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Quantum walks on graphene nanoribbons using quantum gates as coins
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Quantum walks on graphene nanoribbons using quantum gates as coins Ioannis G. Karafyllidis ∗ Department of Electrical and Computer Engineering, Democritus University of Thrace, Kimmeria Campus, 67100, Xanthi, Greece
a r t i c l e
i n f o
Article history: Received 2 March 2015 Received in revised form 5 May 2015 Accepted 26 May 2015 Available online xxx Keywords: Quantum gates Quantum walks Graphene Quantum computing
a b s t r a c t Both discrete and continuous quantum walks on graphs are universal for quantum computation. We define and use discrete quantum walks on the graphene honeycomb lattice to investigate the possibility of using graphene armchair and zigzag nanoribbons to implement quantum gates. The probability distribution of the quantum walker location represents the particle (electron) density distribution on the graphene lattice. We use a universal set of quantum gates as coins that drive the quantum walk and show that different quantum gates result in distinguishable particle distributions on the graphene lattice. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Quantum walks were initially introduced as quantizations of classical random walks [1,2]. Quantum walks represent quantum evolution in continuous spaces, discrete lattices and graphs by the motion of a quantum walker under the action of unitary operators, derived from the specific Hamiltonians [2,3]. Quantum walk theory has been developing rapidly and has become a powerful model for quantum system evolution with direct connections to Feynman propagators and quantum cellular automata [4–9]. Among others, quantum walks have been applied to solve decision problems in terms of quantum walks on decision trees, to model breakdown of electric field driven systems, to study nanotubules and to develop new quantum algorithms [10–13]. Quantum walks have been proven to be a universal model for quantum computation. Continuous quantum walks on graphs can encode any quantum computation with quantum gates implemented by scattering processes [14,15]. Discrete quantum walks have been proven to implement the same universal quantum gate set and thus are able to implement any quantum algorithm [16]. Quantum walks can also be encoded as quantum circuits, which is also a universal model for quantum computation [17,18]. Although both continuous and discrete quantum walks on graphs are universal models for quantum computation the authors of [14–16] do not suggest or explain how a physical quantum
∗ Tel.: +30 2541079548. E-mail address: [email protected]
computing system can be built based on the mathematical graph structures they propose. In the proposed quantum walk models, graphs and wires represent computational quantum basis states and not qubits and the models do not represent a physical quantum computer architecture. Mapping the quantum walk universal quantum computation model on a physical system is not an easy task, but if such a mapping can be achieved, the building of programmable quantum computing systems may be possible. Graphene is a sheet of carbon atoms arranged in a honeycomb lattice. Because of its remarkable electronic properties, it has been proposed that graphene can serve as a physical platform for implementing quantum gates and circuits [19–20]. The possibility of implementing quantum dots, spin qubits, valley filters and valley valves has been studied with encouraging results [21–23]. In this paper, we investigate the possibility of using graphene nanostructures, such as nanoflakes and nanoribbons, to physically implement the universal quantum walk model. We define the discrete quantum walk on the two-dimensional hexagonal graphene lattice. In this discrete quantum walk, the quantum walker represents a particle (electron) moving in the lattice and the probability distribution of the walker location represents the electron density distribution on the graphene lattice. Instead of the usual Grover and quantum Fourier transform (QFT) coins, we use quantum gates as coins to drive the quantum walk. These coins (i.e. the quantum gates) represent physical actions of magnetic fields, electric fields and laser pulses on graphene nanostructures. We use a universal set of quantum gates, namely the Hadamard (H) gate, the phase-shift (P) gate and the controlled-not (CNOT) gate.
http://dx.doi.org/10.1016/j.jocs.2015.05.006 1877-7503/© 2015 Elsevier B.V. All rights reserved.
Please cite this article in press as: I.G. Karafyllidis, Quantum walks on graphene nanoribbons using quantum gates as coins, J. Comput. Sci. (2015), http://dx.doi.org/10.1016/j.jocs.2015.05.006
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Y (Angstroms)
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5 Fig. 3. Motion of the quantum walker and the corresponding quantum coin amplitudes a0 , a1 , a2 and a3 . The indices i and j give the position of the atom in the lattice. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
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X (Angstroms) Fig. 1. A graphene nanoflake The honeycomb graphene lattice comprises two sublattices, indicated by red and blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Quantum walks on infinite lines and infinite regular graphs have been solved analytically [24,25], but it would be very difficult or even impossible to obtain analytical solutions for the various shapes of graphene nanostructures, mainly because of the non-infinite boundaries. Therefore, we develop an algorithm that simulates the quantum walk on finite hexagonal graphene lattices and compare the various particle distributions on different graphene nanostructure lattices. We conclude that these distributions are distinguishable and that it is worthy to further study graphene as a physical quantum computing platform, using both theoretical and experimental methods.
different sublattices are not equivalent, because the lattice looks different from these atom sites. Fig. 2 shows two graphene nanoribbons. The shape of the nanoribbon border determines the carrier velocity in graphene. Fig. 2a shows a zigzag and Fig. 2b shows an armchair nanoribbon. Fig. 3 shows two atoms belonging to the two sublattices, the blue (i,j) and the red (i,j + 1), and their neighbors. Note that atoms in the same zigzag line along the x-axis are indicated with the same j index. We associate a quantum coin to each atom in the graphene lattice. The quantum coin state |c spans a Hilbert space, the coin space Hc, and has four probability amplitudes:
⎡
⎤
⎢ ⎥ ⎢ a1 ⎥ ⎥ ⎥ ⎣ a2 ⎦
|c = ⎢ ⎢
(1)
a3
2. Definition of the quantum walk on the graphene lattice Graphene lattice is not a Bravais lattice and its unit cell comprises two carbon atoms. The repetition of these atoms form two sulattices which are shown in Fig. 1. In the graphene nanoflake shown in this figure, the atoms belonging to different sulattices are indicated with different colors, red and blue. The distance between two neighboring atoms is 1.42 Å. Two carbon atoms belonging to
a0
These four amplitudes are associated with the direction of motion of the quantum walker (particle) from the current atom to each one of its neighbors, as shown in Fig. 3. Amplitude a3 is the amplitude associated with the particle staying at the atom where it is currently located. The atom positions define another basis for the quantum walk in which the state |i, jrepresents the quantum walker positioned at the atom (i, j). These states span the position Hilbert space
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Fig. 2. (a) A zigzag graphene nanoribbon. (b) An armchair graphene nanoribbon.
Please cite this article in press as: I.G. Karafyllidis, Quantum walks on graphene nanoribbons using quantum gates as coins, J. Comput. Sci. (2015), http://dx.doi.org/10.1016/j.jocs.2015.05.006
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Hp. The basis states of the quantum walk |QW are the tensor product of position and coin states: |QW = |i, j ⊗ |c
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and the quantum walk evolves in a Hilbert space, H, given by the tensor product of the position and coin spaces: H = Hc⊗Hp. Two operators drive the discrete quantum walk on the graphene lattice: the coin operator and the shift operator. The coin operator acts on the coin state at each lattice site at each time step and produces a new coin state. The new coin state gives the probability amplitudes related to the particle motion during the next time step. Since there are four coin basis states, the coin operator is given by a four by four matrix. We will use the universal quantum gate set {I, H, P, CNOT} to synthesize the coin operator. The gate I is the do nothing operator, and its matrix representation is the unit matrix. Ten coin operators, C1 to C10 , can be synthesized by the tensor products of the quantum gates in this set:
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C1 =I⊗I
Fig. 4. Probability distribution at the 14nth time step of a quantum walk with C8 as quantum coin. The particle probability at each lattice site is represented by the size of the red and blue circles representing the atoms at the same sites. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
C2 =I⊗H C
15
3
=H⊗I
4
=I⊗P
5
=P⊗I
6
=H⊗P
7
=P⊗H
where I is the identity operator and the index n = 1, 2, ..., 10 in the coin operator indicates one of the ten possible operators of Eq. (3). The evolution of the quantum walk on the graphene lattice is given by:
8
=H⊗H
|QW (t + 1) = (|i(t + 1), j(t + 1) ⊗ |c (t + 1)) =
9
=P⊗P
C C C
C C C C
10
(3)
(7)
= U|QW (t) = S (Cn ⊗ I) (|i(t), j(t) ⊗ |c (t))
where QW(t + 1) and QW(t) are the quantum walk states at two successive time steps. In the general case where different coins are used in different time steps, the coin operator depends also on time:
= CNOT
We can apply the same quantum coin operator in all the quantum walk steps, or we can apply different coin operators in different quantum walk steps. Since quantum gates and, therefore, the quantum coin operators represent physical actions, we can apply different coin operators at different graphene lattice regions in different quantum walk steps. The shift operator, S, acts on the tensor product of the position and coin states of the quantum walker and shifts the particle in superposition in directions which are controlled by the coin state. The action of S on the carbon atoms of the blue sublattice is given by (see Fig. 3):
⎛
⎡
|QW (t + 1) = (|i(t + 1), j(t + 1) ⊗ |c (t + 1)) =
3. Discrete quantum walks on graphene nanoribbons The quantum walk on the graphene lattice, as described by Eqs. (7) and (8), cannot be solved analytically on a graphene nanostructure lattice. To solve this quantum walk we developed an algorithm
⎤⎞
a0
⎜ ⎢ ⎜ ⎢ a1 ⎢ S (|i, j ⊗ |c) = S ⎜ |i, j ⊗ ⎜ ⎢ a ⎝ ⎣ 2
⎡ ⎤ ⎡0 a0 ⎥⎟ ⎥⎟ ⎢ ⎥ ⎢a ⎥⎟ = |i, j + 1 ⊗ ⎢ 0 ⎥ + |i − 1, j ⊗ ⎢ 1 ⎥⎟ ⎣ 0 ⎦ ⎣ 0 ⎦⎠ 0
a3
(8)
= U (t) |QW (t) = S (Cn (t) ⊗ I) (|i(t), j(t) ⊗ |c (t))
⎤
⎡0
⎤
⎡
0
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⎥ ⎢0 ⎥ ⎢0 ⎥ ⎥ + |i + 1, j ⊗ ⎢ ⎥ ⎢ ⎥ ⎦ ⎣ a2 ⎦ + |i, j ⊗ ⎣ 0 ⎦
0
(4)
a3
0
The action of S on the carbon atoms of the red sublattice is given by (see Fig. 3):
⎛ ⎜ ⎝
S (|i, j + 1 ⊗ |c) = S ⎜ |i, j + 1
⎡a ⊗
⎢ ⎢ ⎣
0
a1 a2 a3
⎤⎞ ⎥⎟ ⎥⎟ = ⎦⎠
⎡a ⎤
⎡0 ⎤
0
|i, j
⊗
⎢ 0 ⎥ + |i + 1, j + 1 ⎣ ⎦ 0
⎢ a1 ⎥ + |i − 1, j + 1 ⎣ ⎦ 0 0
0
In Eqs. (4) and (5), the zero amplitudes will be replaced by the amplitudes stemming from the other neighboring atoms.One step of the quantum walk is described by action of the unitary operator U on the position and coin states: U = S (Cn ⊗ I)
⊗
(6)
⎡0 ⊗
⎤
⎢ 0 ⎥ + |i, j + 1 ⎣ a2 ⎦ 0
⎡ ⊗
0
⎤
⎢0 ⎥ ⎣ 0 ⎦
(5)
a3
the inputs of which are the size and shape of the graphene nanostructure, the initial position of the particle and the initial coin state. The coin operator can be chosen to be any of the ten possible operators of Eq. (3). The algorithm produces as output the probability distribution of the particle location on the graphene lattice at any time step.
Please cite this article in press as: I.G. Karafyllidis, Quantum walks on graphene nanoribbons using quantum gates as coins, J. Comput. Sci. (2015), http://dx.doi.org/10.1016/j.jocs.2015.05.006
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Fig. 5. Probability distribution at the 14nth time step of a quantum walk with C8 as quantum coin. (a) On a zigzag and (b) on an armchair nanoribbon. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
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Fig. 6. Probability distribution at the 14nth time step of a quantum walk with C6 as quantum coin. (a) On a zigzag and (b) on an armchair nanoribbon.
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Fig. 7. Probability distribution at the 14nth time step of a quantum walk with C8 and C10 as quantum coins. (a) On a zigzag and (b) on an armchair nanoribbon. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
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The visualization of the algorithm output is shown in Fig. 4. In this figure, the size of the red and blue circles, representing the carbon atoms, is analogous to the probability of finding the particle at the specific site at the specific time step. It is worth noting that in Fig. 4, the particle travels faster along the zigzag direction (along x-axis), as is the fact in electron transport in graphene. Below we will show simulation results for quantum walks on both zigzag and armchair graphene nanoribbons using different coins from Eq. (3). The unitary evolution prevents any quantum walk to reach a steady state [1–3,9]. Because of that we will show the results for the same time step in all cases, for the sake of comparison. We choose this time step to be the 14nth. Since all probabilities must add up to one at any time step, the probabilities become very small after a number of time steps and comparing the circle sizes becomes cumbersome. The 14nth time step is large enough for the probabilities to spread on the lattice and small enough to produce comparable circle sizes. Fig. 5 shows the particle probability distributions after 14 time steps of quantum walk on a graphene zigzag (Fig. 5a) and a graphene armchair nanoribbon (Fig. 5b). The quantum coin used in this walk is the C8 of Eq. (3). The probability spreads faster on the zigzag nanoribbon. The initial position of the particle is on the site indicated by the big red circle in this figure. The probability accumulates at the upper (blue) edge of the zigzag nanoribbon. The inverse happens if the initial position is on a blue carbon atom. In this case the probability accumulates at the down (red) edge. Because of the linearity of the quantum evolution if two particles start form two neighboring red and blue atoms the probability accumulates at the zigzag edges. Experimental data have shown that the carrier transport in zigzag nanoribbons in energies near the Fermi level, takes place along the nanoribbon edges [26,27].Fig. 6 shows the particle probability distributions after 14 time steps of quantum walk on graphene zigzag and armchair nanoribbons using the C6 quantum coin. The phase angle of the quantum phase gate is /4. The probability in both cases spreads along the direction defined by the phase angle. Next, we simulate the quantum walk using the coin C10 . This coin is a CNOT gate and its results in a particle motion in a narrow region around its initial position. We used the C8 coin for the first three time steps to create superposition and spread the particle and then we used the C10 coin for the remaining eleven steps. The results are shown in Fig. 7. In the zigzag nanoribbon the probability spreads to the right on the blue sublattice and to the left on the red sublattice, as shown in Fig. 7a. The opposite happens in the case of the armchair nanoribbon of Fig. 7b. The successive use of the C8 and C10 coins resulted in a separation of the probability distribution on the two sublatices, a very interesting result for which we currently have no explanation. We can only assume that it might be related to the graphene pseudospin. Pseudospin in graphene arises from the use of the Dirac equation to compute the dispersion relations around the Dirac points in the first Brillouin zone. Pseudospin depends on the direction of carrier motion and indicates how the wavefunctions are distributed on the two sublattices. 4. Conclusions We defined the discrete quantum walk on the graphene lattice and used a universal quantum gate set to synthesize the quantum coins that drive the quantum walk. These coins represent physical actions on the graphene nanostructures and the probability distribution of the walker location represents the electron density distribution on the graphene lattice. Our aim was to examine if different
5
quantum coins result in distinguishable particle distributions. Simulation results showed that the quantum walk on the graphene lattice using quantum gates as coins produced a rich variety of distinguishable particle distributions, only a small part of which was presented here. The different probability distributions are a result of the unitary evolution described by the Schrödinger equation. Each coin corresponds to a unitary operator that drives the evolution. We believe that there is no reason why graphene nanostructures cannot be used to implement quantum gates and circuits. The development of such an implementation method will not be an easy task and quantum walks can greatly help in this effort. References [1] Y. Aharonov, L. Davidovich, N. Zagury, Quantum random walks, Phys. Rev. A 48 (1992) 1687. [2] A.M. Childs, E. Farhi, S. Gutmann, An example of the difference between quantum and classical random walks, Quantum Inf. Process. 1 (2002) 35. [3] D. Solenov, L. Fedichkin, Continuous-time walks on a cycle graph, Phys. Rev. A 73 (2006) 12313. [4] V.A. Karmanov, On the derivation of the electron propagator from a random walk, Phys. Lett. A 174 (1993) 371–376. [5] L.H. Kauffman, H.P. Noyes, Discrete physics and the Dirac equation, Phys. Lett. A 218 (1996) 139. [6] D.A. Meyer, From quantum cellular automata to quantum lattice gases, J. Stat. Phys. 85 (1996) 551. [7] I.G. Karafyllidis, Definition and evolution of quantum cellular automata with two qubits per cell, Phys. Rev. A 70 (2004) 44301. [8] F. Barra, P. Gaspard, Transport and dynamics on open quantum graphs, Phys. Rev. E 65 (2001) 16205. [9] S.E. Venegas-Andraca, Quantum walks: a comprehensive review, Quantum Inf. Process. 11 (2012) 1015. [10] E. Farhi, S. Gutmann, Quantum computation and decision trees, Phys. Rev. A 58 (1998) 915. [11] T. Oka, N. Konno, R. Arita, H. Aoki, Breakdown of an electric-field driven system: a mapping to a quantum walk, Phys. Rev. Lett. 94 (2005) 100602. [12] I.G. Karafyllidis, D.C. Lagoudas, Microtubules as mechanical force sensors, BioSystems 88 (2007) 137–146. [13] I. Sinayskiy, F. Petruccione, Efficiency of open quantum walk implementation of dissipative quantum computing algorithms, Quantum Inf. Process. 11 (2012) 1301. [14] A.M. Childs, Universal computation by quantum walk, Phys. Rev. Lett. 102 (2009) 180501. [15] N.B. Lovett, S. Cooper, M. Everitt, M. Trevers, V. Kendon, Universal quantum computation using the discrete-time quantum walk, Phys. Rev. Lett. 81 (2010) 42330. [16] A.M. Childs, D. Gosset, S. Webb, Universal computation by multiparticle quantum walk, Science 339 (2013) 791. [17] B.L. Douglas, J.B. Wang, Efficient quantum circuit implementation of quantum walks, Phys. Rev. A 79 (2009) 52335. [18] I.G. Karafyllidis, Quantum computer simulator based on the circuit model of quantum computation, IEEE Trans. Circuits Syst. I 52 (2005) 1590. [19] L. Brey, H.A. Fertig, Electronic states of graphene nanoribbons studied with the Dirac equation, Phys. Rev. B 73 (2006) 235411. [20] V. Fal’ko, Quantum information on chicken wire, Nat. Phys. 3 (2007) 151. [21] P.G. Silvestrov, K.B. Efetov, Quantum dots in graphene, Phys. Rev. Lett. 98 (2007) 16802. [22] B. Trauzettel, D.V. Bulaev, D. Loss, G. Burkard, Spin qubits in graphene quantum dots, Nat. Phys. 3 (2007) 192. [23] A. Rycerz, J. Tworzydlo, C.W.J. Beenakker, Valley filter and valley valve in graphene, Nat. Phys. 3 (2007) 172. [24] N. Konno, Quantum random walks in one dimension, Quantum Inf. Process. 1 (2002) 345. [25] D. Aharonov, A. Ambainis, J. Kempe, U. Vaziriani, Quantum walks on graphs, in: Proceedings of the 33rd Annual ACM STOC, NY, 2001, p. 50. [26] K. Wakabayashi, Electronic transport properties of nanographite ribbon junctions, Phys. Rev. B 64 (2001) 125428. [27] L. Brey, H.A. Fertig, Electronic states of graphene nanoribbons studied with the Dirac equation, Phys. Rev. B 73 (2006) 235411. Ioannis G. Karafyllidis received the Dipl. Eng. and Ph.D. degrees in Electrical Engineering from the Aristotle University of Thessaloniki, Greece. In 1992, he joined the Department of Electrical and Computer Engineering, Democritus University of Thrace, Greece, as a faculty member, where he is currently a Professor. His current research emphasis is on quantum computing, modeling and simulation of nanoelectronic devices and circuits, and biological networks modelling. He is a Fellow of the Institute of Nanotechnology, a Founding Member of the American Academy of Nanomedicine and a member of the Technical Chamber of Greece (TEE).
Please cite this article in press as: I.G. Karafyllidis, Quantum walks on graphene nanoribbons using quantum gates as coins, J. Comput. Sci. (2015), http://dx.doi.org/10.1016/j.jocs.2015.05.006
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Contents lists available at ScienceDirect
Journal of Computational Science journal homepage: www.elsevier.com/locate/jocs
Quantum walks on graphene nanoribbons using quantum gates as coins Ioannis G. Karafyllidis ∗ Department of Electrical and Computer Engineering, Democritus University of Thrace, Kimmeria Campus, 67100, Xanthi, Greece
a r t i c l e
i n f o
Article history: Received 2 March 2015 Received in revised form 5 May 2015 Accepted 26 May 2015 Available online xxx Keywords: Quantum gates Quantum walks Graphene Quantum computing
a b s t r a c t Both discrete and continuous quantum walks on graphs are universal for quantum computation. We define and use discrete quantum walks on the graphene honeycomb lattice to investigate the possibility of using graphene armchair and zigzag nanoribbons to implement quantum gates. The probability distribution of the quantum walker location represents the particle (electron) density distribution on the graphene lattice. We use a universal set of quantum gates as coins that drive the quantum walk and show that different quantum gates result in distinguishable particle distributions on the graphene lattice. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Quantum walks were initially introduced as quantizations of classical random walks [1,2]. Quantum walks represent quantum evolution in continuous spaces, discrete lattices and graphs by the motion of a quantum walker under the action of unitary operators, derived from the specific Hamiltonians [2,3]. Quantum walk theory has been developing rapidly and has become a powerful model for quantum system evolution with direct connections to Feynman propagators and quantum cellular automata [4–9]. Among others, quantum walks have been applied to solve decision problems in terms of quantum walks on decision trees, to model breakdown of electric field driven systems, to study nanotubules and to develop new quantum algorithms [10–13]. Quantum walks have been proven to be a universal model for quantum computation. Continuous quantum walks on graphs can encode any quantum computation with quantum gates implemented by scattering processes [14,15]. Discrete quantum walks have been proven to implement the same universal quantum gate set and thus are able to implement any quantum algorithm [16]. Quantum walks can also be encoded as quantum circuits, which is also a universal model for quantum computation [17,18]. Although both continuous and discrete quantum walks on graphs are universal models for quantum computation the authors of [14–16] do not suggest or explain how a physical quantum
∗ Tel.: +30 2541079548. E-mail address: [email protected]
computing system can be built based on the mathematical graph structures they propose. In the proposed quantum walk models, graphs and wires represent computational quantum basis states and not qubits and the models do not represent a physical quantum computer architecture. Mapping the quantum walk universal quantum computation model on a physical system is not an easy task, but if such a mapping can be achieved, the building of programmable quantum computing systems may be possible. Graphene is a sheet of carbon atoms arranged in a honeycomb lattice. Because of its remarkable electronic properties, it has been proposed that graphene can serve as a physical platform for implementing quantum gates and circuits [19–20]. The possibility of implementing quantum dots, spin qubits, valley filters and valley valves has been studied with encouraging results [21–23]. In this paper, we investigate the possibility of using graphene nanostructures, such as nanoflakes and nanoribbons, to physically implement the universal quantum walk model. We define the discrete quantum walk on the two-dimensional hexagonal graphene lattice. In this discrete quantum walk, the quantum walker represents a particle (electron) moving in the lattice and the probability distribution of the walker location represents the electron density distribution on the graphene lattice. Instead of the usual Grover and quantum Fourier transform (QFT) coins, we use quantum gates as coins to drive the quantum walk. These coins (i.e. the quantum gates) represent physical actions of magnetic fields, electric fields and laser pulses on graphene nanostructures. We use a universal set of quantum gates, namely the Hadamard (H) gate, the phase-shift (P) gate and the controlled-not (CNOT) gate.
http://dx.doi.org/10.1016/j.jocs.2015.05.006 1877-7503/© 2015 Elsevier B.V. All rights reserved.
Please cite this article in press as: I.G. Karafyllidis, Quantum walks on graphene nanoribbons using quantum gates as coins, J. Comput. Sci. (2015), http://dx.doi.org/10.1016/j.jocs.2015.05.006
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5 Fig. 3. Motion of the quantum walker and the corresponding quantum coin amplitudes a0 , a1 , a2 and a3 . The indices i and j give the position of the atom in the lattice. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
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X (Angstroms) Fig. 1. A graphene nanoflake The honeycomb graphene lattice comprises two sublattices, indicated by red and blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Quantum walks on infinite lines and infinite regular graphs have been solved analytically [24,25], but it would be very difficult or even impossible to obtain analytical solutions for the various shapes of graphene nanostructures, mainly because of the non-infinite boundaries. Therefore, we develop an algorithm that simulates the quantum walk on finite hexagonal graphene lattices and compare the various particle distributions on different graphene nanostructure lattices. We conclude that these distributions are distinguishable and that it is worthy to further study graphene as a physical quantum computing platform, using both theoretical and experimental methods.
different sublattices are not equivalent, because the lattice looks different from these atom sites. Fig. 2 shows two graphene nanoribbons. The shape of the nanoribbon border determines the carrier velocity in graphene. Fig. 2a shows a zigzag and Fig. 2b shows an armchair nanoribbon. Fig. 3 shows two atoms belonging to the two sublattices, the blue (i,j) and the red (i,j + 1), and their neighbors. Note that atoms in the same zigzag line along the x-axis are indicated with the same j index. We associate a quantum coin to each atom in the graphene lattice. The quantum coin state |c spans a Hilbert space, the coin space Hc, and has four probability amplitudes:
⎡
⎤
⎢ ⎥ ⎢ a1 ⎥ ⎥ ⎥ ⎣ a2 ⎦
|c = ⎢ ⎢
(1)
a3
2. Definition of the quantum walk on the graphene lattice Graphene lattice is not a Bravais lattice and its unit cell comprises two carbon atoms. The repetition of these atoms form two sulattices which are shown in Fig. 1. In the graphene nanoflake shown in this figure, the atoms belonging to different sulattices are indicated with different colors, red and blue. The distance between two neighboring atoms is 1.42 Å. Two carbon atoms belonging to
a0
These four amplitudes are associated with the direction of motion of the quantum walker (particle) from the current atom to each one of its neighbors, as shown in Fig. 3. Amplitude a3 is the amplitude associated with the particle staying at the atom where it is currently located. The atom positions define another basis for the quantum walk in which the state |i, jrepresents the quantum walker positioned at the atom (i, j). These states span the position Hilbert space
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Fig. 2. (a) A zigzag graphene nanoribbon. (b) An armchair graphene nanoribbon.
Please cite this article in press as: I.G. Karafyllidis, Quantum walks on graphene nanoribbons using quantum gates as coins, J. Comput. Sci. (2015), http://dx.doi.org/10.1016/j.jocs.2015.05.006
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Hp. The basis states of the quantum walk |QW are the tensor product of position and coin states: |QW = |i, j ⊗ |c
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(2)
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and the quantum walk evolves in a Hilbert space, H, given by the tensor product of the position and coin spaces: H = Hc⊗Hp. Two operators drive the discrete quantum walk on the graphene lattice: the coin operator and the shift operator. The coin operator acts on the coin state at each lattice site at each time step and produces a new coin state. The new coin state gives the probability amplitudes related to the particle motion during the next time step. Since there are four coin basis states, the coin operator is given by a four by four matrix. We will use the universal quantum gate set {I, H, P, CNOT} to synthesize the coin operator. The gate I is the do nothing operator, and its matrix representation is the unit matrix. Ten coin operators, C1 to C10 , can be synthesized by the tensor products of the quantum gates in this set:
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C1 =I⊗I
Fig. 4. Probability distribution at the 14nth time step of a quantum walk with C8 as quantum coin. The particle probability at each lattice site is represented by the size of the red and blue circles representing the atoms at the same sites. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
C2 =I⊗H C
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=H⊗I
4
=I⊗P
5
=P⊗I
6
=H⊗P
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=P⊗H
where I is the identity operator and the index n = 1, 2, ..., 10 in the coin operator indicates one of the ten possible operators of Eq. (3). The evolution of the quantum walk on the graphene lattice is given by:
8
=H⊗H
|QW (t + 1) = (|i(t + 1), j(t + 1) ⊗ |c (t + 1)) =
9
=P⊗P
C C C
C C C C
10
(3)
(7)
= U|QW (t) = S (Cn ⊗ I) (|i(t), j(t) ⊗ |c (t))
where QW(t + 1) and QW(t) are the quantum walk states at two successive time steps. In the general case where different coins are used in different time steps, the coin operator depends also on time:
= CNOT
We can apply the same quantum coin operator in all the quantum walk steps, or we can apply different coin operators in different quantum walk steps. Since quantum gates and, therefore, the quantum coin operators represent physical actions, we can apply different coin operators at different graphene lattice regions in different quantum walk steps. The shift operator, S, acts on the tensor product of the position and coin states of the quantum walker and shifts the particle in superposition in directions which are controlled by the coin state. The action of S on the carbon atoms of the blue sublattice is given by (see Fig. 3):
⎛
⎡
|QW (t + 1) = (|i(t + 1), j(t + 1) ⊗ |c (t + 1)) =
3. Discrete quantum walks on graphene nanoribbons The quantum walk on the graphene lattice, as described by Eqs. (7) and (8), cannot be solved analytically on a graphene nanostructure lattice. To solve this quantum walk we developed an algorithm
⎤⎞
a0
⎜ ⎢ ⎜ ⎢ a1 ⎢ S (|i, j ⊗ |c) = S ⎜ |i, j ⊗ ⎜ ⎢ a ⎝ ⎣ 2
⎡ ⎤ ⎡0 a0 ⎥⎟ ⎥⎟ ⎢ ⎥ ⎢a ⎥⎟ = |i, j + 1 ⊗ ⎢ 0 ⎥ + |i − 1, j ⊗ ⎢ 1 ⎥⎟ ⎣ 0 ⎦ ⎣ 0 ⎦⎠ 0
a3
(8)
= U (t) |QW (t) = S (Cn (t) ⊗ I) (|i(t), j(t) ⊗ |c (t))
⎤
⎡0
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⎥ ⎢0 ⎥ ⎢0 ⎥ ⎥ + |i + 1, j ⊗ ⎢ ⎥ ⎢ ⎥ ⎦ ⎣ a2 ⎦ + |i, j ⊗ ⎣ 0 ⎦
0
(4)
a3
0
The action of S on the carbon atoms of the red sublattice is given by (see Fig. 3):
⎛ ⎜ ⎝
S (|i, j + 1 ⊗ |c) = S ⎜ |i, j + 1
⎡a ⊗
⎢ ⎢ ⎣
0
a1 a2 a3
⎤⎞ ⎥⎟ ⎥⎟ = ⎦⎠
⎡a ⎤
⎡0 ⎤
0
|i, j
⊗
⎢ 0 ⎥ + |i + 1, j + 1 ⎣ ⎦ 0
⎢ a1 ⎥ + |i − 1, j + 1 ⎣ ⎦ 0 0
0
In Eqs. (4) and (5), the zero amplitudes will be replaced by the amplitudes stemming from the other neighboring atoms.One step of the quantum walk is described by action of the unitary operator U on the position and coin states: U = S (Cn ⊗ I)
⊗
(6)
⎡0 ⊗
⎤
⎢ 0 ⎥ + |i, j + 1 ⎣ a2 ⎦ 0
⎡ ⊗
0
⎤
⎢0 ⎥ ⎣ 0 ⎦
(5)
a3
the inputs of which are the size and shape of the graphene nanostructure, the initial position of the particle and the initial coin state. The coin operator can be chosen to be any of the ten possible operators of Eq. (3). The algorithm produces as output the probability distribution of the particle location on the graphene lattice at any time step.
Please cite this article in press as: I.G. Karafyllidis, Quantum walks on graphene nanoribbons using quantum gates as coins, J. Comput. Sci. (2015), http://dx.doi.org/10.1016/j.jocs.2015.05.006
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Fig. 5. Probability distribution at the 14nth time step of a quantum walk with C8 as quantum coin. (a) On a zigzag and (b) on an armchair nanoribbon. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
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Fig. 6. Probability distribution at the 14nth time step of a quantum walk with C6 as quantum coin. (a) On a zigzag and (b) on an armchair nanoribbon.
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Fig. 7. Probability distribution at the 14nth time step of a quantum walk with C8 and C10 as quantum coins. (a) On a zigzag and (b) on an armchair nanoribbon. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
Please cite this article in press as: I.G. Karafyllidis, Quantum walks on graphene nanoribbons using quantum gates as coins, J. Comput. Sci. (2015), http://dx.doi.org/10.1016/j.jocs.2015.05.006
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The visualization of the algorithm output is shown in Fig. 4. In this figure, the size of the red and blue circles, representing the carbon atoms, is analogous to the probability of finding the particle at the specific site at the specific time step. It is worth noting that in Fig. 4, the particle travels faster along the zigzag direction (along x-axis), as is the fact in electron transport in graphene. Below we will show simulation results for quantum walks on both zigzag and armchair graphene nanoribbons using different coins from Eq. (3). The unitary evolution prevents any quantum walk to reach a steady state [1–3,9]. Because of that we will show the results for the same time step in all cases, for the sake of comparison. We choose this time step to be the 14nth. Since all probabilities must add up to one at any time step, the probabilities become very small after a number of time steps and comparing the circle sizes becomes cumbersome. The 14nth time step is large enough for the probabilities to spread on the lattice and small enough to produce comparable circle sizes. Fig. 5 shows the particle probability distributions after 14 time steps of quantum walk on a graphene zigzag (Fig. 5a) and a graphene armchair nanoribbon (Fig. 5b). The quantum coin used in this walk is the C8 of Eq. (3). The probability spreads faster on the zigzag nanoribbon. The initial position of the particle is on the site indicated by the big red circle in this figure. The probability accumulates at the upper (blue) edge of the zigzag nanoribbon. The inverse happens if the initial position is on a blue carbon atom. In this case the probability accumulates at the down (red) edge. Because of the linearity of the quantum evolution if two particles start form two neighboring red and blue atoms the probability accumulates at the zigzag edges. Experimental data have shown that the carrier transport in zigzag nanoribbons in energies near the Fermi level, takes place along the nanoribbon edges [26,27].Fig. 6 shows the particle probability distributions after 14 time steps of quantum walk on graphene zigzag and armchair nanoribbons using the C6 quantum coin. The phase angle of the quantum phase gate is /4. The probability in both cases spreads along the direction defined by the phase angle. Next, we simulate the quantum walk using the coin C10 . This coin is a CNOT gate and its results in a particle motion in a narrow region around its initial position. We used the C8 coin for the first three time steps to create superposition and spread the particle and then we used the C10 coin for the remaining eleven steps. The results are shown in Fig. 7. In the zigzag nanoribbon the probability spreads to the right on the blue sublattice and to the left on the red sublattice, as shown in Fig. 7a. The opposite happens in the case of the armchair nanoribbon of Fig. 7b. The successive use of the C8 and C10 coins resulted in a separation of the probability distribution on the two sublatices, a very interesting result for which we currently have no explanation. We can only assume that it might be related to the graphene pseudospin. Pseudospin in graphene arises from the use of the Dirac equation to compute the dispersion relations around the Dirac points in the first Brillouin zone. Pseudospin depends on the direction of carrier motion and indicates how the wavefunctions are distributed on the two sublattices. 4. Conclusions We defined the discrete quantum walk on the graphene lattice and used a universal quantum gate set to synthesize the quantum coins that drive the quantum walk. These coins represent physical actions on the graphene nanostructures and the probability distribution of the walker location represents the electron density distribution on the graphene lattice. Our aim was to examine if different
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quantum coins result in distinguishable particle distributions. Simulation results showed that the quantum walk on the graphene lattice using quantum gates as coins produced a rich variety of distinguishable particle distributions, only a small part of which was presented here. The different probability distributions are a result of the unitary evolution described by the Schrödinger equation. Each coin corresponds to a unitary operator that drives the evolution. We believe that there is no reason why graphene nanostructures cannot be used to implement quantum gates and circuits. The development of such an implementation method will not be an easy task and quantum walks can greatly help in this effort. References [1] Y. Aharonov, L. Davidovich, N. Zagury, Quantum random walks, Phys. Rev. A 48 (1992) 1687. [2] A.M. Childs, E. Farhi, S. Gutmann, An example of the difference between quantum and classical random walks, Quantum Inf. Process. 1 (2002) 35. [3] D. Solenov, L. Fedichkin, Continuous-time walks on a cycle graph, Phys. Rev. A 73 (2006) 12313. [4] V.A. Karmanov, On the derivation of the electron propagator from a random walk, Phys. Lett. A 174 (1993) 371–376. [5] L.H. Kauffman, H.P. Noyes, Discrete physics and the Dirac equation, Phys. Lett. A 218 (1996) 139. [6] D.A. Meyer, From quantum cellular automata to quantum lattice gases, J. Stat. Phys. 85 (1996) 551. [7] I.G. Karafyllidis, Definition and evolution of quantum cellular automata with two qubits per cell, Phys. Rev. A 70 (2004) 44301. [8] F. Barra, P. Gaspard, Transport and dynamics on open quantum graphs, Phys. Rev. E 65 (2001) 16205. [9] S.E. Venegas-Andraca, Quantum walks: a comprehensive review, Quantum Inf. Process. 11 (2012) 1015. [10] E. Farhi, S. Gutmann, Quantum computation and decision trees, Phys. Rev. A 58 (1998) 915. [11] T. Oka, N. Konno, R. Arita, H. Aoki, Breakdown of an electric-field driven system: a mapping to a quantum walk, Phys. Rev. Lett. 94 (2005) 100602. [12] I.G. Karafyllidis, D.C. Lagoudas, Microtubules as mechanical force sensors, BioSystems 88 (2007) 137–146. [13] I. Sinayskiy, F. Petruccione, Efficiency of open quantum walk implementation of dissipative quantum computing algorithms, Quantum Inf. Process. 11 (2012) 1301. [14] A.M. Childs, Universal computation by quantum walk, Phys. Rev. Lett. 102 (2009) 180501. [15] N.B. Lovett, S. Cooper, M. Everitt, M. Trevers, V. Kendon, Universal quantum computation using the discrete-time quantum walk, Phys. Rev. Lett. 81 (2010) 42330. [16] A.M. Childs, D. Gosset, S. Webb, Universal computation by multiparticle quantum walk, Science 339 (2013) 791. [17] B.L. Douglas, J.B. Wang, Efficient quantum circuit implementation of quantum walks, Phys. Rev. A 79 (2009) 52335. [18] I.G. Karafyllidis, Quantum computer simulator based on the circuit model of quantum computation, IEEE Trans. Circuits Syst. I 52 (2005) 1590. [19] L. Brey, H.A. Fertig, Electronic states of graphene nanoribbons studied with the Dirac equation, Phys. Rev. B 73 (2006) 235411. [20] V. Fal’ko, Quantum information on chicken wire, Nat. Phys. 3 (2007) 151. [21] P.G. Silvestrov, K.B. Efetov, Quantum dots in graphene, Phys. Rev. Lett. 98 (2007) 16802. [22] B. Trauzettel, D.V. Bulaev, D. Loss, G. Burkard, Spin qubits in graphene quantum dots, Nat. Phys. 3 (2007) 192. [23] A. Rycerz, J. Tworzydlo, C.W.J. Beenakker, Valley filter and valley valve in graphene, Nat. Phys. 3 (2007) 172. [24] N. Konno, Quantum random walks in one dimension, Quantum Inf. Process. 1 (2002) 345. [25] D. Aharonov, A. Ambainis, J. Kempe, U. Vaziriani, Quantum walks on graphs, in: Proceedings of the 33rd Annual ACM STOC, NY, 2001, p. 50. [26] K. Wakabayashi, Electronic transport properties of nanographite ribbon junctions, Phys. Rev. B 64 (2001) 125428. [27] L. Brey, H.A. Fertig, Electronic states of graphene nanoribbons studied with the Dirac equation, Phys. Rev. B 73 (2006) 235411. Ioannis G. Karafyllidis received the Dipl. Eng. and Ph.D. degrees in Electrical Engineering from the Aristotle University of Thessaloniki, Greece. In 1992, he joined the Department of Electrical and Computer Engineering, Democritus University of Thrace, Greece, as a faculty member, where he is currently a Professor. His current research emphasis is on quantum computing, modeling and simulation of nanoelectronic devices and circuits, and biological networks modelling. He is a Fellow of the Institute of Nanotechnology, a Founding Member of the American Academy of Nanomedicine and a member of the Technical Chamber of Greece (TEE).
Please cite this article in press as: I.G. Karafyllidis, Quantum walks on graphene nanoribbons using quantum gates as coins, J. Comput. Sci. (2015), http://dx.doi.org/10.1016/j.jocs.2015.05.006