A direct controlled-phase gate between microwave photons

Abstract

Useful quantum information processing ultimately requires operations over large Hilbert spaces, where logical information can be encoded efficiently and protected against noise. Harmonic oscillators naturally provide access to such high-dimensional spaces and enable hardware-efficient, error-correctable bosonic encodings. However, direct entangling operations between oscillators remains an outstanding challenge. Existing strategies typically rely on parametrically activating interactions that populate the excited states of an ancillary nonlinear element. This induces an effective interaction between the oscillators, at the expense of introducing additional dissipation channels and potential leakage from the encoded manifold. Here, we engineer a Raman-assisted cross-Kerr interaction between microwave photons hosted in two superconducting cavities, without exciting the nonlinear element, thereby suppressing coupler-induced decoherence.This approach generates a direct coupling between microwave photons that is exploited to implement a controlled-phase gate within the single- and two-photon subspaces of two oscillators, directly entangling them. Finally, we harness this dynamics to map the photon-number parity of a storage cavity onto an auxiliary oscillator rather than a nonlinear element, enabling error detection while protecting the storage mode from measurement-induced decoherence. Our work expands the bosonic circuit quantum electrodynamics (cQED) toolbox by enabling coherence-preserving direct photon-photon interactions between oscillators. This realizes an entangling gate that operates entirely within a bosonic code space while suppressing decoherence from nonlinear ancilla excitations, providing a key primitive for fault-tolerant bosonic quantum computing.

Figures (14)

• Figure 1: Driven cross-Kerr protocol and experimental device. (a) Conceptual illustration of the engineered direct cross-Kerr, $g_{ab}$, between two harmonic oscillators mediated by a nonlinear coupler, where a photon in one mode shifts the energy levels of the other without physically populating the coupler. (b) Schematic of the experimental device, consisting of two 3D superconducting cavities, each coupled to its own side transmon used for state preparation and tomography. A central asymmetric SQUID acts as the coupler and is flux-biased by a superconducting coil via pick-up loop transformer. (c) Frequency arrangement used in this work. A strong pump is placed at $\omega_d$ with a detuning $\Delta/2\pi$ of few MHz from the resonance exchange $|1\rangle|0\rangle|g\rangle \leftrightarrow |0\rangle|1\rangle|e\rangle$ occurring at $\omega_e\, \mathord{=}\, 5.20\,$GHz. Higher photon resonances are shifted down in frequency by multiples of $\chi_a$ and $\chi_b$.
• Figure 2: Calibration of the cross-Kerr protocol parameters. (a) Coherent exchange of population between $|10g\rangle$ and $|01e\rangle$ modes when $\Delta\,\mathord{=}\,0$ obtained by monitoring the vacuum population of Alice (yellow) and Bob (purple). Oscillation frequency corresponds to $g_1/2\pi \,\mathord{=}\, 1.024\,\mathord{\pm}\,0.004$ MHz. The data (circles) agrees well with master equation simulations (lines) with the independently measured cavity and coupler decoherence. (b) Strength of engineered cross-Kerr (red), extracted from cavity Ramsey experiments, and residual coupler excitations (green) as a function of $\Delta$. Solid line is a Floquet simulation with experimentally extracted Hamiltonian parameters sm.
• Figure 3: Controlled-phase gate in the single-photon manifold. (a) Phase accumulated by Alice, initialized in $|+\rangle \,\mathord{=}\,(|0\rangle+|1\rangle)/{\sqrt{2}}$, as a function of the evolution time under $H_{\text{eff}}$, with Bob in state $|0\rangle$ (green) or $|1\rangle$ (turquoise). A full CZ gate is implemented in 5.244 $\mathord{\pm}$ 0.008 $\mu$s. (b) Sampled Wigner functions of Alice (yellow) and Bob (purple) at specific times corresponding to a controlled-S,T and Z gates. (c) Protocol used to obtain a maximally-entangled state, where cavities are prepared in $|\text{+}\rangle|\text{+}\rangle$. The CZ gates, $C_\pi^{n_an_b}$, are applied $N$-times, followed by state tomography by sampling the populations of the oscillators in $|1\rangle$ after at optimized displacement points, $\alpha$, in phase space sm. (d) Pauli basis representation of the reconstructed two-cavity state after applying $N\,\mathord{=}\,1,3,5$ gates to $|\text{+}\rangle|\text{+}\rangle$. Inset shows the Wigner functions of the statistical mixture in the oscillators when they are measured independently after one CZ gate. (e) Repeated application of $C_\pi$ across a set of initial states. Dots are raw experimental data without any scaling. First points indicate that fidelity is limited by state preparation and measurement fidelity. Control experiment (black triangles) creates $|\text{+}\rangle|\text{+}\rangle$ without drive, has a infidelity of 0.7$\,\mathord{\pm}\,0.3\%$ per gate. When the gate is activated, $|\text{+}\rangle|\text{+}\rangle$ (red squares) introduces an infidelity of 4.6$\,\mathord{\pm}\,0.3\%$ per gate. Similarly, states $|\text{+}\rangle|0\rangle$ (green crosses) and $|1\rangle|\text{+}\rangle$ (blue circles) suffer an infidelity of 3.1$\,\mathord{\pm}\,0.3\%$ and 3.7$\,\mathord{\pm}\,0.2\%$ per gate, respectively. Shaded regions represent master equation simulations of dynamics given by Eq. \ref{['Eq: engineered cross_kerr']}. Spread corresponds to $\mathord{\pm}10\%$ variation of cavity coherences observed in experiments.
• Figure 4: CPHASE on a biased-erasure code. (a) Measured Wigner functions with Alice in $|0\rangle$ or $|2\rangle$, and Bob initialized in $\frac{|0\rangle+|2\rangle}{\sqrt{2}}$ for three different drive durations, corresponding to the controlled-I, S and Z gates. (b) Repeated application of $C_\pi$ when the control is in $|0\rangle$ (red) or $|2\rangle$ (turquoise and green). Data are offset by the initial state preparation infidelity. Dots are raw experimental data, and triangles are data post-selected on the cavities being in even photon number after the protocol. When driving the gate, states acquire an infidelity of 3.9$\,\mathord{\pm}\,$0.5% (turquoise), 1.2$\,\mathord{\pm}\,$0.5% (green), and 0.6$\,\mathord{\pm}\,$ 0.2% (red) per gate. Shaded region represents master equation simulations of dynamics given by Eq. \ref{['Eq: engineered cross_kerr']}. Spread corresponds to $±10\%$ variation of coherences. (c) Sequence to perform parity check via the engineered cross-Kerr. Alice, the storage mode, is initialized in $|2\rangle$, and allowed to decay for a variable duration up to 2 ms. Bob, the ancillary mode, is then prepared in $(|0\rangle+|2\rangle)\sqrt{2}$, and a $C_{2\pi}$ gate is applied. The state of Bob is then mapped onto the ground/excited state of its auxiliary transmon with a SNAP pulse sm. Finally, we perform full state tomography (gray) on Alice to verify the performance of the bosonic parity check. (d) Probability of measuring excited state of Bob's auxiliary transmon. Open circles are the experimental data, with error bars smaller than the markers, and solid line the theoretical prediction calculated using the independently measured single-photon decay time of Alice, $T_1\,\mathord{=}\,800\,\mathord{\pm}\,10$$\mu$s. (e) Population of Fock state $|2\rangle$ in Alice extracted from the reconstructed density matrix. A linear fit shows the decay constant with (red) and without (turquoise) post-selecting on the bosonic parity check result are 763$\,\mathord{\pm}\,11\,\mu$s and 981$\,\mathord{\pm}\,25\,\mu$s, respectively. (f) Wigner plots of the reconstructed states in Alice at two different times marked, with (bottom) and without (top) post-selection based on the bosonic parity check via Bob.
• Figure S1: Tunable SQUID coupler. (a) Layout of the SQUID coupler chip, with the pick-up loop transformer (left), the SQUID (middle), and its readout resonator (right). (b) Readout frequency as a function of the current applied to bias the SQUID. Solid line shows the fit used to extract the current-flux relation. (c) SQUID frequency as a function of current. Solid line shows the fitted frequency.