Using Superconducting Qubits in Quantum Computing

A quantum computer may be able to succeed in computational problems that a traditional computer cannot solve in practice^1,2. A device such as this must fulfill certain criteria³, for example, the ability to initialize the state of its basic building blocks – the qubits – with a high fidelity. Using an active reset method for superconducting qubits, a 10-fold improvement for the final state preparation fidelity is demonstrated along with a 10-fold speed-up in reset for a given fidelity.

This method comprises of a single-shot measurement of the state of the qubit and then a conditional single qubit gate operation which rotates the qubit into the ground state if it was found in the excited state⁴. The method is compared with the simpler alternative for state initialization, waiting passively for the qubit to decay, and quantifying the fidelity and speed advantage of the active method. Speed is important in order to obtain high experimental repetition rates.

Generally, the achievable feedback latency limits the speed of the active method. The speed of the passive method is restricted by the lifetime of the qubit which makes it impractical in view of the advances in fabrication which lead to continually improving qubit lifetimes. The passive method has a limit set by the system temperature and this can be exceeded by the fidelity of the active method.

The main problem when implementing active qubit reset is the requirement for conditional signal generation and low-latency signal processing. The UHFLI from Zurich Instruments allows feedback on the scale of a microsecond. This is due to it incorporating an arbitrary waveform generator, a fast digital lock-in amplifier, and a cross-domain trigger engine to join the two. The UHFLI means that active qubit reset is available without the cost of developing and then maintaining a customized digital signal processing solution⁴.

Setup and Sample

Figure 1(a) shows the sample utilized: it consists of aluminum- and niobium- based resonant circuits on a sapphire substrate⁵. The qubit shown in the inset is a nonlinear resonator which is formed of an on-chip capacitor (orange in the picture) and a small superconducting quantum interference device (SQUID) acting as a nonlinear inductor. The qubit is tunable over a frequency range from 4 to 7.5 GHz due to the SQUID design which applies a small magnetic field to the device.

External control signals are guided to the qubit, which is attached to a coplanar waveguide resonator (blue), by a coplanar waveguide (green)⁶ with a frequency of 4.78 GHz. The readout resonator is connected to a pair of coaxial cables that provide the interface for reading out the quantum state of the qubit through an on-chip Purcell filter. A schematic of the measurement setup, in which the sample is placed in a dilution refrigerator to provide a low-temperature well isolated environment to protect the quantum properties of the sample, is seen Figure 1(b).

Figure 1. (a) Optical micrograph of the sample showing the readout resonator (blue) and the qubit capacitor plates (orange). The resonator is coupled to input and output lines via a Purcell filter [7] (cyan). Insets show magnified views of the qubit and the coupler between resonator and Purcell filter. (b) Simplified diagram of the experimental setup based on the UHFLI instrument integrating a lock-in amplifier and an AWG. The superconducting qubit sample is cooled in a dilution refrigerator.

Quadrature modulation of a microwave signal using two AWG channels creates control pulses. By feeding one of the AWG marker channels to the gate input of a microwave frequency signal generator, the pulses for qubit readout are generated. Then, the pulses are improved using a parametric amplifier followed b room-temperature and low-noise cryogenic amplifiers.

Next, the signal is down-converted in the analog domain to an intermediary frequency f_IF of 28.125 MHz. The signal is further down-converted in the digital domain by employing one of the UHFLI lock-in amplifier channels. The in-phase and quadrature component signals that result are then digitized using the UHF-DIG Digitizer module of the UHF apparatus.

To allow the identification of the optimum set of parameters of the qubit π control pulse used to excite and reset the qubit, Qubit Rabi oscillation measurements were carried out⁸. A first-order DRAG pulse shape was used, as described in Ref 9.

Active Qubit Reset

Single-Shot Qubit Measurement

The beginning of the active qubit reset cycle is marked by measuring the original qubit state. The qubit state is obtained by using the lock-in amplifier to measure the down-converted pulsed readout signal and likening the acquired quadrature voltage most sensitive to the qubit state change to a threshold value. A signal-to-noise ratio that is large enough to discriminate the qubit states in a single shot without averaging over repeated experiments can be obtained with the high-performance amplification chain based on the parametric shift amplifier.

Active qubit reset requires the capability to achieve a single-shot readout. The histogram of the measured signal quadratures over 40,000 repetitions of the same experiment is displayed in Figure 2. No control pulse was used before the state measurement in Figure 2(a) and so the qubit is expected to be in thermal equilibrium close to the ground state (passive reset).

A π pulse was applied immediately before the state measurement in Figure 2(b), and two local maxima matching the qubit ground and first excited states are identified. There is also a weaker maximum in the bottom half of the plot, perhaps due to the population of the second excited state. Most of the contrast for the state measurement in contained within the in-phase (I) signal component. The threshold for state discrimination was set to 1 mV (red vertical lines).

Figure 2. Histograms of integrated signal quadratures in repeated single-shot state measurement. (a) shows the histogram when measuring the qubit in its thermal equilibrium ground state after waiting a time much longer than the qubit lifetime before taking an individual measurement. (b) shows the histogram of measurements taken after having applied a π pulse to the qubit. The two maxima in the histogram in (a) and (b) are identified with the qubit’s ground and excited state, respectively. The red numbers are the relative fraction of counts in the areas delimited by red lines which represent the state discrimination threshold used for active qubit reset. The population of the ground state in (b) is mainly caused by relaxation during the readout.

Implementing Feedback with the UHG-AWG

In order to perform the qubit state discrimination, the internal cross-domain trigger of the UHFL was configured. Following this discrimination, the digital signal is used to define a sequence prancing point in the UHF-AWG sequence program and this determines whether the AWG will output a dual-channel p pulse or no signal. The LabOne AWG Sequencer enables the formulation of the corresponding hardware instructions in high-level, easily readable programming language called SeqC. Figure 3 shows the core part of that program.

The playback of the waveform w­_read is initiated by the first instruction in the program. This waveform has a digital marker that is used as a gate signal in order to generate a readout microwave pulse. After the playback, the sequencer waits during wait_time until the state discrimination operation has been performed by the cross-domain trigger and the readout signal is obtainable. The sequencer then reads the binary result of the discrimination and stores it in the run_time variable qb_state.

The resulting switch statement contains two branches and one of these is executed conditionally on the value of qb­_state (0 to 1). One branch links to the playback of the zero-valued waveforms w_zero_1 and w_zero_2 and the other to the dual-channel playback of the waveforms w_pi_1 and w_pi_2 (the qubit p pulse) of the same length. The waveforms in this part of the program are defined, using mathematical functions assess by the complier, in the same program (not shown).

Figure 3. Core part of the sequence program in the SeqC language used to control the conditional feedback action.

Repeated Active Qubit Reset

Repeating the feedback cycle described above can increase the efficiency of the qubit reset by allowing a sub-percent-level excited-state population to be achieved. This is better than what would be possible using a single feedback cycle. Lopping over the code segment, shown above, enables the repetition of the cycle.

The development of the qubit state during a qubit reset protocol consisting of 23 repetitions of a feedback cycle is illustrated by the blue curve in Figure 4. The excited-state population averaged over 40,000 repetitions of the experiment are shown by the blue squares. A control experiment was completed where the qubit state was repeatedly read out in the same way as with active reset enabled, but without a p pulse applied to the qubit, in order to compare the decay curve with one observed without active reset.

Figure 4. Time evolution of the averaged qubit excited-state occupation with active and passive qubit reset, respectively. Before the start of the protocol, the qubit is initialized by applying a π pulse. Every 1.48 microseconds, a state measurement is taken. After each measurement, a conditional π pulse is applied to initiate an active reset, or no pulse is applied for passive reset.

The decay curve with repeated measurements but without feedback is identical to a decay curve measured with a conventional T₁ measurement, as described in Reference 8.

The qubit excited state population dropped to approximately 12% after the first feedback cycle or 1.48 ms. Achieving this level using the passive method takes about 14 ms. The initially fast decay gradually slows after four to five replications, but the excited-state population continues to decrease. It has dropped below 0.3% after 23 feedback repetitions, but, using the passive method the residual occupation converges to 3% after a long waiting time, also compared to the measurement in Figure 2(a).

Feedback Latency

In order to deduce the fidelity of the active qubit reset, feedback latency is critical. Having a small latency lessens the spontaneous qubit decay occurring between the readout and the p pulse, which is the main cause of error during each feedback cycle. This mechanism accounts for about 19% of error from the lifetime of the qubit T₁ of 7.1 ms and the period of 1.48 ms of one entire feedback cycle.

Furthermore, a small latency indicates that more reset cycles can be accomplished in the same amount of time. Measurements of the feedback latency in the configuration used for the experiment discussed above are discussed in the following.

The setup used for those measurements is shown in Figure 5. The wiring is altered compared to the setup in Figure 1, so that the readout return signal and the qubit control signals are redirected to an oscilloscope. Therefore, the UHFLI does not receive a readout signal and the control signals that the UHF-AWG generates do not reach the qubit. However, as the signal processing latency in the UHFLI instrument is not affected by these changes, this setup allows the observation of the same latency that would apply to the actual measurement. This is referenced to the plane of the UHFI Signal Input and Output connectors.

Figure 5. Setup used to measure the feedback latency of the active qubit reset. The qubit control signals and the qubit readout signals are routed with cables of equal length from the UHFLI to an oscilloscope. The gate signal generated by the UHF-AWG marker channel 1 is connected to the oscilloscope via a T-piece for triggering. The UHFAWG Sequencer executes the same program as in the experiment, which makes the timing in this measurement equal to the timing in the actual experiment.

The qubit readout and control signals measured in the configurations are shown in Figure 6. In order to make the timings clearly visible despite the noise, the readout signal (top) at the intermediate frequency is averaged over multiple scope acquisitions. A latency of 1.12 ms was measured from the beginning of the readout signal burst to the start of the qubit control pulses generated by the AWG.

Some of this time is the 0.37 ms integration time that allowed a high enough SNR to obtain the single-shot readout. Other signal processing such as A/D and D/A conversion or demodulation accounts for the remainder of about 0.75 ms, and this signifies the minimum achievable latency with the UHF instrument.

Conclusion

The benefits of the active qubit reset method are shown by the measurements which determine the ease of implementation with the UHFLI Lock-In Amplifier and the UFH-AWG Arbitrary Waveform Generator. A ten-fold improvement for the final state preparation fidelity and a ten-fold speed-up in reset for a given fidelity was achieved when compared to the passive reset method.

These results build on low-latency, powerful digital signal processing tools that are available on the Zurich Instruments UHF Instrument. It is possible to achieve demodulation at multiple frequencies with the UHF-MR option and this facilitates the flexible extension to multi-qubit measurement and control systems. Additionally, multiple AWG sequence branches allow for more complex feedback protocols, e.g. taking detection of higher excited states into account.

Acknowledgments

Zurich Instruments would like to thank Prof. Wallraff and the members of his Quantum Device Lab at ETH Zurich, Switzerland, where these measurements were carried out. Special thanks go to Michele Collodo. Thanks go to Theodore Walter for providing the qubit sample and Yves Salathé, Simone Gasparinetti, and Philipp Kurpiers for support with the measurements and for discussions. This work was supported by the Swiss Federal Department of Economic Affairs, Education and Research through the Commission for Technology and Innovation (CTI).

References

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This information has been sourced, reviewed and adapted from materials provided by Zurich Instruments.

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