Implementation of a Toffoli gate using an array of coupled cavities in a single step

The Toffoli gate (controlled-controlled-NOT gate) is one typical three-qubit gate, it plus a Hadamard gate form a universal set of gates in quantum computation. We present an efficient method to implement the Toffoli gate using an array of coupled cavities with one three-level atom in each cavity. The large detuning between atoms and classical (quantum) fields plays an important role and the gate is implemented in one-step. The quantum information is encoded into the low-lying states of identical atoms and it is convenient to address qubit individually. Based on the Markovian master equation, it is shown that the scheme to implement the Toffoli gate is robust against the decoherence.

toff where the target qubit swaps its information | 〉 ⇔ | 〉 0 1 3 3 if and only if two control qubits are in |01〉 12 . Note that it is equivalent to the standard form of a Toffoli gate upon a local unitary transformation. Coupled cavity arrays describes a series of optical cavities, each of which contains one or more qubits or atoms, and photons can hop between two neighboring cavities. This model can overcome the problem of individual addressability and has emerged as a fascinating alternative for simulating quantum many-body phenomena. Theoretical works on quantum information processing and quantum computing have been proposed with using the atom-cavity interaction in coupled cavity arrays [33][34][35][36][37][38][39][40][41] . The merit of our scheme is that the Toffoli gate is implemented in one-step without any single-qubit or two-qubit operation, which can significantly simplify the experimental realization and shorten the operation time. Meanwhile, it is easy to control and measure qubit separately because there is one three-level atom in each cavity. Furthermore, we encode the quantum information into the low-lying states of three identical atoms without any ancillary level compared with ref. 42 .

Results
In Fig. 1 we consider three coupled cavities with one three-level atom in each cavity. The k-th (k = 1, 2, 3) atom has two ground states |0 k 〉 and |1 k 〉 and one excited state |e k 〉 with energies ω a , ω b and ω e , respectively. Each | 〉 ↔ | 〉 e 0 k k transition is coupled to its corresponding cavity mode with the coupling strength g k , detuned by Δ. Meanwhile, the transitions | 〉 ↔ | 〉 e 0 3 3 and | 〉 ↔ | 〉 e 1 3 3 for the target atom are driven by a pair of classical fields with the Rabi frequencies Ω a and Ω b respectively, detuned by the same parameter Δ. In addition, the cavities are coupled via the exchange of photons with the coupling constant J. The system Hamiltonian takes the following form (ħ = 1)  where † a a ( ) j j is the annihilation (creation) operator of the j-th cavity mode, ω l 1 and ω l 2 are the frequencies of two classical fields, and ω c is the frequency of the cavity. By changing to the interaction picture, and performing a rotation with the frame defined by = Δ ∑ | 〉〈 | = U i t e e exp ( )  where we have assumed g k = g for simplicity.
In the following, we will discuss the scheme to implement the Toffoli gate based on the large detuning case. Here we consider that the two classical optical pumping lasers are both sufficiently weak (i.e. the Rabi frequencies Ω a and Ω b are both very small compared with {J, g, Δ}), and the excited states of the atoms and the excited cavity field modes are not initially populated, the highly excited level can be neglected [33][34][35]43,44 . Based on the interaction form of the Hamiltonian (4), the qubit basis {|0〉 1 , |1〉 1 , |0〉 2 , |1〉 2 , |0〉 3 , |1〉 3 } a with cavities in vacuum states can be divided into four subspaces. For the first subspace

we first diagonalize the strong interaction described by atom-cavity Hamiltonian in Eq. (4). Based on the new basis
(1) the atom-cavity Hamiltonian reads: The effective Hamiltonian in the first subspace can be evaluated explicitly (see Methods) Similar to the analysis of the first subspace, we consider the second subspace (2) the atom-cavity Hamiltonian reads: The effective Hamiltonian in the second subspace reduces to (see Methods) [ 000 000 001 001 ( 000 001 100 000 )] , which means that the qubit states |000〉 and |001〉 remain unchanged during the whole evolution time. For the atom-cavity Hamiltonian is given by:  3 The effective Hamiltonian in the third subspace is (see Methods) the atom-cavity Hamiltonian reads: The effective Hamiltonian in the fourth subspace is (see Methods) where m = Ω 2 /Δ with the parameters Ω b = −Ω a = Ω. Adjust the evolution period T = πΔ/Ω 2 , we obtain the three-qubit Toffoli gate which takes the form of Eq. (1).
In what follows, we check the accuracy of the effective Hamiltonian compared to the original Hamiltonian with the populations of three qubit states {|000〉, |001〉, |010〉, |011〉, |100〉, |101〉, |110〉, |111〉} a |000〉 c . In Fig. 2, (a) plots the conversion of states |010〉 a |000〉 c and |011〉 a |000〉 c when the system reserves the single excitation. The population can achieve 0.9981 at the period time. (b) depicts the populations of states |000〉 a |000〉 c , |001〉 a |000〉 c , |100〉 a |000〉 c , |101〉 a |000〉 c , |110〉 a |000〉 c and |111〉 a |000〉 c for the system reserving the single excitation. The minimum data of the population is 0.9938 during the evolution time. Since the system do not conserve the total number of excitations, and in front we neglect the highly excited level under the weak excitation case, here we further consider the numerical simulation for the system reserving the double excitation in (c) and (d). Compare plots (c) with (a), (d) with (b), it is found that the results for the double excitation case are in accord with the results for the single excitation case. These numerical results reveal that the effective Hamiltonian is excellently close to the original Hamiltonian under the given parameters. To make our results more clearly, Fig. 3 gives the truth table of the Toffoli gate at the period time for the single excitation case. The fidelity for the Toffoli gate in the ideal case is , with U(T) being the final evolution operator based on the original Hamiltonian (4) and U Toffoli being the ideal Toffoli gate. Thus a Toffoli gate is implemented with high fidelity. Furthermore, we numerically discuss the case that the atom-cavity coupling strengths g 1 , g 2 and g 3 are different with g 1 = g + δ, g 2 = g − δ and g 3 = g, the Toffili gate can be implemented as well, as shown in Fig. 4. When the parameter δ ≤ 0.2, the fidelity contains higher than 95%.

Discussion
In the coupled-cavity arrays, the main decoherence effects in our scheme are the decay of cavities and the spontaneous emission of atoms. In this section, we numerically show how the decay of cavities and the spontaneous emission of atoms affect the fidelity of the resulting gate. The master equation for the whole system in the Markov approximation is governed by the following Lindblad equation 45 :  where κ represents the cavity decay rate, γ j el denotes the spontaneous emission rate of atoms from the level |e〉 j to |l〉 j for the j-th atom (j = 1, 2, 3) and we assume γ for convenience. To quantify the robustness of our logical gate, we adopt the gate fidelity defined as the Bures-Uhlmann fidelity   where ρ(t) is the mixed output system state (obtained from the joint system-bath evolution after a partial trace over the bath) and ρ id is the density operator for target state. Here we choose the initial state as . The corresponding density operator for target state is ρ = |Ψ′〉〈Ψ′| id , with the target state |Ψ′〉 = | 〉 + | 〉 − | 〉 + | 〉+ ( 000 001 010 011 1 8 | 〉 + | 〉 + | 〉 + | 〉 | 〉 100 101 110 111 ) 000 a c . In Fig. 5 we depict the fidelity F of the Toffoli gate for the large detuning model as a function of the decoherence parameter κ/g and interaction time t/T. The fidelity F remains higher than 91%, which shows the Toffoli gate is robust against decoherence. Recently, the coupled cavity arrays can be constructed in several kinds of physical systems, such as photonic crystal defects 46 , toroidal microcavity arrays 47 , and superconducting stripline resonators 48 . Ref. 47 investigated the suitability of toroidal microcavities for strong-coupling cavity quantum electrodynamics with the parameters π ∼ × g 2 750 MHz, γ π ∼ × . 2 262 MHz, κ π ∼ × . 2 35 MHz. And ref. 49 has shown the large-scale ultrahigh-Q coupled nanocavity arrays based on photonic crystals corresponding to the  parameters ∼ . × g 2 5 10 9 Hz, γ ∼ . × 1 6 10 7 Hz, κ ∼ × 4 10 5 Hz. The fidelity of the Toffoli gate can achieve 95.43% and 98.14% for the above two different kinds of parameters (g, γ, κ), respectively. In the multi-qubit quantum computing networks the fidelities are relatively high.
In summary, we have proposed an efficient method to implement the Toffoli gate using an array of coupled cavities with one three-level atom in each cavity. The large detuning between atoms and classical (quantum) fields plays an important role. The Toffoli gate is implemented in one-step without any single-qubit or two-qubit operation, which can significantly simplify the experimental realization and shorten the operation time. Meanwhile, it is easy to control and measure qubit separately because there is one three-level atom in each cavity. Furthermore, we encode the quantum information into the low-lying states of three identical atoms without any ancillary level.

Methods
The effective Hamiltonian in the first subspace. In the case that the two classical optical pumping lasers are both sufficiently weak (i.e. the Rabi frequencies Ω a and Ω b are both very small compared with {J, g, Δ}), and the excited states are not initially populated, the excited states of the atoms and the excited cavity field modes can be adiabatically eliminated. The resulting effective dynamics will describe three two-level systems. To second order in perturbation theory, the dynamics are then given by the effective operators 44 :  where η = ± Δ ± g J 2 2 . Based on Eq. (20), the effective Hamiltonian in the second subspace is given by