## New Research In

### Physical Sciences

### Social Sciences

#### Featured Portals

#### Articles by Topic

### Biological Sciences

#### Featured Portals

#### Articles by Topic

- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology

# Gate fidelity and coherence of an electron spin in an Si/SiGe quantum dot with micromagnet

Contributed by Susan N. Coppersmith, August 28, 2016 (sent for review March 1, 2016; reviewed by Guido Burkard and John J. L. Morton)

## Significance

A quantum computer is able to solve certain problems that cannot be solved by a classical computer within a reasonable time. The building block of a quantum computer is called a quantum bit (qubit), the counterpart of the conventional binary digit (bit). A qubit unavoidably interacts with its environment, leading to errors in the qubit state. This article reports on the qubit performance of an electron spin in a silicon/silicon-germanium (Si/SiGe) quantum dot, and examines the dominant error mechanisms. We demonstrate that this qubit can be electrically controlled with sufficient accuracy so that remaining errors could, in principle, be corrected using known protocols, even without isotopically purified silicon. This qubit also offers a quantum memory that lasts for almost 0.5 ms.

## Abstract

The gate fidelity and the coherence time of a quantum bit (qubit) are important benchmarks for quantum computation. We construct a qubit using a single electron spin in an Si/SiGe quantum dot and control it electrically via an artificial spin-orbit field from a micromagnet. We measure an average single-qubit gate fidelity of ∼99% using randomized benchmarking, which is consistent with dephasing from the slowly evolving nuclear spins in the substrate. The coherence time measured using dynamical decoupling extends up to ∼400 μs for 128 decoupling pulses, with no sign of saturation. We find evidence that the coherence time is limited by noise in the 10-kHz to 1-MHz range, possibly because charge noise affects the spin via the micromagnet gradient. This work shows that an electron spin in an Si/SiGe quantum dot is a good candidate for quantum information processing as well as for a quantum memory, even without isotopic purification.

The performance of a quantum bit (qubit) is characterized by how accurately operations on the qubit are implemented and for how long its state is preserved. For improving qubit performance, it is important to identify the nature of the noise that introduces gate errors and leads to loss of qubit coherence. Ultimately, what counts is to balance the ability to drive fast qubit operations and the need for long coherence times (1).

Electron spins in Si quantum dots are now known to be one of the most promising qubit realizations for their potential to scale up and their long coherence times (2⇓⇓⇓⇓⇓⇓⇓–10). Using magnetic resonance on an electron spin bound to a phosphorus impurity in isotopically purified ^{28}Si (5) or confined in a ^{28}Si metal–oxide–semiconductor (MOS) quantum dot (3), ∼0.3-MHz Rabi frequencies, gate fidelities over 99.5%, and spin memory times of tens to hundreds of milliseconds have been achieved. Also, electrical control of an electron spin has been demonstrated in a (natural abundance) Si/SiGe quantum dot, which was achieved by applying an AC electric field that oscillates the electron wave function back and forth in the gradient magnetic field of a local micromagnet (7). The advantage of electrical control over magnetic control is that electric fields can be generated without the need for microwave cavities or striplines and allows better spatial selectivity, which simplifies individual addressing of qubits. However, the magnetic field gradient also makes the qubit sensitive to electrical noise, so it is important to examine whether the field gradient limits the spin coherence time and the gate fidelity.

In our previous work (7), the effect of electrical noise on spin coherence and gate fidelity was overwhelmed by transitions between the lowest two valley-orbit states. Because different valley-orbit states have slightly different Larmor frequencies, such a transition will quickly randomize the phase of the electron spin. If valley-orbit transitions can be (largely) avoided, then the question becomes what limits coherence and fidelities instead.

Here we measure the gate fidelity and spin echo times for an electron spin in an Si/SiGe quantum dot in a regime where the electron stably remains in the lowest valley-orbit state for long times, and where the corresponding resonance condition is well separated from that associated with the other valley-orbit state. To learn more about the dominant noise sources in this new regime, we use dynamical decoupling experiments to extract the noise spectrum in the range of 5 kHz to 1 MHz, and we compare this spectrum with spectra derived from numerical simulations for various noise sources. We also study the influence of the various noise sources on the gate fidelity.

## Device and Measurement Setup

A single electron spin is confined in a gate-defined quantum dot in an undoped Si/SiGe heterostructure (6⇓–8) (Fig. 1). The sample is attached to the mixing chamber (MC) stage of a dilution refrigerator with base temperature of ∼25 mK, and subject to a static external magnetic field of 794.4 mT along the direction as indicated in Fig. 1, *Inset*. Spin rotations are achieved by applying microwave excitation to one of the gates, which oscillates the electron wave function back and forth in the magnetic field gradient produced by two cobalt micromagnets fabricated on top of the device. The device used in this work is the same as in the previous work (7), but the applied gate voltages are set differently to obtain a higher valley-orbit splitting.

The measurement scheme consists of four stages: initialization, manipulation, readout, and emptying, as shown in ref. 7. Differently from ref. 7, the four-stage voltage pulse is applied to gate 8, and the microwave excitation is applied to gate 3. The initialization and readout stages take 4 ms to 5 ms, and the manipulation and emptying stages last 1 ms to 1.5 ms.

Because the experimental details of the setup are important for the results shown in this paper, we here summarize the key components. A voltage pulse applied to gate 8 is generated by an arbitrary waveform generator (Tektronix AWG 5014C). Phase-controlled microwave bursts are generated by an Agilent microwave vector source E8267D with the I (in-phase) and Q (out-of-phase) components controlled by two channels of the AWG. The on/off ratio of the I/Q modulation is 40 dB. If the microwave power arriving at the sample is not sufficiently suppressed in the “off” state, the control fidelity is reduced and the effective electron temperature increases, which, in turn, will result in lower readout and initialization fidelities. Reduced fidelities were indeed observed when applying high-power microwave excitation (>15 dBm at the source) using I/Q modulation only. As a solution, we use digital pulse modulation (PM) in series with the I/Q modulation, which gives a total on/off ratio of ∼120 dB. A drawback of PM is that the switching rate is lower. Therefore, the PM is turned on 200 ns before the I/Q modulation is turned on (Fig. 1, *Inset*). We also observe that the total microwave burst time applied to the sample per cycle affects the readout and initialization fidelities (*SI Appendix*). To keep the readout and initialization fidelities constant, we apply an off-resonance microwave burst (with microwave frequency detuned by 30 MHz from the resonance frequency) 2 μs after the on-resonance microwave burst, so that the combined duration of the two bursts is fixed. To achieve this rapid shift of the microwave frequency, we used frequency modulation (FM) controlled by another channel of the AWG. FM is turned on 1 μs after the on-resonance burst is turned off (Fig. 1, *Inset*).

The electron spin state is read out via spin-to-charge conversion by aligning the Fermi level of the reservoir between the spin-down and spin-up states and below the spin-up state combined with real-time charge detection (11). The probability that the current exceeds a predefined threshold during the readout stage is interpreted as the spin-up probability of the electron (7). The analysis of the real-time traces and the statistical analysis of the readout events are done on the fly using a field-programmable gate array (FPGA) as depicted in Fig. 1; this allows us to measure faster without waiting for the transfer of real-time traces to a computer. Data points were taken by cycling through the various burst times, spin echo waiting times, or randomized gate sequences, and repeating these entire cycles 250 to 1,000 times. This order of the measurements helps to suppress artifacts in the data caused by slow drift in the setup or sample.

## High-Quality Rabi Oscillations

Rabi oscillations are recorded by varying the burst time and the microwave frequency. With the present gate voltage settings, the spin resonance frequencies corresponding to the two lowest valley-orbit states are separated by ∼5 MHz (at *A*). This difference of ∼5 MHz results mainly from slightly different electron *g* factors between the two valley-orbit states. The population of the valley-orbit ground state is estimated to be ∼80% from Fig. 2*A*, which is higher than in our previous work (7), and implies a higher valley-orbit splitting. (Due to many unknown parameters, it is difficult to obtain a reliable estimate of the valley-orbit splitting. Nevertheless, assuming that the values of these parameters are equal between the previous work (7) and this work, the higher population of the ground valley-orbit state implies a higher valley-orbit splitting. See *SI Appendix* for more details.) Fig. 2*B* shows a Rabi oscillation of a single spin with the electron in the ground valley-orbit state. The Rabi frequency extracted from the data is 1.345 MHz. The decay of the oscillation is what we would expect assuming a statistical distribution of resonance conditions with a line width of 0.63 MHz (FWHM), which is the number extracted from the continuous wave response (7). This line width corresponds to ^{29}Si spins in the substrate, similar to ref. 7. Here there is no evidence of additional decay mechanisms. In particular, we do not see any indication of intervalley switching with or without spin flip, or the combined effects of electrical noise and the magnetic field gradient.

## Dynamical Decoupling

Next we examine the spin memory time of this electrically controlled spin qubit. In our previous work (7, 12), due to switching between the two valley-orbit states, the Hahn echo decay was exponential with coherence time of ∼40 μs. Furthermore, we were unable to extend the coherence time using multiple echo pulses. Due to the difference in Larmor frequency between two valley-orbit states, as soon as a switch from one to the other valley-orbit state occurred, phase information could not be recovered by echo pulses. In this work, we observe significantly extended coherence times, presumably because the switching between valleys is slower in the present gate voltage configuration.

We study the spin memory characteristics using two types of two-axis dynamical decoupling sequences, based on the XY4 (13), (XY4)^{n} [sometimes called vCDD (14)], and XY8 (15) protocols. Fig. 3 *A* and *B*, *Insets* show the (XY4)^{n} and XY8 pulse sequences for 16 π pulses. We use *X* and *Y* to denote π rotations about *A* and *B*, we show the data, normalized to the echo amplitude at ^{n} and XY8, respectively.

To analyze these decay curves, we adopt a semiclassical approach, in which the decay curve of the echo amplitude is written as*C* shows **4** to the decay curves. The longest *SI Appendix*). We fitted *C*.

We can derive the noise spectrum from the decay curves in Fig. 3 *A* and *B* using the fact that the filter function is narrow around *SI Appendix*). The circles in Fig. 3*D* show the noise spectrum extracted from six decay curves in Fig. 3*B*. The colors of the circles in Fig. 3*D* correspond to the colors used in Fig. 3*B* for different *D* is based on Eq. **3**, with *α* obtained from the fit (green line) to the data in Fig. 3*C*; its decay is close to a

To capture both the flat and decaying parts of the spectrum and obtain more insight into the nature of the noise spectrum, we now write the noise spectrum in the form**1** to the six decay curves (leaving out *B* simultaneously, using also Eqs. **2** and **5**, with *A* and **3**, but the fits deviate from the measured echo decays (see *SI Appendix*, Fig. S5*E*). A better fit to the echo decay data using Eq. **5** is obtained for *B*), in which case Eq. **2** can be expressed analytically (20). The fits in Fig. 3*B* yield ^{2}⋅s^{−1} and *D*, shows reasonable agreement with

Extrapolating the fitted noise spectrum to frequencies below 5 kHz, where we do not have experimental data, the noise spectral density looks flat; this would result in an exponential Ramsey decay with *D* (*SI Appendix*).

We now turn to the noise mechanisms and examine whether the hyperfine coupling of the electron spin with the evolving nuclear spins can explain the observed noise spectrum. Nuclear spin dynamics have two main mechanisms, hyperfine-mediated and dipole−dipole interactions between nuclear spins. Decoherence due to the hyperfine-mediated interactions is negligible in Si at *SI Appendix*). However, magnetic dipole−dipole induced nuclear spin dynamics cannot be neglected. We performed numerical simulations of the spectrum of the nuclear spin noise and of the Hahn echo decay for a dot with 4.7% of ^{29}Si nuclei (natural abundance) within the coupled pair-cluster expansion (28) for several choices of the quantum dot parameters. A calculated spectrum is approximated to an analytical expression: *D*. The measured Gaussian-shape Ramsey decay with ^{29}Si nuclear spins dominate the noise at low frequencies (7, 22) (see *SI Appendix* for details); they also dominate the gate fidelities discussed in *Randomized Benchmarking*. However, at higher frequencies, even though the noise spectrum calculated from the nuclear spin dynamics has the same shape as Eq. **5** with *α* = 3, the amplitude and the correlation time are significantly different from the noise spectrum measured by the dynamical decoupling. With the calculated correlation time and the amplitude for the nuclear spins, the Hahn echo decay time

We therefore conclude that the noise spectrum consists of at least two contributions: nuclear spin noise at low frequencies and another mechanism at higher frequencies. At higher frequencies, the noise spectrum decays as *D* [Eq. **5** with ^{2}⋅s^{−1}, and

## Randomized Benchmarking

We measured the average gate fidelity using standard randomized benchmarking (SRB), which is known as an efficient way to measure the gate fidelity without suffering from initialization and readout errors (30, 31). The specific procedure is as follows. After initializing the electron to the spin-down state, we apply randomized sequences of *m* Clifford gates and a final Clifford gate *p* reflects the imperfection of the average of 24 Clifford gates. Under certain assumptions, for *m* successive Clifford gates, the depolarization parameter is

We measure the spin-up probability both for the case where spin-up is the target state, *m*, and varying *m* from 2 to 220. The difference of the measured spin-up probability for these two cases, *A*. Theoretically, *a* is a prefactor that does not depend on the gate error. As seen in Eq. **6**, differently from quantum process tomography (6, 34, 35), the measurement of the gate fidelity is not affected by the initialization and readout infidelities, assuming these infidelities are constant throughout the measurement. To keep the readout and initialization fidelities constant for different *m*, we kept the total microwave burst time

Fig. 4*A* shows that the measured decay does not follow a simple exponential **5** using ^{2}⋅s^{−1}, and *B* and show good agreement with experiment.

To evaluate explicitly the relative contribution of *SI Appendix*). This dominant contribution of the (quasi-)static noise to the gate error is consistent with an earlier report (36), in which it was also shown that ensemble averaging over individual exponential decays can lead to a nonexponential decay. Repeated measurements in the presence of low-frequency noise effectively lead to such ensemble averaging.

Because the measured SRB decay is not of the form

We also characterized the fidelity of individual gates using interleaved randomized benchmarking (IRB). In this procedure, a specific gate is interleaved between randomized Clifford gates. The depolarizing parameter now becomes bigger than in SRB, due to the imperfections of the interleaved gate. From the difference in the depolarizing parameter between SRB and IRB, the fidelity of the interleaved gate is extracted. In Fig. 4*A*, the blue circles show the case where the Hadamard gate is the interleaved gate. The Hadamard gate is implemented by a π rotation around the

## Discussion and Conclusion

We have shown that the average single gate fidelity for a single electron spin confined in a ^{nat}Si/SiGe quantum dot approaches the fault-tolerance threshold for surface codes (1). The low-frequency noise that limits gate fidelity is well explained by the nuclear spin randomness given the natural abundance of ^{29}Si. Therefore, we can increase gate fidelities by reducing the abundance of ^{29}Si using isotopically enriched ^{28}Si (3, 5) or by using composite pulses (38). Also, the readout fidelity can be boosted to the fault-tolerance threshold by using Pauli spin blockade readout (39) and RF reflectometry (40). The longest coherence time measured using dynamical decoupling is ∼400 μs. We revealed that the noise level is flat in the range of 5 kHz to 30 kHz and decreases with frequency in the range of 30 kHz to 1 MHz. In this frequency range (5 kHz to 1 MHz), the measured noise level is higher than expected from the dynamics of the ^{29}Si nuclear spins. Instead, charge noise in combination with a local magnetic field gradient may be responsible. If charge noise is dominant, dynamical decoupling decay times can be further extended by positioning the electron spin so that the gradient of the longitudinal component of the magnetic field gradient vanishes, while keeping the transverse component nonzero as needed for driving spin rotations. At that point, we can reap the full benefits from moving to ^{28}Si enriched material for maximal coherence times as well.

## Acknowledgments

We acknowledge R. Hanson, G. de Lange, M. Veldhorst, S. Bartlett, and L. Schreiber for useful discussions, and R. Schouten and R. Vermeulen for technical support. This work was supported in part by Army Research Office (W911NF-12-0607), the Dutch Foundation for Fundamental Research on Matter (FOM), and a European Research Council (ERC) Synergy grant; development and maintenance of the growth facilities used for fabricating samples is supported by Department of Energy (DOE) (DE-FG02-03ER46028). E.K. was supported by a fellowship from the Nakajima Foundation. This research used National Science Foundation-supported shared facilities at the University of Wisconsin-Madison. Work at the Ames Laboratory (analysis of nuclear spin noise and decoherence) was supported by the DOE-Basic Energy Sciences under Contract DE-AC02-07CH11358.

## Footnotes

↵

^{1}Deceased February 27, 2015.- ↵
^{2}To whom correspondence may be addressed. Email: snc{at}physics.wisc.edu or l.m.k.vandersypen{at}tudelft.nl.

Author contributions: E.K., T.J., P.S., M.F., S.N.C., M.A.E., and L.M.K.V. designed research; E.K., T.J., P.S., D.R.W., D.E.S., M.G.L., V.V.D., M.F., S.N.C., M.A.E., and L.M.K.V. performed research; E.K., T.J., P.S., V.V.D., M.F., S.N.C., M.A.E., and L.M.K.V. analyzed data; and E.K., P.S., V.V.D., M.F., S.N.C., M.A.E., and L.M.K.V. wrote the paper.

Reviewers: G.B., University of Konstanz; and J.J.L.M., University College London.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1603251113/-/DCSupplemental.

Freely available online through the PNAS open access option.

## References

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- de Lange G,
- Wang ZH,
- Ristè D,
- Dobrovitski VV,
- Hanson R

- ↵
- ↵
- ↵
- ↵
- ↵.
- Bluhm H, et al.

- ↵.
- de Sousa R

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Shulman MD, et al.

- ↵

## Citation Manager Formats

### More Articles of This Classification

### Physical Sciences

### Related Content

- No related articles found.

### Cited by...

- No citing articles found.