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A Race Track Trapped-Ion Quantum Processor

S. A. Moses, C. H. Baldwin, M. S. Allman, R. Ancona, L. Ascarrunz, C. Barnes, J. Bartolotta, B. Bjork, P. Blanchard, M. Bohn, J. G. Bohnet, N. C. Brown, N. Q. Burdick, W. C. Burton, S. L. Campbell, J. P. Campora, C. Carron, J. Chambers, J. W. Chan, Y. H. Chen, A. Chernoguzov, E. Chertkov, J. Colina, J. P. Curtis, R. Daniel, M. DeCross, D. Deen, C. Delaney, J. M. Dreiling, C. T. Ertsgaard, J. Esposito, B. Estey, M. Fabrikant, C. Figgatt, C. Foltz, M. Foss-Feig, D. Francois, J. P. Gaebler, T. M. Gatterman, C. N. Gilbreth, J. Giles, E. Glynn, A. Hall, A. M. Hankin, A. Hansen, D. Hayes, B. Higashi, I. M. Hoffman, B. Horning, J. J. Hout, R. Jacobs, J. Johansen, L. Jones, J. Karcz, T. Klein, P. Lauria, P. Lee, D. Liefer, C. Lytle, S. T. Lu, D. Lucchetti, A. Malm, M. Matheny, B. Mathewson, K. Mayer, D. B. Miller, M. Mills, B. Neyenhuis, L. Nugent, S. Olson, J. Parks, G. N. Price, Z. Price, M. Pugh, A. Ransford, A. P. Reed, C. Roman, M. Rowe, C. Ryan-Anderson, S. Sanders, J. Sedlacek, P. Shevchuk, P. Siegfried, T. Skripka, B. Spaun, R. T. Sprenkle, R. P. Stutz, M. Swallows, R. I. Tobey, A. Tran, T. Tran, E. Vogt, C. Volin, J. Walker, A. M. Zolot, J. M. Pino

arXiv:2305.03828v2quant-ph

TL;DR

Scaling trapped-ion quantum computers requires larger systems without sacrificing performance. This paper characterizes and benchmarks a 32-qubit QCCD processor with a race-track trap design, finding maintained or improved fidelities and system-level performance consistent with component benchmarks.

  • Problem

    The paper addresses how trapped-ion QCCD systems can increase qubit count and reduce physical resources per qubit while maintaining high circuit fidelity.

  • Method

    The authors characterize a new race-track QCCD processor through component, system-level, and application benchmarks.

  • Results

    The 32-qubit system maintains or exceeds prior fidelity metrics, with system-level benchmarks consistent with component-measured errors.

  • Takeaways & Limitations

    Broadcast electrode control, under-trap RF routing, and MOT loading support the QCCD architecture’s viability as a route toward large-scale trapped-ion quantum computing.

  • Takeaways & Limitations

    Two-qubit gates remain the dominant error source, despite slight fidelity improvements in the new generation.

Abstract

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We describe and benchmark a new quantum charge-coupled device (QCCD) trapped-ion quantum computer based on a linear trap with periodic boundary conditions, which resembles a race track. The new system successfully incorporates several technologies crucial to future scalability, including electrode broadcasting, multi-layer RF routing, and magneto-optical trap (MOT) loading, while maintaining, and in some cases exceeding, the gate fidelities of previous QCCD systems. The system is initially operated with 32 qubits, but future upgrades will allow for more. We benchmark the performance of primitive operations, including an average state preparation and measurement error of 1.6(1)$\times 10^{-3}$, an average single-qubit gate infidelity of $2.5(3)\times 10^{-5}$, and an average two-qubit gate infidelity of $1.84(5)\times 10^{-3}$. The system-level performance of the quantum processor is assessed with mirror benchmarking, linear cross-entropy benchmarking, a quantum volume measurement of $\mathrm{QV}=2^{16}$, and the creation of 32-qubit entanglement in a GHZ state. We also tested application benchmarks including Hamiltonian simulation, QAOA, error correction on a repetition code, and dynamics simulations using qubit reuse. We also discuss future upgrades to the new system aimed at adding more qubits and capabilities.

I. INTRODUCTION … B. Ion loading and state preparation

The paper presents H2, a race-track QCCD trapped-ion processor designed to improve scaling through a new trap architecture, electrode broadcasting, RF tunnels, and MOT loading. The 32-qubit system combines upgraded hardware and operations with component, system-level, and algorithmic benchmarking, while retaining a path to larger qubit counts.

  • I. INTRODUCTION: H2 increases qubit number and reduces physical resources per qubit while matching or surpassing the circuit fidelity of the previous-generation system.The paper characterizes H2 as a new QCCD-based trapped-ion quantum computer.
  • I. INTRODUCTION: Linear trap geometries create severe scaling challenges because rearrangement time for arbitrary circuit connectivity increases poorly with qubit number.The paper identifies 2D traps as a likely future direction, while noting their unresolved engineering challenges.
  • I. INTRODUCTION: H2 introduces RF tunnels, voltage broadcasting, and MOT loading to reduce control complexity and initialization time while enabling a new trap design.The upgraded operations also include higher-performance and more efficient gating primitives, with component, system-level, and algorithmic benchmarks reported.
  • A. Trap design: H2 uses a race-track geometry with concentric RF electrodes, eight gate zones, and conveyor-belt regions for ion storage and transport.The RF electrodes operate at approximately 200 V and 42 MHz, creating an RF-null 70 µm from the surface.
  • A. Trap design: Voltage broadcasting ties repeating groups of conveyor-belt electrodes to three external voltage signals while supporting 20 wells on each side.The repeating electrode pattern is {a, b, c, a, b, c, ...}.
  • A. Trap design: 376 electrodes are connected to 268 independent voltage sources and one RF drive, reducing electrical feedthroughs per qubit as system size grows.The trap connects its electrodes to DC control signals through a 280-pin ceramic pin grid array.
  • B. Ion loading and state preparation: 1.2 ms is the best-case loading time for one 171Yb+, while 40 ms is the best-case loading time for one 138Ba+; loading 32 deterministic 171Yb+-138Ba+ pairs takes about 3–4 minutes.Under normal operating conditions, the MOT beam does not affect processor behavior.
  • B. Ion loading and state preparation: 8 qubits are prepared at a time in the DG zones, requiring four rounds to prepare all 32 qubits in the |0⟩ state.Preparation uses optical pumping into the hyperfine clock-state qubit subspace of 171Yb+.

C. Quantum gates … F. Classical programming and CPU-QPU interactions

The processor combines Raman-based gates, fluorescence measurement, physically transported ions, and real-time classical control. Its architecture supports scalable gate-zone operation, mid-circuit measurement, compiler-directed transport, and hybrid quantum-classical programs.

  • C. Quantum gates: Raman 1Q gates use co-propagating beams, while 2Q gates use Δk coupling to the axial motional mode.These operations are implemented in dedicated DG gate zones.
  • C. Quantum gates: UZZ(θ) gates combine a phase-sensitive Mølmer-Sørensen interaction with 1Q wrapper pulses, with θ controlled by detuning, duration, and Rabi rate.Modeling predicts infidelity decreasing roughly linearly with θ toward a finite offset of ≈5 × 10^-4 as θ approaches zero.
  • C. Quantum gates: Four 2Q laser beam pairs currently operate four gate zones, while one additional pair could enable four more zones.The 2Q beams impose the strictest requirements and consume a large share of the laser power budget.
  • D. Measurement: State-dependent resonance fluorescence enables measurement in DG zones, with a photomultiplier tube array detecting all eight gate zones independently.Measurement operations are currently implemented only in the DG zones.
  • D. Measurement: Mid-circuit measurement and reset preserve quantum information on other qubits but introduce small crosstalk from stray measurement and reset light.Conveyor-belt ions experience similar crosstalk levels to ions in gate zones, without micromotion hiding applied there.
  • E. Ion transport: Arbitrary qubit connectivity is achieved by transporting ions through split/combine operations, linear shifts, swaps, and collective batch shifts.In 32-qubit operation, ions form four batches of eight distributed across gate and storage regions.
  • E. Ion transport: The compiler layers gates subject to participation and ordering constraints, then schedules transport to position interacting qubits together while minimizing transport time.Cooling sheet beams cover all 138 Ba+ ions during transport, with about 25% center-to-edge intensity variation that does not limit performance.
  • F. Classical programming and CPU-QPU interactions: Programs compile from different frameworks and languages to OpenQASM 2.0 or QIR, supporting real-time classical operations, measurement-conditioned feed-forward, and error-correction recovery.Syndrome results can be decoded classically to update quantum circuits in real time.

III. COMPONENT OPERATIONS AND BENCHMARKS · IV. SYSTEM-LEVEL BENCHMARKS

The paper benchmarks component-level errors using targeted experiments on gate zones and all 32 qubits, then evaluates whether system-level circuit performance aligns with those measurements. System-level predictions generally agree with inferred effective errors, while memory-error dependence on circuit structure limits exact agreement.

  • III. COMPONENT OPERATIONS AND BENCHMARKS: Component benchmarking isolates gate and SPAM errors in eight qubits across four dynamical-gate zones, while memory errors are measured across all 32 qubits.Quantum operations are performed only in the dynamical-gate zones; interleaved transport randomized benchmarking probes errors during circuits.
  • III. COMPONENT OPERATIONS AND BENCHMARKS: The benchmark suite measures SPAM, single-qubit, two-qubit, SU(4), parameterized-gate, crosstalk, leakage, and transport-related errors.These experiments use state-preparation and measurement tests, randomized benchmarking variants, depumping experiments, leakage-detection gadgets, and interleaved transport benchmarking.
  • III. COMPONENT OPERATIONS AND BENCHMARKS: 1.83(5)×10^-3 is the measured average infidelity per 2Q gate across all four gate zones.A leakage-detection gadget additionally measures population leaving the computational subspace during two-qubit randomized benchmarking.
  • III. COMPONENT OPERATIONS AND BENCHMARKS: Leakage detection uses an ancilla to flag whether a qubit has left the computational subspace, most likely through spontaneous emission during stimulated-Raman gates.The gadget distinguishes leaked from non-leaked qubits through the ancilla measurement outcome.
  • IV. SYSTEM-LEVEL BENCHMARKS: Component benchmarks provide fine-grained error estimates, but crosstalk and non-Markovian errors can cause them to mischaracterize device performance.This motivates benchmarking more complex multi-qubit circuits.
  • IV. SYSTEM-LEVEL BENCHMARKS: System-level analyses assume non-SPAM errors arise from uniform depolarizing noise attached to each 2Q gate, incorporating intervening single-qubit and memory errors.Predicted effective errors combine the parameterized 2Q-gate contribution with twice the Transport 1Q RB error.
  • IV. SYSTEM-LEVEL BENCHMARKS: Predicted and inferred effective errors generally agree, although memory-error estimates can overstate errors when circuit timing differs from the Transport 1Q RB assumption.The paper attributes imperfect agreement to circuit-dependent memory errors, while viewing the overall correspondence as reasonable.

A. Mirror benchmarking · B. Quantum volume · C. Random circuit sampling

The processor was evaluated with mirror benchmarking, quantum volume, and classically verifiable random circuit sampling. It achieved QV = 2^16, while the sampling results are expected to challenge classical simulation.

  • A. Mirror benchmarking: Mirror benchmarking used randomized mirrored circuits with all-qubit 1Q layers and native UZZ(π/2) gates between randomly paired qubits.Experiments were performed on H2 with N = 20, 26, and 32 qubits.
  • A. Mirror benchmarking: The effective two-qubit error in mirror benchmarking did not increase with qubit number.The benchmark included N = 20, 26, and 32 qubits.
  • B. Quantum volume: QV = 2^16 was the highest measured quantum-volume value.Quantum volume uses random SU(4) circuits repeated for N rounds and evaluates heavy-output probabilities.
  • B. Quantum volume: 68.2% heavy-output probability exceeded the 2/3 threshold with greater than two-sigma confidence.The dataset used 200 randomly generated circuits, 100 shots per circuit, and an average of 296 parameterized 2Q gates.
  • C. Random circuit sampling: Random circuit sampling used a two-dimensional nearest-neighbor grid tiling for fair comparison, although this constraint is not imposed by H2 hardware.The circuits followed the classically verifiable repeating EFGHEFGH pattern.
  • C. Random circuit sampling: SWAP operations were implemented in software by relabeling and transporting qubits, enabling the fSim(π/2, π/6) gate with exactly one 2Q gate.The circuits were generated with the Sycamore gate and compiled to H2 using UZZ(5π/12) plus 1Q gates.
  • C. Random circuit sampling: Linear cross-entropy benchmarking fidelity FXEB quantified sampling success for classically verifiable random circuits on the 32-qubit processor.Each plotted data point combined 10 circuits, each executed with 100 shots.
  • C. Random circuit sampling: The cross-entropy benchmark results are expected to pose serious challenges to classical simulations.H2’s 32-qubit capability places these experiments within the classically feasible regime used for verification.

D. N-partite entanglement certification in GHZ states

The H2 processor prepares large GHZ states up to 32 qubits using log-depth unitary and constant-depth adaptive circuits. For N = 32, the estimated fidelities are 0.82(1) and 0.74(1) without SPAM correction, respectively.

  • Motivation: GHZ states provide a demanding coherence test because they are maximally sensitive to global dephasing and enable comparison across quantum hardware.GHZ fidelities have been widely measured across different platforms.
  • State preparation: The processor prepares log-depth GHZ states for N=20, 26, and 32, plus an N = 32 state using a constant-depth adaptive circuit.The adaptive construction uses mid-circuit measurement and feed-forward to create long-range entanglement.
  • Fidelity estimation: N + 1 measurement bases suffice for the complete GHZ fidelity estimation protocol, using population measurements and global parity measurements after 1Q rotations.Parity measurements probe equatorial spin axes θk = kπ/N.
  • Results: 0.82(1) and 0.74(1) are the N = 32 GHZ fidelities for unitary and adaptive preparation without correcting for SPAM errors.Each N measurement circuit and each population-measurement circuit used 50 shots.

V. APPLICATION BENCHMARKS · A. Hamiltonian simulation

The paper evaluates four complementary application benchmarks, including a 32-site transverse-field Ising model simulation. The Hamiltonian simulation agrees reasonably well with the exact dynamics through Jt = 7 without post-processing or error mitigation.

  • V. APPLICATION BENCHMARKS: Four application benchmarks target distinct demands: Hamiltonian simulation, QAOA, large-distance repetition codes, and holographic quantum dynamics simulation.Hamiltonian simulation emphasizes two-qubit gate error, while QAOA emphasizes qubit connectivity.
  • A. Hamiltonian simulation: The Hamiltonian benchmark simulates a 32-site one-dimensional transverse-field Ising model, a classically challenging many-body dynamics problem.The model uses periodic boundary conditions.
  • A. Hamiltonian simulation: The experiment prepares |+⟩^⊗L at h/J = ∞, quenches to h/J = 0.2, and evolves the state to Jt = 7.The observable is the all-qubit average Pauli-X expectation value, ⟨X⟩ = 1/L Σ_j⟨X_j⟩.
  • A. Hamiltonian simulation: First-order Trotterization uses native X rotations and parameterized UZZ(θ) gates, with Trotter steps chosen to keep ⟨X⟩ errors below 0.01.The threshold keeps Trotterization errors at or below the expected approximately 1% statistical scale.
  • A. Hamiltonian simulation: Reasonably good agreement with the exact solution persists through Jt = 7, indicating coherent simulation of dynamics to a nontrivial time.The data were not post-processed or error-mitigated; a completely depolarized state would have ⟨X⟩ = 0.
  • A. Hamiltonian simulation: The benchmark compares direct native UZZ(θ) implementation with a decomposition into two UZZ(π/2) Clifford gates and single-qubit rotations.Each plotted data point averages 100 circuit shots.

B. QAOA · C. Error correction: repetition code

The processor demonstrates QAOA optimization on large and deeper MaxCut instances, including an exact best cut for a 32-qubit p = 2 problem. Its repetition-code experiments show improved logical fidelity with increasing code distance and demonstrate components needed for scalable real-time quantum error correction.

  • B. QAOA: QAOA solves unweighted 3regular MaxCut by variationally optimizing alternating mixing and cost unitaries.The 2p parameters βn and γn are searched using a classical optimization algorithm.
  • B. QAOA: N = 130, p = 1 QAOA uses 32 physical qubits through qubit-reuse compilation, with convergence within the first ten circuits.The experiment uses 100 shots per circuit.
  • B. QAOA: 42: N = 32, p = 2 QAOA found the best possible max cut for the graph.Experimental p = 2 energies consistently outperform the best possible p = 1 circuit and nearly saturate the p = 2 ground-state energy.
  • C. Error correction: repetition code: Logical fidelity increases with code distance for a fixed number of syndrome-extraction rounds, while additional rounds inject noise and reduce fidelity.The bit-flip code has higher logical fidelity than the phase-flip code at a given distance, consistent with biased noise.
  • C. Error correction: repetition code: Repetition codes encode logical information across data qubits and use ancillas for syndrome measurements, correcting either bit-flip or phase-flip errors.Stabilizer-code syndromes discretize errors into manageable error types.
  • C. Error correction: repetition code: d = 31: the system uses 31 data qubits and one ancilla to implement the largest tested repetition code.The implementation uses all 32 qubits and relies on qubit reuse capabilities.
  • C. Error correction: repetition code: Online minimum-weight perfect matching decodes syndromes after logical measurement, while real-time syndrome processing uses Wasm calls during the circuit.Experiments varied syndrome-extraction rounds for both d = 31 bit-flip and phase-flip codes and reconstructed smaller odd-distance codes.

D. Holographic quantum dynamics simulation (HoloQUADS) · VI. A SUMMARY OF THE RESULTS AND OUR OUTLOOK

HoloQUADS uses a 32-qubit trapped-ion processor to simulate dual-unitary quantum dynamics, with experimental data closely matching ideal noiseless results. The summary reports H2’s scalable QCCD technologies, robust benchmark performance, and a 32-qubit design intended for further expansion.

  • D. Holographic quantum dynamics simulation (HoloQUADS): Dual-unitary brick-work circuits combine typical-circuit properties with analytically tractable entanglement and correlation dynamics.They exhibit quantum chaos and ballistic entanglement growth while allowing analytical determination of quantities such as entanglement entropy and correlation functions.
  • D. Holographic quantum dynamics simulation (HoloQUADS): An initial matrix product state is prepared with physical–ancilla bond gates and evolved using the self-dual kicked Ising model.After t layers, smoothed correlation functions are measured on the time-evolved state.
  • D. Holographic quantum dynamics simulation (HoloQUADS): 32 qubits simulate t = 0, 8, 16, 24 layers for matrix product states with lengths L = 128, 136, 144, 152.The results are compared with exact theoretical calculations.
  • D. Holographic quantum dynamics simulation (HoloQUADS): Experimental HoloQUADS data show close agreement with ideal noiseless results, indicating low enough mid-circuit and memory-related errors for sizeable dynamics simulations.The relevant errors include mid-circuit measurement, mid-circuit reset, crosstalk, and memory errors; their effects remain circuit dependent.
  • VI. A SUMMARY OF THE RESULTS AND OUR OUTLOOK: H2 significantly upgrades H1 by maintaining or exceeding many previous fidelity metrics while operating on more qubits.System-level benchmarks are consistent with errors measured by component benchmarks, supporting robust scaling of the QCCD architecture.
  • VI. A SUMMARY OF THE RESULTS AND OUR OUTLOOK: H2 demonstrates ion transport controlled by broadcast electrode signals, RF signals routed under the trap surface, and fast MOT-based loading.These are identified as key technological milestones toward scaling.
  • VI. A SUMMARY OF THE RESULTS AND OUR OUTLOOK: The system initially operates with 32 qubits but is designed to accommodate more, strengthening the case for QCCD-based large-scale trapped-ion quantum computing.Future development includes truly two-dimensional trapping structures for fast ion sorting.

Appendix A: Details of component benchmarks · 1. Randomized benchmarking parameters and data · 2. Leakage detection gadget

The appendix specifies randomized-benchmarking procedures across gate zones and explains how leakage detection complements computational-subspace error estimates. It also documents fitting methods, combined-data analysis, benchmark parameters, and leakage-detection limitations.

  • 1. Randomized benchmarking parameters and data: Component benchmarks were repeated for gate zones DG01-DG04, while transport 1Q RB used all 32 qubits with random rearrangements.Sequences were independently generated for each qubit or qubit pair, and Table V lists sequence lengths, repetitions, and shots.
  • 1. Randomized benchmarking parameters and data: RB decay was fit to the standard first-order RB function with a fixed asymptote.The fit uses survival probability, SPAM, depolarizing rate, and qubit-count parameters; reported error is average infidelity.
  • 2. Leakage detection gadget: RB captures computational-subspace errors only, whereas leakage errors are measured with a leakage detection gadget.The appendix separately defines the leakage rate as population leaving the computational subspace under a process Λ.
  • 1. Randomized benchmarking parameters and data: Reset and measurement crosstalk decays were fit using functions derived from operation-specific error models.The fitted scattering rates were converted to average infidelity.
  • 1. Randomized benchmarking parameters and data: Combined component estimates were obtained by analyzing datasets pooled across all measured qubits.For 1Q RB, each qubit measurement was treated as a sequence randomization, producing 8 × 40 random sequences for each length.
  • 2. Leakage detection gadget: Leakage detection events were fit to a model using identity operators on the leakage and computational subspaces.The resulting leakage-detection analysis is shown in Fig. 4b and Fig. 16b.
  • 2. Leakage detection gadget: Leakage-gadget gate errors can produce false-positive or false-negative events, but affect only the length-independent fit parameter A.This behavior is analogous to the SPAM parameter in randomized benchmarking.
  • 2. Leakage detection gadget: Table VI reports component-benchmark results as average infidelity in units of ×10−4.For 1Q RB, 1Q leakage rate, measurement and reset crosstalk, and SPAM, brackets give average infidelity for each side of the gate zone.

3. 2Q parameterized randomized benchmarking

This section introduces direct randomized benchmarking for parameterized two-qubit UZZ(θ) gates, using native-gate sequences and specialized treatment of the θ = 0 limit. The method extracts average infidelity from survival-probability decay while accounting for the reduced gate-set structure at θ = 0.

  • Direct RB construction: The native-gate sequences generate a group whose survival probability approximately decays exponentially, with the decay parameter linearly related to average fidelity.This enables direct estimation of average infidelity from fitted decay curves.
  • Direct RB construction: Direct RB repeatedly applies fixed-angle UZZ(θ) gates interleaved with Haar-random SU(2) gates on each qubit.For θ > 0, the native gate set generates SU(4).
  • Infidelity extraction: For θ ∈ {π/8, π/4, 3π/8, π/2}, decay curves fit p(ℓ) = Ar^ℓ + 1/4, and average infidelity is computed from the fitted model.The θ-dependent results are shown in Fig. 3.
  • θ = 0 treatment: At θ = 0, the gate set becomes SU(2)⊗SU(2), so single-exponential RB theory no longer applies.A near-zero experiment at 2 × 10^-4 measures baseline error from MS wrapper pulses and memory accumulated during cooling pulses.
  • θ = 0 treatment: The θ = 0 fidelity estimate uses SU(2)⊗SU(2) irreducible-representation decay components, with qubit-symmetric errors assumed through r_IZ = r_ZI = r_1.The resulting entanglement fidelity is converted to average infidelity for trace-preserving errors.

Appendix B: Details of system-level benchmarks · 1. Mirror benchmarking

Mirror benchmarking evaluates survival decay to characterize effective two-qubit performance. The appendix describes Pauli-based analysis, depolarizing-channel extraction, and reports a quantum-volume measurement on H2.

  • 1. Mirror benchmarking: Average survival probability versus sequence length ℓ is fit to a decay model to obtain the mirror-benchmarking parameter u.The fit connects measured survival probabilities with the reported decay parameter.
  • 1. Mirror benchmarking: The Pauli fidelity of an N-qubit error channel E is defined using the N-qubit Pauli operators {Pi}i, with P0 = I.This definition supplies the fidelity quantities used in the mirror-benchmarking analysis.
  • 1. Mirror benchmarking: For a constant depolarizing error channel on each 2Q gate, an analytic formula converts fitted u into ϵ2Q.The formula is identified as Eq. (C4) in Ref. [52].
  • 1. Mirror benchmarking: Table VII reports mirror-benchmarking survival probabilities, fit parameter u, and effective 2Q gate average infidelity ϵ2Q.These quantities summarize the experiment’s measured decay and inferred two-qubit performance.
  • 1. Mirror benchmarking: QV = 215 is reported for a quantum-volume measurement on H2.The figure plots average heavy-output frequency against circuit index with a two-sigma confidence interval.

2. Quantum volume … 1. Trotter steps of Hamiltonian simulation experiment

The supplementary benchmarks report QV = 216 and QV = 215 tests, estimate two-qubit error from random-circuit data, and specify a convergence-based procedure for choosing Trotter steps. The Trotter selection targets a tolerance of 0.0025 without being noise-aware.

  • 2. Quantum volume: QV = 215 was measured using 100 random circuits, 50 shots per circuit, and an average of 243 parameterized 2Q gates.The measured heavy output probability was 70.9%, above threshold with over two-sigma confidence.
  • 2. Quantum volume: QV = 215 achieved a 70.9% heavy output probability above threshold with over two-sigma confidence.Confidence was calculated using semi-parametric bootstrap resampling.
  • 2. Quantum volume: The QV analysis converts heavy-output probability into circuit fidelity, then scales it using SPAM error and average 2Q-gate count.The conversion follows Eq. 13 in Ref. and Eq. B6.
  • 3. Random circuit sampling: Linear cross-entropy benchmarking measures correlation between empirical and ideal output distributions but requires exact classical simulation of random circuits.This simulation requirement is identified as a major scalability obstacle.
  • 3. Random circuit sampling: 1 −F2Q = 2.4(2) × 10−3 was obtained by fitting model B6 to H2 data, corresponding to ϵ2Q = 1.9(2) × 10−3.F2Q denotes effective two-qubit entanglement or process fidelity, while ϵSPAM is the component-benchmarking SPAM error.
  • 1. Trotter steps of Hamiltonian simulation experiment: Trotter steps are selected by relative convergence, requiring neighboring noiseless-circuit X expectations to differ by at most 0.0025.The cutoff r satisfies |⟨X⟩r′+1 −⟨X⟩r′| ≤0.0025 for r′ ≥r, with expectations computed exactly using a discrete-time Jordan-Wigner transformation in the Heisenberg picture.
  • 1. Trotter steps of Hamiltonian simulation experiment: The Trotter-step choices and errors are reported alongside the differences between experimental data and exact values.The selection was not noise-aware, and further reducing Trotter steps may improve results in the presence of gate errors.

2. The QAOA optimization landscape · 3. Details of HoloQUADS experiment

The QAOA optimization trace for a 130-node, p = 1 MaxCut instance converged to the energy minimum on a periodic landscape. The HoloQUADS experiment used a bond-dimension-2 matrix product state, SDKI dynamics, qubit reuse, and leakage detection on H2.

  • 2. The QAOA optimization landscape: Bootstrap reverse-percentile uncertainties quantify shot noise in ⟨HC⟩ for H2, excluding physical machine noise.Fig. 12 and Fig. 13 points correspond to different β and γ values.
  • 3. Details of HoloQUADS experiment: The HoloQUADS experiment specifies matrix elements through W = exp[−i(KxXX + KyY Y + KzZZ)].The matrix-element expression uses the displayed bra-ket construction involving physical and bond-qubit states.
  • 2. The QAOA optimization landscape: N = 130 and p = 1 define the MaxCut QAOA instance whose optimization trace converged to the minimum energy.The trace began near the landscape maximum and entered the potential well; evaluated circuits are marked by stars, and the periodic trajectory wraps around the plot.
  • 3. Details of HoloQUADS experiment: The HoloQUADS initial state is a χ = 2 matrix product state prepared using one ancilla bond qubit and physical-qubit gates.The state uses (Kx, Ky, Kz) = (0.3, 0.5, 1.25).
  • 3. Details of HoloQUADS experiment: The state evolves under the SDKI model, formulated as a dual-unitary circuit with two-qubit gates and h = 0.05.This value is close to, but not exactly at, the integrable h = 0 point where correlation functions do not decay.
  • 3. Details of HoloQUADS experiment: Qubit reuse with mid-circuit measurement and reset constructs the circuit while using H2’s four gate zones as much in parallel as possible.The implementation uses MCMR techniques.
  • 3. Details of HoloQUADS experiment: 2%, 2%, 5%, and 7% bond-qubit leakage occurred at t = 0, 8, 16, and 24, respectively, and leaked results were discarded.Leakage detection used the gadget shown in Fig. 5.
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