Quantum computation with graphene nanoribbon

We propose a scalable scheme to implement quantum computation in graphene nanoribbon. It is shown that electron or hole can be naturally localized in each zigzag region for a graphene nanoribbon with a sequence of Z-shaped structure without exploiting any confined gate. An one-dimensional graphene quantum dots chain is formed in such graphene nanoribbon, where electron or hole spin can be encoded as qubits. The coupling interaction between neighboring graphene quantum dots is found to be always-on Heisenberg type. Applying the bang-bang control strategy and decoherence free subspaces encoding method, universal quantum computation is argued to be realizable with the present techniques.

Electron spin is one of the leading candidates for the realization of a practical solid qubit [1]. The coherent manipulation of electron spins in GaAs quantum dots has been efficiently realized [2,3]. However due to the interaction with the environment, the decoherence time is often in nanoseconds scale in GaAs quantum dots [2,4,5]. Even by applying the complex technique to prepare nuclear state, the dephasing time for spin qubits is just about 1µs [6]. The decoherence is one of the most challenges in the way to quantum computer in GaAs quantum dots. Due to the weak spin-orbit coupling and hyperfine interactions in carbon, graphene is argued to be an excellent candidate for quantum computation [7].
However, due to the special band structure of graphene [8], the low-energy quasiparticles in graphene behave as Dirac fermions, and the Klein tunneling and Chiral effect lead to the fact that it is non-trivial to form good quantum dot (localized electron states) in graphene.
It has been shown that the massless Dirac fermions in graphene can ben confined by using suitable transverse states in graphene nanoribbons (GNR) [7,9], by combining single and bilayer regions of graphene [10,11] or by using inhomogeneous magnetic fields [12]. Recently, there was an experiment report that GNR with well defined zigzag or armchair edge structures can be chemically produced [13]. It has also been discovered that localized states exist in the zigzag region in Z-shaped GNR [14].
Here we present a scalable quantum computation scheme based on Z-shaped GNR quantum dot system without exploiting any confined gates. The localized particle can be chosen to be electron or hole by adjusting the back gate even in the room temperature. The qubit is encoded on the electron (hole) spin states localized in the zigzag region of the GNR with a sequence of Z-shaped structure. The interaction between qubits is determined by the GNR geometrical structure and found to be Heisenberg form. By exploiting bang-bang (BB) control strategy and decoherence free subspaces (DFSs) encoding method, universal quantum gates are shown to be realizable in this system with the present techniques.
Based on the π orbital tight-binding model, the local density of state (LDOS) and the band structure of the zigzag region in a GNR with a sequence of Z-shaped structure can be obtained by the direct diagonalization of the single particle Hamiltonian H 0 = ij τ ij |i j|, where the hopping matrix element τ ij = −τ if the orbits |i and |j are nearest neighboring on the honeycomb lattice, otherwise τ ij = 0 [15,16]. From the calculated band structure, we can see that there are several localized states with electron-hole symmetry around the zero energy point as shown in Fig. 1a. Thus we can choose to get one localized electron or hole in the zigzag region by adjusting the Fermi level through the back gate. The electron ground state energy and the energy gap between the ground state and the first excitation state are very sensitive to the size of the zigzag region, as shown in Fig 2. It has been known that the width of the armchair GNR (N unit cells) decides whether the system is metallic or semiconducting [15,16]. If N = 3m − 1 (m being an integer), the armchair GNR is metallic, otherwise it is semiconducting. In addition, for the present Z-shaped structure the boundaries along the ribbon of armchair region is unsymmetrical when N is even. Actually, in our calculation we find there is no confined state in the zigzag region of Z-shaped GNR when N = 3m − 1 or N = 2m as shown in Fig. 2. On the other hand, when N is 7 and the length of the zigzag region L (unit cells) (see Fig. 1c) is 3, 4, 5, 6, 7 and N = 9, L = 3, 4 both the ground level and energy gap are above 0.1eV. Thus we can confine electron (hole) to form quantum dot even in the room temperature. Fig. 1b shows the spatial distribution of local density of ground state for a GNR with two Z-shaped structure in series. Each zigzag region (quantum dot) confines one electron and the quantum dots are coupled by the exchange coupling J 1 . We can obtain J 1 by calculating where ϕ 1 ( r) and ϕ 2 ( r) are the wavefunction of neighboring graphene quantum dots. We can also calculate the next nearest neighboring exchange coupling J 2 by the same method. Obviously, the exchange To carry out quantum computation, we have to form the logical qubit and realize universal quantum gate. It has been shown that single qubit rotations combined with two-qubit operations can be used to create basic quantum gates [17]. The spin of the localized electron or hole can be used as the physical qubit and the GNR with a sequence of Z-shaped structure forms an one-dimensional qubit chain as shown in Fig. 1d. The neighboring qubits in this chain have an always-on Heisenberg interaction H = J 1 S 1 · S 2 . Here S 1 and S 2 are the spin operator of the neighboring localized electron (hole). It has been known that BB control strategy and DFSs encoding method do not require directly controlling the physical qubit 1 interaction between qubits [18,19]. The quantum information in qubits can be protected from decoherence induced by the environment and undesired disturbance induced by the inherent qubit-qubit interaction with these strategies.
For a sequence of Z-shaped structure GNR with N = 7, L = 6, D = 18, the Hamiltonian  of the system can be expressed as where σ x,y,z i,j are the spin Pauli operators of the localized electron (hole) in the quantum dots, i and j represent two neighboring dots. Here we have neglected the interaction between non-neighboring dots, which has been shown to be 5 orders smaller than the neighboring interaction.
To avoid the spin qubits to entangle with the environment, we can apply a BB operation U z = exp(−iσ z π/2) to each quantum dot region. Such rotation operations can be realized if a pulsed magnetic field could be applied exclusively [1]. To counteract phase decoherence, we can use DFSs encoding [20]. For a simply DFSs encoding, two physical qubits can encode a logical qubit: As shown in Fig. 1c, we use localized electron in the two neighboring zigzag regions to form a logical qubit.
In order to protect quantum information in the logical qubits, we must decouple the always-on Heisenberg interaction between two physical qubits within a logical qubits and interaction between two neighboring logical qubits. A nonsynchronous BB pulse operations and a special encoding method can be exploited to eliminate these interactions [19]. Here we propose an architecture in which the one-dimensional GNR chain form a periodic structure L 1 L 2 L 3 L 1 L 2 L 3 · · · with three logical qubits as a unit, as shown in Fig. 1d. L 1 represents a logical qubit encoded as Eq.(2). L 2 is a logical qubit encoded as And L 3 is a logical qubit encoded as With this periodic architecture, we have to apply nonsynchronous BB pluse operations Then we obtain a quantum computation system with entirely decoupled logical qubits. Now we show how to carry out universal quantum gates of the logical qubits defined above. Logical operationsX andZ can generate all SU(2) transformations of logical qubit.
.X can be easily achieved by recoupling qubits 1 and 2 by adjusting the BB pulses of both qubits to be synchronous [19].  [14].
To get high fidelity forX operation, we should avoid the long-range disorder and irregular edge. Actually, if we know the coupling J 1 between different dots exactly, inhomogeneity of J 1 can not effect the fidelity ofX gate when corresponding inhomogeneous operation times are used. In addition, we find the effect of the J 1 inhomogeneity or fluctuation to the fidelity of theX gate is small as shown in Fig. 4. Because the nuclear field would change the evolution of the spin states, the fidelity of theZ gate is dominated by the nuclear field [3].
The fidelity of theZ gate can be very high due to small nuclear field in graphene system.
Similarly, high fidelity operationX andZ can be also realized for logical qubits L 2 and L 3 .

By performing Hadamard transformation
to the two physical qubits of the second logical qubit L 2 and changing the BB control pulse to be the same with L 1 , we can recouple the two neighboring logical qubits and implement Similar to the above discussion forX andZ operation, we can find that the fluctuation or inhomogeneity of J 1 and the nuclear field have trivial effect to the fidelity of the CNOT gate in the present protocol.
The major decoherence sources of spin qubits in solid state system have been identified as the spin-orbit interaction and hyperfine interaction. The weak spin-orbit coupling have been predicted in carbon material due to the low atomic weight [22]. Since the primary component of natural carbon is the zero spin isotope 12 C, the very long coherence time given by hyperfine coupling has been theoretically argued [7]. Assuming the abundance of 13 C is about 1% as in the nature carbon material, the decoherence time has been predicted to be more than 10µs in the graphene quantum dot [7,23]. This decoherence time is 4 orders longer than the gates operation time of the present protocol. In addition, the decoherence time can be much longer if the percentage of 13 C is decreased by isotopic purification.
In this paper we have presented a scalable scheme of quantum computation based on GNR with a sequence of Z-shaped structure. No confined gates is needed to localize the particle, which can be chosen to be electron or hole by adjusting back gate. The qubit is encoded in electron or hole spin states, which is naturally localized in the zigzag region of GNR even in room temperature. The neighboring qubits are found to have an alwayson Heisenberg interaction and the dynamical decoupling techniques with DFSs is exploited to achieve universal quantum computation in this system. Due to recent achievement in production of graphene nanoribbon, this proposal may be implementable within the present techniques. We