Quantum Computation under Micromotion in a Planar Ion Crystal

We propose a scheme to realize scalable quantum computation in a planar ion crystal confined by a Paul trap. We show that the inevitable in-plane micromotion affects the gate design via three separate effects: renormalization of the equilibrium positions, coupling to the transverse motional modes, and amplitude modulation in the addressing beam. We demonstrate that all of these effects can be taken into account and high-fidelity gates are possible in the presence of micromotion. This proposal opens the prospect to realize large-scale fault-tolerant quantum computation within a single Paul trap.

S calable quantum computation constitutes one of the ultimate goals in modern physics 1,2 . Towards that goal, trapped atomic ions are hailed as one of the most promising platforms for the eventual realization 3,4 . The linear Paul trap with an one-dimensional (1D) ion crystal was among the first to perform quantum logic gates [5][6][7] and to generate entangled states [8][9][10] , but in terms of scalability, the 1D geometry limits the number of ions that can be successfully trapped 11,12 . Another shortcoming of the 1D architecture is that the error threshold for fault-tolerant quantum computation with short-range gates is exceptionally low and very hard to be met experimentally [13][14][15] .
Generic ion traps, on the other hand, could confine up to millions of ions with a 2D or 3D structure [16][17][18] . More crucially, large scale fault-tolerant quantum computation can be performed with a high error threshold, in the order of a percent level, with just nearest neighbor (NN) quantum gates [19][20][21][22] . This makes 2D or 3D ion crystals especially desirable for scalable quantum computation. Various 2D architectures have been proposed, including microtrap arrays 23 , Penning traps 16,[24][25][26] , and multizone trap arrays 27,28 . However, the ion separation distance in microtraps and penning traps is typically too large for fast quantum gates since the effective ion-qubit interaction scales down rapidly with the distance. In addition, fast rotation of the ion crystal in the Penning trap makes the individual addressing of qubits very demanding. Distinct from these challenges, Paul traps provide strong confinement; however, they are hampered by the micromotion problem: fast micromotion caused by the driving radio-frequency (rf) field cannot be laser cooled. It may thus create motion of large amplitudes well beyond the Lamb-Dicke regime 29,30 , which becomes a serious impediment to high-fidelity quantum gates.
In this paper, we propose a scheme for scalable quantum computation with a 2D ion crystal in a quadrupole Paul trap. We have shown recently that micromotion may not be an obstacle for design of high-fidelity gates for the two-ion case 31 . Here, we extend this idea and show that micromotion can be explicitly taken into account in the design of quantum gates in a large ion crystal. This hence clears the critical hurdle and put Paul traps as a viable architecture to realize scalable quantum computation. In such a trap, DC and AC electrode voltages can be adjusted so that a planar ion crystal is formed with a strong trapping potential in the axial direction. In-plane micromotion is significant, but essentially no transverse micromotion is excited due to negligible displacement from the axial plane. We perform gates mediated by transverse motional modes and show that the in-plane micromotion influences the gate design through three separate ways: (1) It renormalizes the average positions of each ion compared to the static pseudopotential equilibrium positions. (2) It couples to and modifies the transverse motional modes. (3) It causes amplitude modulation in the addressing beam. In contrast to thermal motion, the fluctuation induced by micromotion is coherent and can be taken into account explicitly. Several other works also studied the effect of micromotion on equilibrium ion positions and motional modes [32][33][34] , or used transverse modes in an oblate Paul trap to minimize the micromotion effect 35 . Here, by using multiple-segment laser pulses [36][37][38] , we demonstrate that high-fidelity quantum gates can be achieved even in the presence of significant micromotion and even when many motional modes are excited. Our work therefore shows the feasibility of quadrupole Paul traps in performing large scale quantum computation, which may drive substantial experimental progress.
A generic quadrupole Paul trap can be formed by electrodes with a hyperbolic cross-section. The trap potential can be written as W(x, y, z) 5 W DC (x, y, z) 1 W AC (x, y, z), where It contains both a DC and an AC part, with U 0 being the DC voltage, and V 0 being the AC voltage forming an electric field oscillating at the radiofrequency V T . The parameter d 0 characterizes the size of the trap and c controls the anisotropy of the potential in the x-y plane.
We choose c to deviate slightly from zero, so that the crystal cannot rotate freely in the plane, i.e. to remove the gapless rotational mode. The AC part, on the contrary, is chosen to be isotropic in the x-y plane. We let U 0 , 0 such that the trapping is enhanced along the z direction in order to form a 2D crystal in the x-y plane. Disregarding the Coulomb potential first, the equations of motion of ions in such a trap can be written in the standard form of Mathieu equations along each direction: where n g {x, y, z}, and the dimensionless parameters are . Neglecting micromotion, one could approximate the potential as a time-independent harmonic pseudopotential with secular trapping frequencies v n 5 b n V T /2, with b n < ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi a n zq 2 being the characteristic exponents of the Mathieu equations 39,40 .

Results
Dynamic ion positions. Adding Coulomb interactions back, the static equilibrium positions can be found by minimizing the total pseudopotential 25,41 , or use molecular dynamics simulation with added dissipation, which imitates the cooling process in experiment 42,43 . In our numerical simulation, we start with N 5 127 ions forming equilateral triangles in a 2D hexagonal structure. We then solve the equations of motion with a small frictional force to find the equilibrium positionsr 0 ð Þ~r t??
which is the starting point for the expansion of the Coulomb potential. Micromotion is subsequently incorporated by solving the decoupled driven Mathieu equations (see supplementary materials). The average ion positionsr 0 ð Þ are found selfconsistently, which differ slightly from the pseudopotential equilibrium positions (an average of 0.03 mm shift). Dynamic ion positionsr t ð Þ can be expanded successively as Numerically, we found thatr 1 ð Þ <{ q 2r 0 ð Þ andr 2 ð Þ < q 2 32r 0 ð Þ , where the expression forr 1 ð Þ is consistent with previous results 31,32,42 . Micromotion thus only results in breathing oscillations about the average positions. Fig. 1(a) shows the average ion positionsr 0 ð Þ in the planar crystal. The distribution of NN distance is plotted in figure 1(b). We choose the voltages U 0 and V 0 such that the ion distance is kept between 6.5 mm and 10 mm. This ensures that crosstalk errors due to the Gaussian profile of the addressing beam are negligible, at the same time maintaining strong interaction between the ions. As micromo-tion yields breathing oscillations, the further away the ion is from the trap center, the larger the amplitude of micromotion becomes. With the furthest ion around 52 mm from the trap center, the amplitude of micromotion is 2q/2 3 52 < 1.4 mm, which is well below the separation distance between the ions but larger than the optical wavelength (see supplementary materials for the distribution of the amplitude of micromotion).
Normal modes in the transverse direction. With the knowledge of ion motion in the x-y plane, we proceed to find the normal modes and quantize the motion along the transverse (z) direction. As ions are confined in the plane, micromotion along the transverse direction is negligible. The harmonic pseudopotential approximation is therefore legitimate. Expanding the Coulomb potential to second order, we have is the 3D distance and is the planar distance between ions i and j. To the second order, transverse and in-plane normal modes are decoupled. Note that coupling between the in-plane micromotion and the transverse normal modes has been taken into account in this expansion as the Coulomb potential is expanded around the dynamic ion positionsr t ð Þ. With significant in-plane micromotion, distances between ions are time-dependent, which in turn affects the transverse modes. We can expand the quadratic coefficients in series: The time-averaged coefficients 1 .
can be used to compute the transverse normal modes. The next order containing cos(V T t) terms can be considered as a time-dependent perturbation to the Hamiltonian. It contributes on the order of O qv 2 in the rotating wave approximation, where v k is the transverse mode .
is the ion distance computed withr 0 ð Þ without considering micromotion (see supplementary materials). Here, the micromotion effect is an overall renormalization in the term 1 . r 3 ij , so it does not modify the normal mode structure. Instead, it slightly shifts down the transverse mode frequencies (in the order of O(q 2 )). Numerically, we found an average reduction of around 0.4 kHz in each transverse mode frequency with our chosen parameters. Although mode structure is not altered by this overall renormalization, the discrepancy in equilibrium positions compared to the pseudopotential approximation will modify both the normal mode structure and mode frequencies.
High-fidelity quantum gates. After obtaining the correct transverse normal modes, we now show how to design high-fidelity quantum gates with in-plane micromotion. Since NN gates are sufficient for fault-tolerant quantum computation in a planar crystal, we show as a demonstration that high-fidelity entangling gates can be achieved with a pair of NN ions in the trap center and near the trap edge. One may perform the gate along the transverse direction by shining two laser beams on the two NN ions with wave vector difference Dkẑ and frequency difference m (see Fig. 2) 38,44 . The laser-ion interaction Hamiltonian is 37 H~X 2 j~1 hV j cos Dk : dz j zmt À Á s z j , where V j is the (real) Raman Rabi frequency for the jth ion, s z j is the Pauli-Z matrix acting on the pseudospin space of internal atomic states of the ion j, and dz j is the ion displacement from the equilibrium position.
Quantize the ion motion, dz j~X k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi where x j t ð Þ~ hV j sin mt ð Þ, g k j~gk b k j and the Lamb-Dicke parameter g k~D k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi h=2mv k p =1. The evolution operator corresponding to the Hamiltonian H I can be written as 37,38,45 where the qubit-motion coupling term x j t ð Þe iv k t dt and the two-qubit To realize a conditional phase flip (CPF) gate between ions j and n, we require a k j <0 so that the spin and phonons are almost disentangled at the end of the gate, and also w jn (t) 5 p/4. It is worthwhile to note that in deriving Eq. (7), we dropped single-qubit operations as we are interested in the CPF gate. These fixed single-qubit operations can be explicitly compensated in experiment by subsequent rotations of single spins. (see supplementary materials for more detailed derivation and analysis).
As the number of ions increases, transverse phonon modes become very close to each other in frequencies. During typical gate time, many motional modes will be excited. We use multiple-segment pulses to achieve a high-fidelity gate 36,37 . The total gate time is divided into m equal-time segments, and the Rabi frequency takes the form j being the controllable and constant amplitude for the ith segment ((i 2 1)t/m # t , it/m). Due to the in-plane micromotion, the laser profile V G j t ð Þ seen by the ion is time-dependent. In our calculation, we assume the Raman beam to take a Gaussian form, with V G j t ð Þẽ , where w is the beam waist and x Detuning m can be used as an adjusting parameter in experiment to find the optimal results. All transverse phonon modes are distributed between 0.85v z and v z . We optimize the gate near either end of the spectrum since optimal results typically occur there. Blue solid lines indicate the optimal results with micromotion and red dashed lines show the results for a genuine static harmonic trap, which are almost identical in (a), (b) and (c). It implies that micromotion can almost be completely compensated, but with a stronger laser power for the edge pair. If we apply the optimal result for the static trap to the realistic case with micromotion, the fidelity will be lower as indicated by the black dash-dot lines. This is especially so for the edge pair, where the fidelity is lower than 85% at any detuning. It is therefore critical to properly include the effect of micromotion. With corrected pulse sequences, a fidelity F . 99.99% can be attained with jVj max /2p < 12 MHz (jVj max /2p < 22 MHz) for the center (edge) ions. The Rabi frequencies can be further reduced by a slower gate and/or more pulse segments.
Noise estimation. Micromotion of any amplitude does not induce errors to the gates as it has been completely compensated in our gate design. We now estimate various other sources of noise for gate implementation. In considering the effect of in-plane micromotion to the transverse modes, we are accurate to the order of q 2 , so an error of q 3 < 10 24 is incurred. The actual error is smaller since the Coulomb potential is an order of magnitude smaller than the trapping potential along the transverse direction. The cross-talk error probability due to beam spillover is P c~e With v x,y /2p < 0.2 MHz, there is a mean phonon number n 0 <50 in the x-y plane. It gives rise to thermal motion with average fluctuation in positions, dr < 0.23 mm, which can be estimated as in Ref. 46. The resultant gate infidelity is dF 1 < (p 2 /4)(dr/w) 4 < 10 24 . Lastly, we estimate the infidelity caused by higher-order expansion in the Lamb-Dicke parameter. The infidelity is dF 2 <p 2 g 4 z n 2 z z n z z1=8 À Á <2|10 {4 , where n z <5 is the mean phonon number in the transverse direction 37 . Other than the effects considered above, micromotion may also lead to rf heating when it is coupled to thermal motion. However, simulation has shown that at low temperature T , 10mK and small q parameters, rf heating is negligible 42,47 . Heating effect due to rf phase shift and voltage fluctuation should also be negligible when they are well-controlled 42 .

Discussion
It is worthwhile to point out that although we have demonstrated the feasibility of our gate design via a single case with N 5 127 ions, the proposed scheme scales for larger crystals. The intuition is that through optimization of the segmented pulses, all phonon modes are nearly disentangled from the quantum qubits at the end of the gate. However, as the number of ions further increases, one would presumably need more and more precise control for all the experimental parameters (, 1% fluctuation in voltage for example). rf heating may also destabilize a much larger crystal 48 , and more careful studies are necessary for larger crystals.
One may also notice that in Ref. 31, we considered gates mediated by the longitudinal phonon modes, so the effect of micromotion is a phase modulation. Here, we utilize transverse modes so the amplitude of the laser beam is modulated. There are a few advantages in using the transverse modes: first, it is experimentally easier to access the transverse phonon modes in a planar ion crystal; second, in a planar crystal, the transverse direction is tightly trapped, so micromotion along that direction can be neglected; third, the transverse phonon modes do not couple to the in-plane modes and the in-plane micromotion affects the transverse modes via the time-dependence of the equilibrium positions, the effect of which is again suppressed due to tight trapping in the transverse direction.
In summary, we have demonstrated that a planar ion crystal in a quadrupole Paul trap is a promising platform to realize scalable quantum computation when micromotion is taken into account explicitly. We show that the in-plane micromotion comes into play through three separate effects, and each of them can be resolved. This paves a new pathway for large-scale trapped-ion quantum computation.