Universal Quantum Computing with Spin and Valley

We investigate a two-electron double quantum dot with both spin and valley degrees of freedom as they occur in graphene, carbon nanotubes, or silicon, and regard the 16-dimensional space with one electron per dot as a four-qubit logic space. In the spin-only case, it is well known that the exchange coupling between the dots combined with arbitrary single-qubit operations is sufficient for universal quantum computation. The presence of the valley degeneracy in the electronic band structure alters the form of the exchange coupling and in general leads to spin-valley entanglement. Here, we show that universal quantum computation can still be performed by exchange interaction and single-qubit gates in the presence of the additional (valley) degree of freedom. We present an explicit pulse sequence for a spin-only controlled-NOT consisting of the generalized exchange coupling and single-electron spin and valley rotations. We also propose state preparations and projective measurements with the use of adiabatic transitions between states with (1,1) and (0,2) charge distributions similar to the spin-only case, but with the additional requirement of controlling the spin and the valley Zeeman energies by an external magnetic field. Finally, we demonstrate a universal two-qubit gate between a spin and a valley qubit, allowing universal gate operations on the combined spin and valley quantum register.


Introduction
Since Loss and DiVincenzo [1] proposed quantum computing with electron spins in double quantum dots, there has been a substantial experimental progress in the field of coherent spin manipulation in semiconductors [2,3,4,5,6]. The majority of these experiments has been performed in gallium arsenide (GaAs) where the electron spin suffers from decoherence due to its coupling to a typically large number of nuclear spins, as well as spin relaxation due to spin-orbit coupling.
In carbon materials such as graphene or carbon nanotubes (CNTs), the hyperfine interaction is much weaker because 13 C is the only naturally occurring carbon isotope carrying a nuclear spin and the amount of 13 C in natural carbon is merely ∼ 1%. Similar considerations hold for quantum dots based on silicon (Si) and germanium (Ge), where less than 5 % (8 %) of all naturally occurring Si (Ge) atoms carry a nuclear spin. In graphene, the spin-orbit coupling is also expected to be weak [7].
However, the situation for quantum dots in graphene and CNTs compared to GaAs is complicated by the presence of an additional orbital degree of freedom, the so-called valley iso-spin [8,7], with basis states |K and |K , denoting the two inequivalent Dirac points in the first Brioullin zone in the graphene band structure. Experimentally, spin states in graphene quantum dots have been identified by transport measurements [9] but valley states have not been observed yet, whereas in CNTs, a fourfold grouping of electronic states due to spin and valley degree of freedom have already been observed for a decade in transport measurements [10,11,12]. The relaxation and dephasing times of two valley-and spin-degenerate electrons in a CNT double quantum dot have been studied experimentally [13] by using both transport measurements in the Pauli blockade regime [14], as well as pulsed-gate measurements [15,16].
Interestingly, the situation for quantum dots in Si/SiGe heterostructures is similar since the six-fold valley degeneracy in bulk silicon is partially lifted in strained systems [17], giving rise to a remaining two-fold valley degeneracy. The confining potential can lead to a further splitting of the remaining two valley states, which ultimately leads back to spin-only qubits and operations [18]. In recent experiments with siliconbased quantum dots, coherent spin manipulation with the exchange interaction has been performed successfully [19], and some control over the valley splitting has been demonstrated [20]. Both in Si [21] and in graphene [22,8,23] there have been speculations that the valley degree of freedom might serve as an additional resource for classical or quantum information processing, i.e. as a classical bit for valleytronics [22,8] or as a qubit [21,23]. However, the presence of an orbital (e.g., valley) degeneracy leads to the following difficulty for quantum computing. The additional degree of freedom modifies the form of the exchange interaction which is based on the Pauli exclusion principle. E. g., a spin triplet in the (1,1) charge configuration may not be blocked from tunneling to a (0,2) state if the two electrons reside in different valleys. Here, (m, n) stands for m electrons in the left and n electrons in the right quantum dot. Such a valley-dependent spin-exchange leads to spin-valley entanglement and implies that the Figure 1. Schematic of a double quantum dot formed by a confinement potential V (x) and filled with two electrons (red dots). In the presence of valley and spin degeneracy there are 16 states with one electron in each dot, i.e. in the (1,1) charge configuration. In the example shown here, the two-electron state is |s 1 , s 2 , τ 1 , τ 2 = | ↑, ↓, K, K . The hopping (tunneling) matrix element between the dots and the inter-dot bias energy are denoted by t and ε.
controlled-NOT (CNOT) gate cannot be performed in the same way as proposed in [1] as long as the valley degeneracy is present [8].
Therefore, proposals for graphene quantum dots have attempted to avoid the valley degeneracy [7] by using armchair boundary condition for quantum dots in a graphene nanoribbon [8] or by applying a magnetic field perpendicular to the graphene sheet for quantum dots defined by electrostatic gates [24]. In a recent proposal, Wu et al. suggest to use only the valley degree of freedom as a qubit and fix the spin degree of freedom by a strong in-plane magnetic field [25].
In this paper, we consider a double quantum dot with two electrons and regard both spin and valley degrees of freedom as potential qubits. This leads to a 16dimensional logic space consisting of two spin and two valley qubits (see Fig. 1). We show that it is possible to perform a CNOT gate as a universal two-qubit gate exclusively on the spin or the valley qubits if the exchange interaction and single-qubit manipulations can be implemented. For singlet-triplet qubits the exchange interaction directly produces a CNOT gate, up to single-qubit operations. Furthermore, we investigate how state preparation and measurements can be carried out by adiabatically changing the asymmetry between the dots with the use of the appropriate gate voltage control. An external magnetic field turns out to be important for both preparation and measurement. The field allows one to break the six-fold degeneracy of the states with both electrons in the same dot, (2,0) and (0,2), and thus allows for the selective preparation of one such state in the initialization process. The magnetic field also selects the states that are driven from a symmetric (1,1) back to this asymmetric (2,0) or (0,2) charge state. For quantum state read-out, the resulting charge state can then be measured with a charge detector, e.g., a nearby quantum point contact [6]. We explain below how a projective measurement on one specific state can be achieved by three charge measurements under different configurations of the magnetic field or, alternatively, with a constant magnetic field and the help of single-qubit operations.
This paper is organized as follows. In Sec. 2, we introduce the model Hamiltonian for the tunnel-coupled double quantum dot with two electrons, and derive the general form of the exchange interaction without a magnetic field (Sec. 2.1), and including a magnetic field (Sec. 2.2). Sec. 3 contains a pulse sequence for the CNOT gate. Our considerations and results concerning state preparation and measurement are reported in Sec. 4. In Sec. 5, we describe how a quantum register using spin and valley qubits may be constructed by using singlet-triplet qubits in two quantum dots and usual singleelectron spin and valley qubits in the other dots. Conclusions are drawn and an outlook towards possible further investigations is given in Sec. 6.

Model
We consider two electrons in a double quantum dot described by the Hamiltonian where the two quantum dots with one orbital each are described by with ε denoting the difference between the energy levels of the two dots, controllable by gate voltages (Fig. 1). The additional Coulomb energy of two electrons in the same dot is denoted by U . The number operatorsn j (j = 1, 2) include a sum over the spin s =↑, ↓ and the valley degree of freedom τ = ± ≡ K, K , whereĉ ( †) j,sτ annihilates (creates) an electron in the jth quantum dot with spin and valley quantum numbers s and τ . In the spin-only case, the Hilbert space for this model of a double quantum dot consists of four states with a (1, 1) charge distribution, one (0, 2) and one (2, 0) charge state [1], where (n, m) denotes a state with n electrons in the left and m electrons in the right dot. No further states with two electrons in one dot with a single orbital are permitted by the Pauli principle. Including the valley degree of freedom, we end up with 16 (1, 1) states, six (0, 2) states, and six (2, 0) states.

Exchange interaction
The two quantum dots are coupled by the spin-and valley-preserving hopping (tunneling), where t denotes the tunneling matrix element. We first consider the case without a magnetic field, H B = 0, and the parameters in the regime |t| |U ± ε| where the (1,1) charge states are approximate eigenstates of the Hamiltonian (1). The Pauli principle implies that only those (1,1) states are coupled to (0,2) and (2,0) states which are antisymmetric in the combined spin and valley space. In spin space, there is one antisymmetric state for two electrons, the spin singlet, and there are three symmetric states, the spin triplet states; for the valley space alone, the situation is analogous. To study the symmetric and antisymmetric states in the combined spin and valley space, we introduce vectors of Pauli matrices for the spin and valley of the electron in the first (j=1) or second (j=2) quantum dot, as s j = (s jx , s jy , s jz ) T and τ j = (τ jx , τ jy , τ jz ) T , and express the projection on the singlet (upper index S) and the triplet (upper index T ) sector as follows, These operators fulfill the usual relation for projectors, (P q F ) 2 = P q F and P S F + P T F = 1, where F = spin, valley and q = S, T . The projection operator on the antisymmetric states of the combined spin and valley space is given by P as = P S spin P T valley + P T spin P S valley and defines the effective low-energy Hamiltonian for the (1,1) states, The exchange coupling J is given by J = 4t 2 U/(U 2 − ε 2 ), which can be determined by a Schrieffer-Wolff transformation on H in the same way as it is used in the spin-only case [26], see Appendix A. The eigenvalues of H eff are −J and 0 with a six-and a ten-dimensional eigenspace, respectively (see also [15]). The projection on the subspace with eigenenergy −J is given in terms of spin and valley operators but for the exchange coupling the origin of the degeneracy is irrelevant. Hence, the result we obtained here is true for any four-fold degeneracy of the electron, provided that tunneling conserves this four-valued internal quantum number [27]. We can consider the reduced Hilbert space of the (1,1) states belonging to H eff as a four-qubit space with the spins in the first and in the second quantum dot as the first and second qubit and the valley iso-spins as the qubits number three and four, with | ↑ ≡ |0 , | ↓ ≡ |1 , |+ ≡ |0 , |− ≡ |1 . Using the four Bell states, as basis states in spin and valley space, and building a product basis, the corresponding matrix of H eff becomes diagonal. Obviously, we can identify |ψ − with the singlet and the other three vectors with the triplet space of the spin or the valley. We call Eq. (9) the double Bell basis.

Magnetic field
The influence of a magnetic field on the spin and valley is given by where the number operators are defined asn js = τĉ † jsτĉ jsτ andn jτ = sĉ † jsτĉ jsτ . The parameter h Sj denotes the spin Zeeman energy in the jth quantum dot, where the spin quantization axis is chosen along the direction of the magnetic field. The valley degeneracy in each dot is broken by the magnetic-field component parallel to the axis of a CNT or orthogonal to the graphene sheet. This splitting is expressed by h V j which we refer to as the valley Zeeman energy. It has been shown experimentally for a CNT [28] and theoretically for graphene quantum dots [24] that the valley Zeeman splitting due to this component of the magnetic field is much larger than the corresponding spin Zeeman splitting. On the other hand, the in-plane components in graphene and the components orthogonal to the axis of a CNT mainly influence the spin Zeeman energy. Therefore, the values of h Sj and h V j can be set nearly independently by an external magnetic field.
We neglect here that the magnetic fields in the dots can have different directions, which would lead to additional avoided crossings in the spectrum of H. Under this condition, we still can apply the Schrieffer-Wolff transformation used in [26] to obtain an effective Hamiltonian for the 16 (1,1) states, see Appendix A. We define The magnetic field might be a resource for tuning the exchange interaction, particularly in situations where the gradient is large and the linear approximation given here is not valid (for a more general expression, see Appendix A). Nevertheless, we consider quantum gates created by the exchange coupling without a magnetic field in the following. More precisely, we assume that ∆h S and ∆h V are negligible while the exchange coupling is applied. This can be achieved if J as a function of time is tuned by varying ε. The parts of the Hamiltonian H eff,B which depend on h S and h V commute with H eff and can therefore be regarded as single-qubit operations performed before or after the exchange coupling is applied.

CNOT gate on spin qubits
In this section, we show that it is possible to perform a controlled-NOT (CNOT) gate on the spin qubits alone, by applying the exchange interaction Eq. (7), supplemented with single-qubit operations on both the spin and valley qubits. Note that in Eq. (12), the matrices are represented in the product basis of the qubit states (not in the Bell basis). Because CNOT gates can be combined with single-qubit gates to form arbitrary unitaries on any number of qubits [29,30], our result below implies that universal quantum computing in the subspace of the spin qubits can be realized with the exchange interaction and single-qubit gates, despite the presence of the valley degeneracy. For an explicit construction of a CNOT gate, we define the time-evolution operator U (φ) of the exchange interaction as where φ = te 0 dt J(t ) is the time-integrated exchange coupling and H eff is the exchange Hamiltonian defined in Eq. (7). In the absence of the valley degeneracy, e.g., τ 1 = τ 2 = K and thus τ 1 · τ 2 = 1 in Eq. (7), the exchange interaction directly generates a √ SWAP gate for φ = π/2, for the spin qubits, which can be applied twice in combination with single-spin rotations to generate CNOT [1]. Here, the SWAP gate simply exchanges the states of the two spin qubits. While the SWAP gate itself can also be obtained from the exchange interaction, it is not sufficient to construct CNOT.
In the presence of the valley degeneracy, a gate that interchanges the spin and valley qubits independently can be obtained similarly as in [1], as U (±π) = SWAP ⊗ SWAP, or explicitly, U (±π)|s 1 , s 2 , τ 1 , τ 2 = |s 2 , s 1 , τ 2 , τ 1 . However, U (±π/2) = √ SWAP ⊗ √ SWAP; instead, we find In addition to producing the required entanglement between the two spins (and between the two valley iso-spins), this gate simultaneously also produces entanglement between spin and valley. To perform CNOT on the spin (or valley) alone, we thus need a modified pulse sequence. We find the following solution, from which we can construct a spin-only CNOT, using the result of Ref. [1], where the signs of the spin rotations about the z axis are opposite if one uses another root of SWAP, given as √ SWAP which reflects the symmetry of the gate under permutation of the Pauli matrices τ 1x , τ 1y , and τ 1z . Note that Eq. (18) can easily be checked because U (φ), τ 1β U (π/4)τ 1β (β = x, y, z), and √ SWAP spin are diagonal in the double Bell basis (9). Equation (17) describes a CNOT gate for the spin qubits that does not affect the valley states. The fact that a CNOT gate exclusively on the spin qubits can be performed as in Eq. (17) by using the new sequence Eq. (16) in a valley-degenerate system is the first main result of this article. By simply exchanging the single-qubit spin and valley operators (s ↔ τ ) in the equations above, we also find a CNOT gate in valley space which does not affect the spins. Here, we have assumed that arbitrary single-qubit operations in spin and valley space are available. The implementation of valley rotations within nanosecond time scales using electron valley resonance in a CNT has been proposed in [31]. Finally, we note that full valley coherence is not required by the "valley-assisted" spin-qubit gate √ SWAP spin , Eq. (16), and thus for CNOT, because the spin and valley operations ultimately factorize. Even if the initial valley state is mixed, the valley iso-spin will be disentangled by the end of the gate operation, leaving the spin qubit sector coherent. However, there is a somewhat less stringent restriction on valley coherence: Any valley qubit error (bit or phase flip) which occurs during the gate operation can propagate into the spin sector. While we do not have sufficient experimental data on valley coherence to decide whether this condition will be fulfilled, we note that at least this condition is much easier to satisfy than full valley coherence. Starting from the estimated Rabi period for electron valley resonance [31], we expect the relevant gate operation time to be around 10 ns.

State preparation and measurement
Before we describe how state preparation and projective measurements can be performed in a valley-degenerate system, we briefly characterize the situation in the spin-only case, which has already been explored experimentally [2,5,6]. In a double quantum dot without a valley degree of freedom, the Pauli principle allows only one state with a (0,2) charge distribution. In the case ε U this is the ground state of the system. Therefore, state preparation is possible by waiting at a large value of ε until the double quantum dot relaxes to this ground state. Afterwards, ε can be reduced to zero adiabatically which drives the system to one specific (1,1) charge state, selected by the magnetic field (for B = 0, the spin singlet). Reading out a qubit state can be achieved by increasing ε adiabatically, thus allowing a projective measurement on the one specific (1,1) state that is connected to the (0,2) state, while all other states remain in a (1,1) charge distribution. The charge distribution can then be measured with a charge sensor, e.g., a quantum point contact.
In the presence of the valley degree of freedom, the situation is more complicated because there are six linearly independent (0,2) states. In order to prepare the system in a well known initial state by a relaxation process, this sixfold degeneracy has to be lifted. This can be done using the spin or valley Zeeman term, i.e., by applying a magnetic field. Measuring the charge state after increasing the value of ε realizes a projection on a six-or a ten-dimensional subspace, when the system goes over to a (0,2) charge state or stays in a (1,1) state, respectively. To achieve a projective measurement on a single quantum state, several charge measurements can be performed in series. By applying a proper external magnetic field it is possible to influence which states are connected to a (0,2) state by the adiabatic transition described above. Assuming that for ε = 0 the exchange interaction J = 4t 2 /U is small compared to h F and ∆h F with F = S, V , the states |s 1 , s 2 , τ 1 , τ 2 with s j =↑, ↓ and τ j = ± are approximate eigenstates of the Hamiltonian (1). Fig. 2 shows the eigenenergies as a function of ε for the situation ∆h S > ∆h V > 0. The six states that are converted to (0,2) states by increasing ε show a nearly linear dependence on ε for ε > U . Which states develop into a (0,2) state depends on the signs of ∆h S , ∆h V , and |∆h S | − |∆h V |, giving rise to 2 3 = 8 different configurations to be distinguished (Table 1). In the following, we explicitly describe two procedures for implementing a projective measurement onto one specific state.
For the first procedure we additionally presume that the magnetic field can be changed in order to reach different configurations for the charge measurement as given in Table 1. This means that after the first charge measurement at ε > U , which projects the state onto a six-or a ten-dimensional subspace, and subsequently reducing ε to zero, it is possible to change the magnetic field, perform a new adiabatic transition, and make a new measurement of the charge distribution. We now consider the example of three charge measurements with the following three different configurations of the magnetic field: (i) ∆h S > ∆h V > 0; (ii) ∆h S > 0 > ∆h V , ∆h S > |∆h V |; (iii) ∆h V > 0 > ∆h S , ∆h V > |∆h S |. By considering these three cases in Table 1, one finds that only the state | ↓, ↑, −, + belongs in all three cases to the six-dimensional subspace corresponding to a measurement of a (0,2) charge state. Therefore, the three charge measurements with outcome (0,2) amount to a projection on the state.
For the second procedure we use a time-independent magnetic field, for example in Here, the magnetic field fulfills ∆h S > ∆h V > 0 and the exchange energy at ε = 0 is small compared to h F and ∆h F with F = S, V . The six darker (blue) lines indicates which states are connected to the (0,2) space by an adiabatic transition, while the brighter (red) lines denote states that remain in the (1,1) space even at large asymmetries. Note that the center dark (blue) line is two-fold degenerate in the limit of large ε.
the configuration ∆h S > ∆h V > 0. Instead of changing the magnetic field, we change the state by single-qubit operations applied when ε = 0. In our example we may apply e iπs 1x /2 , flipping the first spin, after the first charge measurement and e iπs 2x /2 , flipping the second spin, after the second charge measurement. The state | ↑, ↑, −, + is the only state which is mapped after the first and after the second spin flip to the six-dimensional subspace which corresponds to (0,2) states after the adiabatic transition, thus measuring three times a (0,2) charge configuration is again a projection on one specific state. If single-qubit operations for all qubits are feasible, any |s 1 , s 2 , τ 1 , τ 2 can be mapped to | ↓, ↑, −, + or | ↑, ↑, −, + . Therefore, a projection on any of these sixteen states can be done in this way.

Quantum register combining spin and valley qubits
So far, we have shown that universal two-qubit gates between spin qubits or between valley qubits can be implemented. Now we consider the situation where both, valley  and spin, serve as qubits. Note that using valley qubits in a quantum register requires valley coherence times which are sufficiently long to allow for quantum error correction. This a stricter requirement than in the situation where valley operations are only needed to achieve spin manipulation (see Sec. 3). If both kinds of qubits are to be combined in the same quantum register it is necessary to find a two-qubit gate between a spin and a valley qubit. Here we show that this can be done by using singlet-triplet qubits in spin and valley space in one double quantum dot. For these singlet-triplet qubits the exchange interaction leads directly to a universal two-qubit gate, as explained in Sec. 5.1 below. Then, in Sec. 5.2, we show how to connect these qubits to the usual single-electron spin and valley qubits. This leads effectively to a chain of qubits where nearest neighbors are connected by universal two-qubit gates, as shown in Fig. 3. If N is the number of quantum dots, the number of qubits in this register is given by 2(N − 1).

Singlet-triplet qubits
In this subsection, we briefly investigate a different qubit implementation, in which the singlet state |ψ − ≡ |0 and the triplet state |ψ + ≡ |1 (see Eq. (8)) in spin and valley space are used as the qubit basis states. Hence, we only consider a subspace of all (1,1) charge states as the logic space. Since only one out of three triplet states is part of this logic space, the effective Hamiltonian in the basis {|ψ − , |ψ + } spin ⊗ {|ψ − , |ψ + } valley assumes the simple diagonal form H eff = diag(0, −J, −J, 0). Using the Makhlin invariants [32], it is now easy to show that the unitary evolution U (π/2) = diag(1, i, i, 1) generated by this Hamiltonian is equivalent to a CNOT gate, i.e. it equals CNOT up to single-qubit operations. Therefore, in this subspace we are able to connect a spin and a valley qubit with a universal two-qubit gate by applying the exchange interaction. We define σ (k) β (β = x, y, z) as the Pauli matrices in the singlet-triplet basis for spin (k = 1) and valley (k = 2). Single-qubit operations can then be performed as follows. A magnetic field gradient between the dots acts in the singlet-triplet basis as a singlequbit rotation σ (k) x as any difference in the Zeeman splitting between the first and the second spin or valley correspond to a rotation in the singlet-triplet basis. The gates σ (k) z can be realized by applying the exchange interaction and valley or spin rotations as exp(iθσ (1) z ) = e iθ τ 1x U (−2θ)τ 1x and analogously for σ (2) z by replacing τ 1x with s 1x . These single-qubit gates together with the universal two-qubit gate allow universal quantum computing in this two-qubit space.

Two-qubit gate between a single-electron and a singlet-triplet qubit
In Sec. 3, we have shown that any two-qubit gate can be applied between two neighboring spin or valley qubits. We now consider three quantum dots where spin and valley in the dots number 1 and 2 are prepared in states which are linear combinations of |ψ + and |ψ − whereas the spin and the valley of the third dot can be in any possible state (Fig. 3). To couple the single-spin qubit in dot 3 to the singlet-triplet spin qubit in dots 1 and 2, we apply a CPHASE gate between the spins of the electrons in the third and the second dot where the spin of the third dot is the control qubit. The spin state of the first and the second quantum dot remains in the subspace {|ψ + , |ψ − } after this operation. As s 2z represents a change in the relative phase between spins 1 and 2, thus exchanging singlet and triplet states, it acts as a σ (1) x gate in the singlet-triplet basis, thus this CPHASE gate between the spins is a CNOT gate in terms of the qubits if they are defined as a usual spin up/spin down qubit in the third quantum dot and a singlettriplet qubit in the first two dots. A CNOT gate for the valley can be implemented analogously. Consequently, any two-qubit gate between a usual single-electron and a singlet-triplet qubit can be applied.

Conclusions and Outlook
In this paper, we have shown that in the presence of valley degeneracy, a CNOT gate on spin qubits in a double quantum dot can be constructed from a sequence of single-qubit operations and the exchange interaction. A CNOT gate on the valley qubits can be generated analogously. For initialization and measurement, an inhomogeneous external magnetic field is necessary. A projection on one specific state can be constructed from three charge measurements either under different configurations of the magnetic field or by using single-qubit gates. We could show that adding one double quantum dot in the singlet-triplet mode allows for a universal quantum gate (e.g., CNOT) between a spin and a valley qubit. This connection between the spin and the valley qubits in a quantum register implies that universal quantum computing based on spin and valley qubits stored in the same quantum dots is possible in principle. Nevertheless, the realization of coherent manipulation of spin and valley qubits in carbon materials is certainly a big challenge. An important precondition would be that the valley degree of freedom has a sufficiently long coherence time, which is currently unknown. An alternative way to create a spin-valley quantum register may lie in extending the singlet-triplet architecture with spin and valley degrees of freedom beyond two qubits, e.g., along the lines of [33,34] for spin-only qubits.
In this work, we have neglected the influence of the spin-orbit interaction, although it can have important effects in CNT quantum dots [28,35]. It will be a very interesting task to develop a theory for quantum computing with full orbital and spin degree of freedom in a regime dominated by spin-orbit coupling. Despite the proof-of-principle results provided here, there are, obviously, some remaining open problems regarding the construction of quantum gates with the exchange interaction with two degrees of freedom (spin and valley). It is presently not clear whether there is a shorter sequence for the √ SWAP gate on spin qubits than Eq. (16). Also, we did not find a direct CNOT (or SWAP) gate, i.e., without use of singlet-triplet qubits, applied between one singleelectron spin and one single-electron valley qubit, although the exchange interaction also couples spins and valleys. Further efforts could go into finding simpler or even optimal gate implementation for a spin-valley qubit register. The time-evolution operators acting on a four-qubit Hilbert space are, if we fix the irrelevant global phase, elements of the special unitary group SU (16), which is a 16 2 − 1 = 255-dimensional space whereas a unitary operations on a two-qubit space lies in SU (4), which has only 15 dimensions, and its two-qubit part can even be described by three real parameters [32,36]. The sequence for the √ SWAP spin gate given in Eq. (16) follows from Eq. (18), which is relatively easy to find as it is constructed as a product of unitary operations which are diagonal in the double Bell basis. We now face the more general task of finding a desired quantum gate for a given sequence of exchange interactions and single-qubit gates where the pulse lengths (gate times) are free parameters to be determined. This can be attempted numerically by minimizing a scalar function which quantifies the difference between the desired gate and the gate obtained for a given set of parameters [37]. If the desired gate is an element of SU (4) ⊗ SU (4) ⊂ SU (16), e.g., a two-qubit gate between one spin and one valley iso-spin, we can quantify the deviation from this subspace and use the Makhlin invariants to describe only the two-qubit part in both SU (4) factors. This reduces the dimension to 231. Nonetheless, the search for quantum gates constructed with a four-qubit interaction and single-qubit operations remains a challenging problem.
The Hamiltonian (1) in the presence of a magnetic field with the same direction in both dots (see Sec. 2.2) can be written as a 28 × 28 matrix consisting of 7 independent 4 × 4 submatrices, by using the following basis set: We call the 7 submatrices H 1 , . . . , H 7 and find which is not affected by the exchange interaction as block 1 only contains triplet states, and T , which describes the hopping, couples the subspaces which are symmetric and asymmetric in charge. Hamiltonians written in such a matrix form occur already in the spin-only case and have been considered in Ref. [26], where a Schrieffer-Wolff transformation is used to derive an effective Hamiltonian for the 16 (1,1) states. When we omit the index j for better readability, the Schrieffer-Wolff transformation can be written as The approximation holds for |t| |U ± B|. We can use the result from Ref. [26], and find thatH is in lowest order given by two independent 2 × 2 matrices. The matrix describing the subspace with nearly (1,1) charge distribution has the form (A.14) In the case of small gradients in the magnetic field and thus small differences in the Zeeman splitting between the dots, we can expand these terms and find in lowest order of ∆h V and ∆h S (the index refers again to the blocks in the basis set) andÃ ≈ A. Expressed with the Pauli matrices for spin and valley this gives the effective Hamiltonian of Eq. (11) for the states which have approximately (1,1) charge distribution. Note that we did not use the double Bell basis (see Sec. 2.1) in this Appendix as this does not provide the matrix form of the Hamiltonian with independent 4 × 4 submatrices in the presence of a magnetic field. Without a magnetic field the result for the splitting due to exchange interaction isJ = 4t 2 U/(U 2 − ε 2 ) = J for all blocks 2, . . . , 7. In this case, the double Bell basis can be obtained by linear combinations of the basis vectors used here only within the degenerate six-and ten-dimensional subspaces. Therefore, the effective Hamiltonian is diagonal in the double Bell basis if no magnetic field is applied.