Quantifying error and leakage in an encoded Si/SiGe triple-dot qubit

TL;DR

This paper demonstrates a silicon-based encoded three-spin qubit with low error rates, using a novel benchmarking protocol to quantify both computational and leakage errors, advancing scalable quantum computing.

Contribution

It introduces a silicon triple-dot qubit encoded in a subsystem, employing a modified benchmarking protocol to measure errors, including leakage, with only voltage-controlled exchange interactions.

Findings

Total error rate of 0.35% for unitary operations

Leakage driven by substrate nuclear spins at 0.17%

Operational benefits of encoded subsystems demonstrated

Abstract

Quantum computation requires qubits that satisfy often-conflicting criteria, including scalable control and long-lasting coherence. One approach to creating a suitable qubit is to operate in an encoded subspace of several physical qubits. Though such encoded qubits may be particularly susceptible to leakage out of their computational subspace, they can be insensitive to certain noise processes and can also allow logical control with a single type of entangling interaction while maintaining favorable features of the underlying physical system. Here we demonstrate a qubit encoded in a subsystem of three coupled electron spins confined in gated, isotopically enhanced silicon quantum dots. Using a modified "blind" randomized benchmarking protocol that determines both computational and leakage errors, we show that unitary operations have an average total error of 0.35%, with 0.17% of that coming from leakage driven by interactions with substrate nuclear spins. This demonstration utilizes only the voltage-controlled exchange interaction for qubit manipulation and highlights the operational benefits of encoded subsystems, heralding the realization of high-quality encoded multi-qubit operations.

Equations

$$y_0=A+B(1-p{)}^N+C(1-q{)}^N,$$

$$y_1=A-B(1-p{)}^N+C(1-q{)}^N.$$

$$\epsilon =\mathrm{Tr}\left\{P_0[\rho (0)-\rho (1)]\right\}/(2B)=p/2+Cq/(2B).$$

$$\Gamma =\mathrm{Tr}\left\{[P_L[\rho (0)-\rho (1)]\right\}/(2B)=Cq/B,$$

Full text

Quantifying error and leakage in an encoded Si/SiGe triple-dot qubit

R. W. Andrews

  

C. Jones

  

M. D. Reed

  

A. M. Jones

  

S. D. Ha

  

M. P. Jura

  

J. Kerckhoff

  

M. Levendorf

  

S. Meenehan

  

S. T. Merkel

  

A. Smith

  

B. Sun

  

A. J. Weinstein

  

M. T. Rakher

  

T. D. Ladd

  

M. G. Borselli

HRL Laboratories, LLC, 3011 Malibu Canyon Rd., Malibu, CA, 90265, USA

**Quantum computation requires qubits that satisfy often-conflicting criteria, including scalable control and long-lasting coherence Ladd et al. (2010). One approach to creating a suitable qubit is to operate in an encoded subspace of several physical qubits. Though such encoded qubits may be particularly susceptible to leakage out of their computational subspace, they can be insensitive to certain noise processes Zanardi and Rasetti (1997); Lidar et al. (1998) and can also allow logical control with a single type of entangling interaction DiVincenzo et al. (2000) while maintaining favorable features of the underlying physical system. Here we demonstrate a qubit encoded in a subsystem of three coupled electron spins confined in gated, isotopically enhanced silicon quantum dots DiVincenzo et al. (2000); Eng et al. (2015). Using a modified “blind” randomized benchmarking protocol that determines both computational and leakage errors Wallman et al. (2016); Wood and Gambetta (2018), we show that unitary operations have an average total error of $\mathbf{0.35\%}$, with $\mathbf{0.17\%}$ of that coming from leakage driven by interactions with substrate nuclear spins. This demonstration utilizes only the voltage-controlled exchange interaction for qubit manipulation and highlights the operational benefits of encoded subsystems, heralding the realization of high-quality encoded multi-qubit operations DiVincenzo et al. (2000); Fong and Wandzura (2011). **

Electrons trapped in silicon heterostructures have many attractive features, including very long coherence times in isotopically enriched material Tyryshkin et al. (2011); Yoneda et al. (2018) and compatibility with standard fabrication techniques. Single-spin qubits have recently demonstrated high-fidelity RF-controlled single-qubit operations Kawakami et al. (2016); Yoneda et al. (2018) and two-qubit gates using the exchange interaction Zajac et al. (2017); Watson et al. (2018); Huang et al. . However, using RF signals for single-qubit control requires a large, stable magnetic field and introduces challenges with crosstalk. Fortunately, electron spins are particularly well-suited to forming encoded qubits. Two coupled electron spins can be operated at near-zero magnetic field as a “singlet-triplet” qubit Levy (2002); Petta et al. (2005). That qubit is insensitive to uniform magnetic field fluctuations but still requires a magnetic field gradient for universal control. Three coupled electrons Russ and Burkard (2017) can form a qubit with a tunable electric dipole moment, which could enhance RF selectivity, or the exchange-only qubit, which can be universally controlled using only the exchange interaction and does not require synchronization of gate operations with a local oscillator. Exchange is highly local and can be accurately controlled with a large on-off ratio using only fast voltage pulses. The combination of these features makes the exchange-only qubit especially attractive for use in many-qubit systems.

Here we demonstrate high-fidelity operation of a triple-dot, exchange-only qubit. Electrons are trapped in quantum dots formed by the electrostatic potentials of a Si/SiGe heterostructure and the patterned metallic gates shown in Fig. 1a Eng et al. (2015). As depicted in Fig. 1b, we manipulate the chemical potential of the dots by primarily adjusting voltages applied to gates P1, P2, and P3 and the exchange coupling between neighboring electrons by adjusting voltages applied to gates X1 and X2. The three coupled electrons constitute an 8-dimensional spin system that can be used to form an exchange-only qubit DiVincenzo et al. (2000); Russ and Burkard (2017).

Operation of this encoded qubit is conveniently described in terms of the coupled spin states $\left|{S_{12},S,m}\right\rangle$, where $S$ is total spin quantum number, $S_{12}$ is the total spin of the electrons confined beneath gates P1 and P2, and $m$ is the spin projection in an arbitrary direction, such as that of an applied magnetic field. The 8 possible states are enumerated in Fig. 1c. We use energy-selective initialization Petta et al. (2005); Eng et al. (2015) to prepare the first two electrons in a spin singlet state, giving $S_{12}$=0. Since the spin state of the third electron remains random, the qubit is initialized in an incoherent mixture of $\left|{0,\frac{1}{2},\frac{1}{2}}\right\rangle$ and $\left|{0,\frac{1}{2},-\frac{1}{2}}\right\rangle$. These states collectively comprise encoded 0 (denoted hereafter as $\left|{0}\right\rangle$). In this encoding, the quantum number $m$ does not affect logical operations and the qubit is insensitive to global magnetic fields. We measure the $S_{12}$ quantum number (which distinguishes between $\left|{0}\right\rangle$ and the six other spin states) via spin-to-charge conversion Petta et al. (2005); Eng et al. (2015) and charge-state detection using a dot charge sensor (formed using gates whose leads are visible in the top of Fig. 1a) and associated amplification circuitry Eng et al. (2015); Jones et al. . We denote $P_{0}$ as the projector onto $S_{12}=0$ and the probability of being found in $\left|{0}\right\rangle$ for a density matrix $\rho$ as $\mathrm{Tr}(P_{0}\rho)$. More detail on this subsystem algebra appear in the Supplemental Information.

To manipulate the encoded qubit state, we induce interaction between neighboring pairs of electrons by applying voltage pulses to device gates in order to lower the inter-dot potential barrier (see Fig. 1b). The resulting electron-electron exchange interaction only affects the $S_{12}$ quantum number Loss and DiVincenzo (1998); DiVincenzo et al. (2000); Eng et al. (2015), so only states within the $S=1/2$ subspace are accessible through exchange. However, other physical mechanisms may alter the total spin and drive the qubit into leaked states with $S=3/2$; in our case, leakage is generally caused by magnetic gradients such as those due to hyperfine interactions with nuclear spins. In the Bloch sphere representation, exchange between the two pairs of electrons induces rotations about non-orthogonal axes separated by 120 degrees DiVincenzo et al. (2000); Ladd (2012); Russ and Burkard (2017). As diagrammed in Fig. 1d, we label them as $\hat{z}$ and $\hat{n}$.

To demonstrate high-fidelity encoded operations, we first create a mapping between the voltage of the pulses we apply and the resulting exchange rotation. A rotation about the $\hat{n}$ axis involves voltage pulses on the P2, X2, and P3 gates, labelled in Fig. 1b. We choose the ratios of the voltages applied to each of these gates so that we primarily alter the potential barrier between the electrons while minimizing differences in the chemical potentials of the dots (see Fig. 2a). This “symmetric operation” minimizes sensitivity to charge noise Reed et al. (2016); Martins et al. (2016). The relative gate voltages required for symmetric operation can be found using a procedure outlined in Reed et al. (2016). For $\hat{n}$ rotations in this device, we pulse the voltage along the vector $\text{{V}}(V_{n})=\text{{V}}_{0}+V_{n}\left\{-0.20,1,-0.25\right\}$ in the $\left\{\mathrm{P2},\mathrm{X2},\mathrm{P3}\right\}$ voltage coordinates. Between pulses, gates are biased at an “idle” configuration $\text{{V}}_{0}$ in which all electrons are isolated and exchange rates are negligible. We map out qubit rotation by intializing the qubit, applying a sequence of three consecutive voltage pulses that are 10 ns wide and separated by an idle time of 20 ns (diagrammed in Fig. 2a), and measuring the resulting probability of finding the qubit in $\left|{0}\right\rangle$ (Fig. 2b). For this device, a single-pulse $2\pi$ rotation about the $\hat{n}$ axis occurs at $V_{n}\approx 150$ mV. By appropriately processing the data in Fig. 2b (see Supplemental Information), we create a voltage-to-rotation mapping for a single pulse, as shown in Fig. 2c. We see from this calibration that an increase in $V_{n}$ by 120 mV changes the resulting rotation angle by more than 3 orders of magnitude, exemplifying the large dynamic range available with this control technique. A similar procedure is used for mapping rotations about the $\hat{z}$ axis.

Calibrated rotations about the two axes can be interleaved to generate arbitrary single-qubit operations Lowenthal (1971). For example, an encoded Hadamard gate is applied with a sequence of a  0.96 radian rotation about the $\hat{z}$ axis, a  4.37 radian rotation about the $\hat{n}$ axis, and then a second  0.96 rotation about the $\hat{z}$ axis; this type of sequence is diagrammed in Fig 3b. For the construction used here, the 24 quantum gates in the Clifford group require an average of 2.7 individual rotations about some combination of the $\hat{n}$ and $\hat{z}$ axes, with the shortest Clifford requiring one rotation and the longest four. A complete table appears in the Supplementary Information. This construction is less efficient than possible with systems that have orthogonal rotation axes, where Clifford operations require an average of 1.9 rotations Kawakami et al. (2016). The advantage of the triple-quantum-dot, however, lies in its simplicity: all operations only require nearest-neighbor voltage-controlled exchange interactions. Two-qubit gates can also be implemented using only exchange between two neighboring triple-dot encoded qubits DiVincenzo et al. (2000); Fong and Wandzura (2011).

We demonstrate high-fidelity operation of this encoded qubit using a modified form of randomized benchmarking (RB) Emerson et al. (2007); Knill et al. (2008). A conventional RB experiment (diagrammed in Fig. 3a and b) consists of initializing the qubit to $|0\rangle$, applying a series of $N$ gates randomly chosen from the Clifford group, applying a final “recovery” operation $C_{R}$ that ideally inverts the preceding sequence, and measuring the resulting $y_{0}(N)=\mathrm{Tr}[P_{0}\rho(N)]$, where $\rho(N)$ is the qubit density matrix after $N$ gates and the recovery gate. The average gate error rate $\varepsilon$ is determined by sweeping $N$ and fitting $A+B(1-2\varepsilon)^{N}$ to $y_{0}$. Data resulting from such an experiment are shown as blue data points in Fig. 3c. These data reveal that leakage plays a significant role in qubit operation: The $|0\rangle$ probability approaches 0.25 after many Clifford operations, indicating that the final state is a uniform mixture of the 8 three-electron spin states.

We extend conventional RB to measure both qubit errors and leakage errors by exploiting two features of our qubit. First, all leaked states have $S_{12}=1$ and so yield the same measurement outcome as $\left|{1}\right\rangle$. Second, exchange operations preserve $S$ and do not cause leakage. For each RB sequence, we randomly select the recovery operation to either return the qubit to $\left|{0}\right\rangle$ (as in conventional RB) or $\left|{1}\right\rangle$, keeping track of which recovery was used while still making the experiment “blind” to the correct final state. Similar modifications to the recovery operation have been studied in other leakage-detecting variants of RB Wallman et al. (2016); Helson et al. ; Wood and Gambetta (2018); Xue et al. . The measurement probabilities as a function of $N$ are binned in two groups $y_{0}$ and $y_{1}$, where the subscript denotes the selection of recovery operation, as shown in Fig. 3c. We extract error and leakage using the ansatz (see Supplemental Information for details)

[TABLE]

The problem of fitting a coupled system of equations is made simpler by separately fitting single exponentials to $y_{0}+y_{1}$ and $y_{0}-y_{1}$ to estimate parameters $A,B,C,p$ and $q$ (see Fig. 3d).

To define error and leakage per gate, we must re-examine the effect of state preparation and measurement (SPAM) errors. In conventional RB, the dimensionality of the system is an assumption (e.g. $d=2$ for one qubit) that becomes a normalization factor in determining gate error. With leakage, the situation is more complicated because the number of accessible leaked states might not be known. Moreover, if there are multiple leaked states, then population transfer to these states may occur at different rates since only the qubit subsystem undergoes Clifford twirling to depolarize errors. To address these complications, we define the leakage error $\Gamma$ and total error $\varepsilon$ using the decay rates $p$ and $q$, as well as parameters $B$ and $C$ that are typically associated with SPAM (see Supplemental Information for complete derivation). We assign total error to all of the population leaving subspace $P_{0}$ in the first gate, normalized by SPAM fidelity $2B$:

[TABLE]

If there is no leakage, $C=0$ and $\varepsilon$ coincides with the gate error from conventional RB. The leakage probability is likewise defined as the total population leaking from the encoded qubit subspace after the first gate, again normalized by $2B$:

[TABLE]

where $P_{L}$ is the projector onto leaked states. Applying this analysis to the data depicted in Fig. 3c, we obtain a total error rate $\varepsilon=0.35\%$, where leakage $\Gamma=0.17\%$ contributes approximately half of the error. We identify the non-leak error $(\varepsilon-\Gamma)$ as the “qubit error.” We also extract a SPAM fidelity $F_{\mathrm{SPAM}}=0.5+B=99.2\%$.

To confirm that blind RB correctly extracts leakage, we introduce intentional overrotation errors on the control pulses. Exchange preserves total spin $S=1/2$, and so control imprecision only causes qubit errors and not leakage. We modify each value $\theta_{i}$ in the set of angles used to construct the Clifford operations as $\theta_{i}\rightarrow(1\pm\Delta)\theta_{i}$, where the sign of overrotation is randomized for each angle and $\Delta$ is fixed. The gate voltages necessary to produce these rotations are found using calibrations like the one shown in Fig. 2c. We perform blind RB for values of $\Delta$ from 0%-6%, with each value repeated several times with different random overrotation signs. The resulting leakage and total error rates are shown in Fig. 4c. As expected, the total error increases with $\Delta$ but leakage remains constant at $\Gamma\approx 0.15\%$. We also see that blind RB reliably differentiates between qubit error and leakage error: for $\Delta=6\%$, shown in Fig. 4a and 4b, we resolve a leakage error per Clifford of only 0.15% from a total error of 2.0%, a difference which leads to a distinctly non-exponential decay in $y_{0}$.

We use a combination of blind RB and spin simulations to test our understanding of the error per Clifford gate. We perform blind RB as a function of the idle time ($t_{\mathrm{idle}}$) since hyperfine interactions are anticipated to increase both total error and leakage quadratically with $t_{\mathrm{pulse}}+t_{\mathrm{idle}}$ Ladd (2012). Fig. 4 shows that for $t_{\mathrm{idle}}>30$ ns, error increases as expected and the ratio of leakage to total error approaches 0.5. However, for shorter idle times, qubit error increases relative to leakage. The high-$t_{\mathrm{idle}}$ behavior is well-replicated by detailed simulations of three spins subject to both charge and magnetic noise. Charge noise is modeled as fluctuating exchange with noise power similar to that reported in Ref. Reed et al., 2016 and magnetic noise as independently fluctuating vector magnetic fields at each dot consistent with contact hyperfine coupling to the 800 ppm 29Si and natural 73Ge nuclear environment. Both noise sources are modeled with a $1/f$ noise spectral density Eng et al. (2015) (see the Supplementary Information). We suspect the divergence between the simulations and the data at short $t_{\mathrm{idle}}$ is due to the finite bandwidth of our voltage pulses preventing accurate rotations. Such control errors are similar in nature to overrotation error, do not cause leakage, and become worse with shorter $t_{\mathrm{idle}}$ due to unintentional pulse overlap. Using the same expression relating overrotation to qubit error (Fig. 4c caption), the combined effects of imperfect control and charge noise can be modeled as stochastic overrotation. For total error $\varepsilon=0.35\%$ and leakage $\Gamma=0.17\%$ at the minimum total error point of $t_{\mathrm{idle}}=20$ ns, the root-mean-square overrotation amount is about 1.3%.

Despite the necessity of multi-pulse composite rotations and its susceptibility to leakage out of the encoded subspace, we have shown that our implementation of an exchange-only qubit using isotopically-enhanced silicon is capable of thousands of coherent qubit manipulations. This demonstration, performed at a low magnetic field and requiring only baseband voltage control, bodes well for future demonstrations with larger quantum dot arrays. Improvements in qubit error are still required, however. Through its isolation of leakage error, the blind randomized benchmarking protocol shows that total error is primarily limited by a combination of dephasing from nuclear spins and control error. While various strategies for dynamically correcting for magnetic gradients are known Hickman et al. (2013), the most straightforward path for reducing nuclear dephasing is faster pulsing. Reducing the effects of control errors, particularly when operating faster, will require improvements of control bandwidth, software-based precompilation of pulses to accelerate pulse settling Johnson et al. (2010), or a combination thereof. These engineering concerns will be of particular importance when tackling the longer pulse sequences required for two-qubit exchange-only gates DiVincenzo et al. (2000); Fong and Wandzura (2011) and beyond.

Methods

The device utilized here was fabricated with an overlapping aluminum gate architecture similar to previously published devices Borselli et al. (2015); Zajac et al. (2015); Reed et al. (2016). The device was measured in a dilution refrigerator with a base temperature of 20 mK. High bandwidth coaxial cables allowed for rapid pulsing of the voltages applied to gates P1, X1, P2, X2, P3, necessary for performing randomized benchmarking at the point of symmetric operation. Voltage pulses were sourced by arbitrary waveform generators with a 400 MHz sample rate, the outputs of which were low-pass filtered with a cutoff of 120 MHz and a flat time delay. All experiments were performed with 10 ns pulse time and 20 ns idle time (unless indicated otherwise) and with zero applied magnetic field.

Blind randomized benchmarking was performed by generating a random sequence of Clifford gates of a given length, $N$, with the final Clifford randomly chosen to invert to $\left|{0}\right\rangle$ or $\left|{1}\right\rangle$. This sequence is combined with state preparation and measurement and then averaged over 100 runs before moving to the next sequence length. The length of the Clifford sequence is then swept sixty times, from $N=1$ to $N=2000$. The sixty measurements for each sequence length are combined, and their average and standard deviation are the data points and error bars shown in Fig. 3c. The same averaging conditions were used for the data of Fig. 4.

Measurement of $S_{12}$ was accomplished through spin-to-charge conversion and charge sensing, with high SNR ($\sim$9) single-shot discrimination enabled by cryogenic HEMT amplification, as in Ref. Jones et al., . Each measurement was individually thresholded to map each result to 0 or 1. To avoid populating excited states when transitioning between gate voltages used for measurement and idle configurations, voltages on gates P1 and P2 were slewed adiabatically when near the region of spin-to-charge conversion. The total sequence length of state preparation, adiabatic ramping, and measurement with a single exchange pulse was 137.1 $\mu s$.

Author Contributions

The device was designed, characterized, and fabricated by S.D.H., M.P.J., M.L., M.T.R., and M.G.B. Control theory, coding, and analysis were provided by C.J., A.M.J., S.M., S.T.M., A.S., A.J.W., and T.D.L. Measurements were made by R.W.A., M.D.R., A.M.J., J.K., S.M., B.S., and A.J.W. The manuscript was written by R.W.A., C.J., M.D.R., A.M.J, and T.D.L. with input from all authors. The effort was supervised by M.T.R., T.D.L., and M.G.B.

Ackowledgements

We thank Andy Hunter, Kevin Eng, Mark Gyure, and Bryan Fong for valuable contributions leading to this work.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.