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Digital quantum simulation of spin models with circuit quantum electrodynamics
Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, A. Wallraff
TL;DR
Interacting-spin dynamics are difficult to compute classically, motivating controlled quantum simulation. This work digitally simulates Heisenberg and Ising models with two transmon qubits by decomposing their evolution into the native exchange interaction and single-qubit gates, obtaining characterized two-qubit operations and model dynamics.
Problem
Computing characteristics of even small interacting-spin systems is challenging on conventional computers, motivating quantum simulation of their evolution.
Method
The experiment digitally decomposes Heisenberg and Ising evolution into exchange-mediated XY gates, single-qubit rotations, phase gates, and repeated Trotter steps.
Results
Process tomography measures 86.1% fidelity for the Heisenberg SWAP gate and 83.6% for its identity gate, while XY-gate fidelities reach 97.8% and 95.3%.
Takeaways & Limitations
The experiment demonstrates controlled digital simulation of interacting-spin models using a superconducting circuit-QED platform.
Abstract
from arXiv · showhide
Systems of interacting quantum spins show a rich spectrum of quantum phases and display interesting many-body dynamics. Computing characteristics of even small systems on conventional computers poses significant challenges. A quantum simulator has the potential to outperform standard computers in calculating the evolution of complex quantum systems. Here, we perform a digital quantum simulation of the paradigmatic Heisenberg and Ising interacting spin models using a two transmon-qubit circuit quantum electrodynamics setup. We make use of the exchange interaction naturally present in the simulator to construct a digital decomposition of the model-specific evolution and extract its full dynamics. This approach is universal and efficient, employing only resources which are polynomial in the number of spins and indicates a path towards the controlled simulation of general spin dynamics in superconducting qubit platforms.
Appendix A: Chip architecture and measurement setup
The experiment uses superconducting transmon qubits and coplanar-waveguide resonators on a cryogenic chip, with dedicated control and readout wiring. Q1 and Q2 are the active qubits, while Q3 and Q4 are detuned to avoid interaction.
- The experiment uses two transmon qubits, Q1 and Q2, coupled dispersively to resonator R1, whose fundamental resonance frequency is 7.14 GHz.
- Q1 and Q2 have idle transition frequencies of 5.440 and 5.240 GHz, respectively, with resonator couplings of approximately 120 MHz each.
- Q3 and Q4 are tuned to 4.5 and 6.1 GHz so they do not interact with Q1 and Q2 during the experiment.
- Flux-bias lines generate individual qubit-frequency-control pulses, while sideband-modulated microwave pulses implement single-qubit control.
- The chip contains four aluminium superconducting qubits and four niobium coplanar-waveguide resonators coupled to input and output ports.
Appendix B: Implementation of the XY gate
The XY exchange gate is implemented by tuning Q1 into resonance with Q2 through a flux pulse, while phase compensation controls unwanted dynamic phases. The resulting interaction is calibrated spectroscopically and experimentally corrected for pulse distortions.
- The qubit exchange interaction is represented by J(σx_1σx_2 + σy_1σy_2)/2 and is activated by tuning Q1 into resonance with Q2.
- The interaction time τ is varied from 0 to 60 ns, and the coupling strength is determined as J = −40.4 MHz from an avoided-crossing fit.
- An inverted linear filter compensates flux-pulse overshoots, using room-temperature response measurements and in-situ Ramsey measurements of residual detuning.
- A 16 ns intermediate-frequency buffer is applied before and after the flux pulse to cancel unwanted relative dynamic phase.
- The buffer level is calibrated with Ramsey-type experiments separately for each XY-gate interaction length.
Appendix C: Pulse scheme
The simulation protocols are realized with microwave single-qubit rotations and fast-flux XY gates, followed by waiting and refocusing measures that support modular gate use and tomography.
- Heisenberg and Ising simulations use sequences of microwave and flux pulses applied to Q1 and Q2.
- Single-qubit rotations use 24 ns Gaussian-shaped DRAG microwave pulses, while XY gates use fast flux pulses.
- A 40 ns + Δτ waiting interval follows each flux pulse to suppress residual transient response, with Δτ chosen relative to the 5 ns phase-oscillation period.
- The timing measures make a single XY-gate calibration usable across all gate realizations within the algorithm.
- Two consecutive π pulses on Q2 refocus the residual σz_1σz_2 interaction before dispersive joint two-qubit state tomography.
Appendix D: Process tomography
Process tomography characterizes the implemented XY and isotropic Heisenberg operations at selected quantum phase angles. The measured process fidelities quantify performance for iSWAP, SWAP, and identity operations.
- Standard two-qubit process tomography is performed for the XY gate and simulated isotropic Heisenberg model over varying interaction times.
- 86.1% process fidelity is obtained for the Heisenberg SWAP gate at quantum phase angle π/2.
- 83.6% process fidelity is obtained for the Heisenberg identity gate at phase angle π.
Appendix E: Error contributions
The error analysis identifies two-qubit XY gates as the dominant fidelity limitation and models additional relaxation, dephasing, unwanted interactions, and phase offsets.
- Two-qubit XY gates dominate fidelity loss, with process fidelity Fp,XY = 95.7%, while single-qubit gates reach 99.7%.The observed Heisenberg process fidelity of 86.3% agrees with the 87.1% estimate based on three independent XY-gate errors.
- For Ising simulations, measured state fidelities decrease from 91.7% at n = 1 to 60.7% at n = 5 Trotter steps.The corresponding expected fidelities are 93.1% and 65.6%, respectively.
- The error model includes relaxation, dephasing, state-dependent phase errors, and flux-pulse crosstalk affecting a single-qubit phase gate.
- Fitting the observed states estimates an unwanted interaction angle of approximately 2.3° and a constant phase offset of 4.6°.