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Deterministic Quantum State Transfer and Generation of Remote Entanglement using Microwave Photons
Philipp Kurpiers, Paul Magnard, Theo Walter, Baptiste Royer, Marek Pechal, Johannes Heinsoo, Yves Salathé, Abdulkadir Akin, Simon Storz, Jean-Claude Besse, Simone Gasparinetti, Alexandre Blais, Andreas Wallraff
TL;DR
The paper characterizes microwave-transfer hardware and tomography procedures for superconducting transmons, and verifies prepared multilevel entanglement using the CCNR criterion. The measured state has ccnr = 1.612 ± 0.003, while residual |f⟩ population limits a rigorous two-qubit description.
Problem
The work addresses how to characterize and verify multilevel entangled states and state-transfer experiments in superconducting transmons.
Method
The experiment calibrates transmon parameters and microwave photon-generation drives, performs single-shot readout and quantum tomography, and uses master-equation simulations.
Results
ccnr = 1.612 ± 0.003 for the measured entangled state, unambiguously witnessing entanglement.
Takeaways & Limitations
The CCNR criterion verifies entanglement without restricting the measured state to a physical two-qubit density matrix.
Takeaways & Limitations
A residual 3.5% |f⟩ population means the entangled state cannot be rigorously described by a two-qubit density matrix.
Abstract
from arXiv · showhide
Sharing information coherently between nodes of a quantum network is at the foundation of distributed quantum information processing. In this scheme, the computation is divided into subroutines and performed on several smaller quantum registers connected by classical and quantum channels. A direct quantum channel, which connects nodes deterministically, rather than probabilistically, is advantageous for fault-tolerant quantum computation because it reduces the threshold requirements and can achieve larger entanglement rates. Here, we implement deterministic state transfer and entanglement protocols between two superconducting qubits fabricated on separate chips. Superconducting circuits constitute a universal quantum node capable of sending, receiving, storing, and processing quantum information. Our implementation is based on an all-microwave cavity-assisted Raman process which entangles or transfers the qubit state of a transmon-type artificial atom to a time-symmetric itinerant single photon. We transfer qubit states at a rate of $50 \, \rm{kHz}$ using the emitted photons which are absorbed at the receiving node with a probability of $98.1 \pm 0.1 \%$ achieving a transfer process fidelity of $80.02 \pm 0.07 \%$. We also prepare on demand remote entanglement with a fidelity as high as $78.9 \pm 0.1 \%$. Our results are in excellent agreement with numerical simulations based on a master equation description of the system. This deterministic quantum protocol has the potential to be used as a backbone of surface code quantum error correction across different nodes of a cryogenic network to realize large-scale fault-tolerant quantum computation in the circuit quantum electrodynamic architecture.
Appendix A: Literature Overview
The appendix surveys remote entanglement experiments across multiple physical platforms and interaction schemes.
- Remote entanglement experiments span atomic ensembles, trapped ions, superconducting circuits, quantum dots, and other physical systems.
- The comparison covers entanglement rates, concurrence, and entangled-state fidelity.
Appendix B: Sample Parameters
The devices use separately fabricated superconducting nodes with dedicated readout and transfer circuits, matched transfer-resonator frequencies, and Purcell filtering.
- Each node contains coplanar readout and transfer resonators coupled to a transmon on a sapphire substrate.The resonators and feed-lines are fabricated from etched niobium, while transmon pads and junctions use shadow-evaporated aluminium.
- Device parameters include resonator and filter frequencies, external-line decay rates, and dispersive coupling strengths for readout and transfer circuits.
- Flux through each transmon’s SQUID tunes the frequencies so that the transfer resonators have identical frequencies.
Appendix C: Microwave Drive Schemes
The microwave-drive scheme calibrates Raman-mediated coupling and ac-Stark compensation to generate photons with a prescribed temporal shape while characterizing the drive response.
- The effective coupling between |f, 0⟩ and |g, 1⟩ is induced by a microwave tone at the transition resonance.
- The drive amplitude produces a linear effective coupling and a quadratic ac-Stark shift, fitted from measurements for samples A and B.
- The calibrated maximum effective couplings are ˜gA/2π = 6.0 MHz and ˜gB/2π = 6.7 MHz.
- The drive phase is adjusted using the measured ac-Stark shift to maintain resonance with the driven transition.
- The dynamics use a two-level model with loss, where the effective coupling is constrained by κeff ≤κT and produces the desired single-photon temporal shape.
Appendix D: Three-Level Single-Shot Readout
Three-level single-shot readout maps integrated microwave quadratures to qutrit-state assignments and corrects measured populations for readout errors.
- Figure 7 displays quadrature clusters for qutrits A and B, with dashed thresholds defining state discrimination.
- Readout traces are integrated into quadratures u and v, producing three Gaussian clusters associated with prepared states |g⟩, |e⟩, and |f⟩.
- The u-v plane is partitioned into three regions, and counting assignments estimates the probabilities Rss′ = P(s′| |s⟩).
- The optimized measurement settings yield an approximately 5% total assignment error probability for both qutrits.
- The assignment-probability tables report correct identifications on the diagonal and misidentifications off the diagonal for single and paired qutrit states.
- Measured assignment probabilities satisfy M = R ·⃗ρdiag, and applying R−1 corrects the diagonal populations for single-shot readout errors.
- The correction approach is chosen because expected state-preparation errors are lower than readout errors.
Appendix E: Loss Estimation
Losses are estimated separately for the printed-circuit components, coaxial cables, and microwave circulator connecting the samples.
- 2.5 ± 1% loss is measured on the printed circuit boards, including connectors.
- 4.0 ± 0.1% loss is reported for each 0.4 m coaxial cable.
- 13 ± 2% loss is taken from the microwave circulator manufacturer’s specifications.
Appendix F: Master Equation Simulation
The simulations model two coupled transmon–resonator systems, their microwave-driven effective interaction, dissipation, and cascaded photon transfer through a lossy circulator.
- The transmons are modeled as anharmonic oscillators coupled to transfer resonators, with emitter and receiver labeled A and B.
- The driven Jaynes-Cummings Hamiltonian describes each sample, while readout resonators are omitted because they do not affect photon-transfer dynamics.
- Unitary frame, displacement, and Bogoliubov transformations expose the effective coupling between the |f, 0⟩ and |g, 1⟩ states.
- The combined effective Hamiltonian includes transmon anharmonicity, resonator coupling, and the cascaded connection between the two samples.
- The simulated remotely entangled state is characterized through two-qutrit tomography and compared with the ideal Bell state and numerical prediction.
- The master equation incorporates dissipation through resonator decay, transmon decay rates, and photon loss probability ηc between the samples.
Appendix G: Quantum State and Process Tomography
Quantum state and process tomography reconstruct single- and two-qutrit density matrices and the transfer process from measured populations after calibrated tomography operations.
- Single-qutrit tomography applies specified tomography gates before measuring state populations with single-shot readout.
- Maximum-likelihood reconstruction converts the measured populations into a qutrit density matrix under the assumption of ideal tomography gates.
- Two-qutrit tomography applies pairs of local tomography gates to transmons A and B before extracting joint populations.
- Full quantum process tomography prepares six mutually unbiased qubit basis states at node A and transfers them to node B.
- The process matrix is obtained by linear inversion of the reconstructed density matrices from nodes A and B.
Appendix H: Two-Qutrit Entanglement
The entanglement analysis treats the experimentally reconstructed state as a two-qutrit state because residual |f⟩ population prevents a rigorous two-qubit description, then uses CCNR to certify entanglement.
- 3.5% residual |f⟩ population prevents the post-protocol state from being rigorously represented as a two-qubit density matrix.
- The reported two-qubit matrix ρm retains the two-qubit elements of the reconstructed two-qutrit state ρ3⊗3 and has non-unit trace.
- The reduction preserves Bell-state fidelity while providing a conservative concurrence estimate compared with projection onto physical two-qubit states.
- The CCNR criterion is used because it is well defined for multilevel mixed entangled states.
- 1.612 ± 0.003 is obtained for ccnr, unambiguously witnessing entanglement in the prepared two-qutrit state.