Universal logic with encoded spin qubits in silicon

In this section we explain the structure of the entangling pulse sequences we demonstrate, including the purpose and construction of the LCCZ gate.

These entangling gates act on encoded subspaces of multiple spins which are best described according to angular momentum quantum numbers. A single spin, say spin 1, has $S_{1}=1/2$, and projection $m_{1}=\pm 1/2$. For two spins, labelled 1 and 2, the total angular momentum $S_{12}$ can be $S_{12}=0$, the singlet state for which $m_{12}=0$, or $S_{12}=1$, the triplet of states for which $m_{12}=-1,0,$ or $1$. We notate these four states as $\left|S_{12};m_{12}\right\rangle$. The smallest universally-controllable exchange-only qubit requires adding a third spin, for which the algebra of angular momentum provides the quantum number $S_{123}$, which can be $1/2$ either by adding the single $S_{3}=1/2$ spin to the $S_{12}=0$ singlet, or by adding $S_{3}$ to the $S_{12}=1$ triplet. These two choices for $S_{123}=1/2$ are our encoded state, and the quadruplet of states with $S_{123}=3/2$, which also results from adding $S_{3}$ to the $S_{12}=1$ triplet, are leaked states. We also have a total spin projection $m_{123}=-S_{123},-S_{123}+1,\ldots,S_{123}$. In this section we thus notate these states as $\left|S_{12},S_{123};m_{123}\right\rangle$ and our qubit states are $\left|0,1/2;m_{123}\right\rangle$ and $\left|1,1/2;m_{123}\right\rangle$. The $m_{123}$ degree of freedom is called the gauge, and it is unaffected by exchange operation between any of the three spins. The Pauli spin blockade process on dots 1 and 2 provides initialization and measurement only of the $S_{12}$ quantum number and has no impact on $m_{123}$ [9]. Singlet measurements on spins 1 and 2 therefore measure the probability of $\left|0,1/2;m_{123}\right\rangle$ independent of $m_{123}$, and provide no information to distinguish encoded triplet $\left|1,1/2;m_{123}\right\rangle$ states from leaked states $\left|1,3/2;m_{123}\right\rangle$. To understand single-qubit gates in the single-qubit system, see Ref. 1.

When considering operations on two qubits on dots 1-6, the gauges of the two qubits, $m_{123}$ and $m_{456}$, become important. Notating the total angular momentum of all six spins as $S$ rather than $S_{123456}$ and their projection as $m$ instead of $m_{123456}$ for brevity, the two $S_{123}=S_{456}=1/2$ qubits may combine into an $S=0$ subspace and an $S=1$ subspace. While exchange conserves $m$, still respecting gauge freedom, its action for interqubit operations does depend on $S$, which in turn is set by the relative value and phase of $m_{123}$ and $m_{456}$. The gauge freedom, which can be safely ignored for single-qubit gates, must therefore be carefully considered with two-qubit operations.

In 2000, Ref. 16 provided a 19-pulse entangling gate sequence between two encoded qubits that required the total angular momentum of all six spins to be $S=1$, which was imagined to be accomplished by polarizing the gauge of each qubit. The small spin-orbit effect and long spin relaxation times in silicon leave few good hardware choices for such polarization in our system. A decade later, the Fong-Wandzura (FW) construction [17] was discovered by computational search employing a genetic algorithm. This construction is a gauge-independent CNOT sequence, meaning it correctly performs the same CNOT sequence on the $S_{12}$ and $S_{45}$ degrees of freedom for both $S=0$ and $S=1$. Gauge independence allows each triple-dot EO qubit to be initialized as a pair of spin-singlet states and an unpolarized spin, as we do in the present demonstration.

Half a decade later, Ref. 19 showed that the FW sequence is in fact composed of three repetitions of a shorter primitive composite sequence acting on four spins, shown in Fig. 5. This primitive sequence is a quasi-Fredkin (controlled-SWAP) gate, swapping the gauge $m_{123}$ with $m_{4}$ only if $S_{12}=1$, but not if $S_{12}=0$. Applying this quasi-Fredkin gate to one qubit on spins 1,2, and 3 and alternatingly on spins 4, 5, and then 4 again, a $S_{12}=0$ condition will apply identity three times, while an $S_{12}=1$ condition will swap spins 4 and 5, leaving $m_{123}$ in its initial state (regardless of what that initial state is). These three uses provide a Fredkin gate with a three-spin EO qubit as control and two spins as target. If those two spins are the singlet-triplet pair of an EO-qubit, this controlled-SWAP becomes an encoded CZ, the FWCZ, with no gauge dependence.

Compiling the primitive sequences together, one arrives at an entangling gate using 12 $\pi/2$ exchange pulses (i.e. spin $\sqrt{\text{SWAP}}$ gates) on just five fully-connected spins; assuring a controlled-$Z$ adds two more $\pi/2$ pulses, and adapting to the linear, nearest-neighbor coupled layout we employ here with the measured singlet-triplet pairs on the ends of the array, the FWCZ ends up using all 6 spins with an additional 12 $\pi$ pulses (SWAPs on spins) for a total of 26 pulses, shown in Table 1. A CZ converts to CNOT via the construction shown in Fig. 6(a), adding two more pulses. To highlight the physical implementation of such sequences, we present an illustrative example in Fig. 7 of the 28-pulse FW-CNOT sequence translated into experimentally-accurate voltage waveforms required for device control. The sequences are compiled sequentially, so that no two pulses occur simultaneously.

Unfortunately, the quasi-Fredkin gate on four spins discussed above has an undesired feature: if the EO qubit is in its $S_{123}=3/2$ leaked state, then the gate applies a phase flip to the $S_{1234}=2$ states relative to the $S_{1234}=1$ states. Mathematically, no exchange-only four-spin sequence can avoid this problem. As a result, when $S_{123}=3/2$ and we apply this primitive operation as described for the FWCZ, then when $S_{12345}=1/2$ and $S_{45}=1$, the resulting unitary provides a phase flip, and when $S_{12345}=3/2$, the resulting unitary applies a $\pi$ rotation about an axis tipped an angle $\tan^{-1}(3\sqrt{15}/11)$ from the Bloch-sphere $z$-axis to the singlet-triplet qubit defined by $S_{45}$. These operations in general will leak the EO qubit including spins 4 and 5, and as a result, even when applying the FW gate perfectly, leakage will have spread from one qubit to the next. (Notably, leakage spreads only from the $\left|S_{12},S_{123};m_{123}\right\rangle$ qubit to the $\left|S_{45},S_{456};m_{456}\right\rangle$ qubit; if the $\left|S_{45},S_{456};m_{456}\right\rangle$ is leaked into $\left|1,3/2;m_{456}\right\rangle$ then the gate behaves as intended for $S_{45}=1$ regardless of $S_{456}$). This leakage spreading could be highly detrimental to fault-tolerance, as we are unable to detect leakage directly in general and most quantum error correcting codes are ill-equipped to correct leakage even when it is detected.

The goal of the LCCZ gate is to avoid this leakage spreading when applying encoded CZ gates. The key insights to the LCCZ gate are that the unwanted leakage-induced phase flip or $\pi$ rotation on the $S_{45}$ qubit are both square-roots of an identity operation, and all single-qubit operations are identity on leakage spaces. Therefore, if we apply two FWCZ sequences with a single-qubit gate on dots 1-3 in between, then on the encoded subspaces we have achieved some controlled-$\pi$-rotation gate, where the $\pi$-rotation angle depends on the choice of single-qubit operation and can be converted back to Pauli operators $Z$ or $X$ with single-qubit corrections. In particular, we use the construction shown in Fig. 6(b). The operator $\sqrt{Z}$ (often called $S$, but not to be confused with total spin) on $\left|S_{45};m_{12}\right\rangle$ is simply another $\pi/2$ pulse on these spins, and the single-qubit rotations $R$, $R^{\dagger}$, and $H$ may be readily derived as exchange sequences similar to those in Ref. 1. If the CZ in this construction is the FWCZ and the $S_{12}$ qubit is leaky (i.e. $S_{123}=3/2$), then all single-qubit gates have no action and the two FWCZ gates combine into identity, leaving behind only the correctable single-qubit-encoded $\sqrt{Z}$ on the $S_{45}$ pair, (which is also $\sqrt{Z}$ on a $\left|S_{45},S_{456};m_{456}\right\rangle$ qubit). After some compiling, the resulting sequence is 46 pulses for an LC-CNOT and 44 pulses for an LCCZ; the compiled sequences are shown in Table 1.

The final sequence we demonstrate is SWAP, which is not entangling, but is nonetheless a critical two-qubit gate for moving data and randomized benchmarking. In many qubit modalities, the SWAP is more complex than CNOT, as a typical construction uses three CNOT sequences. However, for exchange-only qubits, SWAP is the one transversal operation of the underlying spins. If our spins were fully connected, three $\pi$ pulses to enact spin swaps would suffice. The 15 pulse sequence shown in Table 1 performs the permutations needed to move spin information in linear, nearest-neighbor coupled architecture with the three-spin structure reflected about the center. The timing of these spin-swap operations is chosen so that each spin spends approximately the same amount of time in each dot. A sequence in which this is done exactly effectively decouples low-frequency magnetic noise [33, 34]; the 15-pulse SWAP shown only partially completes such a permutative dynamical decoupling operation.

Though all RB results in the main text are generated via one-sided readout only, we present here benchmarking results with readout from both sides. As stated in the text, small valley splitting on the P5/P6 dots limits the SPAM fidelity which results in a meaningful reduction in measurement visibility. As shown in Fig. 8, the decay rates measured on either side are consistent, and we observe similar gate performance from measurement on either side of the device.

Additionally, we have explored the RB sequences with and without $X$-gate prerotations, reminiscent of the single-qubit blind RB protocol [1] and of character benchmarking on the larger space of encoded qubits [35]. Unlike the single-qubit case, where linear combinations of the two measurement results can be used to differentiate error in the qubit computational space from leakage error, the two-qubit case navigates through a larger Hilbert space, which requires a greater number of measurements to distinguish those types of errors in a similar manner. Though we have not yet formalized such a technique, we still emphasize that the decay rates on the singlet and triplet branches are consistent within the confidence intervals. We take this as a strong sign that the single exponential decay model for each channel is valid here, lending the possibility that linear combinations of exponential decays will enable more detailed leakage analysis in future efforts.

We also note the different asymptotic singlet probability values at large Clifford numbers for the different RB experiments we present. These may be understood by a relatively simple counting argument, which utilizes the fact that the total electron spin-projection, $m$, is partially conserved in our experiments. This conservation is stronger at higher-magnetic fields, where the mismatch of electron and nuclear Larmor frequencies suppress electron-nuclear flip-flops and pulse sequences are insufficiently fast to drive the spin-system to compensate. Although this conservation law is weaker at low-magnetic field, it is strong enough even in Earth’s field that the following counting argument applies. For 1QRB, there are three states for three spins at constant value of $m_{123}=\pm 1/2;$ in the $\left|S_{12},S_{123};m_{123}\right\rangle$ notation, these are $\left|0,1/2;m_{123}\right\rangle,\left|1,1/2;m_{123}\right\rangle,$ and $\left|1,3/2;m_{123}\right\rangle$. Magnetic noise will, under application of a large number of Clifford sequences, scramble these and result in a probability of 1/3 of measuring $S_{12}=0$ at sequence end. For 2QRB, $m$ is the projection across all six spins, and half of experiments begin with $m=0$, while a quarter begin with $m=1$, and another quarter with $m=-1$, depending on the random gauge of the two initialized qubits. There are 6!/(3!3!)=20 ways to obtain $m=0$, and 6!/(4!2!)=15 ways to obtain $m=\pm 1$. Six of the 20 $m=0$ states have $S_{12}=0$, while 4 of the 15 $m=1$ have $S_{12}=0$, and so if these are again fully scrambled, the expected asymptote is $(6/20)/4+2(4/15)/4=17/60\approx 0.283$. This asymptote is not well observed on M2 in Fig. 8 and is another indication of reduced SPAM performance on this side of the device.

We now discuss our techniques for characterizing two well-understood sources of noise and decoherence in our device: magnetic noise and charge noise. This constitutes a more detailed discussion of Fig. 2 than the main text could contain.

Magnetic gradient fluctuations, caused by the noisy magnetization of ${}^{29}\text{Si}$ and ${}^{73}\text{Ge}$ nuclear spins coupled to the electron spins via the contact hyperfine interaction, provide the most significant contribution to our gate error [20]. While uniform magnetic fields are irrelevant for the DFS encoding, magnetic gradients drive relative precession of electron spin pairs, causing decoherence and leakage out of the encoded subspace. We characterize this decoherence rate by measuring the decay of a singlet prepared between two electrons and left in the idle configuration for a varying amount of time. The ensemble-averaged measurement projection decays to a value associated with the predicted mixture of encoded and non-encoded states: 1/2 probability to the initial singlet $\left|0,1/2;m_{123}\right\rangle$, 1/6 to the encoded triplet $\left|1,1/2;m_{123}\right\rangle$, and 1/3 to the leaked state $\left|1,3/2;m_{123}\right\rangle$ [32]. Spin singlet pairs are initialized on either side of the device then shuttled to and from the desired location via consecutive $\pi$ pulses. Due to the relatively poor SPAM fidelity on the M2 side of the device, spin pairs measured on the M2 side show deviations in the decay asymptote from the predicted value of 1/2. The gaussian envelope decays with a characteristic timescale $T_{2}^{*}$, which we measured to be about 3.5 $\mu$s for six different spin pairs, as shown in Fig. 2(a).

$T_{2}^{*}$ is the most important metric for our error budget, as RB or IRB error scales as $[(t_{\text{idle}}+t_{\text{pulse}})/T_{2}^{*}]^{2}$ unless decoupling mitigations are applied [33, 36, 34]. The $T_{2}^{*}$ values we observe are in good agreement with theoretical predictions for the known concentrations of ${}^{29}\text{Si}$ and ${}^{73}\text{Ge}$ isotopes and the silicon well width. The reduction of this physical noise source would be accomplished either by employing magnetic decoupling pulse sequences [33, 36, 34], growing samples with reduced ${}^{29}\text{Si}$ and ${}^{73}\text{Ge}$ content, or by reducing $t_{\text{idle}}$ and $t_{\text{pulse}}$. Unfortunately, further reduction to $t_{\text{idle}}$ or $t_{\text{pulse}}$ relative to the values we use here causes additional errors due to pulse distortion effects in our signal chain [1].

Another contribution to our gate error is charge noise, here manifest as exchange noise induced by fluctuations in the lateral trapping and tunneling potentials. This noise is due either to noise in the signal chain or fluctuating defects in the gate stack and has a strong $1/f$ spectrum across many decades [37]. This type of noise induces error only during active exchange pulsing, since nearest-neighbor tunnel coupling is suppressed when the associated spins are idling. This idle-error-suppression results from the exponential scaling of the spin-spin exchange interaction with applied barrier potentials in general, and in particular from the large on-off ratios that are available with our tightly-confining SLEDGE design [18]. As shown in Fig. 2(b), we quantify this charge noise contribution to our error budget with an exchange oscillation $Q$-factor at a given $J$ (typically $J/h\sim$100 MHz). The product of $J/h$ with the 1/$e$ duration of the gaussian decay envelope of exchange oscillations gives us a number of oscillations $N_{\text{osc}}$. $N_{\text{osc}}$ is a superior metric to the actual decay time, because the impact of charge noise on $J$ scales as $|dJ/dV|$ for gate voltages $V$. If $J$ were exactly exponential with voltage, $N_{\text{osc}}$would be independent of voltage, but it is not perfectly so as exchange is subexponential with X-gate voltage [15]. Similar to our magnetic noise heuristic, the estimated theoretical gate error from charge noise scales quadratically with this decay envelope, $1/N_{\text{osc}}^{2}$.

We repeatedly measure $N_{\text{osc}}$ as a function of $J$ along the symmetric axis, finding that $N_{\text{osc}}$  reaches $\sim$50 at 100 MHz, the frequency of operation due to experimental convenience. Remarkably, and contrary to pure exponential activation, $N_{\text{osc}}$ rapidly increases from $<100$ at $J/h=1$ GHz to $>1600$ above $J/h\sim 20$ GHz, where we observe a sharp reduction in $|dJ/dV|$ as the potential barrier flattens [Fig. 2(c)]. The accessibility of this operation point in a SiGe accumulation mode device, where the exchange energy asymptotes to the double-dot orbital splitting and loses sensitivity to control voltages and noise [21], can be attributed to the large gate action of the SLEDGE design. To resolve coherent oscillations which occur at exchange frequencies well above the Nyquist frequency of our AWGs (200 MHz), we shift the AWG time basis, which is normally set to 2.5 ns, within the range [2.5,5) ns. A continuous shift of the time basis within this range provides the smooth sampling needed to resolve coherent oscillations without aliasing, and thus extract both the exchange rate, $J/h$, and $N_{\text{osc}}$. While these exchange energies are too high for practical pulsed operation, they may prove valuable for microwave-sensitive EO encodings [38].

We perform an initial validation of our error budget with single-qubit randomized benchmarking in Fig. 2(d). To properly account for leakage out of our computational space, we use the “blind” benchmarking technique described in Ref. 1, where sequences are chosen to compile either to the identity or the Pauli-$X$ operation. We find an average single-qubit Clifford error of $(1.1\pm 0.1)\times 10^{-3}$ with a leakage error of $(3\pm 1)\times 10^{-4}$. This is in approximate agreement with our simulated prediction of error from magnetic and charge noise alone, $5.0\times 10^{-4}$, but which does not include the effects of pulse distortion or other physical contributions such as excited state leakage or spin-orbit interactions.

As a final note, previous accumulation-mode device designs exhibited significant paramagnetic gradients in response to external applied magnetic fields. We attributed such gradients to fringe fields induced by Meissner screening in the aluminum gate stack which, due to the gate stackup complexity, varied in strength and direction over the dot array. The magnitude of paramagnetic gradients observed here is notably smaller, and consistent with expectations from spin-orbit coupling (see Fig. 9) [20]. Reducing gradients is a requisite for high fidelity operation in the B $\approx$ 1-10 mT regime, where magnetic gradients would otherwise drive always-on spin-spin interactions detrimental to this control scheme.