High-fidelity parametric beamsplitting with a parity-protected converter

Abstract

Fast, high-fidelity operations between microwave resonators are an important tool for bosonic quantum computation and simulation with superconducting circuits. An attractive approach for implementing these operations is to couple these resonators via a nonlinear converter and actuate parametric processes with RF drives. It can be challenging to make these processes simultaneously fast and high fidelity, since this requires introducing strong drives without activating parasitic processes or introducing additional decoherence channels. We show that in addition to a careful management of drive frequencies and the spectrum of environmental noise, leveraging the inbuilt symmetries of the converter Hamiltonian can suppress unwanted nonlinear interactions, preventing converter-induced decoherence. We demonstrate these principles using a differentially-driven DC-SQUID as our converter, coupled to two high-Q microwave cavities. Using this architecture, we engineer a highly-coherent beamsplitter and fast (~100 ns) swaps between the cavities, limited primarily by their intrinsic single-photon loss. We characterize this beamsplitter in the cavities' joint single-photon subspace, and show that we can detect and post-select photon loss events to achieve a beamsplitter gate fidelity exceeding 99.98%, which to our knowledge far surpasses the current state of the art.

Fig. 1. The differentially driven SQUID as a parity-protected converter.

a The symmetric DC-SQUID contains two orthogonal modes^,, the common mode (coupler) and the differential mode (actuator). We selectively couple the former to two bosonic modes and the latter to the drives to take advantage of the natural symmetries of the Hamiltonian in Eq. (1). b Implementing the purely differential drive through a 3D buffer post-cavity (figure is exaggerated for illustrative purposes). The natural separation in electric and magnetic fields in the λ/4 mode is used to purely drive the actuator, without exciting the coupler. The sensitive quantum information is stored in two high-Q λ/4 post-cavities (Alice and Bob) that participate in the coupler, enabling parametric beamsplitting between them. The inset shows an optical micrograph of the SQUID device, displaying the purposely offset antenna pad that counters residual drive-asymmetry. c Frequency stack for relevant modes in the system. The difference of the two drive frequencies (Δ_d) is fixed to be equal to the cavity detuning (Δ_ab) for resonant beamsplitting. The drives are placed symmetrically around the buffer mode resonance, which is engineered to be far-red detuned from the coupler frequency.

Fig. 2. Beamsplitting with the differentially-driven SQUID.

a Beamsplitting implements an effective driven Rabi evolution in the Bloch sphere of the dual-rail qubit formed by the single photon subspace Alice and Bob, where decay can be detected by monitoring the vacuum state. b Resonant evolution of a single-photon prepared in Bob. The data is normalized for readout infidelity, and state preparation fidelity is shown as a dashed gray line (Supplementary Note 10). The fast coherent oscillations (black dots) between the cavities are fitted to Eq. (6) (green lines show envelope) to obtain the decay and dephasing time-scales. The evolution for the first 1.5 μs is plotted separately to better illustrate the oscillations, and fit to a sinusoid to extract g_BS. c Sweeping both drive amplitudes simultaneously and repeating experiment (a) lets us quantify g_BS (blue crosses), and the decoherence limit on beamsplitter infidelity (red diamonds) at various drive strengths. We choose a drive strength with simultaneously low infidelity and high beamsplitter rate as our operating point (yellow dashed line). d The coupler’s driven excitation (P_c) after evolving for 10 swaps is directly quantified through a dedicated on-chip readout mode. We observe no monotonic correlation with respect to drive amplitude, and driven populations mostly remain within the range of the undriven population (gray region). e Coupler population as a function of number of swaps at the operational driving point. The heating rate is nearly immeasurable, with a fitted (pink line) slope of (1.2 ± 2.4) × 10⁻⁵ excitation per swap, which is within expectation for our natural thermal background (γ_c,↑ ~ (3.3 ms)⁻¹). The non-zero offset of the fit arises from preparation and readout infidelities. Error bars in both (d) and (e) represent fit errors from the protocol described in ref. .

Fig. 3. Randomized benchmarking with a calibrated beamsplitter pulse.

a Wigner function of Bob after preparing $∣0_a{1}_b⟩$ and implementing 1, 10 and 60 calibrated swaps. Each swap is a combination of two identical beamplitter pulses. b Probability of ending in the target state $∣0_a{1}_b⟩$ after executing the RB protocol with randomly selected gates from G_DR (yellow circles). The curve is normalized to account for state preparation and readout imperfections (Supplementary Note 10), and is in good agreement with a single-exponential, with a decay constant of τ_RB = 1271 ± 4 gates. This `Raw RB' is practically indistinguishable from the sequences post-selected on the coupler ground state (black crosses), with the difference of the two curves shown below the main plot (pink crosses). c Focusing on the first 2250 gates, we use measurements of both cavities to post-select on sequences in which no photon loss event occurred (green diamonds). We compare these sequences to the raw RB in (b) (yellow), showing an improvement in average gate infidelity from 0.078 ± 0.001% to 0.020 ± 0.001%. d The gate-sets required for the above protocols are generated from calibrated beampslitter pulses with tanh-shaped ramps, where different U_BS(φ) are obtained by changing the relative phase of our drives. e The benchmarking sequences consist of randomly generated pulses that, under ideal operation, map $∣0_a{1}_b⟩$ back to itself. After each sequence, we measure whether the coupler is in its ground state, the presence of a photon in Bob, and the presence of a photon in Alice using an additional swap gate. This lets us generate the raw, coupler-selected and leakage-detected RB datasets. All sequences are also conditioned on Bob’s ancilla ending in its ground state, to discount first-order effects of ancilla heating.

Fig. 4. Measured driven Zeeman shift and cavity swaps.

a We measure the coupler frequency through direct spectroscopy under a single drive tone, as a function drive amplitude. Comparing the Zeeman shift for either drive tone (pink, purple) to the prediction from Floquet simulation (gray solid line) allows accurate calibration of the drive strength in terms of the driven junction phase ϕ_d1,2. b Measured population in Bob in the presence of both drive tones as a function of the drive-detuning (Δ_d − Δ_ab) and the time of evolution. A single photon is prepared in Bob, and the drives swap this photon between Alice and Bob under the detuned beamsplitter interaction. c Fitting the oscillations as a function of drive-detuning to the Rabi model allows us to calibrate amplitude-dependent shift in the resonance condition (Δ_Z,ab) and strength g_BS of the beamsplitting interaction.

Fig. 5. Minimizing residual drive asymmetry in the SQUID via fine-tuning the circuit geometry.

a The SQUID device in the buffer cavity package driven by an oscillating B-field. The cylindrical geometry of the buffer cavity (outer wall radius greater than the inner wall) dictates that the B-field has a non-uniform distribution along the radial direction (from left to right). The dashed lines with capacitor symbol represents the capacitance between the antenna pads of the SQUID and the wall of the package. b The lumped-element circuit model of the SQUID device, taking into account the geometry of SQUID and the spatial distribution of the B field. c HFSS simulation of the quality factor of the common mode (red) and drive asymmetry (blue, defined in Eq. (21)) as functions of the top pad displacement from center to right, δ. As a result of introducing this asymmetry in the device geometry, the optimization of the drive asymmetry and the quality factor are simultaneously achieved at δ ≈ 350 μm.

Fig. 6. Experimentally characterizing drive asymmetry.

We directly probe the coupler ‘ge/3’ transition due to any residual common-mode drive. The coupler is prepared in the ground state at t = 0 μs, followed by a single-tone drive through the buffer-mode near ω_c/3. The rate of coherent oscillation of the coupler population (color-scale) compared to the effective Zeeman shift (x-axis) of this resonance frequency bounds the drive asymmetry to <1%.

Fig. 7. Measured driven decoherence.

Decoherence rates are extracted from short sections of the long-time evolution under both drive tones, as a function of drive amplitude. The sideband collision of coupler and Bob (see Supplementary Note 5) clearly limits the fidelity at high amplitudes, and we operate on the boundary of this collision (gray dashed line), where κ_BS is limited by κ₁.