Abstract
Semiconductor spin qubits offer the potential to employ industrial transistor technology to produce largescale quantum computers. Silicon hole spin qubits benefit from fast allelectrical qubit control and sweet spots to counteract charge and nuclear spin noise. However, the demonstration of a twoqubit interaction has remained an open challenge. One missing factor is an understanding of the exchange coupling in the presence of a strong spin–orbit interaction. Here we study two holespin qubits in a silicon fin fieldeffect transistor, the workhorse device of today’s semiconductor industry. We demonstrate electrical tunability of the exchange splitting from above 500 MHz to closetooff and perform a conditional spinflip in 24 ns. The exchange is anisotropic because of the spin–orbit interaction. Upon tunnelling from one quantum dot to the other, the spin is rotated by almost 180 degrees. The exchange Hamiltonian no longer has the Heisenberg form and can be engineered such that it enables twoqubit controlled rotation gates without a tradeoff between speed and fidelity. This ideal behaviour applies over a wide range of magnetic field orientations, rendering the concept robust with respect to variations from qubit to qubit, indicating that it is a suitable approach for realizing a largescale quantum computer.
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Main
Semiconductor quantum dot (QD) spin qubits are prime candidates for future implementations of largescale quantum circuits^{1,2,3}. Currently, the most advanced spinbased quantum processor allows for universal control of six electron spin qubits in silicon (Si)^{4}, closely followed by a fourqubit demonstration with holes in germanium^{5}. In comparison to electron spins, hole spins have the advantage that they can be controlled allelectrically, without the added complexity of onchip micromagnets^{6,7} or the need for orbital degeneracy^{8}, thanks to their intrinsic spin–orbit interaction (SOI). Moreover, holes benefit from a reduced hyperfine interaction^{9} and the absence of valleys^{10}.
Holes in quasionedimensional (1D) nanostructures are highly attractive for implementing fast and coherent qubits. The mixing of heavy and lighthole states on account of the 1Dconfinement results in an unusually strong and electrically tunable direct Rashba SOI, with sweet spots for charge and hyperfine noise^{11,12,13}, enabling ultrafast hole spin qubits^{14,15} with reduced sensitivity to noise^{16}. Conveniently, such a 1Dsystem can be realized using today’s industry standard transistor design known as the fin fieldeffect transistor (FinFET)^{17}. Adapting FinFETs for QD integration^{16,18,19,20,21,22} potentially facilitates quantum computer scaleup by leveraging decades of technology development in the semiconductor industry^{23}. Furthermore, recent research has shown that individual hole spin qubits in a bulkSi FinFET can be operated at temperatures above 4 K (ref. ^{22}), paving the way for FinFETbased quantum integrated circuits that host both the qubit array and its classical control electronics on the same chip^{24,25,26}.
Universal quantum computation requires both singlequbit control and twoqubit interactions. Native twoqubit gates for spins such as the \(\sqrt{{{{\rm{SWAP}}}}}\) (refs. ^{1,27}), the controlled phase^{28,29,30,31} or the controlled rotation (CROT)^{4,5,24,29,32,33,34,35} rely on the exchange interaction that arises from the wavefunction overlap between two adjacent QDs. For electrons in Si, twoqubit gate fidelities have recently surpassed 99% (refs. ^{30,31,34}), but for holes in Si or FinFETs, the demonstration of twoqubit logic is still missing due to the challenges in obtaining a controllable exchange interaction^{36}.
We make an important step towards a FinFETbased quantum processor by demonstrating control over the exchange of two holes in a Si FinFET. While the exchange interaction is crucial for implementing highfidelity twoqubit gates, it is, particularly for hole spins, still largely unexplored. We measure the dependence of the exchange splitting on the magnetic field direction and find large values in some directions but closetozero values in other directions. In addition, we develop a general theoretical framework applicable to a wide range of devices and identify the SOI as the main reason for the exchange anisotropy. From our measurements, we can extract the full exchange matrix and hence accurately determine the Hamiltonian of the two coupled spins, allowing us to predict the optimum operating points for the gates. For holes, unlike electrons, the strong exchange anisotropy facilitates CROTs with both high fidelity and high speed, for an experimental setting that is robust against device variations.
Figure 1a shows the device crosssection along the triangularshaped fin, revealing ultrashort lengths, highly uniform profiles and perfect alignment of the gate electrodes^{19,20}; Fig. 1b presents a threedimensional illustration of the device. The double quantum dot (DQD) hosting qubits Q1 and Q2 is formed beneath plunger gates P1 and P2, and the barrier gate B provides control over the interdot tunnel coupling t_{c} (ref. ^{22}). The distance between the QDs was chosen to match the spin–orbit length^{20,22}. Taking advantage of the strong SOI, allelectrical spin control is implemented by electricdipole spin resonance (EDSR)^{37,38}. For this purpose, fast voltage pulses and microwave (MW) bursts are applied to P1 and a spinflip is detected in the form of an increased spin blockade leakage current. The device is tuned close to the (1,1)–(0,2) charge transition, where (n, m) denotes a state with n (m) excess holes on the left (right) QD. In Fig. 1c, the eigenenergies of the twospin states (\(\,\left\vert \uparrow \uparrow \right\rangle ,\,\left\vert \uparrow \downarrow \right\rangle ,\,\left\vert \downarrow \uparrow \right\rangle ,\,\left\vert \downarrow \downarrow \right\rangle \,\)) in the (1,1) and the singlet ground state S_{02} in the (0,2) charge region are plotted as a function of the detuning ϵ, which describes the energy difference between the (1,1) and (0,2) charge states. While spinconserving tunnelling causes an anticrossing between the S_{02} and the antiparallel twospin states, nonspinconserving tunnelling due to the SOI results in an anticrossing between the S_{02} and the parallel twospin states. As a consequence of the anticrossing with the singlet state, the energy of the antiparallel states decreases by J_{∥}(ϵ)/2, where J_{∥}(ϵ) is the measured exchange splitting between the two spins. The energylevel structure of the two hole system can be probed by performing MW spectroscopy (Fig. 1d): at large negative ϵ, the resonance frequencies of both qubits differ due to the individual gtensor \({\hat{g}}_{i}\) for each QD and are independent of each other. At more positive detunings, closer to the (0,2) region, the exchange interaction splits both resonances by J_{∥}/h (h denotes Planck’s constant), resulting in four conditional transitions. The corresponding EDSR frequencies are denoted by f_{iσ}, where i is the index of the target qubit and σ the control qubit state, \(\left\vert \uparrow \right\rangle\) or \(\left\vert \downarrow \right\rangle\).
We map out the ϵdependence of J_{∥} that, as shown in Fig. 2a, is well described by
valid in the limit of t_{c} ≪ U_{0} − ϵ (refs. ^{39,40,41}). Here, U_{0} is an energy offset of the ϵ axis, J_{0} the bare exchange and \(\cos (2\tilde{\theta })\) an SOIinduced correction factor, which is discussed later. The exchange splitting shows an exponential dependence on the barrier gate voltage V_{B} (Fig. 2b) and reaches values of up to ~525 MHz. At the same time, exchange can be turned off within the resolution limit of our spectroscopy experiment that is given by the EDSR linewidth of ≃2 MHz (refs. ^{29,33,41}). This means, using the two ‘control knobs’ ϵ and V_{B}, we achieve excellent control over the exchange coupling. As \({t}_{\rm{c}}\propto {J}_{\parallel }^{1/2}\), the tunnel coupling is also exponentially dependent on V_{B} and tunable by almost one order of magnitude (Fig. 2c).
In Fig. 3a–e, the dependence of J_{∥} on the magnetic field orientation is shown, revealing a striking anisotropy with vanishing splittings. The highly anisotropic exchange frequency is mainly due to the strong SOI and can be qualitatively understood from the gap size \({\Delta }_{{{{\rm{so}}}}}^{{{{\rm{dd}}}}}\) of the anticrossing between the S_{02} and the parallel twospin states. \({\Delta }_{{{{\rm{so}}}}}^{{{{\rm{dd}}}}}\) is proportional to \(\left \hat{\mathbf{n}}_{\mathrm{so}}\times \,\mathbf{B} \right\), where B is the external magnetic field and \({\hat{\bf{n}}_{\rm{so}}}\) a unit vector pointing in the direction of the spin–orbit field^{42}. We expect \({\hat{\bf{n}}_{\rm{so}}}\,\propto \,\mathbf{k}\times \mathbf{E}\) with momentum operator k and applied electric field E (ref. ^{11}). Therefore, \({\Delta }_{{{{\rm{so}}}}}^{{{{\rm{dd}}}}}\) changes with magnetic field orientation and so do the two hole energy levels (see Fig. 1c). However, we remark that from the dependence of \({\Delta }_{{{{\rm{so}}}}}^{{{{\rm{dd}}}}}\) on B/∣B∣, which is extracted close to zero detuning, the exchange matrix \(\hat{{{{\mathcal{J}}}}}\) at the qubit operation point cannot be extracted due to the voltage dependence of both the gtensors and the SOI.
We derive an equation for \(\hat{{{{\mathcal{J}}}}}\) starting from a Fermi–Hubbard model and including both the SOI and the anisotropic and differing hole gtensors (Methods and Supplementary Section 5). Tuned deeply into the (1,1) charge regime where spin manipulation takes place, the system is approximated by the Hamiltonian
Here, μ_{B} is Bohr’s magneton and σ_{i} the vector of Pauli matrices for each QD. The exchange matrix is given by \(\hat{{{{\mathcal{J}}}}}={J}_{0}{\hat{R}}_{{{{\rm{so}}}}}(2d/{\lambda }_{{{{\rm{so}}}}})\), where \({\hat{R}}_{{{{\rm{so}}}}}(\varphi )\) is the counterclockwise rotation matrix around \({\hat{\bf{n}}}_{\rm{so}}\) by an angle φ, λ_{so} is the spin–orbit length and d is the interdot distance. We use the convention that displacing a spin by πλ_{so}/2 induces a spin rotation of π (ref. ^{43}). The experimentally observed exchange splitting is given by (Methods and Supplementary Section 5):
where \({{{{\hat{\bf{n}}}}}}_{i}={\hat{g}}_{i}\cdot {{{\bf{B}}}}/ {\hat{g}}_{i}\cdot {{{\bf{B}}}}\) denotes the Zeeman field direction. On comparing equations (1) and (3), we find for the previously introduced correction factor \(\cos (2\tilde{\theta })={{{{\hat{\bf{n}}}}}}_{1}\cdot {\hat{R}}_{{{{\rm{so}}}}}(2d/{\lambda }_{{{{\rm{so}}}}})\cdot {{{{\hat{\bf{n}}}}}}_{2}\). Finally, by describing the magnetic field direction using the two angles α and β (Fig. 3), we obtain a fit equation J_{∥}(α, β) with five fitting parameters, namely t_{c}, U_{0}, \({\hat{\bf{n}}}_{\rm{so}}\) and λ_{so}.
Next, we apply this model to the data (black points) shown in Fig. 3a–f and perform a common fit to the full data set, consisting of measurements of J_{∥}(α, β) in five different planes (visualized in Fig. 3g) at constant detuning and for J_{∥}(ϵ) for B pointing in the x direction. There is excellent agreement between theory and experiment for the bestfit parameters: λ_{so} = 31 nm, \(\hat{\bf{n}}_{\rm{so}}\) = (−0.06, 0.41, 0.91), t_{c} = 5.61 GHz and U_{0} = 1.07 meV. The spin–orbit length coincides with the values reported previously^{20,22} and corresponds to a spin rotation angle of 2θ_{so} = 2d/λ_{so} ≈ 0.82π for a hole tunnelling from one QD to the other over d ≈ 40 nm. The direction of the spin–orbit field, represented by (α_{so} = 93°, β_{so} = 23°) is, as expected, perpendicular to the long axis of the fin and thus orthogonal to the hole momentum^{11,13}. The small outofthesubstrateplane tilt can arise on account of strain or electric fields not being perfectly aligned along the y direction. Using the five bestfit parameter values, we can reconstruct the full exchange matrix
Because we also find the gtensors when measuring J_{∥}(α, β) by means of MW spectroscopy, the twospin Hamiltonian (equation (2)) is fully characterized, thus allowing us to optimize twoqubit gate operations, as discussed later. Furthermore, we can analyse the different contributions to the exchange anisotropy with equation (3): by setting θ_{so} to zero, we are left with the effect of the anisotropic gtensors. We find that the gtensor contribution to the J_{∥}anisotropy was minor (dashed blue curves in Fig. 3a–e). Finally, we remark that the observed rotational exchange anisotropy relies on a strong SOI and the presence of an external magnetic field^{44,45}, as opposed to a weaker Isinglike anisotropy that can be found in inversion symmetric hole DQDs^{46} or at zero magnetic field^{47,48}.
We make use of the large exchange splitting to demonstrate a fast twoqubit CROT^{4,5,24,29,32,33,34,35} for holes in Si. This quantum operation is naturally implemented by driving just one of the four EDSR transitions (Fig. 1d), resulting in a rotation of the target qubit conditional on the state of the control qubit. First, we initialize \(\left\vert {{{\rm{Q1}}}},{{{\rm{Q2}}}}\right\rangle\) in the \(\left\vert \downarrow \uparrow \right\rangle\)state by pulsing from ϵ > 0, where the spinblockaded \(\left\vert \downarrow \downarrow \right\rangle\)state is occupied, to ϵ = −2.9 meV, where J_{∥}/h ≈ 80 MHz and MWinduced state leakage is suppressed^{29} (Supplementary Section 4). Subsequently, the state of the control qubit Q2 is prepared by a MW burst of length t_{b2} and frequency f_{2↓}, and finally a CROT of the target qubit Q1 is triggered by the subsequent pulse with t_{b1} and f_{1↑} (Fig. 4a). The measurement outcome is presented in Fig. 4b, revealing the characteristic fading in and out of the target qubit’s Rabi oscillations as a function of t_{b2}, that is, the spin state of the control qubit^{5,35}. A controlled spinflip for Q1 is executed in ~24 ns, which is short compared to other realizations with electrons in Si (ref. ^{34}) or holes in Ge (refs. ^{5,35}). We remark that our transportbased readout scheme prevents singleshot spin measurements and severely limits the duration of the qubits’ manipulation stage^{22}, such that randomized benchmarking to determine a twoqubit gate fidelity could not be performed^{49}.
A conditional spinflip provides a natural way of implementing a controlledNOT (CNOT) gate, differing from a CROT only by a phase factor. Two key requirements need to be fulfilled for highfidelity CROT gates. First, to prevent a mixing of the antiparallel spin states (\(\left\vert \uparrow \downarrow \right\rangle ,\left\vert \downarrow \uparrow \right\rangle\)), the Zeeman energy difference between the qubits ΔE_{Z} must be much larger than the ‘perpendicular’ exchange coupling J_{⊥} (J_{⊥}/J_{∥} induces SWAP/controlledphase oscillations^{29,30,40}). Second, either J_{∥} ≫ hf_{Rabi} or \({J}_{\parallel }/\sqrt{15}\,=\,h{f}_{{{{\rm{Rabi}}}}}\) to avoid unwanted rotations of the offresonant states^{34,40}. Hence, for electrons with isotropic exchange (J_{∥} = J_{⊥} = J) the speed of highfidelity CROT gates is limited by hf_{Rabi} ≪ J ≪ ΔE_{Z}. However, for hole spins with highly anisotropic exchange interaction, this limit can be overcome. In fact, J_{∥} = J_{0} while J_{⊥} = 0 is possible, for instance, if the gtensors are isotropic, for θ_{so} = π/2 and B perpendicular to \(\hat{n}_{\rm{so}}\) we remark that the latter condition also ensures fast singlequbit rotations. Consequently, our theory predicts that for holes in comparison to electrons, a CNOT gate with fidelity above 99% can be realized with much shorter gate times (Fig. 4c). The gate fidelities presented in Fig. 4c were numerically calculated in the absence of incoherent noise, that is, the gate infidelities are due to Hamiltonian errors^{31} (Supplementary Section 7). For the controlled rotation operation presented in Fig. 4b the magnetic field orientation (marked by the vertical orange line in Fig. 3b) was chosen such that both a closetoideal exchange configuration (∣J_{∥}∣ = 0.90 J_{0}, ∣J_{⊥}∣ = 0.05 J_{0}) and good readout contrast were achieved. In Fig. 3a–e the red dashed curves show the dependence of J_{⊥} on B/∣B∣, highlighting that the ideal configuration (J_{∥} ≈ J_{0}, J_{⊥} ≈ 0) is stretched over a wide range of directions. The CROT sweet spot is consequently tolerant to device variations, making this concept suitable for large qubit arrays, a point reinforced by the low variability and disorder resulting from industrial manufacturing^{21,50} and the electrical tunability of the SOI^{11,13}.
In summary, we investigated the exchange coupling between two holespins in a Si FinFET and found it to be both highly anisotropic and tunable, allowing for an interaction strength >0.5 GHz. We identify the strong SOI as the main microscopic origin of this anisotropy and propose a simple procedure for determining the exchange matrix. This measurement and analysis scheme applies to a wide variety of devices, for instance, to electron spin qubits with synthetic SOI in the presence of a magnetic field gradient (Supplementary Section 6)^{4,29,34}. By fully characterizing the Hamiltonian of the two coupled spins, the best possible configuration for implementing twoqubit gates can be identified. A strongly anisotropic exchange results in extended sweet spots in magnetic field orientation, where both fast and highfidelity CROTs can be performed. Finally, by choosing a closetoideal configuration we realize a controlled spinflip in just ~24 ns.
Future improvements in device fabrication^{21,50}, assisted by highvolume characterization^{51,52}, are needed to reduce device variability. Lowvariability devices, combined with robust CROT sweet spots, will make twoqubit gate operations with anisotropic exchange highly attractive for largescale qubit arrays. The concepts presented here are, in principle, compatible with elevated temperatures, but experimental confirmation is presently lacking. The advances reported here, if they can be combined with fast readout^{53} and operation above 1 K, would show that industrial FinFET technology has great potential for realizing a universal quantum processor, integrated on the same chip with the classical control electronics.
Methods
Device fabrication
The fin structures are orientated along the [110] crystal direction on a nearintrinsic, natural Si substrate (ρ > 10 kΩ cm and (100) surface) and are covered by an ≃7nmthick, thermally grown silicon dioxide (SiO_{2}) layer. Two layers of titanium nitride gate electrodes, which are electrically isolated by a ≃4.5nmthick SiO_{2} layer deposited by atomic layer deposition, are used for DQD formation. The second gate layer is integrated by a selfaligned process, resulting in a perfect layertolayer alignment. The ptype source and drain regions are made of platinum silicide. Finally, the devices are embedded in an ≃100nmthick SiO_{2} layer and are measured through contact vias filled with tungsten. Further details on the device fabrication are provided in refs. ^{19,20}.
Experimental setup
All measurements are performed using a Bluefors dry dilution refrigerator with a base temperature of ~40 mK and a threeaxis magnet that provides arbitrary control of the magnetic field vector B. The d.c. voltages are supplied by a lownoise voltage source (BasPI SP927) and the fast pulses applied to the P1gate (Fig. 1a) by an arbitrary waveform generator (Tektronix AWG5208), which also controls the I and Q inputs of a vector signal generator (Rohde & Schwarz SGS100A) for generating sidebandmodulated EDSR microwave pulses. The sourcetodrain current is measured with a currenttovoltage amplifier (BasPI SP983c) and a lockin amplifier (Signal Recovery 7265), chopping the microwave signal at a frequency of 89.2 Hz for better noise rejection. Further details are provided in Supplementary Section 1.
Derivation of the fit function for the exchange matrix
Using a Fermi–Hubbard model with a single orbital state \(\left\vert i\right\rangle\) per site i = {1, 2}, our DQD system is described by the Hamiltonian
Here \({a}_{is}^{{\dagger} }\) (a_{is}) creates (removes) a hole on site i and spin \(s=\{\left\vert \uparrow \right\rangle ,\left\vert \downarrow \right\rangle \},\) \({n}_{is}={a}_{is}^{{\dagger} }{a}_{is}\) is the occupation number operator, and U is the charging energy. The singleparticle Hamiltonian \(\tilde{H}\) is given by
and contains spinconserving interdot tunnelling \({t}_{\rm{c}}\cos ({\theta }_{{{{\rm{so}}}}}){\tau }_{x}\) and an SOIinduced spinflip hopping term \({t}_{\rm{c}}\sin ({\theta }_{{{{\rm{so}}}}}){\tau }_{y}{{{{\hat{\bf{n}}}}}}_{{{{\rm{so}}}}}\cdot {{{\mathbf{\upsigma }}}}\). Here we use the convention that the gap size of the anticrossing of two tunnelcoupled states is given by \(2\sqrt{2}{t}_{\rm{c}}\). Moreover, (τ_{x}, τ_{y}, τ_{z}) are the Pauli matrices for the orbital degree of freedom, for example \({\tau }_{z}=\left\vert 1\right\rangle \left\langle 1\right\vert \left\vert 2\right\rangle \left\langle 2\right\vert\), and σ is the vector of Pauli matrices acting on the spin degree of freedom. In the laboratory frame, as defined in Fig. 1, the gtensors \({\hat{g}}_{1}\) and \({\hat{g}}_{2}\) are symmetric (Supplementary Section 3). Finally, \(\tilde{\epsilon }\) is the energy difference for a hole occupying the left or the right QD and is expressed in terms of the detuning energy ϵ between the (1,1) and (0,2) charge states by \(\tilde{\epsilon }=\epsilon +U{U}_{0}\).
We perform a transformation from the laboratory frame to the socalled ‘spin–orbit frame’ and find
In the spin–orbit frame, nonspinconserving tunnelling is gauged away by the unitary transformation \({U}_{{{{\rm{so}}}}}=\exp (i{\theta }_{{{{\rm{so}}}}}{\tau }_{z}{{{{\hat{\bf{n}}}}}}_{{{{\rm{so}}}}}\cdot {{{\mathbf{\upsigma }}}}/2)\), and the gtensors are given by \({\hat{g}}{\,\!}_{1}^{\,{{{\rm{so}}}}}={\hat{g}}_{1}\cdot {\hat{R}}_{{{{\rm{so}}}}}({\theta }_{{{{\rm{so}}}}})\) and \({\hat{g}}{\,\!}_{2}^{\,{{{\rm{so}}}}}={\hat{g}}_{2}\cdot {\hat{R}}_{{{{\rm{so}}}}}({\theta }_{{{{\rm{so}}}}})\). Here \({\hat{R}}_{{{{\rm{so}}}}}(\varphi )\) denotes a counterclockwise rotation around \(\hat{\bf{n}}_{\rm{so}}\) by an angle φ. As our DQD system is operated close to the \(\left\vert {S}_{02}\right\rangle\)\(\left\vert S\right\rangle\) anticrossing, the Hamiltonian H_{FH} can be represented in the basis \(\{\left\vert {S}_{02}\right\rangle ,\left\vert S\right\rangle ,\left\vert {T}_{}\right\rangle ,\left\vert {T}_{+}\right\rangle ,\left\vert {T}_{0}\right\rangle \}\)
where the average and gradient Zeeman fields \(\bar{b}={\mu }_{\rm{B}}{{{\bf{B}}}}\cdot (\,{\hat{g}}{\,\!}_{1}^{\,{{{\rm{so}}}}}+{\hat{g}}{\,\!}_{2}^{\,{{{\rm{so}}}}})/2\) and \(\delta {{{\rm{b}}}}={\mu }_{\rm{B}}{{{\bf{B}}}}\cdot (\,{\hat{g}}{\,\!}_{1}^{\,{{{\rm{so}}}}}{\hat{g}}{\,\!}_{2}^{\,{{{\rm{so}}}}})/2\) were introduced. In the spin–orbit frame, the singlet subspace \(\{\left\vert {S}_{02}\right\rangle ,\left\vert S\right\rangle \}\) is coupled by the total tunnel coupling t_{c} and the hybridized singlets S_{±} have energies \({E}_{{S}_{+}}={U}_{0}\epsilon +{J}_{0}\) and \({E}_{{S}_{}}={J}_{0}\) with \({J}_{0}=\sqrt{2}\tan (\gamma /2)=({U}_{0}\epsilon )[1\sqrt{1+8{t}_{\rm{c}}^{2}/{({U}_{0}\epsilon )}^{2}}]/2\) and mixing angle \(\gamma =\arctan [\sqrt{8}{t}_{\rm{c}}/({U}_{0}\epsilon )]\). Furthermore, we remark that \({J}_{0}\approx 2{t}_{\rm{c}}^{2}/({U}_{0}\epsilon )\) in the limit of t_{c}/(U_{0} − ϵ) ≪ 1. Because S_{+} couples only weakly to the triplet states, our Hilbert space can be restricted to the four levels \(\{\left\vert {S}_{}\right\rangle ,\left\vert {T}_{}\right\rangle ,\left\vert {T}_{+}\right\rangle ,\left\vert {T}_{0}\right\rangle \}\) and we obtain
Hole spin manipulation is performed deep in the (1,1) charge stability region, allowing us to introduce the localized spin operators \({{{{\mathbf{\upsigma}}}}}_{1}^{{{{\rm{so}}}}}\) and \({{{{\mathbf{\upsigma}}}}}_{2}^{{{{\rm{so}}}}}\). The Hamiltonian (9) can then be written as
revealing that the exchange interaction is isotropic in the spin–orbit frame. To find an expression for the experimentally measured values, we first rewrite equation (10) in the lab frame:
Here \(\hat{{{{\mathcal{J}}}}}={J}_{0}{\hat{R}}_{{{{\rm{so}}}}}(2{\theta }_{{{{\rm{so}}}}})\) represents the exchange matrix in the lab frame, \({{{{\mathbf{\upsigma }}}}}_{1}={\hat{R}}_{{{{\rm{so}}}}}({\theta }_{{{{\rm{so}}}}})\cdot {{{{\mathbf{\upsigma }}}}}_{1}^{{{{\rm{so}}}}}\) and \({{{{\mathbf{\upsigma }}}}}_{2}={\hat{R}}_{{{{\rm{so}}}}}({\theta }_{{{{\rm{so}}}}})\cdot {{{{\mathbf{\upsigma }}}}}_{2}^{{{{\rm{so}}}}}\). In addition, independent rotations \({\hat{R}}_{1}\) and \({\hat{R}}_{2}\) are applied to Q1 and Q2, such that the singleparticle terms of the Hamiltonian (11) become diagonal:
where \({E}_{Z,i}{{{{\bf{e}}}}}_{z}^{{{{\rm{Q}}}}}={\mu }_{\rm{B}}{\hat{R}}_{i}\cdot {\hat{g}}_{i}\cdot {{{\bf{B}}}}\) is the ith site’s Zeeman splitting, \({{{{\bf{e}}}}}_{z}^{{{{\rm{Q}}}}}\) the spin quantization axis and \({\hat{{{{\mathcal{J}}}}}}^{{{{\rm{Q}}}}}={J}_{0}{\hat{R}}_{1}\cdot {\hat{R}}_{{{{\rm{so}}}}}(2{\theta }_{{{{\rm{so}}}}})\cdot {\hat{R}}{\,\!}_{2}^{T}\) the exchange matrix in the socalled ‘qubit frame’, wherein the exchange splitting J_{∥} is experimentally observed. To obtain an expression for J_{∥} we rewrite the Hamiltonian of equation (12) in matrix form using the twoqubit basis \(\{\left\vert \uparrow \uparrow \right\rangle ,\left\vert \uparrow \downarrow \right\rangle ,\left\vert \downarrow \uparrow \right\rangle ,\left\vert \downarrow \downarrow \right\rangle \}\)
Here we neglect every coupling that would contribute to the eigenvalues in \({{{\mathcal{O}}}}(\;{J}_{0}^{2}/{E}_{Z})\) and introduce \({J}_{\perp }=[\;{J}_{xx}^{\,{{{\rm{Q}}}}}+{J}_{yy}^{\,{{{\rm{Q}}}}}+i(\,{J}_{xy}^{\,{{{\rm{Q}}}}}{J}_{yx}^{\,{{{\rm{Q}}}}})]/2,{E}_{Z}=\left({E}_{Z,1}\right.\) \(\left.+{E}_{Z,2}\right)/2\) and ΔE_{Z} = E_{Z,1} − E_{Z,2}. The eigenenergies of equation (13) are
with \(\Delta {\tilde{E}}_{Z}=\sqrt{\Delta {E}_{Z}^{2}+\, {J}_{\perp }{ }^{2}}\). We thus find for the exchange splitting, which is defined as the energy difference between the two transitions flipping the same spin, \({J}_{\parallel }={E}_{\uparrow \uparrow }{E}_{\widetilde{\uparrow \downarrow }}({E}_{\widetilde{\downarrow \uparrow }}{E}_{\downarrow \downarrow })={J}_{zz}^{\,{{{\rm{Q}}}}}\). The matrix element \({J}_{zz}^{\,{{{\rm{Q}}}}}\) is in turn given by
equation (15) is the fit function employed to describe the observed exchange anisotropy, where the effect of both SOI and the anisotropy of the gtensors is accounted for. We note that an explicit dependence on the magnetic field direction arises from \({{{{\hat{\bf{n}}}}}_{i}={\hat{g}}_{i}\cdot {{{\bf{B}}}}/ {\hat{g}}_{i}\cdot {{{\bf{B}}}}}\). Further details of the derivation are found in Supplementary Section 5.
Numerical calculation of the CNOT gate fidelity
The CROT gate operation is modelled by numerically evaluating the Hamiltonian’s time evolution
Here \({{{\mathcal{T}}}}\) denotes timeordering, t_{π} is the spinflip time, and the timedependent Hamiltonian \({H}_{(1,1)}^{\,{{{\rm{Q}}}}}(t)\) results from equation (13) after adding the drive \(h{f}_{{{{\rm{Rabi}}}}}\,\sin (2\uppi {f}_{1\uparrow }\,t)\,{\sigma }_{x,1}\), where the Rabi frequency fulfils the condition \(h{f}_{{{{\rm{Rabi}}}}}={J}_{\parallel }/\sqrt{15}\) to suppress offresonant driving^{34,40}. Finally, the CNOT gate fidelity is determined by \({{{\mathcal{F}}}}=\frac{1}{4} \,{{\mbox{Tr}}}\,\) \(\left[{{{\mbox{CNOT}}}}_{{{{\rm{num}}}}}\,{{{\mbox{CNOT}}}}^{{\dagger} }\right]\) where CNOT is the ideal gate matrix and CNOT_{num} is obtained by applying singlequbit phase corrections to equation (16). For more details, see Supplementary Section 7.
Data availability
The data supporting the plots within this paper are available via Zenodo at https://doi.org/10.5281/zenodo.7547764 (ref. ^{54}). Source data are provided with this paper.
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Acknowledgements
We acknowledge the support of the cleanroom operation team, particularly U. Drechsler, A. Olziersky and D. D. Pineda, at the IBM Binnig and Rohrer Nanotechnology Center, and technical support at the University of Basel by S. Martin and M. Steinacher. In addition, we thank T. Berger for providing us with a 3D render of the FinFET device. This work was partially supported by the NCCR SPIN, the Swiss NSF (grant no. 179024) and the EU H2020 European Microkelvin Platform EMP (grant no. 824109). L.C.C. acknowledges support by a Swiss NSF mobility fellowship (P2BSP2_200127).
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S.G. and A.V.K. conceived and performed the experiments, with inputs from L.C.C., R.E., R.J.W., A.F. and D.M.Z. A.V.K. and S.G. designed and fabricated the device, with support by A.F. B.H., S.B. and D.L. developed the theory model. S.G., A.V.K., B.H. and S.B. analysed the data and wrote the manuscript, with inputs from all the authors. A.V.K. managed the project with support from R.J.W. and D.M.Z.
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Geyer, S., Hetényi, B., Bosco, S. et al. Anisotropic exchange interaction of two holespin qubits. Nat. Phys. 20, 1152–1157 (2024). https://doi.org/10.1038/s41567024024815
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DOI: https://doi.org/10.1038/s41567024024815
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