Fast High-Fidelity Quantum Nondemolition Qubit Readout via a Nonperturbative Cross-Kerr Coupling

Qubit readout is an indispensable element of any quantum information processor. In this work, we experimentally demonstrate a nonperturbative cross-Kerr couplingbetween a transmon and polariton mode which enables an improved quantum nondemolition (QND) readout for superconducting qubits. The new mechanism uses the same experimental techniques as the standard QND qubit readout in the dispersive approximation, but due to its nonperturbative nature, it maximizes the speed, the single-shot fidelity, and the QND properties of the readout. In addition, it minimizes the effect of unwanted decay channels such as the Purcell effect. We observe a single-shot readout fidelity of 97.4% for short 50-ns pulses and we quantify a QND-ness of 99% for long measurement pulses with repeated single-shot readouts.

Popular Summary

The measurement of a quantum bit (qubit) must compromise between two apparently contradictory facts: We want a reliable measurement, and thus the qubit should be strongly coupled to its detector, but the fragile quantum information must be protected from sources of noise, including the detector itself. Here, we propose and experimentally demonstrate a new readout scheme for superconducting qubits that preserves the underlying quantum states while enabling fast measurement times.

The standard readout scheme relies on a transverse coupling between a qubit and a readout cavity, however, this interaction does not preserve the probabilities of the qubit’s ground and excited states. When the readout cavity is highly off-resonant from the qubit, it approximately preserves these probabilities, but readout speed and fidelity are sacrificed. Our readout scheme, based on a new realization of a native energy-energy interaction, or “cross-Kerr coupling,” allows us to maximize all of these characteristics. In a first experimental demonstration of our technique, we measure 99% preservation of probabilities, a readout fidelity of 97.4%, and a short measurement time of 50 ns.

The refinement of our new readout scheme may allow measurements with a high signal-to-noise ratio even without amplification of the signals. This may reduce the overhead and complexity of superconducting quantum chips and thus facilitate the scaling up of these quantum technologies to large sizes, as well as the implementation of quantum error correction and fault-tolerant quantum computation in the future.

Figure 1

Schematics of the circuit QED setup with the transmon molecule used for a high-fidelity and fast qubit QND readout. (a) A cavity mode $\widehat{c}$ is strongly and transversely coupled to an ancilla system $\widehat{a}$, which in turn couples diagonally to the qubit ${\widehat{\sigma }}_z$ as $\sim g_{zz}{\widehat{\sigma }}_z{\widehat{a}}^{†}\widehat{a}$. (b) The strong hybridization between cavity and ancilla is manifested by two orthogonal polariton modes ${\widehat{c}}_u$ and ${\widehat{c}}_l$, which couple to the qubit with a nonperturbative cross-Kerr coupling $\sim {\widehat{\sigma }}_z({\chi }_u{\widehat{c}}_u^{†}{\widehat{c}}_u+{\chi }_l{\widehat{c}}_l^{†}{\widehat{c}}_l)$ (see text). This allows us to infer the state $|g⟩$ or $|e⟩$ of the qubit by measuring the resonance shifts of the polaritons at the cavity transmission output.

Figure 2

Quantum circuit with nonperturbative cross-Kerr coupling. (a) Picture of the two parts of the Copper-OFHC 3D cavity with the input-output pin connectors. The sample is placed at the center of the cavity. (b) The electric field distribution of the first EM mode of the cavity in the center plane is sketched in red. The cavity directions (${\mathbf{a}}_{\mathbf{c}}$, ${\mathbf{b}}_{\mathbf{c}}$, ${\mathbf{c}}_{\mathbf{c}}$) and sample directions (${\mathbf{a}}_{\mathbf{s}}$, ${\mathbf{b}}_{\mathbf{s}}$, ${\mathbf{c}}_{\mathbf{s}}$) are represented. (c) Lumped element of the transmon molecule circuit. (d) Optical microscope and SEM pictures of the transmon molecule sample. The Josephson junctions are highlighted in red. The SQUID Josephson junctions implementing the coupling inductance $L_a$ are highlighted in green.

Figure 3

Polariton spectroscopy via the transmitted amplitude of the cavity as a function of the driving frequency ${\omega }_d$ at $\mathrm{\Phi }=5{\mathrm{\Phi }}_0$. The resonances at lower and higher frequency correspond to the lower and upper polariton modes, respectively. In addition, both polariton resonances are cross-Kerr shifted depending on the prepared qubit state (ground $|g⟩$ in blue and excited $|e⟩$ in red). The highlighted red and blue lines correspond to the theoretical prediction in Eq. (7) with the parameters given in Table 4, valid for $\mathrm{\Phi }=5{\mathrm{\Phi }}_0$.

Figure 4

Typical individual measurement records as a function of time, using the pulse sequence sketched in (a). We show typical quantum trajectories of the qubit in the presence (b) and absence (c) of a quantum jump. Blue and red points refer to the case the qubit is initially prepared in states $|g⟩$ and $|e⟩$, respectively ($t=0$). The readout pulse with amplitude ${\overline{n}}_l=2$ starts at $t=0\mathrm{ns}$ and stops at $t=1000\mathrm{ns}$. Each point is measured with a 30-ns integration, corresponding to the resonator rising time $2{\kappa }_l^{-1}$. An average over 1000 measurement records is plotted in solid blue and red lines, as well as their standard deviation represented by corresponding shaded areas.

Figure 5

(a) Pulse sequence sketch. (b) Histograms of 50-ns single-shot measurement for qubit prepared in ground state (blue points) and excited state (red points) with heralding. The solid blue and red lines are fits with a double Gaussian model. Black line is a single Gaussian fit. The green area depicts the overlap error ${\epsilon }_o=0.8\%$. The blue and red areas indicate the remaining error ${\epsilon }_{r,g}=0.6\%$ and ${\epsilon }_{r,e}=3.9\%$, respectively. It leads to a readout fidelity of 97.4%.

Figure 6

Qubit relaxation time $T_1$ versus flux. Black points and error bars are the extracted values of $T_1$ from Gaussian means and standard deviations, respectively. The various types of red points correspond to computed values of Purcell-limited $T_1$, assuming a one-mode cavity, the parameters described in Appendix pp6-s4, and various imperfections. For instance, the red diamond points consider only the asymmetry in Josephson energies, the star points consider only the misalignment, and the circle points consider both imperfections.

Figure 7

Schematic of the experimental setup. Abbreviations: arbitrary waveform generator (AWG), low-pass filter (LPF), band-pass filter (BPF), local oscillator (LO), vector network analyzer (VNA), and analog to digital converter (ADC).

Figure 8

(a) Single-tone transmission $S_{21}$ measurements in arbitrary units (arb. units) as function of driving frequency and flux (coil current). (b) Two-tone measurement, where the corresponding transmission $S_{21}^n$ is normalized by its value without second tone. (c) Extracted resonance frequencies of qubit ${\omega }_q^{\prime }$ (blue), lower polariton ${\overline{\omega }}_l$ (orange), and upper polariton ${\overline{\omega }}_u$ (purple) as a function of the applied flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$. The dashed black lines correspond to the theoretical predictions from the numerical diagonalization of the circuit model in Sec. pp6-s1.

Figure 9

(a) The lower (orange) and upper (purple) polariton resonant frequencies as function of integer quantum flux. They are fitted (black lines) using the numerical model discussed in Appendix pp2. The gray dashed lines correspond to the bare cavity and bare ancilla frequencies. An avoided crossing between ancilla and cavity can thus be seen. (b) Cross-Kerr strengths between qubit and lower (orange) and upper (purple) polaritons. Black lines are the expected cross-Kerr coupling using ${\chi }_l=-g_{zz}{\sin }^2(\theta )$ and ${\chi }_u=-g_{zz}{\cos }^2(\theta )$ with $g_{zz}/(2\pi )=34.5\mathrm{MHz}$. The gray diamonds are simulated points computed using black box quantization [53] with EM simulation.