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standard quantum computing @xcite is based on encoding , manipulating , and measuring quantum information encoded in the state of a collection of quantum two - level systems - qubits . spin-@xmath0 is an ideal implementation of a qubit , since it is a natural two - level system , and every pure state of a spin-@xmath0 corresponds to a state of a qubit . for this reason , spins have been considered as carriers of quantum information in a variety of proposals @xcite . the initial proposal @xcite called for spins in single - electron quantum dots electrically manipulated by the exchange interaction and local time - dependent zeeman fields . a variety of other encoding schemes and manipulation techniques @xcite rely upon encoded qubits . in these schemes , the simplicity of qubit states and minimal number of physical carriers of quantum information are traded for less stringent requirements for experimental implementations . on one hand , the alternative setups protect from the most common types of errors by decoupling the computational degrees of freedom from the most common sources of noise , and therefore allow for longer gating times . on the other hand , in some alternative setups the manipulation without fast switching of the local magnetic fields becomes possible . the optimization in the encoding and manipulation protocols is always guided by the state of the art in the experiments . recent results suggest that spin qubits can reside in a variety of material hosts with novel properties . quantum dots in graphene @xcite and carbon nanotubes @xcite are less susceptible to the decoherence due to nuclei and spin - orbit interaction . spins in nanowires show very strong confinement in two spatial directions , and the gating is comparably simple @xcite . in hole systems , the carriers have distinct symmetry properties , and show coupling to the nuclear spins of a novel kind @xcite . recently , the experiments in silicon @xcite have demonstrated coherent manipulation of spins similar to the one achieved in the gaas - based nanostructures . within these hosts , the manipulation techniques that use exchange interaction , spatially inhomogeneous time independent zeeman splitting , and nuclear hyperfine interactions are within reach . despite these developments , gaas remains a promising route to spin qubits due to the highly advanced experimental techniques developed for this material .    here , we study the implementation of the quantum gates on the encoded two - spin singlet - triplet ( st ) qubits @xcite using resources that closely resemble the ones available in the current experimental setups . there , the application of the time - dependent electric fields through the gates fabricated into a structure @xcite are preferred to time - dependent local magnetic fields . in addition , the nuclear spins @xcite and inhomogeneous magnetic fields @xcite are possible resources for spin control . in the setups based on semiconductors , the electrons or holes in the quantum dots are influenced by the spin orbit interaction ( soi ) , which can contribute to the control @xcite .    in this work we present a scheme for control of st - qubits which uses switching of the exchange interaction as a primary resource . we consider the scheme that is optimized for the application of quantum gates in the network of quantum dots . the construction of the cnot gate uses pulses of the exchange interaction as the only parameter that is time dependent . the exchange interaction itself is not sufficient for the universal quantum computation over the st - qubits due to its high symmetry . the additional symmetry breaking is provided by nonuniform , but static magnetic fields . these fields describe the influence both of magnetic fields , provided by the nearby magnets , and of the coupling to the nuclear spins in the host material via hyperfine interaction . depending on the scheme used for the application of the quantum gates , the optimal geometry is either the one in which the magnetic fields point parallel to the axis of symmetry of the soi or perpendicular to it . one major problem in the realization of two - qubit quantum gates ( in particular , we consider here the cnot gate based on conditional phase gates ) , is the possibility of leakage errors where the spin states defining the logical qubit leave the computational space . these errors move the state of four spins from the @xmath1-dimensional computational space of two qubits into some other portion of the @xmath2-dimensional hilbert space of four spin-@xmath0 particles . we consider two ways of addressing this problem . one scheme possesses the axial symmetry due to the fact that the soi vector and magnetic fields are parallel . for this ` parallel scheme ' we are able to construct a perfect cnot gate , if we are able to control all the available parameters . having in mind 2d architectures , we also consider the cnot gate between two qubits in the case when the soi vector and magnetic fields are perpendicular to each other . here we can not prevent the leakage out of the computational space , however , we show that it is suppressed by a ratio between the soi and zeeman energy coming from a strong external magnetic field . all our constructions assume that the controlled interactions are switched in time by rectangular pulses . any deviations from this form of time dependence lead to additional corrections and affect the fidelity of the gate . the paper is organzied as follows . in sec . [ sec : model ] , we introduce the model for the double dots and effective hamiltonians for field gradients and exchange and spin orbit interactions . in sec . [ sec : parallel ] , we consider the parallel geometry and derive the cnot gate via the conditional phase gates and swap gates , all based on exchange . there we also give estimates for the switching times . the scheme for the perpendicular geometry is then addressed in sec . [ sec : perp ] , and we conclude in sec . [ sec : conclusion ] . parallel geometry : four quantum dots ( yellow discs ) aligned along the x - axis in the presence of an external magnetic field @xmath3 that is applied parallel to the soi vector @xmath4 ( red arrow ) , which must be perpendicular to the line of the dots and which we take to be the @xmath5-direction . at each dot , there is a local magnetic field @xmath6 ( blue arrows ) , also assumed to be parallel to @xmath3 , but with alternating orientations as indicated . the direction of @xmath3 defines the spin quantization axis . the dots are defined electrostatically by metallic gates ( light green structures ) . each dot contains a spin-1/2 , and the exchange ( @xmath7 ) and the soi - induced ( @xmath8 ) interactions between the spins can be controlled by changing the electrostatic potential between the dots . the dots 1 and 2 ( from left to right ) define the first st - qubit , and the dots 3 and 4 the second st - qubit . , width=302 ]    we consider in the following singlet - triplet ( st ) qubits that are implemented by two electrons confined to a double quantum dot system @xcite , see figs . [ fig : setupallz ] and  [ fig : setup ] . such st - qubits have been realized successfully in several labs @xcite , and single and two - qubit operations have also been demonstrated recently @xcite . there are several schemes for the fundamental cnot gate , which can be divided into two classes , schemes which make use of exchange interaction and schemes which do not , but instead rely on coupling of dipole moments @xcite . the latter schemes has the disadvantage to be rather slow and also to be affected by charge noise rather strongly . here , we focus on exchange - based schemes specifically adapted to quantum dots in iii - v semiconducting materials , that have the soi , such as in gaas or inas quantum dots . although the soi is typically small compared to the level spacing of the dots , it needs to be taken into account in order to achieve high fidelities in gate operations . we focus now on two such st - qubits and assume them to be encoded in four quantum dots that are arranged in a row , see figs . [ fig : setupallz ] and  [ fig : setup ] . the external magnetic field @xmath9 is assumed to give the largest energy scale and determines the spin quantization axis @xmath5 . the hilbert space of four spins-@xmath0 is spanned by @xmath10 basis states . the total spin of the system is given by @xmath11 , where @xmath12 is a spin-@xmath0 operator acting on the spin in a dot @xmath13 . due to axial symmetry , the z - component , @xmath14 , becomes a good quantum number , and the eigenstates corresponding to @xmath15 span a six - dimensional subspace . we define the computational basis of the two st - qubits in this subspace as @xmath16 denotes the st - qubit space , ` @xmath17 ' and ` @xmath18 ' denote states of the quantum dot spins corresponding to the projection @xmath19 on the @xmath5 axis . the remaining two states , latexmath:[\ ] ]    with @xmath153 , where we assume @xmath101 [ see eq . ( [ eq : t23 ] ) ] to be even . the total gate acting on the qubits as defined by eq . ( [ phase ] ) can be written in the compact form @xmath154 where the pauli matrix @xmath155 acts on the first st - qubit ( formed by dot 1 and 2 ) and @xmath156 on the second one ( formed by dot 3 and 4 ) with @xmath157 . choosing @xmath158 where @xmath159 is a positive integer , we obtain the @xmath56-gate , @xmath160 both eqs . ( [ eq : t23 ] ) and ( [ t_231 ] ) should be satisfied simultaneously . for example , if @xmath161 and @xmath162 , we get @xmath163 from this we can estimate the total switching time to perform the @xmath56-gate that is a sum of the switching times at each step [ see eq . ( [ gate_consequence ] ) ] . for the scheme discussed in subsecs . [ sec_c ] and [ sec_pi ] the slowest part is given by the swap gates @xmath60 and @xmath61 , whose switching time is limited by field gradients ( due to nuclear spins @xcite and/or micromagnets ) , @xmath164 , which corresponds to @xmath165 . the gate can be faster if the rotations around the @xmath5-axis are performed not on the equator but more closely to the poles . this allows us to decrease the switching time of the swap gate to @xmath166 ; however , this would require a more precise control over the pulses . the same trick can be used to decrease the switching time of the conditional phase gate @xmath59 ( compare figs . [ fig:23_rotation ] and [ fig:23_rotation_alt ] ) . if field gradients larger than @xmath167 are used , the switching rates will be proportionately larger . an alternative scheme for the conditional phase gate @xmath59 ( see also fig . [ fig:23_rotation ] ) . instead of the rotation with the pulse defined by eqs . ( [ eq : t23 ] ) and ( [ t_231 ] ) we consider a sequence of three pulses . during the first and the third pulses ( blue arcs ) the state precesses quickly acquiring the @xmath56-phase , ( [ t_231 ] ) . during the second pulse , @xmath107 , the state precesses around the @xmath5-axis over a shorter path than the one in fig . [ fig:23_rotation ] . as a result , the switching is faster.,width=302 ]    using the @xmath56-gate , we construct the controlled phase flip ( cpf ) gate @xmath168 ( see footnote [ 13 ] in @xcite ) as @xmath169 finally , we obtain the cnot gate , @xmath170 by using the cpf gate and performing a basis rotation on qubit 2 ( a single - qubit rotation by @xmath171 about the @xmath172-axis ) , @xmath173 in summary , the full sequence of operations for the cnot gate , @xmath174 , is given by @xmath175 e^{-i\frac{\pi}{4}(\sigma_1^z + \sigma_2^z)}e^{-i \frac{\pi}{4 }    \sigma_2^y}.\ ] ]    we note again that this result has been derived under the assumption of rectangular pulse shapes . this is certainly an idealization , and in practice we expect deviations from this shape to cause errors for the gates and to affect the gate fidelity . the study of this issue , being very important for practical purposes , requires a separate investigation and is beyond the scope of this work . in the previous section we have discussed the parallel geometry for which we were able to construct a perfect cnot gate under the assumption that we have a complete control over the parameters . the cnot gate , together with single - qubit gates , allows us to simulate any other quantum gate and its implementation is a crucial step toward the realization of a quantum computer @xcite . in a next step , many such elementary gates need to be connected into a large network . in recent years , the surface code @xcite has emerged as one of the most promising platforms for this goal due to its large threshold of about @xmath176 for fault tolerant error correction @xcite . this platform requires a two - dimensional ( 2d ) geometry and can be implemented in semiconductors of the type considered here . @xcite . schematic setup for 2d architecture . two dots define an st - qubit ( black dotted ellipses ) . an external magnetic field @xmath3 and local magnetic fields ( blue arrows ) are parallel . in the case of the coupling between two qubits from the same row ( red solid ellipse ) the soi vector @xmath177 is parallel to the magnetic field @xmath3 , corresponding to the parallel geometry ( see fig . [ fig : setupallz ] ) . in the case of coupling between two qubits from two neighboring rows ( red dashed ellipse ) the soi vector @xmath178 is perpendicular to the magnetic field @xmath3 , corresponding to the perpendicular geometry ( see fig . [ fig : setup]).,width=226 ]    the basic 2d scheme is illustrated in fig . [ fig:2d ] . there , we show an array of quantum dots where two neighboring dots in a given row represent one st - qubit . these quantum dots are embedded in a semiconductor where the soi is of the same type in the entire structure . as a typical example we mention rashba and/or dresselhaus soi that both depend on the momentum of the electron . as a result , the direction of the soi vector @xmath4 is always perpendicular to the line along which two quantum dots are coupled by exchange interaction @xcite . thus , for the coupling of two such qubits in the _ same _ row , the soi vector @xmath4 is parallel to the external magnetic field @xmath3 , so we can use the scheme designed for the parallel geometry in the previous section . in contrast , if we want to couple two qubits from _ neighboring _ rows , we should also consider a ` perpendicular geometry ' ( see figs . [ fig : setupallz ] and [ fig:2d ] ) in which the soi vector @xmath4 is perpendicular to the magnetic field @xmath3 . this perpendicular geometry is characterized by several features . as was mentioned before , the axial symmetry in this case is broken by the soi , leading to the coupling between computational ( @xmath179 ) and non - computational ( @xmath180 ) space . if the magnetic field @xmath181 is sufficiently large to split the triplet levels @xmath182 far away from the computational space ( @xmath183 ) , we can neglect this leakage . we estimate for gaas @xmath184 . at the same time , the soi does not couple states within the computational space , so for the realization of the phase gate @xmath59 and the swaps gates @xmath60 and @xmath61 we can use the same scheme as in sec . [ sec : parallel ] only with @xmath185 . we have studied the implementation of the cnot gate for st - qubits in a model that is appropriate for current experiments @xcite . the setup consists of an array of quantum dots with controlled growth direction and the relative orientation of the dots . pairs of neighboring dots build the st - qubits , where the quantization axis is determined by an externally applied magnetic field @xmath3 . moreover , we introduce an inhomogeneity in magnetic fields , @xmath6 , by local micromagnets or by the hyperfine field . the resources used for time - dependent control are the exchange interaction @xmath31 , the field gradients @xmath186 , and the soi vector @xmath187 .    if the soi vector @xmath4 is parallel to the external ( @xmath3 ) and local magnetic fields ( @xmath6 ) , we are able to construct a perfect scheme for the cnot gate based on the @xmath56-phase gate , @xmath57 , consisting of four basic steps . two of the steps involve interaction of spins that belong to different qubits , and open the possibility of leakage errors . under condition of total control over system parameters , we show that the leakage can be eliminated . in the other two steps , the tuning of exchange interaction enables perfect swap gates even in the presence of field gradients and soi . motivated by recent results on the surface code , we shortly comment also on the 2d architecture . here we encounter a situation in which the soi vector @xmath4 and the magnetic fields , @xmath3 and @xmath6 , are perpendicular . in this case , the leakage to the non - computational space with @xmath188 is inevitable . however , it can be made very small as long as @xmath189 . depending on the pulsing scheme , the switching times for the conditional phase gate are shown to lie in the range @xmath62-@xmath190 ns for typical gaas parameters . compared to the experimentally established decoherence times of about @xmath191 @xmath192s @xcite , this switching is sufficiently fast and shows that a cnot gate based on exchange is a promising candidate for experimental realizations . we acknowledge discussions with andrew doherty , charles marcus , izhar neder , and mark rudner . this work is supported by iarpa / mqco program , darpa / quest program , the swiss nsf , nccr nanoscience , and nccr qsit .
we propose a scheme for implementing the cnot gate over qubits encoded in a pair of electron spins in a double quantum dot . 
 the scheme is based on exchange and spin orbit interactions and on local gradients in zeeman fields . 
 we find that the optimal device geometry for this implementation involves effective magnetic fields that are parallel to the symmetry axis of the spin orbit interaction . 
 we show that the switching times for the cnot gate can be as fast as a few nanoseconds for realistic parameter values in gaas semiconductors . 
 guided by recent advances in surface codes , we also consider the perpendicular geometry . in this case , 
 leakage errors due to spin orbit interaction occur but can be suppressed in strong magnetic fields .