1 Introduction

Quantum computation is built by using quantum mechanical principles and performs under the logic different from the Boolean logic. Because of using quantum parallelism, quantum computation has been shown to be able to solve many problems more efficiently than classical computation [1–5]. To achieve these advantages, a central requirement is to realize quantum gates with sufficiently high fidelities. However, while quantum technology has been improved significantly, realizing high-fidelity quantum gates is still a challenging work. The challenges usually come from two aspects. On one hand, the control of a quantum system can not be totally accurate and this is known as control errors. On the other hand, a quantum system inevitably interacts with its environment, causing decoherence. Both control errors and decoherence reduce the fidelity of quantum computation seriously, and to suppress them, various robust quantum gates have been proposed and they attract continuous attention.

Geometric gates are one kind of robust quantum gates and they are built by using geometric phases. Geometric phases have the feature that they do not depend on the evolution details but only on the global nature of the evolution, making them robust against control errors. Based on geometric phases, geometric gates inherit the geometric robustness against control errors and therefore have the potential to be high-fidelity. Geometric gates were first developed by using adiabatic Abelian geometric phases [6] and such kind of geometric gates are adiabatic geometric gates [7]. Soon, adiabatic geometric gates were extended to adiabatic holonomic gates [8, 9], which are based on adiabatic non-Abelian geometric phases [10]. The common feature of adiabatic geometric gates and adiabatic holonomic gates is the requirement of adiabatic evolutions. Then to relax such a requirement, nonadiabatic geometric gates were proposed [11, 12] and such geometric gates are based on nonadiabatic Abelian geometric phases [13]. Recently, nonadiabatic holonomic gates [14, 15], which are based on nonadiabatic non-Abelian geometric phases [16], were proposed. Nonadiabatic holonomic gates share the geometric robustness against control errors. Meanwhile, they are able to be performed at high speed. Due to these features, nonadiabatic holonomic gates have received considerable attention and various nonadiabatic holonomic schemes have been put forward, although they were proposed not long before [17–51]. In particular, nonadiabatic holonomic gates have been demonstrated experimentally in various platforms, including circuit quantum electrodynamics [27, 42, 44–46], nuclear magnetic resonance [25, 28, 38], and nitrogen-vacancy centers in diamond [29, 30, 39, 40].

A practical quantum computer contains a large number of qubits. Thus multi-qubit gates are as necessary for the computer as one-qubit and two-qubit gates. Among multi-qubit gates, multi-qubit controlled gates play an important role and are frequently used in various quantum information tasks. Although such gates can be built with gates from the universal set of one- and two-qubit gates, this procedure typically becomes very demanding as the number of such gates rapidly grows with the size of the computational problem. Particularly, quantum information is entering noisy intermediate-scale quantum era and in this era, quantum computers do not have enough resources for full fault tolerance and therefor can only support the computation with a short duration. Because of the above two facts, finding a way to realize multi-qubit controlled gates with fewer steps is of importance. The reduction of steps can save the time for computation and therefore shorten the exposure time to the environment. This reduces the errors caused by decoherence and increases the precision of the computation.

In this paper, we put forward the realization of multi-qubit controlled nonadiabatic holonomic gates with connecting systems. We divide the qubits into different blocks and make the operations acting on different blocks be able to be performed in parallel. Because of this feature, our proposal can efficiently reduce the operation steps, thereby reducing the affection from decoherence and increasing the precision of the computation. Our proposal can be implemented with trapped ion systems, which are a promising candidate for realizing various quantum information tasks. So, it is useful to realize efficient and robust quantum information processors.

The structure of our paper is organized as follows. In Section 2, we demonstrate the realization of multi-qubit controlled nonadiabatic holonomic gates with four-level connecting systems. In Section 3, we generalize the idea one step further to use connecting systems with more levels for the realization. Section4 is the conclusion.

2 Realizing gates by using four-level connecting systems

We now illustrate our proposal. We consider N ions confined in a linear trap and each ion encodes a logical qubit. Our aim is to realize controlled nonadiabatic holonomic gates acting on these N qubits. In the following, we specifically demonstrate the realization of controlled phase nonadiabatic holonomic gates. The generalization to other controlled nonadiabatic holonomic gates is direct. It is known that for controlled phase gates, any qubit can be treated as the target qubit. Because of this, we do not specify which one is the target qubit in the following.

To realize the gates efficiently, we divide the N ions into m blocks, with m≥1. How to divide the N ions depends on the value of N and particularly when N is small, we divide all the N ions into m=1 block. We let each block contain at least three ions. Without loss of generality, we label the ions in block k by 1,…,nk and specify to which block an ion belongs when necessary, where k∈{1,…,m}. Clearly, one has the relation \(\sum _{k=1}^{m}n_{k}=N\). For each ion except the last one of each block, we use three of its internal states for the realization and suppose they form a Λ structure, to ensure the holonomic feature. We use two of the three internal states as logical states |0〉 and |1〉, and the third one as an auxiliary state |a〉. The transitions between states |0〉⇔|a〉 and |1〉⇔|a〉 are allowed to be separately driven by laser fields. For the last ion of each block, e.g., ion nk of block k, we use it as a connecting system, to connect different blocks. For the connecting system, we use four of its internal states for the realization and these four internal states form a tripod structure in Fig. 1. From the figure, one can see that these four internal states are used as logical states |0〉 and |1〉, and auxiliary states |a〉 and |b〉, where the transitions |0〉⇔|a〉,|1〉⇔|a〉, and |b〉⇔|a〉 are allowed to be separately driven by laser fields. Note that the connecting systems are also used to store logical information.

Fig. 1
figure 1

The level configuration of the four-level connecting system. Two lower levels of the system are used as logical states |0〉 and |1〉. The other two are used as auxiliary states |a〉 and |b〉. For the system, the transitions between states |0〉⇔|a〉,|1〉⇔|a〉, and |b〉⇔|a〉 are allowed to be separately driven by laser fields

Full size image

To realize the controlled phase nonadiabatic holonomic gate, we first perform operations separately on each block. The operations performed on block one are as follows. For block one, the ions in it are labeled by 1,…,n1 for simplicity and we first implement the transition |11〉12→|aa〉12 of ions 1 and 2 by using the Hamiltonian

$$ H_{12}(t)=\Omega_{12}(t)|{aa}\rangle_{12}\langle{11}|+\Omega^{\ast}_{12}(t)|{11}\rangle_{12}\langle{aa}|. $$
(1)

In the above, Ω12(t) describes the coupling strength and to implement the desired transition, the evolution time T12 needs to satisfy \(\phantom {\dot {i}\!}\int _{0}^{T_{12}}|\Omega _{12}(t)|dt=\pi /2\). After performing the above operation, we sequentially perform operation j on block one with j being from 2 to n1−1. Specifically, for the operation j, we implement the transition |a1〉j,j+1→|1a〉j,j+1 of ions j and j+1 by using the Hamiltonian

$$ {}H_{j,j+1}(t)=\Omega_{j,j+1}(t)|{1a}\rangle_{j,j+1}\langle{a1}| +\Omega^{\ast}_{j,j+1}(t)|{a1}\rangle_{j,j+1}\langle{1a}|. $$
(2)

In the above, Ωj,j+1(t) describes the coupling strength and the corresponding evolution time Tj,j+1 satisfies the condition \(\phantom {\dot {i}\!}\int _{0}^{T_{j,j+1}}|\Omega _{j,j+1}(t)|dt=\pi /2\). After the above operations, we then perform operation n1. Specifically, we implement the transition \(\phantom {\dot {i}\!}|{a}\rangle _{n_{1}}\rightarrow |{b}\rangle _{n_{1}}\) by resonantly coupling the states |a〉 and |b〉 of ion n1, and the corresponding Hamiltonian reads

$$ H_{n_{1}}(t)=\Omega_{n_{1}}(t)|{a}\rangle_{n_{1}}\langle{b}|+\Omega^{\ast}_{n_{1}}(t)|{b}\rangle_{n_{1}}\langle{a}|, $$
(3)

with the evolution time satisfying \(\phantom {\dot {i}\!}\int _{0}^{T_{n_{1}}}|\Omega _{n_{1}}(t)|dt=\pi /2\).

The operations separately performed on block k are similar to that separately performed on block one, where k∈{2,…,m}. Similarly, the ions in block k are labeled by 1,…,nk, and we specify to which block an ion belongs when necessary. For block k, we first implement the transition |11〉12→|aa〉12 of ions 1 and 2 by using the Hamiltonian

$$ H^{\prime}_{12}(t)=\Omega^{\prime}_{12}(t)|{aa}\rangle_{12}\langle{11}| +\Omega^{\ast\prime}_{12}(t)|{11}\rangle_{12}\langle{aa}|. $$
(4)

with the evolution time \(T^{\prime }_{12}\) satisfying \(\int _{0}^{T^{\prime }_{12}}|\Omega ^{\prime }_{12}(t)|dt=\pi /2\). Then we sequentially perform operation j′ on block k, with j′ being from 2 to nk−1. Specifically, for operation j′, we implement the transition \(|{a1}\rangle _{j^{\prime },j^{\prime }+1}\rightarrow |{1a}\rangle _{j^{\prime },j^{\prime }+1}\) of ions j′ and j′+1 of block k by using the Hamiltonian

$$ H^{\prime}_{j^{\prime},j^{\prime}+1}(t)=\Omega^{\prime}_{j^{\prime},j^{\prime}+1}(t) |{1a}\rangle_{j^{\prime},j^{\prime}+1}\langle{a1}| +H.c., $$
(5)

with evolution time satisfying \(\int _{0}^{T^{\prime }_{j^{\prime },j^{\prime }+1}}|\Omega ^{\prime }_{j^{\prime },j^{\prime }+1}(t)|dt=\pi /2\).

After performing the above operations separately on each of the m blocks, we perform operations on the connecting systems, i.e., connecting operations, to connect the m blocks.

We first connect the first m−1 blocks by sequentially implementing the transition \(\phantom {\dot {i}\!}|{ba}\rangle _{n_{k},n_{k+1}}\rightarrow |{ab}\rangle _{n_{k},n_{k+1}}\) of ion nk of block k and ion nk+1 of block k+1, with k being from 1 to m−2. To realize such transitions, one can use the Hamiltonian

$$ H_{n_{k},n_{k+1}}(t)=\Omega_{n_{k},n_{k+1}}(t)|{ab}\rangle_{n_{k},n_{k+1}}\langle{ba}| +H.c., $$
(6)

and the corresponding evolution time needs to satisfy the condition \(\phantom {\dot {i}\!}\int _{0}^{T_{n_{k},n_{k+1}}}|\Omega _{n_{k},n_{k+1}}(t)|dt=\pi /2\). After performing the above connecting operations, we perform an operation that acts on ion nm−1 of block m−1 and ion nm of block m. This operation not only connects blocks m−1 and m, but also generates a π phase, which is a necessary component to the realization of controlled phase gates. Specifically, we implement the transition \(\phantom {\dot {i}\!}|{ba}\rangle _{n_{m-1},n_{m}}\rightarrow -|{ba}\rangle _{n_{m-1},n_{m}}\) with the help of the intermediate state \(\phantom {\dot {i}\!}|{ab}\rangle _{n_{m-1},n_{m}}\). To this end, we use the Hamiltonian

$$ H_{n_{m-1},n_{m}}(t)=\Omega_{n_{m-1},n_{m}}(t)|{ab}\rangle_{n_{m-1},n_{m}}\langle{ba}| +H.c., $$
(7)

with evolution time satisfying \(\phantom {\dot {i}\!}\int _{0}^{T_{n_{m-1},n_{m}}}|\Omega _{n_{m-1},n_{m}}(t)|dt=\pi \).

After performing the operation generated by the Hamiltonian in Eq. (7), we perform the reversed operations. Both the connecting operations and the operations separately performed on each of the m blocks are reversed, making the logical states evolve back while leaving the previously generated π phase to state |1…1〉.

The reversed operations are performed as follows. We first reverse the connecting operations. We sequentially perform the connecting operations generated by Eq. (6), but with both the implementation order and the implemented transitions reversed. Then we reverse the operations separately performed on each block. Specifically, we perform the operations generated by Hamiltonians in Eqs. (1) to (5) again, but with both the implementation order and the implemented transitions reversed. This completes the whole realization procedure. In the following, we demonstrate that the realized gate is a controlled phase nonadiabatic holonomic gate.

To show the holonomic feature, it is instructive to recapitulate what evolution operator of an P-dimensional quantum system plays a holonomy transformation in an L-dimensional subspace. For a P-dimensional quantum system defined by H(t), if there is a time-dependent L-dimensional subspace \(\mathcal {S}(t)\) spanned by the orthonormal vectors \(\{ |{\phi _{l}(t)}\rangle \}_{l=1}^{L}\) that satisfy \(i|{\dot \phi _{l}(t)}\rangle =H(t)|{\phi _{l}(t)}\rangle \), i.e., |ϕl(t)〉=UH(t)|ϕl(0)〉 with \(U_{H}(t) = \mathbf {T} \exp {i\int _{0}^{t}H(t')dt'}\), and if |ϕl(t)〉 satisfies the two conditions: \( \text {(i)} \sum _{l=1}^{L} |\phi _{k}(T)\rangle \langle \phi _{k}(T)| = \sum _{l=1}^{L} |\phi _{k}(0)\rangle \langle \phi _{k}(0)|\), and \(\text {(ii)} \langle \phi _{l}(t)|H(t)|\phi _{l^{\prime }}(t)\rangle =0,\ l,l^{\prime }=1,...,L,\) then the unitary transformation UH(T) is a holonomy matrix on the L-dimensional subspace \(\mathcal {S}(0)\) spanned by \(\{ |{\phi _{l}(0)}\rangle \}_{l=1}^{L}\). Condition (i) indicates that the evolution of the subspace \(\mathcal {S} (t)\) is cyclic, while condition (ii) guarantees that there is no dynamical component in the cyclic evolution.

In our realization, except the operation generated by the Hamiltonian in Eq. (7), each operation has a corresponding reversed operation, which is also performed in the realization. About the operation generated by the Hamiltonian in Eq. (7), it only generates a π phase. Combining the above two features, the logical subspace will evolve back to itself at the end of the evolution. This means condition (i) is satisfied. On the other hand, each of the used Hamiltonians commutates with its own evolution operator. Moreover, each Hamiltonian couples states inside L(t) with that outside L(t), where L(t)=U(t)L(0) with U(t) being the evolution operator and L(0) being the initial logical subspace. The above two features make sure that condition (ii) is satisfied and therefore the evolution is purely geometric. Since both conditions (i) and (i) are satisfied, the realized gate is a nonadiabatic holonomic gate. By checking the functions of the performed operations, one can further conclude that the realized gate is a controlled phase nonadiabatic holonomic gate.

In our realization, besides the connecting operations and the reversed connecting operations, the other operations are performed separately on different blocks. Since different blocks contain different ions, the operaions performed on different blocks can be implemented in parallel. This reduces the operation steps for the realization efficiently. For example, we consider five blocks, of which each contains five ions. By using our proposal, it only needs 17 steps to realize the controlled phase nonadiabatic holonomic gate acting on these ions.

Before ending this section, we further make some discussions about the above realization procedure. We first discuss the physical realization of the used two-body Hamiltonians. We second discuss the change of the realization procedure in the case of dividing all ions into one block. We third discuss the generalization of the realize procedure to other controlled nonadiabatic holonomic gates.

We first discuss the physical realization of the used two-body Hamiltonians. The Hamiltonians in Eqs. (1) and (4) can be realized by using the bichromatic laser mechanism and the specific procedure can be briefly illustrated as follows. Consider two ions and for simplicity denote them as ions 1 and 2 respectively. We drive the transition |1〉⇔|a〉 of ion 1 by a red detuned laser with detuning −(ν+δ) and Rabi frequency ω1(t) and a blue detuned laser with detuning (ν−δ) and Rabi frequency ω2(t). Meanwhile, we drive |0〉⇔|a〉 of ion 2 by a blue detuned laser with detuning ν+δ and Rabi frequency ω3(t), and |1〉⇔|a〉 of ion 2 by a red detuned laser with detuning −(ν−δ) and Rabi frequency ω4(t). Here, ν is the phonon frequency and δ is an additional detuning. In this case, one can realize the Hamiltonian H(t)=iηω1(t)e−iδta†|a〉11〈1|+iηω2(t)e−iδta|a〉11〈1|+iηω3(t)eiδta|a〉22〈0|+iηω4(t)eiδta†|a〉22〈1|, where a and a† are the annihilation and creation operators of the vibrational mode, and η is the Lamb-Dicke parameter that satisfies η2(nν+1)≪1 with nν being the quantum number of the vibrational mode. If the large detuning condition δ≫ηωj(t) is satisfied, the above Hamiltonian can be reduced to \(\tilde {H}(t)=\Omega _{0}(t)|{aa}\rangle \langle {10}|+\Omega _{1}(t)|{aa}\rangle \langle {11}| + \text {H.c.}\), where Ω0(t)=−η2ω1(t)ω3(t)/δ and Ω1(t)=η2ω2(t)ω4(t)/δ. Clearly, the Hamiltonians in Eqs. (1) and (4) are special cases of the above Hamiltonian \(\tilde {H}(t)\). The other used two-body Hamiltonians can also be realized by using the bichromatic laser mechanism. Specifically, consider two ions and label them as 1 and 2 for simplicity. Apply two lasers, one of which is with detuning −(ν+δ) and another is with (ν−δ), to drive the transition |1〉⇔|a〉 of ion 1, and two lasers, one of which is with detuning −(ν+δ) and another is with (ν−δ) to respectively drive the transitions |0〉⇔|a〉 and |1〉⇔|a〉. Then under the large detuning condition, one can have the following Hamiltonian \(\tilde {H}^{\prime }(t)=\Omega ^{\prime }_{0}(t) |{1a_{0}}\rangle \langle {a_{0}0}|+\Omega ^{\prime }_{1}(t)|{1a_{0}}\rangle \langle {a_{0}1}|+\text {H.c.}\). One can see that the other used two-body Hamiltonians are special cases of the Hamiltonian \(\tilde {H}^{\prime }(t)\).

We second discuss the change of the realization procedure in the case of dividing all ions into one block. When N is small, it is suitable to divide the N ions into m=1 block, i.e., all the ions are contained in one block. We label the ions by 1,2,⋯,n1 for simplicity. Since connecting systems are used to connect different blocks and we now only have one block, we do not need connecting systems in this case. Thus for each of ions 1,2,⋯,n1, we use three of its internal states and these three internal states are respectively used as |0〉,|1〉,|a〉 and meanwhile form a Λ structure. To realize the controlled phase nonadiabatic holonomic gate, we first implement the transition |11〉12→|aa〉12 of ions 1 and 2 by using

$$\begin{array}{@{}rcl@{}} H_{12}(t)=\Omega_{12}(t)|{aa}\rangle_{12}\langle{11}|+\Omega^{\ast}_{12}(t)|{11}\rangle_{12}\langle{aa}|, \end{array} $$
(8)

with the evolution time T12 satisfying \(\phantom {\dot {i}\!}\int _{0}^{T_{12}}|\Omega _{12}(t)|dt=\pi /2\). We next sequentially implement the transition |a1〉j,j+1→|1a〉j,j+1 of ions j and j+1 by using the Hamiltonian

$$ {}H_{j,j+1}(t)=\Omega_{j,j+1}(t)|{1a}\rangle_{j,j+1}\langle{a1}| +\Omega^{\ast}_{j,j+1}(t)|{a1}\rangle_{j,j+1}\langle{1a}|, $$
(9)

with j being form 2 to n1−2 and the corresponding evolution time Tj,j+1 satisfying \(\phantom {\dot {i}\!}\int _{0}^{T_{j,j+1}}|\Omega _{j,j+1}(t)|dt=\pi /2\). Then we implement the transition \(\phantom {\dot {i}\!}|{a1}\rangle _{n_{1}-1,n_{1}}\rightarrow -|{a1}\rangle _{n_{1}-1,n_{1}}\) by using the Hamiltonian

$$\begin{array}{@{}rcl@{}} {}H_{n_{1}-1,n_{1}}(t)=\Omega_{n_{1}-1,n_{1}}(t)|{1a}\rangle_{n_{1}-1,n_{1}}\langle{a1}| +H.c., \end{array} $$
(10)

with the evolution time satisfying \(\phantom {\dot {i}\!}\int _{0}^{T_{n_{1}-1,n_{1}}}|\Omega _{n_{1}-1,n_{1}}(t)| dt=\pi \). Finally, we implement the first n1−2 operations again but with both the implementation order and the implemented transitions reversed. This completes the realization procedure. It is noteworthy that when the number of ions is three, i.e., n1=3, the operations generated by the Hamiltonians Hj,j+1(t) are not needed. One only needs to first perform the operation generated by the Hamiltonian H12(t) in Eq. (1), then perform the operation generated by the Hamiltonian \(\phantom {\dot {i}\!}H_{n_{1}-1,n_{1}}\) in Eq. (10), and finally reverse the operation generated by the Hamiltonian H12(t) in Eq. (1).

We third discuss the generalization of the realize procedure to other controlled nonadiabatic holonomic gates. In this case, we divide all the qubits except the target qubit into different blocks. For simpicity, we show the case that all the qubits except the target one are divided by two blocks. We first perform operations separately on blocks one and two. For block one, we label its ions by 1,…,n1 and first implement the transition |11〉12→|aa〉12 of ions 1 and 2; we then sequentially implement the transition |a1〉j,j+1→|1a〉j,j+1 of ions j and j+1, with j being from 2 to n1−1; then we implement the transition \(\phantom {\dot {i}\!}|{a}\rangle _{n_{1}}\rightarrow |{b}\rangle _{n_{1}}\) by resonantly coupling the states |a〉 and |b〉 of ion n1. For block two, we label its ions by 1,…,n2 and first implement the transition |11〉12→|aa〉12 of ions 1 and 2; we then sequentially implement the transition |a1〉j,j+1→|1a〉j,j+1 of ions j and j+1, with j being from 2 to n1−1. After performing operations separately on blocks one and two, we connect these two blocks by implementing the transition \(\phantom {\dot {i}\!}|{ba}\rangle _{n_{1}n_{2}}\rightarrow |{ab}\rangle _{n_{1}n_{2}}\) of ions n1 of block one and n2 of block two. Then we couple ion n2 of block two with the ion encoding the target qubit by using the Hamiltonian H(t)=Ω(t)|aa〉〈b0|+Ω′(t)|aa〉〈b1|+H.c.. The realization of this Hamiltonian is similar to that of \(\tilde {H}(t)\) whose realization has been discussed. One can see that the states |b0〉,|b1〉, and |aa〉 form a Λ structure. It is noteworthy that states |b0〉,|b1〉 are now logical states while state |aa〉 is an auxiliary state. Thus by using H(t), one can perform a holonomic operation on the subpsace spanned by states |b0〉,|b1〉. After performing the operation generated by the Hamiltonian H(t), we perform the reversed operations. We reverse all the previously performed operations except that generated by the Hamiltonian H(t). When performing the reversed operations, both the order of the previously performed operations and the previously implemented transitions need to be reversed. This completes the realization procedure.

3 Realizing gates by using connecting systems with more levels

In the above section, we have demonstrated the use of the four-level connecting systems for realizing multi-qubit controlled nonadiabatic holonomic gates. Because we divide the qubits into different blocks and make the operations acting on different blocks be able to be performed in parallel, our proposal reduces the operation steps of the realization and thereby improves the efficiency of the computation. In this section, we take the connecting system idea one step further to consider using connecting systems with more levels. This is particularly suitable to the case that the number of ions is big. In the following, we demonstrate the realization of controlled phase nonadiabatic holonomic gates with connecting systems described by Fig. 2.

Fig. 2
figure 2

The level configuration of the five-level connecting system. Two lower levels of this system are used as logical states |0〉 and |1〉. The other three levels are used as auxiliary states |a〉,|b〉, and |c〉

Full size image

In the realization, we divide the N ions into x upper blocks. Moreover, the ions of ith upper block are further divided into yi blocks, with i=1,…,x. For the last ion of each block except the last ion of the last block of each upper block, i.e., the last ion of each upper block, we use it as a connecting system and suppose it has the tripod structure in Fig. 1. For the last ion of each upper block, we also use it as a connecting system and suppose it has the structure in Fig. 2. While the four-level connecting systems described by Fig. 1 are used to connect the blocks, the five-level connecting systems described by Fig. 2 are used to connect the upper blocks. For other ions, we suppose each of them has a Λ structure. In the following, we demonstrate the realization of controlled phase nonadiabatic holonomic gates with the above division.

For clarity, we use a specific realization procedure to demonstrate the principle. We consider the case of x=3 and y1=y2=y3=2, i.e., we have three upper blocks and each upper block has two blocks. In this case, we have six blocks in total. Without loss of generality, we suppose the first upper block contains the first two blocks, the second upper block contains the third and fourth blocks, and the third upper block contains the fifth and sixth blocks. For simplicity, we label the ions in the i-th block by 1,…,ni, where ni is the number of the ions in the block and i∈{1,…,6}. We specify to which block an ion belongs when necessary. Ions n1 of block one, n3 of block three, and n5 of block five are connecting systems with the structure in Fig. 1. Ions n2 of block two, n4 of block four, and n6 of block six are connecting systems with the structure in Fig. 2.

In our realization, we first perform operations separately on each block. The operations performed on blocks 1, 3, and 5 are similar. Specifically, for block μ with μ∈{1,3,5}, we first implement the transition |11〉12→|aa〉12 of ions 1 and 2; then we sequentially implement the transition |a1〉j,j+1→|1a〉j,j+1 of ions j and j+1, with j being from 2 to nμ−1; then we implement the transition \(\phantom {\dot {i}\!}|{a}\rangle _{n_{\mu }}\rightarrow |{b}\rangle _{n_{\mu }}\) by resonantly coupling the states |a〉 and |b〉 of ion nμ. On the other hand, the operations performed on blocks 2, 4, and 6 are similar. Specifically, for block ν with ν∈{2,4,6}, we first implement the transition |11〉12→|aa〉12 of ions 1 and 2; then we sequentially implement the transition |a1〉j,j+1→|1a〉j,j+1 of ions j and j+1, with j being from 2 to nν−1. It is noteworthy that operations acting on different blocks can be implemented in parallel.

We next separately connect the blocks in each upper block. To connect blocks s and s+1, we perform the operation acting on ions ns of block s and ns+1 of block s+1, with s∈{1,3,5}. The operation realizes the transition \(\phantom {\dot {i}\!}|{ba}\rangle _{n_{s}n_{s+1}}\rightarrow |{ab}\rangle _{n_{s}n_{s+1}}\). Then we implement the transition \(\phantom {\dot {i}\!}|{b}\rangle _{n_{\nu }}\rightarrow |{c}\rangle _{n_{\nu }}\) by resonantly coupling the states |b〉 and |c〉 of ion nν of block ν, with ν∈{2,4,6}. It is noteworthy that operations acting on different upper blocks can also be implemented in parallel.

Then we connect the three upper blocks by performing operations on the connecting systems described by Fig. 2. Specifically, we first implement \(\phantom {\dot {i}\!}|{cb}\rangle _{n_{2}n_{4}}\rightarrow |{bc}\rangle _{n_{2}n_{4}}\) of ions n2 of block two and n4 of block four; then we implement the transition \(\phantom {\dot {i}\!}|{cb}\rangle _{n_{4}n_{6}}\rightarrow -|{cb}\rangle _{n_{4}n_{6}}\) of ions n4 of block four and n6 of block six with the help of the intermediate state \(\phantom {\dot {i}\!}|{bc}\rangle _{n_{4}n_{6}}\). The last operation generates a π phase, which is necessary for controlled phase gates.

Then we perform the reversed operations. We reverse all the previously performed operations except the operation that generates the π phase. Moreover, for the reversed operations, both the order of the previously performed operations and the previously implemented transitions need to be reversed. Specifically, we first reverse the connecting operation acting on ions n2 of block two and n4 of block four. Then we reverse the operations that separately connect the blocks in each upper block. Finally, we reverse the operations separately performed on each block. This completes the realization procedure. While we use a specific realization procedure to demonstrate the principle, the connecting systems in Fig. 2 is particularly suitable to the case that the number of ions is big, i.e., many upper blocks with each upper block containing many blocks. One can generalize the above realization procedure to realize other controlled nonadiabatic holonomic gates. It is noteworthy one can also generalize the connecting system idea further to consider connecting systems with more than five levels.

4 Conclusion

In conclusion, we have demonstrated the realization of multi-qubit controlled nonadiabatic holonomic gates with connecting systems. By dividing the qubits into different blocks and with the connecting systems, we make the operations acting on different blocks be performed in parallel. Because of this feature, our proposal can efficiently reduce the operation steps. The reduction of steps can reduce the exposure time to the environment and thereby reduce the affection from decoherence. We also show how to realize multi-qubit controlled nonadiabatic holonomic gates with connecting systems with more levels. Our proposal is implemented with trapped ion systems, which are a promising candidate for realizing various quantum information tasks. So, it is useful to realize efficient and robust quantum information processors.