Realization of a Universal Quantum Gate Set for Itinerant Microwave Photons

Deterministic photon-photon gates enable the controlled generation of entanglement between mobile carriers of quantum information. Such gates have thus far been exclusively realized in the optical domain and by relying on postselection. Here, we present a nonpostselected, deterministic, photon-photon gate in the microwave frequency range realized using superconducting circuits. We emit photonic qubits from a source chip and route those qubits to a gate chip with which we realize a universal gate set by combining controlled absorption and reemission with single-qubit gates and qubit-photon controlled-phase gates. We measure quantum process fidelities of 75% for single- and of 57% for two-qubit gates, limited mainly by radiation loss and decoherence. This universal gate set has a wide range of potential applications in superconducting quantum networks.

Popular Summary

Large-scale quantum computing will likely employ distributed network architectures in which photons, the fundamental quantum units of light, act as mobile carriers of quantum information to communicate between network nodes. Engineering the necessary interactions between two photons is challenging at optical frequencies but is more readily achievable at microwave frequencies. Here, we experimentally demonstrate an entangling two-qubit gate—a fundamental building block of any quantum algorithm—using controlled interactions between itinerant microwave photons.

Our setup makes use of the strong coupling between microwave photons and superconducting circuit devices. We use one device to generate single-photon wave packets acting as flying qubits, which are then funneled to a second superconducting circuit engineered to perform one of two operations on the received qubit. The receiving circuit can absorb, act on, and reemit a qubit (a single-qubit gate) or it can shift the phase of a qubit depending on the state of another (a controlled-phase gate). These operations represent a universal set of quantum gates of microwave-photon qubits.

We envision this setup being used to generate larger entanglement among many microwave-photon qubits and finding a wide range of applications in superconducting quantum networks.

Figure 1

(a) Schematic representation of the concept of the experiment. We generate photonic qubits $P_1$ (dark green) and $P_2$ (turquoise) from the source qubit (red) via swap gates (swap) mediated by a tunable coupling depicted as a control valve (orange). The photonic qubits propagate through a circulator to the gate qubit (blue), performing a swap gate (swap) or a controlled-phase gate (cphase). We linearly amplify the output field and measure the photonic qubits using heterodyne detection (purple). (b),(c) Equivalent quantum circuits implemented to realize (b) an arbitrary single-qubit gate $U$ and (c) a photon-photon controlled-phase (cphase) gate.

Figure 2

(a) Electrical circuit diagram of the gate qubit (blue) interacting with fields in the transmission line (black) through a tunable coupler (orange) and a converter mode (dark orange, realized as a transmon qubit). (b) Energy-level diagram indicating transitions induced by single-qubit gates (blue), parametric modulation of the flux $\mathrm{\Phi }$ threading the SQUID loop embedded in the coupler (orange), and photon emission via spontaneous decay (black). (c) Measured (dots) and simulated (solid lines) real part of the envelope of the field amplitude $\mathrm{Re}⟨a_{\mathrm{out}}⟩$ vs time $t$ for superposition states $(|0⟩+|1⟩)/\sqrt{2}$ emitted from the source qubit with the converter mode of the gate device being detuned from the field (red), when absorbing and reemitting the field at the gate qubit (green), and when directly preparing and emitting the state at the gate qubit (blue). A red (blue) wire frame shows the timing of qubit pulses with a rotation angle of $\pi /2$ applied to the source (gate) qubit prior to emission. Imaginary parts (not shown) are below $0.07/\sqrt{\kappa }$ in magnitude.

Figure 3

(a) Measured (dots) and simulated (lines) real part of the envelope of the amplitude $⟨a_{\mathrm{out}}⟩$ for the states $|+⟩$ emitted from the source qubit and reflected off of the gate qubit while driving the $|f0⟩\leftrightarrow |e1⟩$ transition of the gate qubit. Level diagrams and traces are shown for the gate qubit in the ground state (turquoise) and in the excited state (orange). The imaginary part (not shown) is below $0.04/\sqrt{\kappa }$ in magnitude. (b) Real part of the measured density matrix ${\rho }_{\mathrm{meas}}$ (solid bars) and the ideal density matrix ${\rho }_{\text{ideal}}=|\psi ⟩⟨\psi |$ (black wire frames) of the two-mode Bell state $|\psi ⟩=(|+,0⟩+|-,1⟩)/\sqrt{2}$. The fidelity is $F=⟨\psi |{\rho }_{\mathrm{meas}}|\psi ⟩=0.69$. Imaginary parts (not shown) are all below 0.04 in magnitude.

Figure 4

Measured process maps with ${\chi }_{\mathrm{int}}$ (blue wire frames) and without ${\chi }_{\mathrm{meas}}$ (filled bars) accounting for state preparation errors for (a) the identity, (b) the $X$ gate, (c) the $Y$ gate, (d) the $T$ gate, and (e) the controlled-phase (cphase) gate. For comparison, we also show the simulated ${\chi }_{\mathrm{sim}}$ (red wire frames) and ideal ${\chi }_{\text{ideal}}$ (black wire frames) process maps. $F=F_{\mathrm{tot}},F_{\mathrm{int}}$, with the total fidelity $F_{\mathrm{tot}}=\mathrm{Tr}(\sqrt{\sqrt{{\chi }_{\mathrm{meas}}}{\chi }_{\text{ideal}}\sqrt{{\chi }_{\mathrm{meas}}}}{)}^2$ and the internal fidelity $F_{\mathrm{int}}=\mathrm{Tr}(\sqrt{\sqrt{{\chi }_{\mathrm{int}}}{\chi }_{\text{ideal}}\sqrt{{\chi }_{\mathrm{int}}}}{)}^2$. Imaginary parts (not shown) are all below 0.04 in magnitude.

Figure 5

False color optical micrograph of the sample of the gate qubit (blue). Depicted are the two coupler paths (orange) and the converter mode (dark orange), coupled to the transmission line (purple). Flux lines (cyan) and charge lines (pink) couple to the gate qubit, converter mode, and the coupler. The readout resonator (green) is coupled to a feed line (yellow) via a Purcell filter (light green).

Figure 6

Experimental setup for operating the source qubit ($S$, red) and the gate qubit ($G$, blue) devices. For details, see the text.

Figure 7

Absolute value of the measured transmission coefficient $|S_{21}|/S_0$, normalized to the transmission at zero detuning $S_0$, when driving the converter mode of the gate qubit (blue) and the source qubit (red, vertically offset by 0.2) via their respective charge lines. Solid lines are Lorentzian fits to the data (see the text), resulting in the indicated linewidths $\kappa$.

Figure 8

Measured power spectral density (PSD) $P$ of the inelastic scattering of a coherent tone resonant with (a) the source qubit and (b) the gate qubit converter mode for the indicated drive rates $\mathrm{\Omega }$. All datasets are scaled with respect to the gate qubit fitted amplitude $P_{0,G}$; see the text. Solid lines are joined fits to all datasets with a single set of fitting parameters. The drive rates $\mathrm{\Omega }$ are fitted independently for each dataset.

Figure 9

(a),(b) Excited state population $P_e$ after preparing the source qubit (red) [gate qubit (blue)] in the excited state and applying a modulated flux pulse with a square envelope and a fixed, normalized amplitude of $A=0.35$. (a) Fixed pulse duration $\tau =283\mathrm{ns}$ (349 ns), plotted vs flux pulse frequency detuning $\delta$ from the $|e0⟩\leftrightarrow |g1⟩$ transition. Solid lines are Gaussian fits to the data. (b) On resonance $\delta =0$, plotted vs flux pulse duration $\tau$. Solid lines are fits to the data; see the text. (c) Effective coupling rate $J$ for the $|e0⟩\leftrightarrow |g1⟩$ transition of the gate qubit ($G$, blue circle) and the source qubit ($S$, red rectangle) vs drive amplitudes $A$, normalized to the respective maximal amplitude. Solid lines are quadratic fits to data.

Figure 10

Simulated mode-matching inefficiency $1-{\eta }_F=1-|\int {\psi }_g^{*}f(t)dt{|}^2$ of time-symmetric photonic qubits with shape $\xi (t)=(\sqrt{\mathrm{\Gamma }}/2)\mathrm{sech}(\mathrm{\Gamma }t/2)$ reflected off of the gate qubit in the ground ${\psi }_g$ or the excited state ${\psi }_e$ vs interaction strength normalized by the photon bandwidth $g/\mathrm{\Gamma }$, for indicated photon bandwidths $\mathrm{\Gamma }$ and a coupling rate $\kappa =g/0.7$. As a filter function for mode matching, we use $f(t)=[({\psi }_g-{\psi }_e)/(\parallel {\psi }_g-{\psi }_e\parallel )]$.

Figure 11

Measured moments $\mathrm{Re}⟨a⟩$ (blue), $\mathrm{Im}⟨a⟩$ (purple), $⟨a^{†}a⟩$ (orange), and $⟨(a^{†}{)}^2{a}^2⟩$ (green), for superpositions $|\psi ⟩=\sin (\theta /2)|0⟩+\cos (\theta /2)|1⟩$ of vacuum and single-photon states emitted from (a) the source qubit and (b) the gate qubit. Lines indicate ideal expectation values, calculated taking into account photon loss between the source and the gate chip.

Figure 12

Measured (blue dots) and simulated (red line) excited state population $P_e$ of the gate qubit for three calibration routines. The inset shows the envelopes of the pulse sequence applied to the gate chip ($G$, blue circle) and the source chip ($S$, red rectangle) during the respective calibration steps. (a) Measured excited state population $P_e$ after emitting a single-photon Fock state $|1⟩$ from the source qubit and absorbing it at the gate qubit vs frequency detuning $\mathrm{\Delta }$ between the converter modes of the source and gate qubit. Photons are emitted or absorbed in resonance with the respective converter mode. (b) Excited state population $P_e$ vs time delay $\tau$ (relative to the optimal time delay found in the experiment) between the emission pulse at the source qubit and the absorption pulse at the gate qubit. (c) Excited state population $P_e$, when emitting and absorbing a photonic superposition state $(|0⟩+e^{i\phi }|1⟩)/\sqrt{2}$ vs phase $\phi$ and applying a final $\pi /2$ rotation along the $Y$ axis to the gate qubit.

Figure 13

Timing diagram of the pulses applied to the gate qubit ($G$, blue circle) and the source qubit ($S$, red rectangle) for quantum process tomography of (a) single-photon gates and (b) the cphase gate. Depicted are the envelopes of the pulses on the charge line of the source (red) and the gate qubit (blue), as well as the envelopes of the flux pulses driving the $|e0⟩\leftrightarrow |g1⟩$ transition with the flux line of the respective coupler (green) and the $|f0⟩\leftrightarrow |e1⟩$ transition via the flux line of the coupler of the gate qubit (purple).

Figure 14

(a),(b) Second excited state population $P_f$ after preparing the gate qubit in the second excited state and applying a modulated flux pulse with a square envelope and a fixed (normalized) amplitude of $A=1$. (a) Fixed pulse duration $\tau =170\mathrm{ns}$, plotted vs flux pulse frequency detuning $\delta$ from the $|f0⟩\leftrightarrow |e1⟩$ transition. The solid line is a Gaussian fit to the data. (b) On resonance $\delta =0$, plotted vs flux pulse duration $\tau$. The solid line is a fit to the data based on a driven two-level system decaying at a rate ${\kappa }_G/2\pi =2.1\mathrm{MHz}$.