Data and code for collision-model Dicke-state preparation: depth-fidelity frontiers, circuit costs, and superconducting-processor measurements

AI summary

Overview

Research area: Quantum information science, specifically multipartite entangled state preparation (Dicke states) with a focus on superconducting-hardware benchmarking and open-data release.

Technical level: Intermediate. The paper is a data-and-code descriptor rather than a theory paper; the underlying physics (partial-swap collision models) is explained from the ground up, but readers benefit from familiarity with entanglement, gate counts and readout tomography.

One-sentence scope: This release documents an open dataset and codebase that maps, for 31 Dicke-state targets from five to fourteen qubits, how noiseless preparation fidelity improves with circuit depth, what each operating point costs in two-qubit gates, and how a selected subset of those circuits actually performs on the 54-qubit IQM Emerald superconducting processor.

What This Paper Is About

Preparing a prescribed Dicke state — an equal superposition of all basis states with exactly K excitations across N qubits — is hard, and existing proposals almost always optimize only the fidelity of an idealized, noise-free circuit while treating gate count as a footnote. Because two-qubit error rates on a real chip can vary by more than an order of magnitude between couplers and coherence times cap total circuit duration, a circuit that is optimal on paper can be useless in practice. This paper releases the full data behind a collision-model preparation protocol so that fidelity, circuit depth and hardware cost can be weighed against each other at every operating point rather than at a single converged one.

Key Contributions

1. A complete depth–fidelity frontier (L1). For each of 31 target states D_N^K (N = 5 to 14, K = 2 to 5) and each maximum allowed depth m = 0 to 200 sub-rounds, the release reports the highest noiseless fidelity found, the collision strengths used, and the sub-round at which it was attained. This yields 6,231 fully populated cells, backed by the entire search surface: 256 candidates × 201 depths = 51,456 values per state per array, or 1,595,136 values per array across all 31 states. Both pre-phase (_raw.npy) and post-phase (_final.npy) arrays are deposited.

2. A circuit-cost layer (L2) absent from the companion paper. One row per frontier cell gives the partial-swap interaction count, the corresponding all-to-all CNOT/CZ-equivalent two-qubit gate count (3 × pSWAP), and two routing-inclusive estimates (n2q_expected, n2q_expected_hi) derived from median and 90th-percentile routing overheads. Interaction counts are exact for 5,687 cells; the remaining 544 carry conservative upper bounds, and the release explicitly flags them via provenance in frontier_source.csv.

3. A full hardware record (L3). Ten circuit arms across four target states were executed on IQM Emerald, a 54-qubit superconducting processor with 85 calibrated two-qubit couplings accessed through Amazon Braket in the eu-north-1 region. The release includes 88,656 raw shots, submitted execution plans, the calibration snapshot, processed fidelities with error bars, a two-variant noise-model refit, and a layout experiment.

4. Validation and reproducibility infrastructure. Equation (3) is verified to hold at all 1,595,136 stored candidate–depth pairs to a largest relative deviation of 6 × 10⁻⁸, making it a usable unit test for independent reimplementations (code/pi_raw_invariant.py). The archive ships with CHECKSUMS.sha256, a pinned requirements.txt, a reproduce.py entry point, and machine-readable metadata.

Main Findings

• Deeper is not automatically worth it. Across the 31 released states, the median gap between the best-found fidelity and the fidelity after one full round is 0.0223, and the largest gap is 0.1382 for D_13^5. For 14 of the 31 states, one full round already achieves the highest fidelity found anywhere in the explored depth range.

• Shallow circuits get most of the way. The median depth required to reach 99% of a state's best-found fidelity is 14 sub-rounds, though the spread is wide, ranging from 2 to 192 sub-rounds.

• The fidelity–cost trade-off is steep. The median difference F_one-round − F_deep is −0.0257, with the largest decrease at −0.1349. Meanwhile the median ratio of the deep-search operating point's two-qubit gate count to that of one full round with γ_in = 0 is 72.9, with a maximum ratio of 173.

• Five operating regimes quantify the ladder. Across the 31 states, median values are: deep-search best found F = 0.8869 at m = 116, pSWAP = 944, 2q = 2832; best found within m ≤ 49 F = 0.8633 at m = 4, pSWAP = 78, 2q = 234; cap-30 reference set (m ≤ 29) F = 0.8559 at m = 4, pSWAP = 64, 2q = 192; one full round with γ_in free F = 0.8504 at m = 3, pSWAP = 55, 2q = 165; one full round with γ_in = 0 F = 0.8446 at m = 3, pSWAP = 22, 2q = 66.

• Hardware retention collapses with depth. For D_5^2, the 26-sub-round reference arm retains F/F_ideal = 0.091, whereas the two one-round arms retain between 0.713 and 0.758. For D_8^3, the reference arm retains 0.041 and the one-round arms 0.204–0.510. The shallowest arms keep roughly three-quarters of their noiseless fidelity, while the deepest approach the fully decohered value.

• Layout choice is a measurable variable. All arms for a given state were assigned to the same set of physical qubits within a single hardware session (440 circuits, 20 tasks, submitted 2026-08-21T00:34:00.279745+00:00, total execution cost $167.35), removing qubit layout as a confounder between arms; a separate layout experiment measures the sensitivity to layout directly.

• The device is highly heterogeneous. In the calibration snapshot recorded at 2026-08-20T05:19:15.989291+00:00, the calibrated two-qubit gate error ranges from 0.153% to 8.206% — a factor of 54 across the device — with a median of 0.349%. Median relaxation and dephasing times are T₁ = 54.7 μs and T₂ = 16.6 μs.

• Estimators can go negative near the noise floor. For D_10^5 / published, the estimated fidelity is F = −0.0014 ± 0.0437 with P_K = 0.2271, close to the maximally mixed value of (10 choose 5)/2^10 = 0.2461 for that subspace. The four replicate estimates are +0.0569, −0.0822, +0.0322 and −0.0276. The authors report the unclipped value deliberately, since clipping would bias fidelities near zero upward. All 10 measured arms satisfy F ≤ P_K.

• Phase-layer ordering matters. Fixing the depth before optimizing the single-qubit phase layer is not equivalent to the released procedure; it changes the result in 8 of the 31 states, by up to +0.068 in fidelity.

• Some reported gates are physically trivial. The search returned γ_in = π for 11 states (D_7^3, D_8^4, D_9^3, D_10^5, D_11^3, D_12^4, D_12^5, D_13^3, D_13^4, D_14^3, D_14^4), which is identity up to a global phase but is costed as if within-register collisions were present. This affects 1,003 of the 6,231 cost rows and 134,958 of the 11,381,241 reported two-qubit gates (1.19%), removable at compilation without changing the state.

Methodology in Plain English

The protocol borrows the language of collision models, in which a system evolves through a sequence of short interactions with ancillas. Here the ancillas are put to constructive use. K of the N qubits act as mobile "shuttles" carrying excitations in; the remaining N − K qubits are split into two registers that are never directly coupled to each other. Each interaction is a partial swap, U(γ) = cos γ I + i sin γ SWAP, controlled by two angles: γ_sh for ancilla–register collisions and γ_in for collisions between neighbouring sites inside a register. A layer of single-qubit phase rotations closes each round.

Depth is counted in "sub-rounds," one sub-round being a single ancilla completing its sweep across both registers (with within-register collisions included when γ_in ≠ 0); a full round is K sub-rounds. Because a nontrivial partial swap has three nonzero canonical coordinates, it compiles to three CZ (or CNOT) entangling gates interleaved with single-qubit rotations — hence the 3 × pSWAP rule for the all-to-all gate count.

The optimization treats the collision strengths as free design parameters. At each depth, 256 refined candidates are compared under box constraints, and the winner is selected on post-phase fidelity across both candidate and depth. Because the number of collisions is a free variable, the search naturally produces operating points spanning many depths. Two independent searches were run — one to m = 200 and one to m = 49 — and combined by a stated selection rule, since neither dominates. Fidelity itself comes from exact statevector simulation with no noise model.

For hardware, ten arms (one parameter choice per state) were transpiled to IQM Emerald's native gate set on a chosen physical-qubit layout and run in verbatim mode. Fidelity was reconstructed by a signed weighted sum of measured Pauli expectation values, with per-qubit readout-error unfolding from the calibration snapshot. The authors also report the target-excitation population P_K; the ratio F/P_K has a clean interpretation as the fidelity that would be obtained after an ideal, nondestructive measurement of excitation number — though they stress that no such heralding measurement was implemented, and that measuring all qubits in the computational basis would destroy the coherences that distinguish the Dicke state from an incoherent mixture.

Why This Matters

Most preparation proposals are judged on noiseless fidelity and gate count, which on real hardware can point to operating points that are far too deep to survive. This release makes the third axis — measured performance and its degradation with depth — available alongside the other two, for every point on the frontier rather than a single converged result. It also establishes a reusable benchmark: alternative optimizers can be run against the same stored problem and compared cell by cell, and noise models can be scored against the deposited depth series, which contains measurements at five circuit depths per state including P_K.

Applications the paper situates the work within:

• Quantum sensing and magnetometry, where a phase-estimation magnetometer based on superconducting fluxonium sensors showed Dicke probes trading peak local sensitivity for a substantially broader dynamical range and greater robustness to noise and bias errors relative to GHZ probes.
• Quantum networking and distributed sensing, where multipartite entanglement is a shared resource.
• Metrology in symmetric subspaces and entanglement detection through collective observables.
• Collective-emission and superradiance studies in cavity and waveguide QED, the original setting in which Dicke states were introduced.

Industry relevance: the benchmark is built on a commercially accessible superconducting platform (IQM Emerald via Amazon Braket, with execution cost reported), and it speaks directly to the practical question of choosing a circuit depth under a gate budget and a known, highly non-uniform error map — a decision any hardware-facing quantum algorithm team has to make.

Future Directions

• Better searches on the same problem. Every L1 cell is a best-found value, not a certified global optimum. The deposited starting points, refined pairs, losses, box constraints and selection rule are explicitly intended so that an alternative optimizer can be compared cell by cell, and _raw.npy versus _final.npy lets the collision-strength stage and the phase stage be improved independently.
• Richer estimators on the same shots. Full permutationally invariant tomography, maximum-likelihood reconstruction, alternative readout unfolding, or bootstrapping over shots (rather than over measurement settings) can each be run on the deposited counts — the last directly addressing the three-degree-of-freedom limitation the authors flag.
• Closing the 544-cell gap. Exact interaction counts for the shallower-search cells require per-candidate collision strengths that were not deposited; supplying them would convert those conservative upper bounds into exact costs, and would also let the four excluded states (D_12^6, D_13^6, D_14^6, D_14^7) re-enter the released set.
• Heralded excitation-number measurement. The ratio F/P_K is an idealized conditional fidelity; an actual nondestructive heralding protocol, with its own error budget, remains an open experimental target. Extending the state range beyond N = 14 and K = 5, and fitting better noise models to the five-depth series, are the other natural next steps.

Target Audience

Quantum-information researchers working on multipartite state preparation and on near-term superconducting hardware will get the most from this, particularly those who need to pick an operating point under a gate budget rather than quote a noiseless optimum. It is equally useful to methodologists building or benchmarking optimizers, to experimentalists who want raw counts and calibration snapshots for re-analysis, and to graduate students looking for a worked example of how fidelity, circuit cost and device noise interact. Readers seeking new theory will find the protocol itself in the companion paper; this article is the data and validation record.

Authors’ abstract

Dicke states are multipartite entangled states in which a fixed number of quantum excitations is coherently shared among many qubits. Originally introduced in the context of cooperative emission and superradiance, they are now important resources for quantum sensing, networking, and collective quantum phenomena. Preparing prescribed Dicke states with high fidelity, however, remains challenging, particularly as the system size and excitation number increase. Here we present an open dataset and accompanying code for preparing Dicke states using a collision-based quantum protocol. The dataset covers systems from five to fourteen qubits over a broad range of excitation numbers and records how the best-found noiseless preparation fidelity changes with circuit depth. It also provides circuit-resource estimates and experimental measurements for selected states on the 54-qubit IQM Emerald superconducting processor. The accompanying code reproduces the processed data and validation checks, providing a reusable benchmark for studying the trade-off between state-preparation fidelity, circuit cost, and hardware noise.