Qubit gates with simultaneous transport in double quantum dots

A single electron spin in a double quantum dot in a magnetic field is considered in terms of a four-level system. By describing the electron motion between the potential minima by spin-conserving tunneling and spin flip caused by a spin-orbit coupling, we inversely engineer faster-than-adiabatic state manipulation operations based on the geometry of four-dimensional (4D) rotations. In particular, we show how to transport a qubit among the quantum dots performing simultaneously a required spin rotation.


INTRODUCTION
Device architecture based on electrons confined in coupled quantum dots [1,2,3,4] is considered as a potential and significant candidate for quantum computation and quantum information processing. The advantages of this architecture are based on the facts that electron spin is a natural qubit with spin-up and spin-down states, mature semiconductor technology may be used, and long coherence times on the scale of microseconds have been achieved in these systems [5,6]. Laboratories use electric, microwave or magnetic fields to manipulate spin states, performing 10 3 ∼ 10 5 operations in the spin dephasing time [5,6,7,8,9,10].
Scalability of quantum information devices is associated with several architectures having the capability to transport qubits. In this paper we theoretically explore a four-level model for a spin in a double quantum dot (DQD) aiming at the possibilities to implement fast qubit transport with simultaneous rotations. We achieve this goal for arbitrary rotations by controlling the synchronized time dependence of interdot tunneling and spin-orbit coupling (SOC). We inverse-engineer these time dependencies based on our recent work [11] on the control of four-level systems. The method separates population control from control of the phases of the bare state basis [12]. Populations can be mapped onto a 4D sphere so their evolution amounts to 4D transformations controlled by the rotation Hamiltonian that may be engineered from the target state (in our case via isoclinic rotations and quaternions). A full Hamiltonian can then be constructed from the rotation Hamiltonian to realize the desired phase changes. Arbitrary state manipulations require full flexibility in the Hamiltonian, i.e., the possibility to implement the different Hamiltonian matrix elements with specific time-dependences. In the systems of interest, however, there are constraints that hinder certain manipulations and transitions. In particular, in this paper we examine the Hamiltonian structure that corresponds to combined tunneling and SOC controllable couplings, and deduce the possible transformations.
Spin-orbit coupling in semiconductors consists of two main contributions due to the Dresselhaus-and the Bychkov-Rashba-effect. The former is due to the bulk inversion asymmetry of material and the latter results from the structure inversion asymmetry, produced, e.g., by the confining potential or an external electric field [13]. The practical advantage of the Rashba coupling is the ability to manipulate it by an external electric field applied across the semiconductor structure [14,15]. The Rashba coupling controlled by a high-frequency ac gate voltage [16] provides an effective method to control the spin states in short times [17,18].
This paper is organized as follows. In Section II, we introduce first the method that parameterizes the time-dependent Hamiltonian and time evolution operator of a four-level system by using isoclinic rotations and quaternions [11]. Then we map the Hamiltonian of the spin in a DQD coupled by SOC and tunneling onto this scheme. In Section III, we apply the method developed in Section II to design the synchronized time dependences of the control parameters to perform different qubit operations, such as the interdot transport combined with spin rotations. Section IV provides discussion of the results and their relation to other systems. Some details on the structure of the Hamiltonian are presented in the Appendix.

4D Hamiltonians and evolution operators
The wave function of a four-level system where c n , ϕ n are real amplitudes and phases (we set ϕ 1 = 0), and 4 n c 2 n = 1, can be decomposed as |ψ(t) = K(t)|ψ r (t) , where is a vector on the surface of a 4D sphere, and the phase information is contained in The states |ψ(t) and |ψ r (t) evolve via evolution operators U(t) and U r (t) related by U r (t) = K † (t)U(t)K(0), where we set the initial time as 0. Accordingly, the rotation-related Hamiltonian in the Hilbert space is defined as and the total Hamiltonian is To engineer H r for a specific rotation, it is convenient to express first a general 4D rotation matrix as a product of two isoclinic rotation matrices [19,20]: where q i and p j are components of two unit quaternions q = q w + q x i + q y j + q z k and p = p w + p x i + p y j + p z k . We shall parameterize them in terms of generalized 4D spherical angles [21,22], where 0 ≤ φ 1,2 ≤ 2π, 0 ≤ θ 1,2 , γ 1,2 ≤ π. Thus by using U(t) = K(t)U r (t)K † (0) and , we find the parameterized forms for the evolution operator U(t) and the Hamiltonian H(t). The explicit expressions are lengthy, and will not be reported here.

Single electron in a double quantum dot
Consider a single electron spin in a semiconductor DQD, for example made of silicon or GaAs, with tunneling and Rashba spin-orbit coupling, as shown in Fig. 1. We use a bare basis of spin up and down states localized in each well, numbered as |ψ L↓ = |1 , |ψ R↓ = |2 , |ψ R↑ = |3 , |ψ L↑ = |4 . Following the derivation in Appendix A, and after an diagonal energy shift of −∆/2, the Hamiltonian of this system (see (A.7)) can be written as Here τ (t) represents the tunneling coupling between the two quantum dots, α(t) is the Rashba coupling, and ∆ is a Zeeman splitting. All these quantities have dimensions of frequency. Following the approach of Mal'shukov et al. [16], we consider the timedependent Rashba coupling in the complex form α(t) = α 0 + α 1 (t)e iωt . The Hamiltonian structure corresponds topologically to a diamond-configuration [11], which, in the parametric expression of H(t) we may impose with the conditionṡ see Fig. 2. Specifically, after substituting (10) in the parameterized form of (6), the Hamiltonian H(t) acquires the corresponding form To make H 0 (t) and H(t) fully consistent, we further fix the angles aṡ Then (11) gives where we have simplified the notation as γ(t) = γ 1 (t), θ = θ 1 . Now we may impose H 0 (t) = H(t), as they have the same structure, to find the following relations between control functions and auxiliary angles, which implies α 0 = 0, α 1 (t) = −γ 1 (t), ω = ∆ (i.e., the external bias is in resonance with the Zeeman frequency), and θ can be considered as a coupling mixing angle. Under the conditions stated in Eqs. (12), the parameterized time-evolution operator becomes We impose the boundary condition γ(0) = 2nπ, n = . . . , −2, −1, 0, 1, 2, . . ., to guarantee U(0) = 1 at the initial time.

Qubit preparation
Assume that the four-level system is initialized in state |1 on the left well and the objective is to prepare from it an arbitrary qubit in the right well encoded in levels |2 and |3 . Besides the conditions in (12) We can transfer |1 to any bare state except |4 , or to arbitrary superpositions of |2 and |3 (i.e., any qubit on the right well) by imposing γ(T ) = (2n + 1)π/2, n = 0, ±1, .... As an example we shall perform a state transfer to b 1 = 0, b 2 = 1/2, b 3 = e iπ/2 √ 3/2, b 4 = 0. Equation (16) with corresponds to the desired final state |ψ(T ) = −i(|2 +e iπ/2 √ 3|3 )/2 within an irrelevant global phase factor. An Ansatz for γ(t) consistent with the above boundary conditions is The resulting tunneling and Rashba SOC are calculated from (14) as  with the characteristic values τ, α ∝ 1/T . We can prepare a qubit with an arbitrary relative phase by adjusting ∆ and the operation time T as long as the tunneling and SOC are experimentally feasible. We plot the time-dependence of the tunneling matrix elements, Rashba SOC, and populations evolution of all bare states in Fig. 3 with parameters corresponding to the g * −factor of electron in GaAs (g * = −0.44) with B = 100 mT, ∆ ≈ 2π × 0.5 GHz, and T = 3π/(2∆) = 1.5 ns [23].

Qubit transport and rotation
Our method may be applied to transport the qubit from one dot to the other applying simultaneously some qubit rotation, i.e., to produce an arbitrary gate. Suppose we have already prepared a qubit in the left dot in an arbitrary superposition of |1 and |4 as ψ(0) = cos χ|1 + e iµ sin χ|4 , where χ is the initial amplitude mixing angle and µ is the initial relative phase. The corresponding general final state with the unitary evolution operator (15) is given by the amplitudes where A = sin γ(T ) √ 2 1 + cos 2χ cos 2θ + cos µ sin 2χ sin 2θ, We can inversely calculate the coupling mixing angle θ under the condition that γ(T ) = nπ/2, n = 1, 3, 5, · · ·, so that the amplitudes b 1 and b 4 vanish, and for given desired final real amplitudes A and B we obtain: θ = ± arccos A 2 cos 2χ + sin 2 χ(1 − 2 cos 2 χ sin 2 µ) + 2 S cos µ sin 2χ 1 − sin 2 2χ sin 2 µ , where S = 4A 2 B 2 − sin 2 µ sin 2 2χ.
Notice that there is still a degree of freedom to control the final relative phase of the qubit on the right. Suppose our target relative phase is λ = ζ B − ζ A − ∆T . By adjusting the operation time as T = (ζ − λ)/∆, where ζ = ζ B − ζ A , the process produces the desired relative phase λ. Now we can consider two examples of application of (22).
Example 1: transport and phase gate. We assume that A = cos χ and B = sin χ and substitute them into (22) to get θ = 0, which means that τ =γ(t) and α = 0. The final state is calculated as By letting γ(t) evolve from 0 to π/2, the qubit is transported from left to right and rotated by a relative phase factor e −i∆T . Example 2: transport and NOT gate. The NOT gate with transport swaps the amplitudes between up and down states, so we set A = sin χ and B = cos χ in (22) to Figure 5. Scheme for qubit operations in a chain of quantum dots. The system is initialized in state | ↓ in Dot 1 at time t = 0; then, from this initial state, a qubit is prepared in Dot 2 at a time t 1 (the duration of the process is T 1 = t 1 ). This qubit is transported to Dot 3 with an additional relative phase π/4 (process duration T 2 = t 2 − t 1 ). Finally, a "NOT gate with transport" operation is applied to flip and transport the qubit to Dot 4 in a process with duration T 3 = t 3 − t 2 . The upper figures represent each qubit on the corresponding Bloch spheres.

Discussion
By applying an approach based on four-dimensional rotations, we studied electron charge and spin motion in tunneling-and spin-orbit coupled quantum dots. By a proper synchronization of their time-dependences, we inversely engineered the tunneling and spin-orbit coupling matrix elements to achieve spin transport with simultaneous single qubit rotations in quantum information transformations such as the qubit preparation and U phase and U NOT gates. In a chain of quantum dots, these transport+rotation operations may be applied sequentially for a long-distance qubit transfer in a multi-dot architecture, where the ability of a coherent spin transfer has been recently demonstrated [24,25]. Figure 3 demonstrates these processes for a particular sequence starting in Dot 1 and ending in Dot 4. We point out that this technique can also be applied to heavy-hole systems, where the control of the hole spin via the tunneling and strong SOC has been demonstrated for silicon-based double quantum dots [26]. In addition, a similar approach can be used to design the spin and mass transport of cold atoms in optically produced potentials [27].
The one-dimensional Rashba spin-orbit coupling is represented as where the α is the corresponding coupling parameter. We define the full four-state basis of a single electron in the DQD as The time dependence α(t) in (A.7) comes from two main sources: time-dependent α due to ac external bias and time-dependent overlap of the wave functions localized near the left (−x 0 ) and right (x 0 ) minimum of the potential V (x).