How Double-Superheterodyne Compares with IQ Mixing in Qubit Experiments

Were you ever curious about how double-superheterodyne (DSH) frequency conversion compares with IQ mixing when tested on real qubits? Johannes Herrmann and coworkers from the Quantum Device Lab at ETH Zurich have done exactly that and have recently made their findings available in a preprint paper [1]. They compared the performance of our SHFSG Signal Generator with that of a high-quality IQ mixing setup similar to the HDIQ IQ Modulator. The most exciting result: randomized benchmarking measurements hint at DSH enabling a higher gate fidelity and lower state leakage rate than a standard IQ mixing approach! Both methods allow for operation very close to the coherence limit of the used qubits, but further measurements are required to demonstrate a sizable improvement with higher coherence qubits.

These results are complemented with  characterization measurements of parameters important for achieving high-fidelity gate operations. Compared with IQ mixing, DSH gets you 20 dB better spurious-free dynamic range, and a comparable phase noise which is even superior in the low-frequency, or long-timescale limit. It also removes the need for mixer calibration, a method that is both time-consuming and susceptible to temperature drift. 

Let’s look at these elements in more detail.

DSH provides better spurious-free dynamic range over a wider frequency band

Spurious-free dynamic range (SFDR) specifies the ratio in dB between a carrier signal amplitude and the largest unwanted peak within the specified bandwidth. In quantum computing, a spectrum free of spurious peaks means less risk of exciting unwanted transitions and less dephasing of the addressed qubit transition. In the relevant operating conditions, Johannes Herrmann and coworkers measured a SFDR of 72 dBc in the DSH approach vs. 52 dBc in the IQ mixing approach. That’s 20 dB of safety margin which provides you with more freedom and peace of mind when choosing your transition frequencies.

How does that come about? A real IQ mixer is not as perfect a multiplier as its textbook mixer equation suggests. One can tune knobs to iron out the imperfections of the analog world. By adding voltage offsets to the I and Q ports one may suppress an undesired signal at the local oscillator frequency - the LO leakage. By adjusting the relative amplitude and phase of the I and Q signals, one may suppress an undesired sideband at the mixer image frequency. But unfortunately, the imperfections don’t stop there: higher-order sidebands are next in line. In the setup used in this paper, a second-order sideband limits the SFDR to 52 dBc even after a fresh calibration. Too many imperfections for the number of available knobs!
 

DSH provides better spectral stability

A mixer calibration doesn’t stay fresh for too long. Upon a temperature change of a few degrees C, which can easily occur in a typical lab setting, the suppressed LO and image components rise in amplitude by 10 to 20 dB. That means there are now 3 spurious components in the 50 to 60 dBc amplitude range within several 100 MHz, not only one! In comparison, the DSH spectrum stays where it is: 72 dBc, independent of time and temperature.

What’s the explanation? Filtering, which is used in DSH to suppress unwanted sidebands, is hardly temperature dependent. In contrast, IQ mixing relies on an interferometric cancellation of signals for that purpose. Already a minimal amplitude drift or phase shift on one of the interfering paths is enough to move away from the optimal point significantly.
 

DSH offers comparable phase noise

Qubit control signals need to have a better phase coherence than the qubits themselves over the full length of a coherent pulse sequence. This puts a tough requirement on the phase noise of the control electronics. 

In practice, the phase noise is determined by the quality of the microwave sources used, and not by the frequency conversion technology. However, the setup must perform as a whole, and phase noise characterization was therefore part of the comparison. 

The authors compared the output signal of the IQ setup based on the Rohde & Schwarz SGS100A, and the output signal of the Zurich Instruments SHFSG Signal Generator. For the latter, they tested two variants: one with the standard local oscillator source, and one with a local oscillator design with lower phase noise. [2] The comparison shows that for both variants, the phase noise of the DSH setup is better than that of the IQ setup for low offset frequencies (<1 kHz or <10 kHz for the standard or the low-phase-noise variant, respectively).

How does this relate to the application? The offset frequency in the phase noise spectrum is the inverse time scale of the phase fluctuations. The longer the pulse sequence of interest, the more important are slow phase fluctuations, and with them the low-frequency part of the spectrum. The longer the pulse sequence of interest, the more important the low-frequency phase fluctuations are. For a quantitative analysis, the authors calculate the process fidelity limit for a Ramsey experiment as a function of the Ramsey pulse delay and find that the DSH setup begins to outperform the IQ setup for delays above ~1 millisecond. Both approaches offer more than 1 order of magnitude headroom over the limits imposed by today’s qubit coherence. Looking forward however, the phase noise performance of the DSH setup offers a healthy outlook for research on new superconducting qubits, which begin to enter this range of coherence times!
 

DSH may improve gate error and leakage

Spectral measurements and simulations are all interesting - but what really matters is the performance in an experiment. The authors ran a randomized benchmarking study with both setups on the same qubit. Specifically, they measured the gate error rates and leakage to the second excited state.

For both these parameters, the values obtained with the DSH scheme are slightly better than those obtained with the IQ scheme, while the uncertainty in the measured values is similar to their difference. The average gate error rate obtained with the DSH scheme, for example, is 20% closer to the coherence limit than that obtained with the IQ mixing setup [3]. Since both methods operate very close to the coherence limit already, the theoretically possible improvement is small, and it’s yet hard to obtain clearer evidence for gate fidelity improvement than the difference of about 1 standard deviation found in these measurements. But these results are exciting, and we are curious to see further comparison measurements!

Don’t you want to test that on your qubits? Call or e-mail us and we’re happy to discuss, or even arrange a test measurement in your lab!

References

1. Johannes Herrmann, Christoph Hellings, Stefania Lazar, Fabian Pfäffli, Florian Haupt, Tobias Thiele, Dante Colao Zanuz, Graham J. Norris, Flavio Heer, Christopher Eichler, Andreas Wallraff, "Frequency Up-Conversion Schemes for Controlling Superconducting Qubits", arXiv:2210.02513.
2. If you would like to know more about the availability of the variant with lower phase noise, please contact us.
3. Error rate with DSH: 0.161+-0.008%; error rate with IQ: 0.169+-0.007%; coherence limit: 0.13%.