Fast, Lifetime-Preserving Readout for High-Coherence Quantum Annealers

We demonstrate, for the first time, that a quantum flux parametron (QFP) is capable of acting as both isolator and amplifier in the readout circuit of a capacitively shunted flux qubit (CSFQ). By treating the QFP like a tunable coupler and biasing it such that the coupling is off, we show that $T_1$ of the CSFQ is not impacted by Purcell loss from its low-$Q$ readout resonator ($Q_e=760$) despite being detuned by only 40 MHz. When annealed, the QFP amplifies the qubit’s persistent current signal such that it generates a flux qubit-state-dependent frequency shift of 85 MHz in the readout resonator, which is over 9 times its linewidth. The device is shown to read out a flux qubit in the persistent current basis with fidelities surpassing $98.6\mathrm{\%}$ with only 80 ns integration, and reaches fidelities of $99.6\mathrm{\%}$ when integrated for 1 $\mu \mathrm{s}$. This combination of speed and isolation is critical to the readout of high-coherence quantum annealers.

Popular Summary

Quickly determining the state of a superconducting qubit is a delicate task. Extracting information from the small signal requires coupling the qubit to the environment, putting its fragile quantum state at risk of decay or decoherence. In this paper, we present a new device and method to circumvent these problems. We use a quantum flux parametron (QFP) to faithfully read out the state of a flux qubit in tens of nanoseconds and simultaneously isolate it from the environment, preventing unwanted state decay.

The direction of persistent current flow in the qubit’s superconducting loop shifts the frequency of a flux-tunable resonator, creating a state-dependent signal that we can detect. Faster readout speeds require a greater coupling, which comes at the cost of increasing the decay rate of the qubit. We place the QFP in between the qubit and the readout resonator, and it acts like a tunable coupling that we control via an external magnetic field. When the coupling is turned off, the QFP provides isolation from the lossy resonator while we perform sensitive quantum operations on the qubit. The QFP also supports much larger persistent currents than the qubit, so it acts like a current transformer that amplifies the state-dependent signal. We show experimentally that the QFP performs fast, accurate readout while preserving the lifetime of the qubit.

This QFP readout technology is a key enabler of the next generation of quantum annealing processors that will possess much longer coherence times. Such processors may demonstrate a quantum speedup in important applications such as optimization. The method presented in this paper ensures that quantum annealing results will be measured accurately while preserving the quantum effects necessary for computational advantage.

Figure 1

(a) Schematic of device (for more details, see Appendix pp2). A four-junction CSFQ is inductively coupled to a QFP via their $z$-flux loops. The QFP is further coupled to a rf SQUID loop attached to the current antinode of a $\lambda /4$ resonator. Relevant fluxes and persistent currents are labeled and color coded. Each loop flux is controllable via an independent external bias line (not drawn). Dispersive diagnostic resonators coupled to the CSFQ and QFP are omitted from the schematic. (b) Example control sequence (top three plots) and the resulting persistent currents modeled in WRspice (bottom two plots). Applying a small amount of ${\mathrm{\Phi }}_z^{\mathrm{qub}}$ causes an initial tilt to the qubit potential (i). We anneal the qubit by increasing ${\mathrm{\Phi }}_x^{\mathrm{qub}}$ to a value of $1{\mathrm{\Phi }}_0$, inducing a nonzero persistent current in the qubit (ii). After the qubit anneal is complete, we subsequently anneal the QFP, which senses $I_z^{\mathrm{qub}}$ and latches to a much larger persistent current $I_z^{\mathrm{qfp}}$ (iii). We can then reset the qubit by lowering its barrier, and the QFP remains latched (iv). Note that the QFP persistent current causes some back action on the qubit, which causes it to return to a steady-state current greater than $0$ while the QFP remains annealed. (c) Representative plot of experimental tunable resonator line shifts, $S_{21}$, versus frequency. The tunable resonator position moves by multiple linewidths when the QFP latches to either a positive (light blue solid) or negative (light blue dashed) persistent current state.

Figure 2

(a) Measured (circles) and fitted (dashed lines) QFP $s$ curves for flux qubits prepared in the left (purple) and right (yellow) circulating current states. Widths extracted from Eq. (4) are found to be $1.38$ and $1.42\mathrm{m}{\mathrm{\Phi }}_0$ for the left and right flux qubit persistent current states, respectively. (b) QFP $s$-curve separation readout fidelity is plotted as $1-F_{\mathrm{sep}}=1-[P_R({\mathrm{\Phi }}_z)-P_L({\mathrm{\Phi }}_z)]$ for clarity. The fidelities are found by subtracting the QFP $s$-curve data (green circles), $s$-curve fits (green dashed), and a collection of theoretical curves with varying $\mathrm{\Delta }{\mathrm{\Phi }}^{\mathrm{qub}}/w$ (solid lines). The $s$-curve separation limit on readout fidelity for the measured device is found to be $99.91\mathrm{\%}\pm 0.095\mathrm{\%}$.

Figure 3

Histograms of single-shot measurements of the qubit prepared in either its $|L⟩$ (blue) or $|R⟩$ (red) state for integration times of 80 ns (left) and 1 $\mu \mathrm{s}$ (right). Thick dashed lines are Gaussian fits to the dominant peaks, used to determine the overlap error. The vertical, black, dashed line indicates the threshold value, as determined by the point of intersection of the Gaussian fits. Fidelity improves from 98.63% $\pm$ 0.04% to 99.65% $\pm$ 0.02% as the integration time is increased from 80 ns to 1 $\mu \mathrm{s}$. The overlap error correspondingly improves from 0.43% to approximately ${10}^{-15}$%, when the histograms become separated by more than $16\sigma$. $t_{int}$; integration times.

Figure 4

Transmission ($S_{21}$) near the maximal tunable resonator frequency versus ${\mathrm{\Phi }}_z^{\mathrm{qub}}$, which tunes the qubit frequency through the resonator. The top plot has ${\mathrm{\Phi }}_x^{\mathrm{qfp}}=0{\mathrm{\Phi }}_0$, turning off the QFP’s isolation. This produces an anticrossing from the qubit-resonator interaction. The dashed white lines are calculated from a full circuit Hamiltonian simulation (see Appendix pp4) and correspond to an effective coupling of $g/2\pi =9.8$ MHz. The bottom plot has ${\mathrm{\Phi }}_x^{\mathrm{qfp}}=0.5{\mathrm{\Phi }}_0$, turning on the isolation provided by the QFP. The anticrossing disappears because the qubit and resonator are no longer coupled.

Figure 5

Qubit lifetime as a function of frequency when the QFP isolates (orange squares) and does not isolate (blue circles) the qubit from the tunable resonator (frequency indicated by the gray dashed vertical line). Lifetime values are determined by averaging over between 80 to 100 individual measurements, and the pictured error bars denote $1\sigma$. The calculated Purcell limited $T_1$ (green dotted trace) is reciprocally summed with the average isolated $T_1$ lifetime (green solid trace), and captures the behavior of the nonisolated flux qubit lifetime as the qubit is tuned toward the resonator.

Figure 6

A schematic diagram of the room-temperature measurement setup and dilution refrigerator wiring.

Figure 7

An illustration of the physical device layout, generated by design software. The light pink regions delineate the high-quality MBE aluminum layer in which qubit capacitors, bias lines, resonators, and the ground plane are patterned. The blue regions depict the shadow-evaporated aluminum layer used to define Josephson junctions and SQUID loops. The purple bands represent aluminum air bridge crossovers used both within SQUID loops, as well as to connect different regions of the ground plane.

Figure 8

(a) Tunable resonator frequency as a function of applied flux, where the frequencies are extracted from $S_{21}$ data. The full range of tunability is roughly 1.2 GHz. The blue cross marks a typical operating point at ${\mathrm{\Phi }}_z^{\mathrm{tres}}={\mathrm{\Phi }}_0/4$. (b) The derivative of the tunable resonator modulation curve in (a) as a function of applied flux, in units of $\mathrm{MHz}/\mathrm{m}{\mathrm{\Phi }}_0$. The flux range has been restricted to show a region within only $1{\mathrm{\Phi }}_0$ of flux tuning. The blue cross marks the same typical operating point as in the top plot.

Figure 9

Transmission and fit for the tunable resonator at its zero-flux bias point. The fit yields a total quality factor of $Q=720\pm 50$.

Figure 10

Circuit schematic used to construct a Hamiltonian for quantum energy level calculations. The QFP is in the middle of the schematic and couples to the flux qubit and rf SQUID section of the tunable resonator by 60 pH mutual inductances at the red inductors. The parallel capacitances of the Josephson junctions and linear inductors are included with the junction and inductor symbols. The two junctions in the flux qubit $z$ loop are approximated by a linear inductor in order to reduce the dimensionality of the Hamiltonian (far left). The quarter-wave transmission line segment of the tunable resonator is approximated by a lumped-element model (far right).

Figure 11

Circuit energy levels calculated using QuTip on the Hamiltonian generated by Circuitizer. (a) Wide view of the lowest 10 energy levels as a function of ${\mathrm{\Phi }}_z$. (b) Zoom-in of the first excited state of the flux qubit and its anticrossings with the fundamental mode of the tunable resonator. The energy levels are colored by energy ordering, not by mode, so the modes switch colors at the anticrossings. The shift of the minimum toward negative ${\mathrm{\Phi }}_z^{\mathrm{qub}}$ is caused by asymmetry in the $x$-loop junctions of the flux qubit.