Systematic error tolerant multiqubit holonomic entangling gates

Quantum holonomic gates hold built-in resilience to local noises and provide a promising approach for implementing fault-tolerant quantum computation. We propose to realize high-fidelity holonomic $(N+1)$-qubit controlled gates using Rydberg atoms confined in optical arrays or superconducting circuits. We identify the scheme, deduce the effective multi-body Hamiltonian, and determine the working condition of the multiqubit gate. Uniquely, the multiqubit gate is immune to systematic errors, i.e., laser parameter fluctuations and motional dephasing, as the $N$ control atoms largely remain in the much stable qubit space during the operation. We show that $C_N$-NOT gates can reach same level of fidelity at a given gate time for $N\leq5$ under a suitable choice of parameters, and the gate tolerance against errors in systematic parameters can be further enhanced through optimal pulse engineering. In case of Rydberg atoms, the proposed protocol is intrinsically different from typical schemes based on Rydberg blockade or antiblockade. Our study paves a new route to build robust multiqubit gates with Rydberg atoms trapped in optical arrays or with superconducting circuits. It contributes to current efforts in developing scalable quantum computation with trapped atoms and fabricable superconducting devices.

In this work, we propose a one-step approach for implementing (N + 1)-qubit holonomic gates [(N + 1)-QHG] in a many-body model where qubits interacting with each other by the Ising interaction. Rydberg atoms, as natural qubits with perfect identity, are a promising platform for realizing the Ising model by means of van der Waals or dipole-dipole interactions among atoms, and recent experiments have shown cryogenic atom cooling, largeprobability atom trapping and loading, defect-free arrangement of atom arrays, and high-fidelity single-atom manipulations [53][54][55]. With Rydberg atoms in an optical array [56][57][58][59][60][61][62], we identify an unconventional regime where the Rydberg antiblockade condition is employed but the blockade phenomenon emerges. In this regime, all N control atoms remain unexcited in the gate. We obtain the parameter region where increasing N does not demote the driving strength of the effective system. Importantly, the error sensitivity to fluctuations in laser parameters and interatomic distances can be largely suppressed through optimal pulse engineering [63][64][65]. Moreover, we show that the multiqubit holonomic gates can be realized alternatively with superconducting circuits. Our study paves a route to achieve optimized holonomic quantum computation with strongly interacting Rydberg atoms and superconducting circuits, and might find applications in quantum computation and simulations based on robust multiqubit gates. This paper is organized as follows: in section II, we introduce the many-body model and show the deviation of effective Hamiltonian for implementing (N + 1)-QHG. In section III, we describe the realization of the manybody model with a Rydberg atom array and show simulations for implementing C N -NOT gates of NHQC. In section IV, we give an optimal pulse scheme for robust (N + 1)-QHG and identify the excellent robustness of the scheme. Finally, a conclusion is given in section V.

A. Hamiltonian with Dicke states
To reveal the working mechanism of the multiqubit gate, we introduce collective states of the control qubits (similar to Dicke states of two-level atoms [66]) |ψ K ≡ |D N −M |B K |A M −K consisting of (N − M ), K, and (M − K) qubits in states |D , |B , and |A , respectively (0 ≤ K ≤ M ≤ N ) [45,67]. The states |ψ K and |ψ K−1 (K ≥ 1) are coupled with collective Rabi frequency √ K Ω c with K = K(M − K + 1) [68][69][70][71][72][73][74] (see Appendix A for details). Then, the composite basis of the collective control and target qubits is de- N A being the binomial coefficient. Consequently, we can rewrite the Hamiltonian H with the composite states (the derivation is given in whereŜ are the collective transition operator and detuning, respectively. (5) for which the schematic diagram of transitions is shown in Fig. 3(b). Here we chooseΩ c /2 Ω t , such that the transitions with M = 0 are effectively suppressed, due to the faster oscillation at frequency λ. We then neglect the terms with M = 0, so a final effective Hamiltonian is obtainedĤ Only when all control qubits populate in |D will the operation field on the target qubit work. Otherwise, the evolution of all the (N + 1) qubits is frozen. It indicates that the controlled (N + 1)-QHG can be achieved as long as a single-qubit holonomic gateÛ t is operated on the target qubit.

III. REALIZATION WITH A RYDBERG ATOM ARRAY
Due to the large polarizability, Rydberg atoms experience strong and long-range van der Waals interaction V jk = C 6 /R 6 jk (corresponding to the Ising interaction in HamiltonianĤ) with C 6 being the dispersion coefficient and R jk the inter-atom distance [55,73,[77][78][79]. We will demonstrate that the strong Rydberg-Rydberg interactions (RRI), precise control of atom positions with optical tweezer arrays [80][81][82][83][84] and of laser pulses allow to achieve high-fidelity multiqubit gates.

A. Effective dynamics
Our setting is a 2D atom array [56] where the control atoms sit on a ring and the target atom in the center, which ensures equivalent control-target interatom couplings, as depicted in Fig. 4(a). A pair of hyperfine ground states of atoms are used to encode qubit states |0 and |1 , and a high-lying Rydberg state |r corresponds to the auxiliary state, i.e., |A = |r . We specify |B j = |0 j and |D j = |1 j for all control atoms so that only |0 of control atoms is transited to the Rydberg state, while for the target atom we follow the general definitions. This yields the many-atom Hamiltonian Ωc 2 |0 j r| + H.c. andĤ t = s=0,1 Ωs 2 e iϕs |s t r| + H.c. = Ωt 2 |B r| + H.c. with Ω t = Ω 2 0 + Ω 2 1 . The space dependent RRI is described bŷ H i = N j>k=1 (V jk |rr jk rr| + V jt |rr jt rr|) with V jk being the coupling strength between the j-th and k-th atoms [55,73,85,86]. Although the interaction depends on distances between the control qubits, we will show high-fidelity gates can be still achieved.
In the Rydberg atom setting, one can choose parameters (as given bellow) such that the interaction between any pair of the control quibts is relatively strong. This allows that maximally one control qubit can be excited to the Rydberg state. This corresponds to the desired situation with K = M in the collective state |ψ K . Working in this restricted Hilbert space, we can obtain the approximation Hamiltonian, with the same form as Eq. (1) Then following similar derivation to obtain the effective Hamiltonian Eq. (6), Rabi frequenciesΩ c and Ω t are chosen appropriately with ω = V jt √ N |Ω c |/4 and |Ω c | |Ω t |, which guarantees that the underlying dynamics is governed by the effective Hamiltonian Eq. (6).
For the realization of the proposed many-body model with Rydberg atoms, to guarantee an important requirement that the target atom has an equal interaction to all of control atoms, the geometry of atom distribution is shown in Fig. 4(a), where a dozen of atoms can be loaded on the ring of radius d i = 3.8 µm and the interatomic separation d c = 2 µm. In addition to the 2D atomic array that can be readily achieved in experiment [56,79], a defect-free 3D array with the control atoms are distributed on a spherical surface may be also accessible [61,62], based on which the available number of control atoms can be greatly increased. We remark here that, according to recent experiments, the neutral atom array with intersite spacing near [79,87] or even less than [88] 2 µm is possible.
We emphasize that it is a ring that the control atoms are distributed on, so it is difficult to drive them ho- mogeneously by a same field without individual addressing. Single-site addressability is needed, and the control atoms can be driven by N independent fields with identical drive parameters, assisted by the devices of laser beam splitter. There is no obstacle to realize such single-site addressing and driving for Rydberg atoms in a 2D array by using drive lasers with waist ≤ 1 µm [72,[79][80][81][82] (see also the review article [73]).
We illustrate that many-atom dynamics of the system indeed can be captured by the effective Hamiltonian (6). Such benchmark is carried out by implementing a C N -NOT gate [43,46]. The figure of merit is the state overlap fidelity is obtained by solving the Schrödinger equation numerically. An important result is that the full HamiltonianĤ I and the effective HamiltonianĤ e produce nearly identical fidelity, as shown in Fig. 4(b). The excellent agreement results primarily from the fact that double or more excitations of the Rydberg state are strongly suppressed, which is verified in Fig. 4(c), validating the approximations used in deriving the effective Hamiltonian.
Note that the physical regime is different from either the Rydberg blockade or antiblockade. In our protocol the first sideband of the modulation field offsets the control-target RRI, where ω = V jt is similar to the Rydberg antiblockade condition [91][92][93][94][95]. For a normal process of Rydberg antiblockade, a doubly-excited state can be achieved from |χ M 1 to |χ M 4 , where the latter includes two Rydberg states. In our gate, however, only the target atom can be excited (i.e., the Rydberg blockade), as the strong control field prevents the excitation of doubly-excited Rydberg states. This feature is particularly beneficial to maintaining high gate fidelity, as state loss and motional dephasing due to Rydberg excitation are intrinsically mitigated. In contrast, other multiqubit gate schemes either encode the logic states in excited states [45], or allow multiple control qubits in the excited-state manifold through, e.g., the three-step, adiabatic process [46], where the qubits could suffer stronger decay from the electronically excited states.

IV. ROBUST MULTIQUBIT HOLONOMIC GATES
A. Gate pulse engineering The NHQC gates are usually sensitive to gate lasers when the laser profile has a simple shape [20,39,40]. To illustrate this dependence, we consider a rectangular pulse with small variation δT (δΩ t ) to the gate time T (Rabi frequency Ω t ). As depicted in Fig. 5(a), the infidelity of the CNOT gate of NHQC grows rapidly with increasing errors in T or Ω t , showing the relatively large sensitivity of NHQC to the systematic errors.
It turns out that the fidelity as well as error tolerance of (N +1)-QHG can be improved through pulse engineering, permitting to carry out optimized holonomic quantum computation (OHQC) [20,40]. Here we achieve OHQC by employing a time-dependent Rabi frequency Ω t (t) and detuning ∆(t) in the transition |B t ↔ |A t . Due to the detuning, Hamiltonian (8) now becomesĤ I + ∆(t) 2Ẑ t withẐ t = |r t r| − |B t B|. This results to an effective Hamiltonian, Ω c /2. The pulse engineering is based on the optimal control technique [63,64], such that the time-dependent operation field is shaped elaborately for minimizing the systematic error sensitivity of OHQC gates (see Appendix C for details) Ω t (t) =α 1 + λ 2 sin 2 α, ∆(t) = −λα cos α −λ sin α + λα cos α 1 + λ 2 sin 2 α .

B. Optimized (N + 1)-QHG
We now thoroughly examine the robustness of OHQC where fluctuations of parameters relevant to current experiments will be considered. First, we show OHQC is largely immune to finite Rydberg lifetime. To take into account of spontaneous decay in the Rydberg state, a many-atom master equation is solved (see Appendix D for details), in which the qubit dephasing (rate γ φ /2π = 1 kHz) is included additionally. Through comparing the fidelity of a CNOT gate of OHQC with that of NHQC in Fig. 5(b), it is found that the gate fidelity of OHQC is robust, and higher when fluctuations of the Rabi frequency ranges within |δΩ t |/2π > 30 kHz. With typical experimental conditions, our extensive calculations furthermore show that fidelities for two-and three-qubit controlled-Û t (θ, φ, γ) gates of OHQC can be better than 0.995 by identifying the average fidelity, as discussed in Appendix D.
Next, we demonstrate that fluctuations of the laser parameters only affect the gate fidelity marginally. For N = 2 and N = 3, we identify the tolerance of C N -NOT gates to the systemic error in Ω 0 in Fig. 6(a), and fidelities are always larger than 0.994 when |δΩ 0 |/2π ≤ 0.1 MHz. Errors in ∆(t), however, need attention, which can alter the gate time and lead to unwanted dynamical phase. The latter can be eliminated through applying spin-echo [34,96], then from Fig. 6(b) we can find the fidelity is barely reduced. On the other hand, the gate fidelity oscillates with varying δω in Fig. 6(c), as nonzero δω gives off-resonant coupling. However the resulting fidelity is rather high, around and above 0.99 when |δω|/2π < 5 MHz. We emphasize that the driving strength Ω t (t) in the effective Hamiltonian (6) is not degraded, compared to the original one. In contrast, the effective driving strengths in other multiqubit gate schemes decrease with increasing number of qubits [95,97]. In fact the fidelity increases with N in certain parameter regions, as shown in Figs. 6(a) and (b). Although it becomes difficult to achieve V jt √ NΩ t when N is too large, high fidelities are still obtained with moderate number of qubits, up to N = 5 according to Figs. 4(b) and 6(d).
Finally, the effect of motional dephasing, due to imperfect positions of atoms in the trap array or thermal motions, can be suppressed, too. The imperfect placement of atoms causes errors to the RRI. To be specific, we assume the RRI fluctuates randomly in [−δV, δV ]. Though the infidelity can be large with increasing δV in Fig. 7(a), we can reduce its impact by increasingΩ c , ensuring the system stays in the regime described by Hamiltonian (6). Thermal motions lead to unwanted detunings of pumping lasers seen by the atoms [81,98,99]. Our numerical simulations show that the thermal motions can be almost suppressed by increasing Ω max , when jointly utilizing spin echo [81,96], as shown in Fig. 7(b). The thermal motion has negligible effects on the fidelity of the multiqubit gate even for N = 5 at T a = 10 µK [inset Fig. 7(b)]. The high fidelity results from the fact that the effective dynamics is largely captured by Eq. (6).

V. CONCLUSION
We have presented a robust scheme for realizing (N + 1)-qubit holonomic gates. Its advantage is that errors due to electronic excitation of the control qubits are mitigated, enabled by the strong inter-spin interaction. Using Rydberg atom arrays, we have revealed the robustness of the multiqubit holonomic gate against errors in the laser parameters, variations of interatomic interaction, and thermal fluctuations. Besides neutral Rydberg atoms, trapped Rydberg ions [100] are another candidate to realize the multiqubit gates due to the strong RRI. Moreover, the (N + 1)-qubit holonomic gates can be implemented with superconducting circuits [101]. The very long-range coupling between superconducting qubits suggests that they are potentially less sensitive to the spatial fluctuation. Our study opens a new route to achieve robust and error-tolerant multiqubit holonomic gates with strongly interacting Rydberg atoms and superconducting circuits, which might find applications in scalable quantum computation and simulation of many-body models.
Despite the decoupled state |D , qubits considered in our work can be regarded as spin-1 2 particles, where |A and |B denote positive and negative spin states, respectively. We introduce spin operatorsŜ Similarly, then we can obtainĤ being the spin operator of the target qubit, soĤ Ising is exactly described by the collective spin oper-atorĴ Using the Dicke states, we re-express the control qubit state |ψ K ≡ |D N −M |B K |A M −K = | M 2 , m = M 2 −K c and the target qubit states, |A t = | 1 2 , 1 2 t and |B t = | 1 2 , − 1 2 t . Here the value of m is calculated according to the numbers of |A and |B in |ψ K , that is, m = (M − K) × 1 2 − K × 1 2 . Then the composite basis of the collective control and target qubits denoted with control qubits in state |A for |χ K 1,2 (|χ K 3,4 ), can be re-expressed, respectively, by which are eigenstates ofĤ Ising with eigenvalues The eigenvalues in Eq. (A4) can be unified as E with ξ = 0 (ξ = 1) for p = 1, 3 (p = 2, 4) and C 2 N A to be the binomial coefficient. Hence one can diagonalize the Ising interaction to beĤ Ising = p |. Now we can calculate the collective coupling strength between |ψ K and |ψ K−1 (1 ≤ K ≤ M ) Accordingly, there is a strengthened collective Rabi frequency √ K Ω c with K = K(M − K + 1) for the coupling between the two Dicke states |ψ K and |ψ K−1 . Therefore, the Hamiltonian of two fields interacting with the (N + 1) qubits can be written aŝ whereŜ (K) pq = |χ K p χ K q | are collective transition operators. Finally we can see the total Hamiltonian, H Ising +Ĥ f q , of the whole system as Eq. (1).
To test the validity of the two-photon processes discussed above, we use the four-level atoms shown in Fig. 8 to perform a CNOT gate based on the OHQC scheme (see Appendix C for details of optimal pulses). We adopt parameters Ω cp =Ω cr = 2π × 400 MHz, δ c = 2π × 2 GHz, Ω 0r = max[Ω 0p (t)] = Ω 1r = max[Ω 1p (t)] = 2π × 60/ √ 2 MHz, δ 0 = δ 1 = 2π × 1.8/ √ 2 GHz, and V = ω = 2π × 285.1 MHz. The fidelity evolution of the target state based on Schrödinger equation simulation is shown in Fig. 9 for OHQC. The fidelity can reach 0.995, which is high, compared with results based on the effective three-level atoms used to illustrate gate performances in our work; see the contents related to the average gate fidelities shown in Fig. 11 based on the effective three-level atoms, where the final average gate fidelities oscillates around 0.995. This shows that the two-photon Rydberg pumping can be safely used to conduct our gate scheme, even when a fast-oscillating driving amplitude Ω cr is applied in obtaining the effective Hamiltonian.

Appendix C: Optimized pulse engineering
Using the target-atom HamiltonianĤ t = [Ω t (t)X t + ∆(t)Ẑ t ]/2 withX = |B r| + |r B| andẐ = |r r| − |B B|, we transfer the full population from |B t first to |r t during t ∈ [0, T /2] and then back to |B t during t ∈ (T /2, T ], with T being the gate time. In the following, we mainly discuss the population transfer from |B t to |r t , because the inverse process is based on the same theory. During the state transfer from |B t to |r t , a solution of the time-dependent Schrödinger equation i∂|ψ /∂t =Ĥ t |ψ can be parametrized as a superposition |ψ 0 (t) = e −iη/2 e iβ/2 cos(α/2)|B t + e −iβ/2 sin(α/2)|r t [64], for which the time dependence symbol "(t)" of α(t), β(t), and η(t) is omitted for simplicity.
There is also an orthogonal solution |ψ 1 (t) = e iη/2 e iβ/2 sin(α/2)|B t − e −iβ/2 cos(α/2)|r t such that ψ 1 (t)|ψ 0 (t) = 0. Inserting |ψ 0 (t) and |ψ 1 (t) into the Schrödinger equation, Ω t (t) and ∆(t) are related to α and β, as At the same time, the global phase η = t 0α (t ) cot β(t )/ sin α(t )dt can be obtained, and η(T ) = 0 is supposed to be satisfied so as to ensure a null dynamical phase. We assume that the evolution follows |ψ 0 (t) , and thus the boundary conditions α(0) = 0 and α(T /2) = π are supposed to be satisfied for completing the state transfer |B t → |r t . We introduce a systematic error δX into an ideal parameter X, yielding an actual parameter (X + δX). Our goal is to design a pair of Ω t (t) and ∆(t) such that the holonomic gates are insensitive to systematic errors in the gate time T and the pulse amplitude Ω t (t). For errors in the gate time, Ω t (t) and ∆(t) can be designed to be softly turned on and off, so that a moderate surplus or deficiency in the pulse duration has little effect on the pulse area. When δΩ t is taken into account, it leads to a perturbation-containing HamiltonianĤ t =Ĥ t +Ĥ r withĤ r = δΩt 2X t . Then using the perturbation theory and keeping the final state to the second order, we obtain a perturbed population of |r t at t = T /2, as . Then we define a quantity S ≡ − 1 2 ∂ 2 Pr ∂δΩ 2 t δΩt=0 to measure the sensitivity of P r to the systematic error in Ω t (t) [63]. Substituting the expressions of |ψ 0 (t) and |ψ 1 (t) into P r (T /2), the systematic error sensitivity can be calculated out For rendering holonomic gates to hold strong tolerance to δΩ t , S is supposed to be as small as possible, which needs pulse engineering with suitable forms of α, β, and η. According to α(0) = 0 and α(T /2) = π, α can be designed with a polynomial ansatz as α = 12πt 2 /T 2 − 16πt 3 /T 3 . Then we choose η = 2α + a 1 sin(2α) + a 2 sin(4α) such that the dynamical phase is absent at t = T /2, and a 1 and a 2 are to be determined so as to minimize S. The forms of α and η give β = cos −1 (λ sin α/ 1 + λ 2 sin 2 α) with λ = 2 + 2a 1 cos(2α) + 4a 2 cos(4α), so the pulse form given in Eq. (9) is obtained.
In order to achieve a small sensitivity to systematic errors and a short gate time, in Figs. 10(a) and (b) we plot numerically S and T max[Ω t (t)]/2, respectively, with varying a 1 and a 2 . We find that in the region of T max[Ω t (t)]/2 < 30 there exists a very small region of S < 0.5, which indicates a trend that a small error sensitivity costs a longer gate time. However, there are still points guaranteeing S < 0.5 and T max[Ω t (t)]/2 < 10. For example, a 1 = 0.28 and a 2 = −0.12 give S = 0.3 and T max[Ω t (t)] = 17.9. This gate time is three times longer than that (2π) of non-adiabatic holonomic gates.
With the trade-off between the robustness and the speed of implementing holonomic gates, we pick up a 1 = 0.28 and a 2 = −0.12 that can ensure a short gate time max[Ω t (t)]T /2π = 2.85 and a small systematic-error sensitivity S = 0.3. Based on Eq. (9), the pulse forms can be determined, as shown in Fig. 10(c), with which the state transfer |B t → |r t can be achieved with an enhanced tolerance against the systematic error in Ω t (t). An identical process can be performed again to transfer the population from |r t back to |B t . In addition to optimal control for engineering pulses, other techniques may also be efficient to enhance gate robustness even with a constant Rabi frequency, for example Landau-Zener-Stückelberg interferometry [126,127]. L js ≡ √ γ s |s j r| andL φj ≡ √ γ φj (|r j r| − |B j B|) describe energy relaxation and dephasing of the j-th atom, respectively, with relaxation rate γ s from |r to |s and dephasing rate γ φj for coherence of |r and |B . For the energy relaxation, an additional ground state |2 is introduced to denote those Zeeman magnetic sublevels out of the computational states |0 and |1 . For convenience, we assume that energy relaxation rates from a Rydberg state of 87 Rb atoms into the eight Zeeman ground states are identical, so γ 0 = γ 1 = Γ/8 and γ 2 = 3Γ/4, where Γ = 1/τ is the total relaxation rate of Rydberg state with τ being the Rydberg lifetime. In the system of laserdriven natural atoms, the dephasing rate is in general much less than the relaxation rate [80,81,83], and then here we set γ φj /2π = 1 kHz.
Here we take the two-and three-qubit holonomic gates as examples, and show the average fidelities to verify the arbitrariness of an initial state. The average fidelity is defined based on a trace-preserving operator, as [128] F (ε,Û ) = 4 N +1 v=1 tr Ûû † vÛ † ε (û v ) + l 2 l 2 (l + 1) , (D2) kσ k is a tensor product of Pauli matricesσ k ∈ {Î,σ x ,σ y ,σ z } on computational states {|0 , |1 }, and l = 2 N +1 for an (N + 1)-qubit gate. ε(û v ) is a trace-preserving quantum operation obtained through solving the master equation. We show average fidelities of various two-and three-qubit controlled-U t (θ, φ, γ) gates in Figs. 11(a) and (b), respectively. The evolutionary trends of the average fidelity for different gates are dependent of the value of γ, and the two-and three-qubit gates can be achieved with average fidelities over 0.995.