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New Design of three-qubit system with three transmons and a single fixed-frequency resonator coupler

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A single fixed-frequency resonator can run CNOT gates among three transmon qubits with average fidelity above 0.98.

desk verdict A plausible three-transmon single-resonator design with solid closed-system simulations, but the fidelity formula is a typo and the hardware-level claim outruns the evidence. read the letter →

arxiv 2412.15629 v1 pith:QY6MXYP6 submitted 2024-12-20 quant-ph

Jeongsoo Kang , Chanpyo Kim , Younghun Kim , Younghun Kwon This is my paper · ORCID
classification quant-ph
keywords three-transmonsystemresonatorcouplercross-resonancegateCNOTaveragefidelitytransmonqubitfixed-frequencyall-microwavecontrol
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a wiring alternative to the usual one-resonator-per-qubit-pair layout: three fixed-frequency transmon qubits all coupled through a single resonator coupler. Using cross-resonance microwave pulses, the authors design CNOT gates for every ordered pair of qubits and report an average gate fidelity above 0.98 and gate times under 200 ns in a closed-system simulation. If this holds in hardware, a single resonator could fan out to more than two qubits, increasing connectivity in transmon processors without adding flux-tunable elements or extra couplers. The claim matters because connectivity and gate fidelity are two main bottlenecks in scaling superconducting quantum computers.

What carries the argument

The mechanism that carries the design is the cross-resonance effect on a shared resonator bus. A microwave pulse applied to one transmon at the frequency of another transmon makes the second qubit rotate in a direction that depends on the first qubit's state; the resonator mediates the interaction without needing a flux-tunable coupler. The paper implements this with a sinusoidal flat-top envelope for the cross-resonance pulse, a Gaussian auxiliary pulse on the target qubit, DRAG correction to suppress population of the transmon's higher levels, and virtual Z gates to fix the frame. The simulation includes transmon levels up to the third excited state and treats the resonator staying in its ground state as the success condition.

What would settle it

Add energy relaxation and dephasing to the Hamiltonian of Eqs. (1)-(4) and re-optimize the pulse parameters; if the average CNOT fidelity drops below 0.98, the closed-system claim does not transfer to real hardware. A direct device measurement of CNOT12, the lowest-fidelity gate in Table 2, would give a sharp test of the model.

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Extended reading notes

Core claim

The paper's central claim is that the transmon-resonator-transmon building block is not the only viable wiring choice for a superconducting processor: a single fixed-frequency resonator coupler can mediate interactions among three transmon qubits at once. Concretely, the authors specify qubit frequencies, charging and Josephson energies, and qubit-resonator couplings (Table 1) and give optimized pulse parameters for six CNOT directions (Table 2). The average fidelity over these six gates exceeds 0.98, and the average basis-state success probability also exceeds 0.98. The protocol uses only local microwave drives: a cross-resonance pulse on the control qubit, an auxiliary Gaussian pulse on the target, DRAG shaping to suppress leakage, and virtual Z rotations. The authors position this as a connectivity improvement over two-qubit-per-coupler layouts and a gate-time improvement over echoed cross-resonance gates.

Load-bearing premise

The load-bearing premise is that the unmodeled effects of relaxation, dephasing, and parasitic crosstalk will not substantially lower the simulated fidelities when the design is built in real hardware.

Editorial extensions

If this is right

  • A single resonator can serve more than two qubits, so transmon processors could increase qubit connectivity without adding a coupler per pair.
  • Because all gates are driven by local microwave pulses with no flux bias, the design avoids flux-noise dephasing that limits tunable-frequency architectures.
  • CNOT gate times under 200 ns are shorter than echoed cross-resonance gates, which could reduce accumulated error in deep circuits.
  • Every ordered qubit pair has a direct CNOT, potentially reducing the number of SWAP gates needed to route algorithms on near-neighbor hardware.
  • The reported average fidelities above 0.98 and basis-state success probabilities provide a concrete benchmark that future hardware implementations can be compared against.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next check is whether the idle qubit's state stays intact for all six gates and all input states; the paper shows the idle Bloch vector static only for CNOT01, and the shared bus may still cause spectator errors in other configurations.
  • The 0.98 headline is an average: CNOT12 (0.964) and CNOT02 (0.971) fall below it, so fault-tolerance thresholds should be evaluated against the worst pair rather than the average.
  • The same optimization pipeline could be applied to four or more transmons per resonator, but mode crowding and parasitic interactions would likely grow; the paper does not quantify that scaling limit.
  • A direct hardware test would be to fabricate the specified device and measure the six CNOT fidelities; the design's viability in a real processor depends on whether the omitted relaxation, dephasing, and crosstalk terms preserve the ordering of the simulated fidelities.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes a three-transmon architecture in which a single fixed-frequency resonator coupler mediates the interaction among all three qubits. The authors design cross-resonance based microwave pulses for CNOT gates on each qubit pair, optimize the pulse parameters numerically against an average fidelity objective, and report average fidelities in Table 2. The central claim is that a single resonator coupler can support more than two transmons while maintaining average two-qubit CNOT fidelities above 0.98.

Significance. The proposed architecture is practically motivated: fixed-frequency transmons with microwave-only control avoid flux-noise-induced dephasing, and a single resonator coupling several qubits could increase connectivity compared with pairwise transmon-resonator-transmon structures. The simulations use a multilevel Hamiltonian (including transmon levels up to |3> and resonator states up to |3>), which is more credible than a purely two-level treatment, and the authors publish their pulse parameters in a repository. If the fidelity metric is corrected and the closed-system limitation is made explicit, the work could provide a useful data point for experimental groups designing fixed-frequency processors. However, the paper's central quantitative claims currently rest on a fidelity formula that is incorrect as written, and the hardware-oriented conclusions are drawn from noise-free simulations.

major comments (3)
  1. [Quantum Gate Optimization (Eq. (12))] The fidelity definition F_ψ = |<ψ|U†U|ψ>| is identically 1 for a unitary target gate U, so it cannot produce the values reported in Table 2. The manuscript needs to define the actual time-evolution operator (e.g., V or U_pulse) and use a standard expression such as F_ψ = |<ψ|U_target† U_pulse|ψ>|^2, with an explicit projection onto the computational three-qubit subspace. Without this correction, the numerical results are not reproducible as stated and the central claim is unverifiable.
  2. [Discussion] The sentence "CNOT gates with a fidelity of at least 0.988 can be achieved" is contradicted by the paper's own Table 2, which lists CNOT12 = 0.9640 and CNOT02 = 0.9713, and by the corresponding basis-state success probabilities of 0.9681 and 0.9720 in Figure 3. If the intended claim is only that for each qubit pair at least one CNOT direction achieves a fidelity above 0.988, that should be stated explicitly; otherwise the statement must be revised.
  3. [Results and Discussion (closed-system simulation)] All fidelities are computed from the closed-system Hamiltonian in Eqs. (1)-(4), with no relaxation, dephasing, or parasitic crosstalk terms. The authors themselves note in Methods that real hardware has relaxation and dephasing, and the gate times are 110-190 ns. With typical transmon T1/T2 values of tens to hundreds of microseconds, the expected decoherence error is on the order of 0.1-0.4%, which is comparable to the margin by which the reported average (0.9839) exceeds 0.98 and is larger than the shortfall of the two sub-0.98 gates. The abstract and Discussion frame the design as "preserving gate performance" in a transmon-based quantum computer; that hardware-level claim requires at least an order-of-magnitude estimate of open-system effects or a Lindblad/dephasing simulation.
minor comments (6)
  1. [Introduction] There are several typos, including "It is appearant" and "SW AP gates"; a careful proofreading pass is needed.
  2. [Table 2] The column header "ΩX Sq, ΩS" is ambiguous; please clarify the meaning of the two entries per gate (pulse amplitude and envelope shape) and their units.
  3. [Table 4] The two rows both labeled "CNOT(sym.) 01" should be distinguished, for example by indicating the control-target direction.
  4. [Figure 3 caption] The caption says "The average of the gate success probability of (a)CNOT01..." but the values listed are averages over the eight computational basis states; please make this explicit.
  5. [Methods (pulse optimization)] The sentence "The shape index q is not variable for the pulse optimization but fixes the shape of the envelope" is unclear; it should specify that q is fixed beforehand and not optimized.
  6. [References] Some references have formatting errors, e.g., "Nakamura, Y .," and inconsistent capitalization of journal names.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CNOT fidelities are outcomes of an explicit pulse-optimization simulation, not quantities defined into existence; the only self-citation is introductory background.

full rationale

The derivation chain is self-contained. The paper fixes a concrete closed-system Hamiltonian (Eqs. 1-4), defines a pulse protocol (Eqs. 14-16), and minimizes the infidelity 1-F (Eq. 13) with the Nelder-Mead algorithm. The fidelities in Table 2 are then the evaluated figures of merit for the optimized pulses. This is a constructive optimal-control demonstration for the stated model: the 0.98 threshold is not inserted into the Hamiltonian or the objective; it is a numerical outcome that the optimization happened to reach for five of six gates (and 0.9839 on average). The two gates below 0.98, CNOT12 (0.9640) and CNOT02 (0.9713), show that the result is not forced by construction. No 'prediction' is made from a fitted subset of data: all pulse parameters were optimized against the full fidelity and are presented as design parameters, not as independent predictions. The only self-citation (Ref. 4) supports background on the Cooper-pair box and carries none of the argument; no uniqueness theorem or prior same-author result is invoked to force the architecture. The paper's own Discussion notes that the shorter asymmetric CNOT time could help in real hardware where relaxation and dephasing are present, implicitly flagging that closed-system fidelities are not hardware guarantees; that is a correctness/validity limitation, not a circular step. The absence of relaxation, dephasing, and crosstalk in the simulation concerns whether the model is realistic, not whether the reported numbers reduce to their own inputs.

Assumptions & free parameters 11 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the closed-system circuit-QED model, a truncated Hilbert space, and a set of pulse parameters that are numerically optimized to the CNOT target. No new physical entities are introduced. The hardware specifications are taken from IBM devices, so they are inputs rather than fitted parameters. The main free parameters are the pulse shape, frequency, duration, amplitude, phase, and rotation angles, all optimized per gate.

free parameters (11)
  • CR pulse frequency f1 = varies per gate, e.g., 4.9783 GHz for CNOT01
    Optimized to maximize average CNOT fidelity; tuned near the target qubit frequency.
  • Auxiliary pulse frequency f2 = varies per gate, e.g., 4.9783 GHz for CNOT01
    Optimized alongside f1; for most gates it equals the target qubit frequency, except CNOT20 where it is detuned to add single-qubit corrections.
  • Auxiliary pulse duration TX = varies per gate (8.5-10 ns in Table 2)
    Optimized to position the target qubit state after the CR pulse.
  • CR pulse duration TS = varies per gate (110-190 ns in Table 2)
    Optimized as the dominant gate time; the paper aims to minimize it.
  • Auxiliary pulse amplitude ΩX = varies per gate (0.0055-0.035 in Table 2)
    Optimized to rotate the target qubit without exciting leakage states.
  • CR pulse amplitude ΩS = varies per gate (0.04-0.08 in Table 2)
    Optimized to produce the conditional rotation while avoiding unwanted control-qubit X rotation.
  • Rising time ratio ρ = 0.2-0.3 in Table 2
    Sets the adiabatic rising time fraction of the CR pulse envelope; optimized to reduce nonadiabatic errors.
  • CR pulse phase γ1 = 0 for all gates in Table 2
    Optimized phase; set to zero in the reported solutions.
  • Auxiliary pulse phase γ2 = varies per gate, e.g., 2.2007 for CNOT01
    Determines the rotation axis of the target qubit; optimized to align with the CNOT mapping.
  • VZ rotation angles θ0, θ1, θ2 = varies per gate, see Table 2
    Frame-rotation angles applied after the physical pulses to correct relative phases; optimized to complete the CNOT unitary.
  • Envelope shape index q = S1 or S2, per gate in Table 2
    Chosen by hand to fix the envelope shape; not optimized during the numerical search.
assumptions (5)
  • domain assumption The circuit-QED Hamiltonian in Eqs. (1)-(4) accurately describes the three-transmon/resonator system
    The paper starts from the standard transmon and resonator Hamiltonians and capacitive coupling, but does not justify the lumped-element model against a full circuit analysis.
  • domain assumption Truncating each transmon to the lowest four levels and the resonator to Fock states 0-3 is sufficient for accurate gate simulation
    Basis S in Eq. (7) truncates the Hilbert space; convergence in truncation level is not demonstrated.
  • domain assumption Closed-system unitary evolution with no decoherence is an appropriate model for estimating achievable gate fidelity
    The Hamiltonian has no Lindblad or noise terms; the paper uses this to claim a hardware-relevant design.
  • standard math The Suzuki-Trotter decomposition with τ=1 ps gives a faithful time evolution
    They cite De Raedt (1987) for the error bound and assume 1 ps is small enough, but no convergence check is reported.
  • standard math The average gate fidelity formula (Nielsen 2002) applies to the reported quantities
    Cited, but the written Eq. (12) is misstated; the actual formula used is not fully specified.

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Pith. "Pith review of New Design of three-qubit system with three transmons and a single fixed-frequency resonator coupler." pith.science (2026). https://pith.science/paper/QY6MXYP6

@misc{pith2026241215629,
  author       = {Pith},
  title        = {Pith review of: New Design of three-qubit system with three transmons and a single fixed-frequency resonator coupler},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QY6MXYP6}},
  note         = {Machine review of arXiv:2412.15629}
}
read the original abstract

The transmon, which has a short gate time and remarkable scalability, is the most commonly utilized superconducting qubit, based on the Cooper pair box as a qubit or coupler in superconducting quantum computers. Lattice and heavy-hexagon structures are well-known large-scale configurations for transmon-based quantum computers that classical computers cannot simulate. These structures share a common feature: a resonator coupler that connects two transmon qubits. Although significant progress has been made in implementing quantum error correction and quantum computing using quantum error mitigation, fault-tolerant quantum computing remains unachieved due to the inherent vulnerability of these structures. This raises the question of whether the transmon-resonator-transmon structure is the best option for constructing a transmon-based quantum computer. To address this, we demonstrate that the average fidelity of CNOT gates can exceed 0.98 in a structure where a resonator coupler mediates the coupling of three transmon qubits. This result suggests that our novel structure could be a key method for increasing the number of connections among qubits while preserving gate performance in a transmon-based quantum computer.

Figures

Figures reproduced from arXiv: 2412.15629 by the authors.

Figure 1
Figure 1. Illustrations of the three-transmon system and conventional transmon-based quantum computers. (a) The three-transmon system, which consists of three transmons and a single resonator. T and R denote the transmon qubit (blue box) and the resonator coupler (orange circle), respectively. The black solid line connecting the transmon and the coupler indicates that a coupling capacitor mediates the interaction between the … view at source ↗
Figure 2
Figure 2. Pulse protocol for CNOT gate implemented in the three-transmon system. The Blue(red) line denotes the offset number ng(t) of the control(target) qubit. The idle qubit is at rest during the CNOT gate. ng(t) of the idle qubit is illustrated as the green line. The CR(auxiliary) pulse is applied by the gate time TS(X) . Trise refers to the rising time of the CR pulse, which is the time it takes from zero pulse to the pl… view at source ↗
Figure 3
Figure 3. The success probability of CNOT gates for the three-qubit system in the computational basis. The average of the gate success probability of (a)CNOT01, (b)CNOT10, (c)CNOT12, (d)CNOT21, (e)CNOT20, and (f)CNOT02 are 0.9975, 0.9967, 0.9681, 0.9882, 0.9928, 0.9720 for the basis states respectively. We investigate the expected values of all the Pauli operators for each qubit while applying the designed CNOT gates. This pr… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Trajectory of the Bloch vector for each qubit during CNOT01. The passage of time flows from blue to red as indicated by the colorbar. The Bloch vector of ‘Transmon 0’(a, d), ‘Transmon 1’(b, e), and ‘Transmon 2’(c, f) are illustrated in the case of the initial state |00…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Construction of new type of CNOT gate using cross-resonance pulse in the transmon-PPQ system

    quant-ph 2025-01 conditional novelty 5.0 of 10

    A cross-resonance pulse sequence can implement a CNOT gate between a transmon and a parity-protected qubit with simulated average fidelity above 0.998.

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