Two-qubit quantum gate and entanglement protected by circulant symmetry

We propose a method for the realization of the two-qubit quantum Fourier transform (QFT) using a Hamiltonian which possesses the circulant symmetry. Importantly, the eigenvectors of the circulant matrices are the Fourier modes and do not depend on the magnitude of the Hamiltonian elements as long as the circulant symmetry is preserved. The QFT implementation relies on the adiabatic transition from each of the spin product states to the respective quantum Fourier superposition states. We show that in ion traps one can obtain a Hamiltonian with the circulant symmetry by tuning the spin-spin interaction between the trapped ions. We present numerical results which demonstrate that very high fidelity can be obtained with realistic experimental resources. We also describe how the gate can be accelerated by using a “shortcut-to-adiabaticity” field.

We discuss the physical implementation of our gate scheme in a linear ion crystal driven by bichromatic laser fields. Such an interaction creates a coupling between the internal states of the trapped ions with the collective vibrational modes. We consider the dispersive regime in which the beatnote laser frequency is far off-resonant to any vibrational mode frequency. In this regime the collective phonons can be traced out leading to an effective spin-spin interaction. Such a regime where the phonons are only virtually excited was studied in the context of high-fidelity two-qubit gate implementation [21][22][23] . We show that by controlling the laser detuning we can perform the desired adiabatic evolution to the quantum Fourier modes.
The paper is organized as follows. In Sec. II we provide the general framework of the circulant-symmetric spin-spin Hamiltonian. In Sec. III we discuss the adiabatic transition to the quantum Fourier modes. The physical realization of the circulant Hamiltonian using a laser driven ion crystal is discussed in Sec. IV. In Sec. V we provide numerical estimation for the two-qubit gate fidelity as well as the fidelity for the creation of entangled states. Finally, the conclusions are presented in Sec. VII.

Model
We begin by considering two interacting spins which are subjected to a magnetic field. The Hamiltonian of the system is given by These four vectors (4a),(4b),(4c),(4d) are the columns of the 4 × 4 quantum Fourier transform matrix. Thus, by preparing the system in the eigenstates of the circulant Hamiltonian one can implement the two-qubit quantum Fourier transform. In order to fulfill the circulant cyclic permutation symmetry we consider two different cases. with p being integer. The first condition requires the spin-spin coupling to be equal to the Rabi frequency on the second spin. The circulant symmetry of the Hamiltonian (2) leaves arbitrariness in the choice of the Rabi frequency on the first spin. Here we have set to zero, Ω 1 = 0. Using this, the Hamiltonian (2) becomes a circulant matrix and can be rewritten as The same conditions as (5) but now with Again the Hamiltonian is circulant and can be expressed as We will show latter on that the additional Rabi frequency Ω 1 in the circulant Hamiltonian in Case 2 can be used to improve significantly the adiabatic evolution even when the spin-spin coupling J is rather small.

Adiabatic transition to fourier Modes
In order to implement the two-qubit Fourier transform we assume that additionally to the circulant Hamiltonian time-dependent frequency shifts are applied such that the total Hamiltonian becomes where Δ k (t) is the time-dependent detuning of the kth spin. Such a term is needed to control the adiabatic transition of the computational spin states to the quantum Fourier states (4a)(4b)(4c)(4d). Let us assume that initially the system is prepared in one of the computational product states ψ As long as at the initial moment t i the detuning Δ 1,2 (t i ) is mich higher than the couplings J(t i ), Ω 1 (t i ), i.e. Δ 1,2 (t i ) ≫ J(t i ), Ω 1 (t i ) the respective eigenstates of the Hamiltonian (9) coincide with the computational spin states, namely ψ Then we adiabatically decrease in time the detunings Δ 1 (t) and Δ 2 (t) to zero, while we increase the couplings J(t) and Ω 1 (t) such that in the end we have → . In the adiabatic limit, the system remains in the same eigenstate of the full Hamiltonian Ĥ t ( ) at all times. With the chosen time behavior of the couplings and the detunings, each such eigenstate is equal to a computational spin state (eigenstate of Ĥ 0 ) in the beginning, ( ) t f p (p = 0, 1, 2, 3). Hence the adiabatic evolution maps each computational spin state onto a Fourier state, thereby producing the quantum Fourier transform in a single interaction step.
The adiabatic evolution requires that the separation between the eigenefrequencies λ ± j ( ) and μ ± j ( ) of Ĥ t ( ) is larger at any instance of time than the nonadiabatic coupling between each pair of the eigenstates λ ± | ⟩ j ( ) and μ ± | ⟩ j ( ) For smoothly varying Hamiltonian parameters adiabatic evolution usually demands that the interaction duration T is large compared to the inverse of the smallest coupling or detuning implying large pulse areas and/or large detuning areas.

Case 1.
Let us consider the eigenspectrum of the total Hamiltonian (9). Consider first the circulant . We find that the eigenfrequencies of Ĥ t ( ) are which correspond to the eigenvectors λ ± | ⟩ (1) and μ ± | ⟩ (1) . Note that in order to drive the adiabatic transition we require that the eigenfrequencies are nondegenerate at any instance of time. Otherwise the system may evolve into a superposition of Fourier states which will spoil the gate implementation. Initially we begin with (2020) 10:5030 | https://doi.org/10.1038/s41598-020-61766-w www.nature.com/scientificreports www.nature.com/scientificreports/ (1) i 1i 2 i . Hence in order to have nondegenerate spectrum we require Δ 1 (t i ) ≠ Δ 2 (t i ). The eigenfrequencies will be equidistant if Δ At the final instance of time where Δ 1,2 (t f ) ≪ J(t f ) the Hamiltonian possesses circulant symmetry. At this final stage of the adiabatic transition the eigenfrequencies becomes ( 3 . As can be seen there exist a finite energy gap for any phase ϕ except for ϕ = nπ/4, (n = 0, ±1, ±2, …) where the spectrum becomes degenerate. The gaps are equal when ϕ = tan( ) 1 3 or 3, i.e. when ϕ π = ≈ . arctan(3) 0 3976 or ϕ π = ≈ . arctan(1/3) 0 1024 . To summarize, the conditions for the scheme to work in this case are Case 2. Alternatively, one can drive the adiabatic transition to the Fourier states using the circulant (2) . In order to get insight of the eigenfrequencies we set the phase to ϕ = π/4 which allows analytical treatment. We find We denote the corresponding instantaneous eigenvectors by λ ± | ⟩ (2) and μ ± | ⟩ (2) . Again initially we start with and resp e c t ively (2) i 1i 2 i . As in Case 1, the condition Δ 1 (t i ) ≠ Δ 2 (t i ) must be fulfilled in order to avoid degeneracy. Equidistant eigenfrequencies occur initially if Δ 1 (t i ) = 3Δ 2 (t i ) or Δ 2 (t i ) = 3Δ 1 (t i ). In the end, J(t f ), Ω 1 (t f ) ≫ Δ 1,2 (t f ), the system arrives in an eigenstate of the circulant Hamiltonian H cir (2) . For any value of ϕ the circulant eigenfrequencies at t f are given by ( 2 . Assuming that Ω 1 ≠ 2J we see that the spectrum is nondegenerate except for ϕ = nπ/2 with n being integer. transitions. Let us now discuss the set of transitions which realize the quantum Fourier transform. For concreteness we focus on the case with Ω 1 ≠ 0 and choose the phase ϕ = π/4, with eigenfrequencies (15a), (15b). Initially, each of the computational spin states coincide with the eigenvectors of the Hamiltonian (9), namely λ = ↑↑ , and μ = ↑↓ . The realization of the quantum Fourier transform relies on the adiabatic following of each of the instantaneous eigenvectors, (2) i f and ∫ β μ (2) i f are the global adiabatic phases which appear due to the adiabatic evolution. As we will show latter on by a proper choice of the detunings Δ 1,2 the adiabatic phases can be tuned to be α 2 = 2pπ and β 2 = 2mπ with p and m being integers. This choice realises the following gate Up to an additional phase factor −π/2 in the second column, the matrix (18) resembles the quantum Fourier transform for two qubits. This phase factor appears due to the determinant invariance during the adiabatic evolution, which imposes the requirement = π G det 1 4 .
Finally, we point out that if we replace ϕ = π/4 by ϕ = − π/4 then two of the circulant eigenfrequencies (16) ( 2) and hence the adiabatic following of the eigenstates implies that can be carried out even without the presence of energy offset described by Eq. (10). Indeed, let's set the phases in (1) to ϕ 2 = φ 2 = ϕ and φ 1 = 2pπ. Then we have ( The Hamiltonian (19) has no circulant symmetry because the condition J = Ω 2 is not fulfilled. However, the adiabatic transition to the Fourier modes can be carried out for example by varying in time the Rabi frequency Ω 2 (t). At the initial moment we begin with Ω 1 , Ω 2 (t i ) ≫ J such that the eigenstates are 1 / 2. Then, adiabatically decrease Ω 2 (t) such that at the final instance of time we have Ω 2 (t f ) = J. Adiabatically following the instantaneous eigenstates transform the initial states into the respective quantum Fourier states (see the Supplement for the derivation). In contrast to the gate realization with nonzero detuning, now the adiabatic transition is carried out between the initial rotating computation spin states and the quantum Fourier states. Finally, we point out that instantaneous eigenvectors of Hamiltonian (19) can be found exactly, which allows to combine the gate scheme with the shortcuts to adiabaticity technique (see the Supplement for more details).

physical implementation with trapped ions
The implementation of our gate scheme can be realized in various quantum optical systems, for example, including superconductiong qubits coupled to transmission lines 25 , as well as using color center in nanodiamonds coupled to carbon nanotubes 26 . Here we consider a trapped-ion realization of the circulant Hamiltonian. Consider a linear ion crystal which consists of N ions with mass M, aligned along the trap axis z with radial and axial trap frequencies Ω x , Ω z . The qubit system typically consists of two metastable levels ↑ | ⟩, ↓ | ⟩ of the trapped ion with energy difference Ω 0 . The small radial vibrations around the equilibrium positions are described by a set of collective vibrational modes with a Hamiltonian = ∑ Ω Ĥ â â † n n n n ph 27 . Here â † n , â n are the phonon creation and annihilation operators of the nth vibrational mode with a frequency Ω n . Including the internal energy of the qubits σ = ∑ Ω Ĥ k k z q 0 /2 the interaction-free Hamiltonian becomes = + Ĥ Ĥ Ĥ 0 q ph . In order to induce an effective spin-spin interaction between spin states we assume that an optical spin-dependent force is applied which couples the internal states of the ions with the collective vibrational modes [28][29][30] . In the following we assume that the desired spin-spin interaction is mediated by the radial phonons which are less sensitive to ion heating and thermal motion 31 . Consider that each ion interacts with two pairs of noncopropagating laser beams along the radial direction with laser frequencies which give rise to a spin dependent force at frequency μ. Here is the small time-dependent laser detuning (Ω 0 , μ ≫ Δ k (t)) of the ac Stark shifted states with respect to Ω 0 which introduce an effective qubit frequency. In order to induce a single-spin transition we assume that the each ion interacts with a pair of copropagating laser beams with a frequency difference Ω k,L = Ω 0 − Δ(t).
Here Ω x , Ω k are the Rabi frequencies, and respectively, ϕ k , φ k are the laser phases. The small radial oscillations of the kth ion can be written in terms of collective normal modes, www.nature.com/scientificreports www.nature.com/scientificreports/ We consider the regime in which the beatnote frequency μ is not resonant with any radial vibration mode and the condition |Ω n − μ| ≫ g k,n is satisfied for any mode n. In that case the radial collective phonons are only virtually excited, thereby they can be eliminated from the dynamics 32 . As a result of that the ion's spin states at different sites become coupled. Finally, by assuming that only the kth and mth ions interact with the bichromatic field we obtain n k n m n n , , 2 2 1 being the spin-spin coupling between the two ions. By imposing the conditions (5) or (7) we realize the desired circulant Hamiltonian. Note that such dispersive spin-phonon interaction was studied in the context of quantum simulation of effective spin models 33 as well as for high-fidelity gate implementation 21 .

Numerical Examples
Here we discuss specific time dependences of the detunings and the couplings which can be used to perform the gate implementation. Consider first the Cases1 and 2 where the adiabatic transition to the quantum Fourier modes can be realized by using an exponential ramp of the detunings, Δ k (t) = Δ k e −γt (Δ k ≫ J, Ω 1 ), with a characteristic rate γ. Such a time dependence captures the asymptotic behaviour of the eigenvectors. Another convenient choice of the time-dependent couplings and detunings, which we use for numerical examples, is where ω is a characteristic parameter which controls the adiabaticity of the transition. The interaction time varies 1 tmax . Finally, the adiabatic transition to the Fourier states using Hamiltonian (19) can be carried out by using Δ 1,2 = 0, eigenfrequencies. In Fig. 1 we plot the eigenfrequencies (12a),(12b) and (15a), (15b) as a function of time.
We see that the eigenfrequencies for both cases are nondegenerate during the time evolution. Approaching the final interaction time the energy separation between the adiabatic levels for the Hamiltonian (6) is determined by the coupling strength J 0 , see Eq. (13). For the circulant Hamiltonian (8) the separation between eigenfrequencies λ ± and μ ± is again determined by J 0 . However, the presence of the single-qubit Rabi frequency Ω 1 (t) leads to higher separation between the eigenfrequencies λ + , μ + , and λ − , μ − , where the energy gap is determined by Ω 1 (Ω 1 ≫ J 0 ), see Fig. 1 Gate fidelity. We numerically simulate the adiabatic transition to the quantum Fourier states (4a),(4b),(4c),(4d) using the time-dependent couplings and detunings (23) as well as (24). In Fig. 2(a) we plot the time evolution of the spin populations assuming that the system is prepared initially in the product state . We observe that even for the relatively small coupling J 0 the adiabatic transition transforms the initial state into the respective quantum Fourier state, namely ψ ↓↓ → | ⟩ | ⟩ 3 . In this case the nonadiabatic transition is suppressed due to the single-qubit Rabi frequency Ω 1 which improves the adiabaticity of the transition. We have found that all other initial computational spin states approach the respective quantum Fourier states according to Eq. (17a),(17b),(17c),(17d). We also show the adiabatic transition ψ −− → | ⟩ | ⟩ 3 using Hamiltonian (19), see Fig. 2(b). We observe that compared to the Case 2 now the adiabatic transition is performed for shorter interaction time.
In Fig. 3 we plot the time evolution of the arguments of the probability amplitudes for the different spin states. The arguments tend toward the respective phases given by Eq. (4a),(4b),(4c),(4d). The same result also is observed for all other initial computational states.
As a figure of merit for the fidelity of the gate implementation we use where s k = ↑ k , ↓ k . Here π G 4 is the desired two-qubit quantum Fourier transform (18) and ′ π G t ( ) 4 is the actual one. In Fig. 4(a) we show the two-qubit fidelity (25) as a function of time where we choose the detunings Δ 1 , Δ 2 such that the adiabatic phases become α 2 = 2kπ, β 2 = 2pπ. As the time progresses the unitary propagator π ′ G 4 converges toward π G 4 . We observe that for spin-spin coupling J 0 /2π = 2 kHz and gate time ≈ .
creation of entangled states. The action of the two-qubit gate on the computational basis creates superposition states which, however, are not entangled. In order to create entangle states one needs to prepare initially the system is a superposition spin state. For example, consider that the initial state is The two-qubit gate (17a),(17b),(17c),(17d) transforms the initial state . We solve numerically the time-dependent Schrödinger equation with c ir (2) assuming the initial condition The parameters are set to J 0 /2π = 2 kHz, Ω 1 /2π = 50 kHz, Δ 1 /2π = 30 kHz, Δ 2 /2π = 10 kHz, ω/2π = 0.2 kHz, ϕ = π/4.
c ir (2) . The parameters are set to Ω 1 /2π = 40 kHz, ϕ = π/4, J 0 /2π = 2 kHz and ω/2π = 0.18 kHz. We choose the detunings Δ 1 /2π = 59.96 kHz and Δ 2 /2π = 27.76 kHz such that the adiabatic phases becomes α 2 = 2kπ, β 2 = 2pπ (k = 40 and p = 20) which realise the two qubit gate (18). (b) Fidelity of the adiabatic transition using Hamiltonian (19). The parameters are set to J 0 /2π = 2 kHz, V 0 /2π = 2.02 kHz, Ω 1 /2π = 146.3 kHz, ω/2π = 0.55 kHz, and ϕ = π/4. where the mixing angle is ξ = Ω tan( ) 2 /J. Using the time-dependent couplings (24) we obtain In Fig. 6 we show the shape of the counterdriving field (28) for various values of ω and J 0 . We see that the countrerdriving field vanishes at t = 0 which preserves the requirement system to begin in the rotating spin states. At t max we have ξ ∂ t ( ) t max such that the system end up in state with circulant symmetry. Importantly, we observe that for the same magnitude of J 0 ~ ∂ t ξ one can reduce the gate time such that ω > J 0 . Consider as an example spin coupling J 0 /2π = 2.0 kHz. For approximately the same maximal magnitude of ∂ t ξ the gate time is approximately a factor of four shorter, ω/2π = 2.5 kHz and μ = t 100 max s, see Fig. 4(b) for comparison.