# Single-step implementation of a multiple-target-qubit controlled phase gate without need of classical pulses

###### Abstract

We propose a simple method for realizing a multiqubit phase gate of one qubit simultaneously controlling target qubits, by using three-level quantum systems (i.e., qutrits) coupled to a cavity or resonator. The gate can be implemented using one operational step and without need of classical pulses, and no photon is populated during the operation. Thus, the gate operation is greatly simplified and decoherence from the cavity decay is much reduced, when compared with the previous proposals. In addition, the operation time is independent of the number of qubits and no adjustment of the qutrit level spacings or the cavity frequency is needed during the operation.

###### pacs:

03.67.Lx, 42.50.DvMultiple qubit gates have many applications in quantum information processing (QIP). A multiqubit gate can be decomposed into two-qubit and one-qubit gates and thus can in principle be constructed using these basic gates. However, the number of basic gates increases dramatically as the number of qubits increases. Thus, it becomes difficult to build a multiqubit gate by using the conventional gate-decomposition protocol. Over the past years, many efficient schemes have been proposed for the direct implementation of a multiqubit controlled-phase or controlled-NOT gate with multiple-control qubits acting on one target qubit (e.g., [1-5]). This type of multiqubit gate plays significant roles in QIP, such as quantum algorithms and error corrections.

We here focus on another type of multiqubit gates, that is, a multiqubit phase gate with one control qubit simultaneously controlling multiple target qubits. This multiqubit phase gate is described by

(1) | |||||

where the subscript represents the control qubit while subscripts and represent the target qubits, and The transformation (1) implies that when the control qubit is in the state nothing happens to the states of each target qubit; however, when the control qubit is in the state a phase flip (from the sign to the sign) happens to the state of each target qubit. This multiqubit gate is useful in QIP such as entanglement preparation, error correction, and quantum algorithms.

Several methods have been proposed for the direct implementation of this multiqubit phase gate, by employing two-level or four-level quantum systems coupled to a single cavity or resonator [6-8]. However, these methods require several steps of operation and application of classical pulses so that the gate operation is complex. Moreover, in these schemes cavity photons are populated during the operation and thus decoherence caused by the cavity decay may pose a problem. In addition, for the methods proposed in [7,8], a higher-energy fourth level was employed, which is experimantally challenging.

In the following, we present a new approach for implementing this multiqubit
phase gate using three-level quantum systems (i.e., qutrits) coupled to a
single cavity or resonator. Compared with the previous proposals, the
proposal has these features: (i) only a single step of operation is needed
and no classical pulse is used, thus the operation is greatly simplified;
(ii) no photon is populated, thus decoherence caused by the cavity photon
decay is much suppressed; and (iii)* *it is unnecessary to employ a
fourth level. In addition, the proposal has the following additional
advantages: the operation time is independent of the number of qubits and no
adjustment of the qutrit level spacings or the cavity frequency is required
during the operation.

Consider qutrits labeled by and . Each qutrit has three levels , , and (Fig. 1). Assume that qutrits and are identical, whose levels spacings are different from those of qutrit The cavity mode is coupled to the transition of each qutrit, but decoupled (highly detuned) from the transition between any other two levels (Fig. 1). These requirements can in principle be met by choosing or designing the qutrits (e.g., the level spacings of artificial atoms, such as superconducting quantum devices, can be readily adjusted by varying the device parameters appropriately). The interaction Hamiltonian in the interaction picture and under the rotating-wave approximation is given by

(2) | |||||

where the subscript represents the th qutrit, () is the photon creation (annihilation) operator of the cavity mode with frequency , is the coupling constant between the cavity mode and the transition of qutrits , while is the coupling constant between the cavity mode and the transition of qutrit In addition, , and with the transition frequency of qutrits and the transition frequency of qutrit

For and there is no energy exchange
between the qutrits and the cavity mode. Then the system dynamics described
by the Hamiltonian of Eq. (2) is approximately equivalent to that determined
by the following Hamiltonian [9,10]* *

(3) | |||||

where and The photon number is conserved
during the interaction. When the cavity mode is initially in the vaccum
state, it will remain in this state. Under this condition the photon number
operator in Eq. (3) can be set to be . Furthermore, when the
condition* ** *is satisfied, qutrit 1 does not exchange energy with the
other qutrits. Under these conditions Hamiltonian (3) can be replaced by the
effective Hamiltonian [11]

(4) | |||||

The last term of Eq. (4) describes the effective coupling of qutrits () arising from the far off-resonant coupling to qutrit 1 described by the last term of Eq. (3). When the level of each of qutrits () is not populated, it will remain unpopulated since the total number of qutrits () being in the level remains unchanged under the effetive Hamiltonian . In this case, reduces to

(5) | |||||

which can be further expressed as

(6) | |||||

We use the asymmetric encoding scheme [5]. Namely, for the gate of Eq. (1), the logic state of each qubit is represented by the level , while is represented by the level for qutrit but by for qutrit (). Since is not involved for qutrit the last term in Eq. (6) can be dropped due to . In addition, because is not involved for qutrit the third term in Eq. (6) can be discarded owing to Hence, the Hamiltonian (6) becomes

(7) |

for which the time-evolution unitary operator is

(8) |

where is an unitary operator acting on qutrit while is a joint unitary operator acting on qutrits and which are given by

(9) | |||||

(10) |

Note that and where (). Thus, the operator leads to the transformation

(11) |

For i.e., setting the transformation (11) can be further written as

(12) |

which shows that when qutrit is in the state nothing happens to the states of each of qutrits ; however, when qutrit is in the state a phase flip (from the sign to the sign) happens to the state of each of qutrits . Hence, a multi-target phase gate described by Eq. (1) is realized with qutrits, i.e., the control qutrit and the target qutrits ().

To see the above more clearly, let us consider three qutrits for implementing a three-qubit phase gate. The three-qubit computational basis corresponds to {}. Based on Eq. (12), one can find that the four states and of the qutrits become and , respectively; while the other four states and remain unchanged.

We now give a general discussion on the fidelity of the operation. After taking the dissipation and dephasing into account, the dynamics of the lossy system is determined by the master equation

where is the Hamiltonian in Eq. (2), and with and .* ** *is the photon decay rate of the
cavity is the relaxation rate of the level of qutrit for the decay path , () is the
relaxation rate of the level of qutrit for the
decay path (), and () is the dephasing rate of the level () of qutrit ()

The fidelity of the operation is given by where is the output state for an ideal system (i.e., without dissipation and dephasing) after the entire operation, while is the final density operator of the whole system when the operation is performed in a realistic physical system.

For the sake of definitiveness, let us consider the experimental feasibility of realizing a three-qubit phase gate. Assume that qutrit is in the state qutrits and are in and the cavity mode is in the vacuum state before the gate operation.

Fig. 2 is plotted for the fidelity versus and , without considering the dissipation and dephasing of the whole system. The middle red-color convex surface in Fig. 2 shows that a high fidelity can be achieved for a wide range of and . Without loss of generality, choose , and . With these parameters and the dissipation and dephasing considered, we solve the master equation numerically. As an example, consider qutrits with a ladder-type level structure (available in natural atoms, quantum dots, superconducting phase qutrits, and transmon qutrits). We set , , and , and plot the fidelity as a function of in Fig. 3. The result shows that when a fidelity higher than can be obtained. The fidelity can be further increased by optimizing the system parameters. For and MHz the qutrit decoherence time is s and MHz, which are readily available for superconducting transom qutrits. Decoherence time can be made to be on the order of s for state-of-the-art superconducting transom devices [12-14], and a coupling constant MHz has been reported for a superconducting transmon device coupled to a resonator [15]. For superconducting qutrits, the typical transition frequency between two neighbor levels is between and GHz. As an example, we take GHz, GHz, MHz, and MHz. The corresponding quality factor of the resonator is (a value much lower than required by [6,7]). Note that superconducting resonators with a loaded quality factor have been experimentally demonstrated [16,17].

In conclusion, we have presented an approach for implementing the multiqubit phase gate. As shown above, the operation is greatly simplified and decoherence caused by the cavity photon decay is much reduced when compared with the previous proposals. This work is quite general, and can be applied to a wide range of physical implementation with natural atoms or artificial atoms (e.g., quantum dots, NV centers, or various superconducting qutrits such as flux, phase, charge, and transmon qutrits) coupled to a cavity or resonator.

This work was supported by the Major State Basic Research Development Program of China under Grant No. 2012CB921601, the National Natural Science Foundation of China under Grant Nos. [11074062, 11374083, 11147186, 11175033, 11374054] and the Zhejiang Natural Science Foundation under Grant No. LZ13A040002. C.P.Y. acknowledges the funding support from Hangzhou Normal University under Grant Nos. HSQK0081 and PD13002004 and Hangzhou City for the Hangzhou-City Quantum Information and Quantum Optics Innovation Research Team.

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