑hedral groups ℤ_q⋊ℤ_p.

Abstract plot depicting joint probabilities of measurement outcomes for different hidden elements, illustrating that two elements in the same orbit produce identical probability distributions, whereas an element from a different orbit does not.
Figure 5. Figure 5 shows that for the forgetful abelian sampling experiment with p=7, q=3, the outcomes for b=1 and b=2 (belonging to the same orbit {1,2,4}) are indistinguishable, while b=3 (different orbit) can be separated. This empirical result confirms the theoretical limitation that orbit members cannot be distinguished.
O'Brien, Oliver · arXiv:2611.06716v1 · Figure 5: · CC BY 4.0

In a concrete test with p = 7 and q = 6, the noisy hardware outperformed an optimised classical birthday algorithm for up to three oracle queries, indicating a query‑complexity advantage for the quantum approach in this small instance.

The experiment implemented the four sampling methods on the IonQ Forte platform for primes p≤7, demonstrating that the theoretical predictions survive practical noise and that circuit synthesis via Walsh‑series techniques yields the same two‑qubit gate counts as the standard Qiskit diagonal‑gate construction.

Table comparing two‑qubit gate counts for our method and Qiskit for different qubit configurations, demonstrating identical results.
qQubits (t+y)OursQiskit
21+333
32+3615
63+31939
Table 2. Table 2 lists the two‑qubit (RZ​Z) gate counts for the oracle diagonal D_b at p=7, showing that our Walsh‑series synthesis achieves the same counts as the standard Qiskit diagonal‑gate construction for every hidden element b.O'Brien, Oliver · arXiv:2611.06716v1 · Table 2: · CC BY 4.0

However, the limitations of weak sampling and forgetful abelian sampling are also evident: weak sampling reveals no information, and forgetful sampling cannot distinguish members within an orbit, capping its success probability. These findings frame the boundaries for efficient quantum algorithms solving non‑abelian hidden subgroup problems.

Source paper

Solving a non-abelian hidden subgroup problem on a trapped-ion quantum computer: the power of the measurement basis

Quantum Physics
View on arXiv