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Entanglement generation in superconducting qubits using holonomic operations
Daniel J. Egger, Marc Ganzhorn, Gian Salis, Andreas Fuhrer, Peter Mueller, Panagiotis Kl. Barkoutsos, Nikolaj Moll, Ivano Tavernelli, Stefan Filipp
TL;DR
Fixed-frequency superconducting qubits need additional controllable two-qubit operations, particularly direct holonomic entanglers. The paper implements a non-adiabatic Λ-system operation using simultaneous microwave drives through a common resonator, creating two-qubit states above 95% fidelity in 213 ns, with charge dispersion identified as the main limitation.
Problem
Fixed-frequency qubits improve coherence but have reduced controllability, while direct holonomic two-qubit operations remain challenging in scalable architectures.
Method
The experiment simultaneously drives the |f0⟩↔|g1⟩ transitions of two qubits coupled to a common resonator, using cyclic evolution in an effective Λ-system.
Results
213 ns square pulses create two-qubit states with 95.2% and 95.3% fidelity for Q1 and Q2 as source qubits, respectively.
Takeaways & Limitations
The operation extends the available two-qubit operations in fixed-frequency architectures and enables entanglement manipulation with all-microwave control.
Takeaways & Limitations
Charge dispersion in the qubit |f⟩ states is identified as the main fidelity limitation, while further extension requires separating the relevant excitation manifolds.
Abstract
from arXiv · showhide
We investigate a non-adiabatic holonomic operation that enables us to entangle two fixed-frequency superconducting transmon qubits attached to a common bus resonator. Two coherent microwave tones are applied simultaneously to the two qubits and drive transitions between the first excited resonator state and the second excited state of each qubit. The cyclic evolution within this effective 3-level $\Lambda$-system gives rise to a holonomic operation entangling the two qubits. Two-qubit states with 95\% fidelity, limited mainly by charge-noise of the current device, are created within $213~\rm{ns}$. This scheme is a step toward implementing a SWAP-type gate directly in an all-microwave controlled hardware platform. By extending the available set of two-qubit operations in the fixed-frequency qubit architecture, the proposed scheme may find applications in near-term quantum applications using variational algorithms to efficiently create problem-specific trial states.
I. INTRODUCTION
The paper motivates an all-microwave, non-adiabatic holonomic entangling scheme for fixed-frequency superconducting qubits, addressing the challenge of realizing holonomic two-qubit operations in a scalable architecture.
- Motivation: Fixed-frequency transmons improve coherence by avoiding flux sensitivity but trade away controllability, so resonators enable microwave-activated two-qubit operations.Existing examples include controlled-NOT operations generated with coherent microwave signals through cross-resonance.
- Contribution: The proposed scheme simultaneously drives transitions between a resonator and two fixed-frequency qubits using a holonomy in a driven three-level Λ-system.It realizes a direct non-adiabatic two-qubit holonomic operation that entangles the qubits.
- Background: Holonomic operations use non-Abelian geometric phases generated by steering a system around a closed loop in Hilbert space.Their potential robustness to certain errors motivates interest, while long-duration adiabatic implementations are limited by decoherence.
II. SETUP AND HOLONOMIC OPERATION
The experiment uses two fixed-frequency superconducting qubits coupled to a common coplanar wave-guide resonator and driven independently by time-dependent microwave amplitudes.
- Hardware: Two fixed-frequency qubits are coupled to a common coplanar wave-guide resonator, with coupling strengths g1/(2π) = 156 MHz and g2/(2π) = 196 MHz.The resonator frequency is ωr/(2π) = 6.272 GHz.
- Operating regime: The qubits operate dispersively from the resonator to avoid direct excitation transfer, with T1 times of 42 µs, 56 µs, and 7 µs for Q1, Q2, and the resonator.The qubit frequencies are 4.896 GHz and 4.689 GHz, with anharmonicities of −330 MHz and −333 MHz.
- Control: Each qubit is driven through an individual capacitively coupled charge-bias line using a microwave signal with time-dependent amplitude Ωi(t).The setup independently controls the two qubit drives while preserving the fixed-frequency architecture.
Entanglement with holonomic operations
The holonomic operation uses two microwave-controlled qubit–resonator transitions to form an effective three-state Λ-system, whose cyclic geometric evolution rotates and entangles the two-qubit subspace.
- Effective Λ-system: The operation drives the |f0⟩↔|g1⟩ transition for each qubit, coupling |fg0⟩ and |gf0⟩ through the intermediate state |gg1⟩.The resonator is only partially populated during transfer and is depleted at the end.
- Drive implementation: The two drives use Ωi(t) = λiηi(t) cos(ωd,it + ϕi), with adjustable frequency, phase, envelope, and complex scaling factors.Single-qubit drives address qubit transitions, while difference-frequency drives activate induced Jaynes–Cummings-type vacuum-Rabi oscillations between |f0⟩ and |g1⟩.
- Holonomic condition: Equal pulse envelopes and fixed relative scaling produce a time-dependent Hamiltonian proportional to a time-independent operator, avoiding dark-state transitions.The cross ac-Stark shifts must be compensated so the intended transitions remain resonant.
- Geometric evolution: A cyclic evolution satisfies parallel transport and is therefore purely geometric, returning states to the {|fg0⟩, |gf0⟩} subspace after the pulse.The cyclic condition is set by the integrated effective coupling, ∫ 0^T ɡ̃(t)dt = π.
- Entanglement control: Changing λ1 and λ2 controls arbitrary rotations between |fg⟩ and |gf⟩, while the drive phase difference φ controls their relative phase at fixed operation time T.The rotation angle θ sets population transfer between the qubits, and φ sets the relative phase.
III. CALIBRATION OF THE SIMULTANEOUS TWO-TONE DRIVE
The calibration compensates drive-induced frequency shifts and equalizes effective qubit–resonator couplings so simultaneous two-tone driving realizes the holonomic operation. Cross ac-Stark shifts are measured under two-tone driving and calibrated as functions of the rotation angle.
- Individual ac-Stark calibration: Individual ac-Stark shifts are calibrated by spectroscopy of the |f0⟩↔|g1⟩ transition across drive amplitudes.The resonance line is fitted to a second-order polynomial in drive amplitude, with zero-amplitude frequencies of 3.196 GHz for Q1 and 2.775 GHz for Q2.
- Calibration steps: 213 ns operation time yields nearly equal effective couplings of 4.69(1) MHz for Q1 and 4.72(1) MHz for Q2.Individual drives are adjusted using Rabi oscillations between each qubit’s |f⟩ state and the resonator |1⟩ state.
- Cross ac-Stark calibration: Two-tone frequency offsets are optimized from population transfer measurements, using a two-dimensional Gaussian fit to compensate cross ac-Stark shifts.Q2 is prepared in |f⟩, both drives are applied, and population in Q1 identifies the offsets producing maximum transfer.
- Cross ac-Stark calibration: The cross ac-Stark shifts differ from single-tone shifts, consistent with drive-induced crosstalk and/or unwanted static qubit–qubit interactions.The difference is absent from the cited simulations and is therefore attributed to these additional interactions.
- Angle-dependent calibration: Cross ac-Stark shifts are recalibrated versus θ because the drive-amplitude ratio controlling θ changes the shifts.Second-order fits provide θ-dependent calibration curves, while residual shifts produce a systematic θ-dependent phase removed from the fidelity data a posteriori.
IV. EXPERIMENTAL RESULTS
The holonomic two-tone operation creates arbitrary superpositions of the two qubit excitations by transferring population between qubits and controlling the relative phase. Bell states are produced in 213 ns with about 95% fidelity, while larger-angle transfers are mainly limited by charge dispersion.
- Control of θ and φ: The operation controls population transfer through θ and the relative phase between |eg⟩ and |ge⟩ through φ.Drive-amplitude scaling sets θ, while changing the Q2 drive phase changes the measured relative phase.
- State-transfer characterization: As θ increases, more population transfers to the target qubit in agreement with theory, while phase-averaged fidelity decreases.The fidelity data average over 30 phase values for each θ.
- State-transfer characterization: Charge dispersion of 1.8 MHz and 3.0 MHz on the two |f⟩ states is identified as the main source of reduced fidelity at larger θ.Numerical simulations confirm that charge dispersion together with T1 explains the observed degradation.
- State-transfer characterization: 94.8% is the average state fidelity across the θ-dependent transfer measurements.The fidelity is averaged over 30 different values of φ at each θ.
- Phase control: The relative phase is set accurately, with a mean measured-minus-expected difference of −2±42 mrad for nonzero target phase.A large systematic phase error occurred in 8.3% of measurements and was attributed to charge noise.
V. SIMULATION RESULTS
Simulations model the holonomic population transfer with realistic device parameters, showing high-fidelity transfer under relaxation but strong sensitivity to charge dispersion. The resonator is transiently populated and returns to its ground state when the drives are calibrated.
- The simulation uses anharmonic four-level qubits, a three-level harmonic resonator, Lindblad dynamics, and a 206 ns flat-top Gaussian pulse with σ = 3.5 ns.
- Figure 7 reports relative-phase errors with mean −2 mrad and standard deviation 42 mrad after correcting θ-dependent cross ac-Stark shifts.
- The resonator reaches 50% population during the θ = π/2 transfer but returns to its initial ground state at the pulse end.
- 99.98% fidelity is reached under unitary dynamics, decreasing to 98.48% when qubit and resonator relaxation are included.With finite qubit relaxation but no resonator relaxation, fidelity remains 99.11%.
- Charge dispersion is detrimental: simulated frequency shifts leave population partially in the resonator and prevent proper qubit-to-qubit transfer.The simulated charge-noise behavior matches the experimental population-transfer data.
VI. CONCLUSION AND OUTLOOK
The experiment demonstrates entanglement creation and manipulation with non-adiabatic holonomic operations, producing high-fidelity two-qubit states in 213 ns. Charge dispersion is the main fidelity limitation, while improved device parameters could enable a SWAP-type gate.
- Two-qubit states with fidelities above 95% are created using 213 ns square pulses.
- Charge dispersion in the qubit |f⟩ states is identified as the main fidelity limitation, while higher resonator T1 could provide further gains.
- Increasing manifold separation or using optimized pulse shapes could extend the operation into a SWAP-type gate without population loss.The demonstrated manifold separation was 5 MHz, and larger separation is proposed as one route forward.
Appendix A: Dynamics
The effective Hamiltonian evolution is purely geometric: an initially defined two-qubit subspace undergoes parallel transport and returns without dynamical contributions.
- Appendix A: Dynamics: The evolution under H′eff is purely geometric because the parallel-transport condition eliminates dynamical contributions.The condition ⟨ψj(t)| ˙ψk(t)⟩ = 0 means infinitesimal evolution is perpendicular to the instantaneous subspace.
- Appendix A: Dynamics: The time evolution is represented in the three-state basis {|fg0⟩, |gg1⟩, |gf0⟩} using a time-ordered exponential.Time ordering becomes unnecessary when the relevant operator is time-independent and commutes with itself at all times.
- Appendix A: Dynamics: Parallel transport holds because the operator A maps the initial qubit states to a vector proportional to |gg1⟩, orthogonal to the qubit subspace.Since U commutes with A, the same orthogonality argument applies throughout the evolution.
Appendix B: Cyclical evolution
The operator algebra reduces the exponential evolution to a simple form whose action is cyclical, returning the state to the original two-qubit subspace.
- Appendix B: Cyclical evolution: The normalization |λ1|^2 + |λ2|^2 = 1 implies A^3 = A, so odd and even powers reduce to A and A^2.Specifically, A^(2n+1) = A and A^(2n) = A^2 for n ≥ 1.
- Appendix B: Cyclical evolution: Applying the Taylor expansion of the exponential, cosine, and sine functions yields the time-evolution operator used for the cyclic dynamics.The ratio λ1/λ2 is parameterized by −e^(iφ) tan(θ/2) to recover the main-text operator.
- Appendix B: Cyclical evolution: The evolution is cyclical because A^2 does not mix the two-qubit subspace {|fg0⟩, |gf0⟩} with |gg1⟩.The resulting operator is U(τ) = I − 2A^2, and the final state returns to its initial subspace.