Abstract
Among the various approaches to quantum computing, alloptical architectures are especially promising due to the robustness and mobility of single photons. However, the creation of the twophoton quantum logic gates required for universal quantum computing remains a challenge. Here we propose a universal twoqubit quantum logic gate, where qubits are encoded in surface plasmons in graphene nanostructures, that exploits graphene's strong thirdorder nonlinearity and long plasmon lifetimes to enable singlephotonlevel interactions. In particular, we utilize strong twoplasmon absorption in graphene nanoribbons, which can greatly exceed singleplasmon absorption to create a “squarerootofswap” that is protected by the quantum Zeno effect against evolution into undesired failure modes. Our gate does not require any cryogenic or vacuum technology, has a footprint of a few hundred nanometers, and reaches fidelities and success rates well above the faulttolerance threshold, suggesting that graphene plasmonics offers a route towards scalable quantum technologies.
Introduction
Quantum computing could efficiently solve many essential problems. However, building a quantum computer is not an easy task. One particularly promising approach is to use singlephotons, whose weak interaction with the environment makes them perfectly suitable for encoding and transmitting quantum information. Nonetheless, this weak interaction strength makes the implementation of photon–photon interactions a significant challenge. While this can be overcome at the cost of extra photons,^{1} the additional overhead makes purely linearoptical schemes difficult to scale up.^{2} Alternatively, singlephotonlevel nonlinearities can be used to directly create deterministic gates.^{3} However, this typically requires complex interactions with atomic systems that cannot readily be miniaturized. Recent work shows that graphene can provide a strong enough nonlinearity without the technical drawbacks of those atomic systems.
Our graphenebased twoqubit logic gate is centered on Franson's quantum Zeno gate,^{4} which is a universal “squarerootofswap” (SWAP^{1/2}) gate.^{5} If two separable singlequbit states ϕ〉 and ψ〉 enter modes 1 and 2, respectively, the gate creates an entangled superposition of these states being swapped and not swapped, i.e.,
where the subscripts indicate the mode. As illustrated in Fig. 1a, such an operation can be achieved by sending two photons to a 50:50 beamsplitter (BS): The gate succeeds when the two photons exit in different modes, generating the state of Eq. (1), while, half of the time, the gate will fail by allowing both photons to exit the same mode (in reality, the situation is even more complicated because of twoparticle interference effects and the logical qubit encoding).
If the SWAP process is made continuous by replacing the 50:50 beamsplitter with coupled waveguides, the quantum Zeno effect^{6} (wherein continuous measurement prevents a quantum system from evolving), can boost the success probability of the gate to 100%.^{4} In this scenario, however, the quantum Zeno effect requires nonlinear twophoton absorption to occur at the singlephotonlevel. To date, such a strong optical nonlinearity has only been achieved via complex interactions with atomic systems,^{7} which lack scalability.
Plasmonpolaritons, formed when light hybridizes with the collective chargecarrier density oscillations in conducting materials, confine electromagnetic energy to deeplysubwavelength scales, and could potentially enable extremely strong optical nonlinearities in nanoscale photonic circuits^{8}—an ideal situation for a scalable quantum logic gate. While plasmons supported by noble metals provide large nonlinear enhancements and are compatible with singlephotonlevel quantum experiments,^{9,10} they suffer from intrinsically high ohmic losses, severely limiting their application to quantum technologies.
Graphene has recently arisen as a robust material platform for plasmonics, capable of sustaining plasmon resonances with extremely long lifetimes^{11,12} that can be tuned actively via electrostatic gating.^{13} Furthermore, its lowdimensionality provides unprecedented levels of optical field confinement,^{14} boosting optical nonlinearities well above those in noble metals,^{15,16,17,18} potentially enabling nonlinearities on the single or fewplasmon level.^{19,20} Here we propose that this system can be used to implement a twoqubit quantum logic gate using nanoplasmonic graphene waveguides.
We will use the socalled singlerail encoding, just as in the original Zenogate proposal,^{4} where the absence of a particle represents a logical 0, and the presence of a particle a logical 1. In other words, 0〉 (1〉) in the Fock basis represents a logical 0〉 (1〉) state of the qubit. Higherorder Fock states fall out of this logical subspace. Although the singlerail encoding has limitations,^{21} it can be transformed into the more wellknown dualrail encoding with linear optical elements.^{22}
Implementing the SWAP^{1/2} gate with a BS is not straightforward (Fig. 2a): If the logical input state is 00〉, 01〉, or 10〉 (encoded by no particles in either mode, or one particle in the first or second, respectively), the gate functions perfectly. In contrast, when one particle is incident in each mode (a logical state 11〉) the correct output is 11〉. Unfortunately, the HongOuMandel (HOM) effect, already observed for single plasmons,^{9,10,23} causes the particles to bunch and exit in the same mode, implying that the gate always fails. Since the HOM effect is independent of the relative phase between the two modes, this holds in general. Even if the particles are made indistinguishable, to circumvent HOM bunching, the gate fails 50% of the times (see Fig. 1a).
In a Zeno gate, the swap between the two modes has to be a continuous process, so that a “Zeno measurement” can be applied as the system evolves. Such a continuous swap can be achieved with a directional coupler (DC). To prevent the system from evolving into a state in which both particles are in the same mode, one must continuously monitor whether both particles are in the same mode. In practice, the presence of a sufficiently strong twophoton absorber can perform this measurement.^{4} At first glance, it appears that in such a DC, when the particles bunch into the same mode, they would be absorbed. However, when the swap probability is much smaller than the twoparticle absorption, the Zeno effect does not even allow the particles to bunch in the first place.
In graphene, this Zeno condition can be easily achieved. When a single plasmon has less energy than the Fermi level, it is not absorbed via electronhole pair excitation. At the same time, a mode containing two plasmon quanta can have enough energy to be absorbed via an interband transition (Fig. 1b). Since the twoplasmon absorption depends on the field strength while the singleplasmon absorption does not, confining the graphene plasmon field to a nanostructure enhances the twoplasmon absorption rate, while leaving the singleplasmon absorption rate unaffected^{20} (Fig. 1c).
Results
As a physical realization of such a graphenebased quantum gate, we envision a system of two graphene nanoribbons that support propagating single plasmons (see Fig. 2). In this work we will assume that the single plasmons are already excited, which could, in principle, be achieved through the emission of a quantum light source.^{24,25,26,27} The two nanoribbons are brought close to each other, so that the plasmons are coupled via Coloumb interaction, forming a graphene plasmon DC, whereby a plasmon starting in one ribbon can couple to the other ribbon. The interaction length, the ribbon width, and the ribbon spacing set the splitting ratio of the DC. At the same time, the ribbon width and the Fermi energy of the nanoribbons determine the twoplasmon absorption rate.
To model this system, we describe each ribbon as a twolevel system with energy ℏω, where ω is the resonant plasmon frequency that depends on the nanoribbon width W and doping level (Fermi energy) E_{F}. As shown in Fig. 2c, we consider a maximum of two plasmons, limiting the Hilbert space to six states. States with an equal number of plasmons are coupled via a Coulomb interaction of strength U. Decay processes are governed by inelastic scattering rate γ, and γ^{(2)} denotes the twoplasmon absorption rate.
We quantify the Coulomb interaction by describing plasmons in semiinfinite graphene nanoribbons within the socalled plasmon wave function (PWF) formalism,^{28} adapted here to include the effect of a nonvanishing optical wave vector k_{} in the direction of the ribbon transversal symmetry. Setting the nanoribbons to be aligned horizontally, and separated by a distance d_{z} in the zdirection (see Fig. 2a), the interaction between N plasmons in one ribbon and N′ plasmons in the neighboring one, both of them propagating with parallel wave vector k_{}, is given by
where the integrals are evaluated over the nanoribbons in a 2D space R = (x, y) and \(\rho _{k_{},N}^{{\mathrm{ind}}}({\mathbf{R}},\omega )\) is the induced charge associated with N plasmons (see Methods and Fig. S1).
Next, we compute γ^{(2)} from the nonlinear conductivity \(\sigma _\omega ^{(3)}\), for which an analytical expression in the local and zerotemperature approximation is obtained quantummechanically in the Dirac cone approximation and reported in ref. ^{29}. As shown in the Methods, the twoplasmon absorption rate is given by
where \(\beta _{q,1}^{(2)}\) and \(\beta _{q,1}^{(4)}\) are the momentumdependent field normalizations, which we consider to be unity for low momentum values. Here Δ characterizes the spatial extent of the propagating plasmon along the direction of transversal symmetry, which we set to be equal to the ribbon width. We set the singleplasmon lifetime to be γ = 500 fs^{−1}, which is a realistic value, measured at room temperature.^{11} Note that this lifetime can be extended by going to cryogenic temperatures; for which lifetimes up to 10 ps have recently been measured.^{12}
We can now calculate the density matrix ρ(t) of the system by solving the timedependent Lindblad master equation, which is the most general type of Markovian and timehomogeneous master equation describing an openquantumsystem evolution that is both tracepreserving and completely positive for any initial condition^{30}
where γ^{(1)} ≡ γ, \(a_m^\dagger\) (a_{m}) denote plasmon creation (annihilation) operators, n is the number of absorbed plasmons and m is the nanoribbon mode. The Hamiltonian of the twonanoribbon system is
where U is the Coulomb interaction given in Eq. (2), while ω is the plasmon frequency of each nanoribbon mode.
We numerically solve Eq. (4) using Mathematica, from which we obtain the required time \(t_{{\mathrm{SWAP}}^{1/2}}\) at which a single plasmon incident in either nanoribbon is placed in an equal superposition of both nanoribbon modes at the output. This time is related to the Coulomb interaction U from Eq. (2) (i.e. stronger Coulomb interaction U resulting in shorter \(t_{{\mathrm{SWAP}}^{1/2}}\)). To calculate \(t_{{\mathrm{SWAP}}^{1/2}}\) we define our initial state to be ρ(t = 0) = ψ_{i}〉〈ψ_{i}, where ψ_{i}〉 = 1〉_{1}0〉_{2}, and let it evolve until the probability of the plasmon being in either of the modes is equal: \(P_{\left {10} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}}) = P_{\left {01} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}})\). We convert this time to a length \(L_{{\mathrm{SWAP}}^{1/2}}\), by computing the plasmon group velocity as shown in Fig. S2. The resulting \(L_{{\mathrm{SWAP}}^{1/2}}\) is plotted in Fig. 3a. For E_{F} > 0.1 eV the required \(L_{{\mathrm{SWAP}}^{1/2}}\) is always less than the singleplasmon decay length, thus showing the potential of longlived graphene plasmons: novel physical effects can manifest before the plasmon decays.
For all the results presented here, we set the spacing between the two nanoribbons to d_{z} = 1 nm. With current technology, such atomically thin spacings can be realized by taking advantage of 2D materials like graphene.^{31} This parameter only affects the Coulomb interaction strength, which will determine \(L_{{\mathrm{SWAP}}^{1/2}}\). The PWF used in our calculations is applicable for these scales, as discussed in detail in ref. ^{28}. Furthermore, for our parameter regime, the Coulomb interaction does not depend very strongly on d_{z} (see Fig. S4 of the Supplementary Information).
Once \(L_{{\mathrm{SWAP}}^{1/2}}\) is determined, we proceed to analyze the system when a single plasmon is input in each mode; that is, ρ(t = 0) = ψ_{i}〉〈ψ_{i} where ψ_{i}〉 = 1〉_{1}1〉_{2}. For this input, the gate functions correctly if there is still one plasmon in each output mode, which occurs with probability \(P_{\left {11} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}})\).
Discussion
In Fig. 3a–c we show the success probability \(P_{\left {11} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}})\), the probability of the plasmons bunching in the same nanoribbon \(P_{\left {20} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}}) + P_{\left {02} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}})\), and the probability for both plasmons to decay \(P_{\left {00} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}})\), for a range of nanoribbon widths W and Fermi energies E_{F}. Notice the similarity of the contour features between these figures and the γ^{(2)}/γ ratio shown in Fig. 1c. In the upper right corner the twoplasmon absorption is much weaker than the singleplasmon absorption, leading to a very weak Zeno effect, so the HOM effect prevails: that is, \(P_{\left {20} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}}) + P_{\left {02} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}}) \gg P_{\left {11} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}})\).
As we decrease both W and E_{F}, γ^{(2)} increases, but not enough to drive a noticeable Zeno effect. Instead, both of the plasmons are likely to be absorbed, which is reflected in \(P_{\left {00} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}}) \gg P_{\left {20} \right\rangle } + P_{\left {02} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}})\).
In the region where \(\gamma ^{(2)}/\gamma \sim 10^4  10^6\), a strong Zeno effect can be realized (light blue area of Fig. 1c). This leads to a large increase in the success probability \(P_{\left {11} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}})\), while \(P_{\left {20} \right\rangle } + P_{\left {02} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}})\) becomes negligible, meaning that the Zeno effect completely suppresses the HOM effect. Despite the large γ^{(2)}, \(P_{\left {00} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}})\) shows a minimum when \(\gamma ^{(2)} \gg \gamma\). In this optimal region, we find a maximum success probability of 87.0% for W = 5 nm and E_{F} = 0.335 eV, which is an increase in the success probability of the SWAP^{1/2} gate from 0 to 87.0%. This already places us well above the gate success probability rate required to generate universal cluster states for quantum computation.^{32} This performance is limited by the single plasmon lifetime. In Fig. 3e we plot the success probability, maximized over the range of W and E_{F} shown in pannels ad, versus the plasmon lifetime given by 1/γ. For lifetimes longer than 7.5 ps the success probability increases above 99%, reaching faulttolerance regimes for surface codes.^{33} Nevertheless, edge imperfections and structural defects would decrease the plasmon lifetime and thus the fidelity of the gate. The predicted nonlinearities, nevertheless, should persist in their presence.
Since singleplasmon decay can also result in logical states changing into other logical states, this process fidelity will be further decreased. To quantify this, we evaluated the process fidelity^{34,35} of our gate by simulating process tomography for the complete range of W and E_{F} under consideration (see Methods). The resulting process matrix for W =5 nm and E_{F} = 0.335 eV with a lifetime of 500 fs is plotted in Fig. 4, and has a fidelity of 93.3%. When the lifetime is increased to 10 ps, the fidelity is 99.6%.
Our proposed gate achieves process fidelities in the faulttolerance regime for relatively reasonable physical parameters. Doping levels as high as 1–2 eV have been achieved,^{36,37} nanoribbon widths in the range of 10–40 nm have been constructed using different means,^{31,38,39,40} and separation distances ≈1 nm are routinely achieved through singleatomic hexagonal boron nitride spacers, which additionally guarantees the preservation of highquality graphene optical response.^{31} Furthermore, by combining ideas from quantum optics with nanoplasmonics, our work opens up an entirely new and promising avenue in the search for singlephoton nonlinearities. While we have studied the application of graphene nanoplasmonics to a quantum logic gate, this could also be used for deterministic optical implementations of quantum teleportation,^{41} cluster state generation,^{42} and singlephoton sources,^{19} underlining the applicability of this platform.
Methods
Classical electrostatic description of plasmons in graphene nanoribbons
We consider a single graphene nanoribbon occupying the R = (x, y) plane that has a finite width W in the xdirection and is infinitelyextended along the ydirection. In the linear approximation, following refs., ^{19,23} the selfconsistent electric field within the ribbon E_{q} produced by an impinging field \({\mathbf{E}}^{{\mathrm{e}}xt}({\mathbf{R}},t) = {\mathbf{E}}_q^{{\mathrm{ext}}}e^{i(k_yy  \omega t)} + {\mathrm{c}}.{\mathrm{c}}.\), i.e., having frequency ω and momentum k_{y} ≡ q/W along y, is given by
where \(\varepsilon _\omega ^{{\mathrm{ab}}} = \left( {\varepsilon _\omega ^{\mathrm{a}} + \varepsilon _\omega ^{\mathrm{b}}} \right)/2\) is the average of the dielectric functions describing media above (\((\varepsilon _\omega ^{\mathrm{a}})\)) and below \((\varepsilon _\omega ^{\mathrm{b}})\) the 2D layer and \(\rho _q^{{\mathrm{ind}}}({\mathbf{R}},\omega )\) is the induced charge. From the continuity equation, we express \(\rho _q^{{\mathrm{ind}}}\) in terms of the local, linear 2D graphene conductivity \(\sigma _\omega ^{(1)}\) as
where we have introduced the occupation factor f_{R}, which is equal to one when −W/2 ≤ x ≤ W/2 and is vanishingly small everywhere else. In practice, we employ the optical conductivity obtained for zero temperature in the local limit (i.e., for vanishing inplane optical momentum) of the randomphase approximation (RPA) as^{18}
where the Fermi energy E_{F} is related to the graphene Fermi velocity v_{F} ≈ c/300, doping chargecarrier density n according to \(E_{\mathrm{F}} = \hbar v_{\mathrm{F}}\sqrt {\pi n}\) and τ = 1/γ is a phenomenological inelastic scattering rate. The first and second terms in Eq. (8) describe the optical response arising from intraband and interband electronic transitions, respectively, with the latter becoming unimportant when \(E_{\mathrm{F}} \lesssim \omega\).^{13} Incidentally, we have neglected inelastic damping in the interband transitions. In terms of normalized coordinates \(\vec \theta \equiv {\mathbf{R}}/W\) and the normalized electric field \(\vec \varepsilon _q(\vec \theta ,\omega ) \equiv W\sqrt {f_{\vec \theta }} {\mathbf{E}}_q(\vec \theta ,\omega )\), Eq. (6) can be expressed as
where \(\eta _\omega ^{(1)} \equiv {\mathrm{i}}\sigma _\omega ^{(1)}/\varepsilon _\omega ^{{\mathrm{ab}}}\omega W\) is a dimensionless parameter characterizing the intrinsic linear optical response of graphene, and
which we identify as a real, symmetric operator that admits a complete set of real eigenvalues. The electric field of Eq. (9) is expanded in eigenmodes of the matrix \(M(\vec \theta ,\vec \theta^\prime )\) as
where the modes \(\vec \varepsilon _{q,m}(\theta _x)e^{{\mathrm{i}}q\theta _y}\) and their corresponding eigenvalues η_{q,m} satisfy
and form an orthonormal set
Inserting Eq. (11) into Eq. (9), we make use of Eqs. (12 and 13) to write \(a_m = b_{q,m}(1  \eta _\omega ^{(1)}/\eta _{q,m})^{  1}\), where
is a coefficient that depends on the form of the external field. In what follows we take \({\mathbf{E}}_q^{{\mathrm{ext}}}\) to be independent of x, so we may write \(b_{q,m}(\theta _y) =  W{\mathbf{E}}_q^{{\mathrm{ext}}} \cdot \vec \xi _{q,m}^{\,\ast} c_q(\theta _y)\), where c_{q}(θ_{y}) contains the ydependence of the external field and \(\vec \xi _{q,m} \equiv  {\int} d \theta _x\vec \varepsilon _{q,m}(\theta _x)\), so that the normalized electric field in Eq. (11) becomes
Electrostatic energy in nanoribbons
The electrostatic energy for identical, parallel ribbons separated by a distance d_{z} in the zdirection is given by
where, from Eq. (7) (taking \(\varepsilon _\omega ^{{\mathrm{ab}}} = 1\) for simplicity), we can express the induced charge in ribbon l as
Inserting the above expression into Eq. (16) and making use of Eq. (15), the electrostatic energy becomes
where
and L → ∞ is the nanoribbon length.
Assuming a plane wave field profile along the ydirection corresponding to \(c_q(\theta _y) = e^{{\mathrm{i}}q\theta _y}\), in a single ribbon (i.e., taking l = l′ and d_{z} = 0), the use of Eqs. (10) and (13) yields I_{q,mm}(0) = −Lδ_{mm′}/Wη_{q,m}, and so the electrostatic energy per unit length in ribbon l is
In practice, we restrict our study to the lowestorder m = 1 mode, and fix the number of plasmon quanta in this mode using the condition \(l\hbar \omega _{\mathrm{p}} = 2\Delta \tilde U_{q,l}\), where Δ is an effective length for the plasmon mode along the ribbon (i.e., the characteristic spatial width of a pulse), leading to
where it is now understood that the indices l and l′ denote the number of plasmons in the first and second ribbon, respectively. Using the above condition, the coupling energy between ribbons containing l and l′ plasmons is obtained directly from Eq. (18), again considering only the m = m′ = 1 contribution.
Plasmon normalization
We normalize the electric field amplitude of the plasmon mode by equating the absorbed and dissipated power at linear order, i.e.,
where l is the number of plasmon quanta, \({\mathbf{j}}_q^{(1)}({\mathbf{R}},\omega ) = \sigma _\omega ^{(1)}{\mathbf{E}}_q({\mathbf{R}},\omega )\), and 〈...〉 denote the timeaverage. Using the result of Eq. (15) with only the m = 1 mode, we obtain
where \(\beta _{q,1}^{(n)} = {\int}_{  1/2}^{1/2} d \theta _x\vec \varepsilon _{q,1}(\theta _x)^n\). For a mode defined as a planewave along the ribbon, such that \(c_y(\theta _y) = e^{{\mathrm{i}}q\theta _y}\) within an effective length Δ, we write the normalization condition for N plasmons as
Twoplasmon absorption rate
Power absorption in a nanoribbon via twoplasmon absorption arises from the nonlinear current \({\mathbf{j}}^{(3)}({\mathbf{R}},t) = j_q^{(3)}({\mathbf{R}},\omega )e^{{\mathrm{i}}k_yy  {\mathrm{i}}\omega t} + {\mathrm{c}}.c.\), where \({\mathbf{j}}_q^{(3)}({\mathbf{R}},\omega ) = \sigma _\omega ^{(3)}\left {{\mathbf{E}}_q({\mathbf{R}},\omega )} \right^2{\mathbf{E}}_q({\mathbf{R}},\omega )\), and is given by
where \({\mathbf{j}}_q^{(3)}({\mathbf{R}},\omega ) = \sigma _\omega ^{(3)}\left {{\mathbf{E}}_q({\mathbf{R}},\omega )} \right^2{\mathbf{E}}_q({\mathbf{R}},\omega )\) and \(\sigma _\omega ^{(3)}\) is the local thirdorder conductivity of extended graphene, for which we adopt the analytical result obtained quantummechanically at zero temperature in the Dirac cone approximation, as reported in ref. ^{29} Using Eq. (15) we write the timeaverage of the absorbed power per unit length as
Equating 〈P_{TPA}〉 with the power dissipated by twoplasmon absorption, 2ℏωγ^{(2)}, we make use of the field normalization condition in Eq. (24) to write the twoplasmon absorption rate for a ribbon containing l = 2 plasmons in the m = 1 mode as
In obtaining the above expression, we have again chosen the field along the ribbon to have the form of a planewave (i.e., \(c_y(\theta _y) = e^{{\mathrm{i}}q\theta _y}\)), and an effective length Δ.
Process tomography
We send a complete set of 16 twoqubit states through our simulation and compute the output states at \(t_{{\mathrm{SWAP}}^{1/2}}\). To deal with failure events, when 2〉_{1}0〉_{2} and 0〉_{1}2〉_{2} terms arise in the output states, we truncate the output density matrix and renormalize the result. Such events only occur when states involving two plasmons are input. We also numerically correct for local singlequbit phases which arise in the output of the simulation. We feed these output states in a leastsquares process tomography routine, generating a process matrix χ_{sim}. This process matrix is defined as,
where ρ_{in(out)} is the input (output) density matrix, and E_{i} are the basis operators constructed from the Kronecker product of the Pauli matrices (labels of Fig. 4. We calculate the process fidelity between these, and the ideal process (given by Eq. (10) of ref. ^{12} as Tr{χ_{sim}χ_{ideal}}.^{34,35}
Numerical solution of the linblad master equation
We use the Lindblad equation introduced in Eq. (4) to describe and solve the density matrix of our system. The first term of the Lindblad equation contains the Hamiltonian given in Eq. (5). This Hamiltonian describes the two identical graphene nanoribbons as a twolevel system, where the coupling between the levels is given by the Coulomb interaction U. We define a 6state Hilbert space that contains a vacuum state (0〉_{1}0〉_{2}), two singleplasmon states (1〉_{1}0〉_{2}, 0〉_{1}1〉_{2}) and three twoplasmons states (1〉_{1}1〉_{2}, 2〉_{1}0〉_{2}, 0〉_{1}2〉_{2}). In this basis, the matrix form of the Hamiltonian is
where ℏω is the energy of the plasmon. The second term of the Lindblad equation contains the loss channels of the system; namely, the singleplasmon absorption γ^{(1)} and the twoplasmon absorption γ^{(2)}. In matrix form, this second term reduces to
where \(\rho _{ijkl} = \left i \right\rangle _1\left j \right\rangle _2\hat \rho \left k \right\rangle _1\left l \right\rangle _2\). So as to obtain the timedependent density matrix of the system, we numerically solve the system of ordinary differential equations in Wolfram Mathematica. We employ the variable stepsize implicit Backward Differentiation Formulas (BDF) or order 5. The WorkingPrecision used in this algorithm was set to the MachinePrecision, which, in our case, corresponds to 16 digits. In addition, the AccuracyGoal and PrecisionGoal options are set to 10. The diagonal elements of this density matrix exactly correspond to the probability of the plasmons being in different modes. For example, ρ_{1111}(t) is the probability that one plasmon is found in each nanoribbon at a given time, ρ_{2020}(t) + ρ_{0202}(t) is the probability that two plasmons are found in a single nanoribbon at a given time, and ρ_{0000}(t) is the probability of not having any plasmon in the system at a given time.
Once the density matrix of our system is found, we proceed to find the required interaction time between the nanoribbons to implement a SWAP^{1/2}. To do so, we set our initial state to be ρ(t = 0) = ψ_{i}〉〈ψ_{i}, where ψ_{i}〉 = 1〉_{1}0〉_{2}, let it evolve in time and find \(t_{{\mathrm{SWAP}}^{1/2}}\) by looking for the time at which the probability of the plasmon being in either of the modes is equal; i.e., \(P_{\left {10} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}}) = P_{\left {01} \right\rangle }(t_{{\mathrm{SWAP}}^{1/2}})\). The solution to this condition was found numerically using Wolfram Mathematica with a minimum accuracy and precision of 10 digits. Once \(t_{{\mathrm{SWAP}}^{1/2}}\) is determined, we define our initial state to be ρ(t = 0) = ψ_{i}〉〈ψ_{i}, where ψ_{i}〉 = 1〉_{1}1〉_{2}, and find the success probability of the gate P_{11} at time \(t_{{\mathrm{SWAP}}^{1/2}}\). Representative timedependent densitymatrix elements are plotted in Fig. S6 in the Supplementary Information.
Data availability
The datasets generated and analysed during the current study are available from the corresponding author if you ask nicely.
Change history
30 October 2019
An amendment to this paper has been published and can be accessed via a link at the top of the paper.
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Acknowledgements
I.A.C. acknowledges support from the University of Vienna via the Vienna Doctoral School. L.A.R. acknowledges support from the Templeton World Charity Foundation (fellowship no. TWCF0194). M.R. acknowledges support from the European Commission through the project QUCHIP (grant no. 641039). P.W. acknowledges support from the European Commission through ErBeSta (No. 800942), the Austrian Research Promotion Agency (FFG) through the QuantERA ERANET Cofund project HiPhoP (no. 731473), from the Austrian Science Fund (FWF) through CoQuS (W 1210N25), NaMuG (P 30067N36) and BeyondC (F 7113N38), the U.S. Air Force Office of Scientific Research (FA2386233 1714011), and Red Bull GmbH. J.D.C., J.R.M.S., and F.J.G.A., acknowledge support from the European Research Council (Advanced Grant No. 789104eNANO), the Spanish MINECO (MAT201788492R and SEV20150522), the European Commission (Graphene Flagship 696656), the Catalan CERCA, and Fundació Privada Cellex. We thank T. Rögelsperger for his artistic input.
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I.A.C., L.A.R., and P.W. proposed the design of the coupled nanoribbons and performed the numerical calculations. I.A.C., and M.R. derived the master equation for our system. J.D.C., J.R.M.S., and F.J.G.A. derived analytical expressions for the graphene parameters. I.A.C., J.D.C., M.R., and J.R.M.S. wrote the simulation code. I.A.C., J.D.C., and L.A.R. wrote the manuscript. All authors read and commented on the manuscript.
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Alonso Calafell, I., Cox, J.D., Radonjić, M. et al. Quantum computing with graphene plasmons. npj Quantum Inf 5, 37 (2019). https://doi.org/10.1038/s4153401901502
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