# Compact quantum gates on electron-spin qubits assisted by diamond nitrogen-vacancy centers inside
cavities^{1}^{1}1Published in Phys. Rev. A 88, 042323
(2013)

###### Abstract

Constructing compact quantum circuits for universal quantum gates on solid-state systems is crucial for quantum computing. We present some compact quantum circuits for a deterministic solid-state quantum computing, including the CNOT, Toffoli, and Fredkin gates on the diamond nitrogen-vacancy centers confined inside cavities, achieved by some input-output processes of a single photon. Our quantum circuits for these universal quantum gates are simple and economic. Moreover, additional electron qubits are not employed, but only a single-photon medium. These gates have a long coherent time. We discuss the feasibility of these universal solid-state quantum gates, concluding that they are feasible with current technology.

###### pacs:

03.67.Lx, 42.50.Ex, 42.50.Pq, 78.67.Hc## I Introduction

Quantum logic gates are the key elements in quantum computing. It is well known that two-qubit entangling gates can be used to implement any -qubit quantum computing, assisted by single-qubit gates book (); universal (). The family composed of controlled-NOT (CNOT) gates and one-qubit gates is the most popular universal set of quantum gates for quantum computing today cnota1 (); cnota2 (); cnota3 (); CNOT2 (); cnota4 (); weioe (); RenLPL (); longpra (); longprl (); cnota5 (); CNOT-Hybrid (); weipra (); CNOT1 (). The simulation of any two-qubit gate requires at least three CNOT gates and 15 single-qubit rotations 3CNOT1 (); 3CNOT2 (); 3CNOT3 (); 3CNOT4 (); small (). Therefore, projects for realizing a CNOT gate in a solid-state system are highly desired for quantum computing in the future.

An optimal unstructured quantum circuit for any multi-qubit gate requires CNOT gates 3CNOT4 (). In the domain of a three-qubit case, people pay much attention to Toffoli Toffoli () and Fredkin gates Fredkin (). {Toffoli (Fredkin) gate, Hadamard gates} is a universal set for multi-qubit quantum computing Toffoli (); Fredkin (). It is usual much more complex and difficult to realize a Toffoli gate or a Fredkin gate with CNOT and one-qubit gates in experiment because it requires at least six CNOT gates Toffolicost () to synthesize a Toffoli gate and it requires two CNOT and three controlled- gates Fredkincost () to synthesize a Fredkin gate. It is particularly interesting to discuss the physical realization of a Toffoli gate and a Fredkin gate in a simpler way.

Quantum gates on solid-state systems have attracted much attention as they have a good scalability, and it has been demonstrated for superconducting qubits superPRL (); superPRB (); super () and quantum dots QD (). Electron-spin qubits in solid-state systems, in particular, associated with nitrogen-vacancy (NV) defect centers, are particularly attractive.

The negatively charged NV defect center occurs in the diamond lattice consisting of a substitutional N atom and an adjacent vacancy, and is one of the most attracting and promising solid-state candidates for quantum information processing, due to the long room-temperature coherent time (1.8 ) coherence1 () that can be manipulated and coupled together in a scalable fashion. The procedures have been established for optical initialing, optical preparing, fast microwave or magnetic manipulating, and optical detecting the long-lived spin triplet state associated with NV centers manipulate2 (); manipulate3 (); manipulate4 (); manipulate5 (); manipulate6 (); ODMR2 ().

Tremendous theoretical and experimental progress has been made on quantum information processing based on NV centers. The schemes for the quantum entanglement generation between a photon and an NV center photon-NV (), and between electrons associated with NV centers NV-NV1 (); NV-NV2 (); NV-NV3 (); NV-NV4 (); NV-NV5 () were proposed. Recently, the schemes for the quantum state transfer between separated NV centers were introduced transfer1 (); transfer2 (); transfer3 (). Multiqubit quantum registers associated with separated NV centers in diamonds have been proposed NV-NV1 (); NV-NV2 (); transfer1 (). Hyperentanglement purification and concentration of two-photon systems in both the spatial-mode and polarization degrees of freedom were investigated renhyperepp () with the assistance of diamond NV centers inside photonic crystal cavities. Yang et al. CCPF () proposed a scheme for implementing the conditional phase gate between NV centers assisted by a high-Q silica microsphere cavity. As the electron spin of the NV defect center couples to nearby C nuclear spins, a high-fidelity polarization and the detection of the single-electron and nuclear-spin states can be achieved, even under ambient conditions detect-nuclear2 (); detect-nuclear3 (); detect-nuclear4 (); detect-nuclear5 (), which allows quantum information transfer register1 (); register2 (); nuclear-nuclear (), entanglement generation between an electron-spin qubit and a nuclear-spin qubit electron-nuclear (); elec-register () and between two nuclear spins nuclear-nuclear (), and the construction of the quantum gate between an electron and a nuclear spin CROT ().

In 2011, Chen et al. NV-NV3 () proposed a composite system, i.e., a diamond NV center with six electrons from the nitrogen and three carbons surrounding the vacancy, which is confined in a microtoroidal resonator (MTR) MTR () with a quantized whispering-gallery mode (WGM). This system allows for an ultrahigh- and a small mode volume of WGM microresonators ultrahigh1 (); ultrahigh2 (); ultrahigh3 (). When the MTR couples to the fiber, the ultrahigh-Q is degraded. The experiments in which a diamond NV center couples to WGMs in a silica microsphere microsphere1 (); microsphere2 (); microsphere3 (), diamond-GaP microdisk microdisk (), or SiN photonic crystal crystal () have been demonstrated. The photon input-output process of a coupled atom and MTR platform has been demonstrated in experiment MTR ().

It is important to construct compact quantum circuits for universal quantum gates because they reduce not only time but also errors. In this paper, we investigate the possibility of constructing compact universal quantum gates for a deterministic solid-state quantum computing, including the CNOT, Toffoli, and Fredkin gates on the diamond NV centers confined in cavities, by some single-photon input-output processes. The qubits of these deterministic gates are encoded on two of the electron-spin triple ground states associated with the diamond NV centers, and they have a long decoherence time even at the room temperature. Our quantum gates on NV centers are obtained by interacting a photon with the NV centers, detecting the emitting photon medium, and applying some proper feedforward operations on the electron-spin qubits associated with NV centers. Our quantum circuits for these gates are compact and economic. The CNOT and Toffoli gates are particularly appealed as the photon medium only interacts with each electron qubit one time. Compared with the synthesis programs, our schemes are simple. In our proposals, auxiliary electron-spin qubits are not required and only one photon medium is employed, which is different from the quantum gates on moving electrons based on charge detection CNOT1 () and the photonic quantum gates based on cross-Kerr nonlinearities CNOT2 (). With current technology, these universal solid-state quantum gates are feasible. If the photon loss, the detection inefficiency, and the imperfection of the experiment are negligible, the success probabilities of our gates are 100%.

This article is organized as follows. In Sec. II, we introduce the photon-matter platform based on the diamond NV center coupled to a resonator and the compact quantum circuit for a deterministic CNOT gate on two separated diamond NV centers. Subsequently, the quantum circuits for constructing three-qubit Toffoli and Fredkin gates on three separated diamond NV centers in a deterministic way are given in Secs. III and IV, respectively. The fidelities and efficiencies of our proposals are estimated in Sec. V. Finally, we discuss the feasibility of our universal quantum gates and give a summary in Sec. VI.

## Ii Two-qubit controlled-not gate on an NV-center system

### ii.1 A diamond NV center coupled to an MTR with a WGM

The electron-spin triple ground states of an NV center are split into (denoted by ) and (denoted by ) by 2.88 GHz with zero-field, due to the spin-spin interactions split (). The structure of the excited states is relatively complex, and it includes six excited states defined by the method of group theory photon-NV (), , , , , , and , owing to NV center’s C symmetry, spin-spin, and spin-orbit interactions in the absence of external magnetic field or crystal strain. Here, , and are the orbital states, and has angular momentum projections along the NV axis.

In our work, the quantum information of the quantum gate is encoded on the spins of the electronic ground triple states and . The -type three-level system (see Fig.1) is realized by employing one of the specific excited state as an ancillary state photon-NV (). The -type system in which optical control is required, can be obtained by using a particular magnetic field to mix the ground states magnetic (). Alternatively, it is possible to find a -type system at zero magnetic field as the inevitable strain in diamond reduces the symmetry and primarily modifies the excited-state structure according to their orbital wave functions. The excited state is separated into two branches strain1 (); strain2 (), , , , and , , at moderate and high strain. Togan et al. photon-NV () demonstrated that the state is robust to low strain and magnetic fields due to the stable symmetric properties, and it decays with an equal probability to the ground-state sublevels through a left circularly polarized radiation () and to through a right circularly polarized radiation (). That is, the zero phonon line (ZPL) was observed after the optical resonant excitation at 637 ( driven by a -polarized photon and driven by a -polarized photon). The mutually orthogonal circular polarization will be destroyed by high strain. The preparation and measurement of the electron spin can be realized by exploiting resonant optical excitation techniques. As illustrated in Ref. photon-NV (), the electron spin can be polarized by first preparing the electron spin to by means of optical pumping with a 532- light, and then transferring the population to either by means of microwave pulses. The spin can be a high-fidelity (93.2%) readout and addressed at low temperature (T=8.6K) based on spin-dependent optical transitions. The state connects , and connects , after spin manipulation by a microwave pulse and resonant excitation transition . The presence or absence of fluorescence decay reveals the spin state photon-NV (); elec-register ().

The Heisenberg equations of the motion for the annihilation operator of the cavity mode and the lowing operator of the NV center operation and the input-output relation for the cavity are given by QObook ()

(1) |

where , , and are the frequencies of the cavity, the single photon, and the NV center, respectively. and are the cavity input and output operators, respectively. is the inversion operator of the cavity. is the decay of the NV center. is the damping rate of the cavity. is the coupling rate. is the vacuum input field felt by the NV center with the commutation relation .

In a weak excitation, i.e., taking , the adiabatical elimination of the cavity mode leads to the reflection coefficient of the NV center confined in the cavity as Hersenberg (); Hu ()

The phase shift and the amplitude of the reflected photon are a function of the frequency detuning , with . For , i.e., when the cavity mode resonant with the NV center interacts with the resonant photon pulse, one can obtain Hersenberg ()

(3) |

Here, is the reflection coefficient of the cold (or the empty) cavity, that is, and the cavity is not coupled to the NV center. is the one for the hot cavity, i.e., . Therefore, the change of the input photon is summarized as NV-NV3 ()

(4) |

The effect of the coupling strength on the amplitude of the reflected photon and that of the frequency detuning on the phase shift have been discussed in NV-NV3 (). Chen et al. NV-NV3 () showed that when with ,

(5) |

That is, Eq. (II.1) becomes

(6) |

From the -type diamond NV-center optical transition depicted by Fig. 1, one can see that it requires a polarization-degenerate cavity mode. Therefore, it is suitable for not only WGM microresonators NV-NV2 (); NV-NV3 (); MTR (); degenerate2 (), but also H1 photonic crystals unpolarized-photon1 (); unpolarized-photon2 (), micropillars unpolarized-pillar1 (); unpolarized-pillar2 (); unpolarized-pillar3 (), and fiber-based fiber-based () cavities.

In our work, all the devices work under the resonant condition . In the following, we first consider the case , that is, , and then we discuss the effect of on the the fidelities and the efficiencies of our universal quantum gates on NV-center systems.

### ii.2 Compact quantum circuit for a two-qubit controlled-not gate on an NV-center system

Our quantum circuit for a CNOT gate on two NV centers is shown in Fig. 2 . The two NV centers are initially prepared in two arbitrary superpositions of the two ground states and ; that is,

(7) |

Here . The subscripts and stand for the control qubit NV and the target qubit NV, respectively. The single-photon medium is initially prepared in the equal superposition of and ; that is,

(8) |

Polarizing beam splitter PBS splits the input single photon into two wave-packets. The component transmits through PBS and then arrives at PBS directly, while the component is reflected to spatial model 2 for interacting with NV, which induces the transformation . Here and the after, the subscript of (or , ) stands for the spatial mode from where the -polarized photon (-polarized photon) emits. After the and the wave packets arrive at PBS simultaneously, the photon emits from spatial mode 4. The specific evolution process of the whole system composed of the input photon and two NV centers can be shown as follows:

(9) | |||||

From Eq. (9), one can see that the balanced Mach-Zehnder (MZ) interferometer composed of PBS, NV, and PBS completes the operation

(10) |

in the basis .

Next, the photon passes through a half-wave plate HWP whose optical axes is set at 22.5 to complete the Hadamard gate () on the polarization photon,

(11) |

That is, after an , the state of the whole system becomes

(12) | |||||

PBS transforms the wave packet into , and transforms into for interacting with NV and then it reaches PBS simultaneously with . Before and after the photon passes though NV, a Hadamard operation is performed on NV, respectively. According to Eq. (10), one can see that the above operations () complete the transformation as

(13) | |||||

Here Hadamard operation completes the following transformations:

(14) |

From Eq. (13), one can see that to complete the CNOT gate on two NV centers, which implements the transformation

(15) | |||||

after the photon is detected by the detector or in the basis , some proper single-qubit operations shown in Tab. 1 should be performed on the control qubit and the target qubit, respectively. Therefore, the quantum circuit shown in Fig. 2 performs the CNOT gate on two NV centers, which flips the state of the target electron qubit in NV if and only if (iff) the control electron qubit in NV is in the state . This gate works with a success probability of 100% in principle.

Feed-forward | ||
---|---|---|

photon | control qubit | target qubit |

## Iii Solid-state Toffoli gate on a three-qubit NV-center system

A Toffoli gate is used to complete a NOT operation on the state of the target qubit when both two control qubits are in the state ; otherwise, nothing is done on the target qubit. The principle for implementing a Toffoli gate on a three-qubit NV-center system is shown in Fig. 3. Suppose the first control qubit in the defect center NV, the second control qubit in the defect center NV, and the target qubit in the defect center NV are prepared in three arbitrary superposition electron-spin states as follows:

(16) |

Here, .

In order to describe the principle of our Toffoli gate on a three-qubit NV-center system explicitly, we specify the evolution of the system as follows.

An input single-photon medium in the equal polarization superposition state passes though a balanced MZ interferometer composed of PBS, NV, and PBS described by Eq. (10), and then an (with ) is performed on it. PBS transforms into , and transforms into . The evolution of the total states induced by the above operations () can be described as follows:

(17) |

Before and after the photon emitting from spatial model 6 (5) passes through a balanced MZ interferometer composed of PBS, NV, and PBS (PBS, NV, and PBS), an is performed on it, respectively. These processes ( and ) complete the transformation . Here

The transformation of can be described by Eq. (10), and can be written as

(19) |

in the basis . When the photon emits from spatial mode 10, it reaches the 50:50 BS directly. When the photon emits from spatial mode 9, before it reaches the 50:50 BS, it passes through a balanced MZ interferometer composed of PBS, NV, and PBS described by Eq. (10), and an is performed on the defect NV before and after the photon transmits through it, respectively. The above operations () complete the transformation as

(20) | |||||

Next, the wave packet emitting from spatial 11 interferes with the wave packet emitting from spatial 10 at the BS, which implements the transformations

(21) |

will be transformed into the state

(22) | |||||

The photon medium is measured in the basis by the detector or . Following with the feedforward operations performed on the NV centers, shown in Table 2, we accomplish the construction of the Toffoli gate on the three NV centers in a deterministic way. That is, the state of the system composed of the three defect NV, NV, and NV becomes

(23) | |||||

From the processes above, one can see that the setup shown in Fig. 3 completes the transformation,

(24) | |||||

That is, the setup shown in Fig. 3 realizes exactly the Toffoli gate on the three-qubit NV-center system, which flips the state of the target qubit iff both the two control qubits are in the state .

Feedforward | |||

photon | qubit | qubit | qubit |

## Iv Solid-state Fredkin gate on a three-qubit NV-center system

A Fredkin gate is used to exchange the states of the two target qubits iff the control qubit is in the state . Our quantum circuit for implementing a Fredkin gate on a three-qubit NV-center system in a deterministic way is shown in Fig. 4. The control qubit encoded on NV center “NV”, the first target qubit encoded on NV center “NV”, and the second target qubit encoded on NV center “NV” are initially prepared in three arbitrary states