Abstract

Electron and nuclear spins of diamond nitrogen-vacancy (NV) centers are good candidates for quantum information processing as they have long coherence time and can be initialized and read out optically. However, creating a large number of coherently coupled and individually addressable NV centers for quantum computing has been a big challenge. Here we propose methods to use high-density diamond NV centers coupled by spin-spin interaction with an average separation on the order of 10 nm for quantum computing. We propose to use a strain gradient to encode the position information of each NV center in the energy level of its excited electron orbital state, which causes a shift of its optical transition frequency. With such strain encoding, more than 100 closely-packed NV centers below optical diffraction limit can be read out individually by resonant optical excitation. A magnetic gradient will be used to shift the electron spin resonance (ESR) frequencies of NV centers. Therefore, the spin state of each NV center can be individually manipulated and different NV centers can be selectively coupled. A universal set of quantum operations for two-qubit and three-qubit system is introduced by careful design of external drives. Moreover, entangled states with multiple qubits can be created by this protocol, which is a major step towards quantum information processing with solid-state spins.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

E. Bersin, M. Walsh, S. L. Mouradian, M. E. Trusheim, T. Schröder, and D. Englund, “Individual control and readout of qubits in a sub-diffraction volume,” npj Quantum Inf. 5(1), 38 (2019).
[Crossref]

2018 (4)

M. H. Abobeih, J. Cramer, M. A. Bakker, N. Kalb, M. Markham, D. J. Twitchen, and T. H. Taminiau, “One-second coherence for a single electron spin coupled to a multi-qubit nuclear-spin environment,” Nat. Commun. 9(1), 2552 (2018).
[Crossref]

V. M. Schäfer, C. J. Ballance, K. Thirumalai, L. J. Stephenson, T. G. Ballance, A. M. Steane, and D. M. Lucas, “Fast quantum logic gates with trapped-ion qubits,” Nature 555(7694), 75–78 (2018).
[Crossref]

P. C. Humphreys, N. Kalb, J. P. J. Morits, R. N. Schouten, R. F. L. Vermeulen, D. J. Twitchen, M. Markham, and R. Hanson, “Deterministic delivery of remote entanglement on a quantum network,” Nature 558(7709), 268–273 (2018).
[Crossref]

X. Li, Y. Ma, J. Han, T. Chen, Y. Xu, W. Cai, H. Wang, Y. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Perfect quantum state transfer in a superconducting qubit chain with parametrically tunable couplings,” Phys. Rev. Appl. 10(5), 054009 (2018).
[Crossref]

2017 (5)

L. Dong, X. Rong, J. Geng, F. Shi, Z. Li, C. Duan, and J. Du, “Scalable quantum computation scheme based on quantum-actuated nuclear-spin decoherence-free qubits,” Phys. Rev. B 96(20), 205149 (2017).
[Crossref]

C. Song, K. Xu, W. Liu, C.-P. Yang, S.-B. Zheng, H. Deng, Q. Xie, K. Huang, Q. Guo, L. Zhang, P. Zhang, D. Xu, D. Zheng, X. Zhu, H. Wang, Y. A. Chen, C. Y. Lu, S. Han, and J.-W. Pan, “10-qubit entanglement and parallel logic operations with a superconducting circuit,” Phys. Rev. Lett. 119(18), 180511 (2017).
[Crossref]

L. Caspani, C. Xiong, B. J. Eggleton, D. Bajoni, M. Liscidini, M. Galli, R. Morandotti, and D. J. Moss, “Integrated sources of photon quantum states based on nonlinear optics,” Light: Sci. Appl. 6(11), e17100 (2017).
[Crossref]

Y. Ma, T. M. Hoang, M. Gong, T. Li, and Z.-Q. Yin, “Proposal for quantum many-body simulation and torsional matter-wave interferometry with a levitated nanodiamond,” Phys. Rev. A 96(2), 023827 (2017).
[Crossref]

T. Schröder, M. Walsh, J. Zheng, S. Mouradian, L. Li, G. Malladi, H. Bakhru, M. Lu, A. Stein, M. Heuck, and D. Englund, “Scalable fabrication of coupled NV center-photonic crystal cavity systems by self-aligned N ion implantation,” Opt. Mater. Express 7(5), 1514–1524 (2017).
[Crossref]

2016 (2)

A. D. Greentree, “Nanodiamonds in fabry-perot cavities: a route to scalable quantum computing,” New J. Phys. 18(2), 021002 (2016).
[Crossref]

K. W. Lee, D. Lee, P. Ovartchaiyapong, J. Minguzzi, J. R. Maze, and A. C. Bleszynski Jayich, “Strain coupling of a mechanical resonator to a single quantum emitter in diamond,” Phys. Rev. Appl. 6(3), 034005 (2016).
[Crossref]

2015 (3)

Y. Wang, F. Dolde, J. Biamonte, R. Babbush, V. Bergholm, S. Yang, I. Jakobi, P. Neumann, A. Aspuru-Guzik, J. D. Whitfield, and J. Wrachtrup, “Quantum simulation of helium hydride cation in a solid-state spin register,” ACS Nano 9(8), 7769–7774 (2015).
[Crossref]

X. Rong, J. Geng, F. Shi, Y. Liu, K. Xu, W. Ma, F. Kong, Z. Jiang, Y. Wu, and J. Du, “Experimental fault-tolerant universal quantum gates with solid-state spins under ambient conditions,” Nat. Commun. 6(1), 8748 (2015).
[Crossref]

B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, “Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526(7575), 682–686 (2015).
[Crossref]

2014 (4)

F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbrüggen, J. Biamonte, and J. Wrachtrup, “High-fidelity spin entanglement using optimal control,” Nat. Commun. 5(1), 3371 (2014).
[Crossref]

M. W. Dohertyet al., “Electronic properties and metrology applications of the diamond nv− center under pressure,” Phys. Rev. Lett. 112(4), 047601 (2014).
[Crossref]

L. Rondin, J.-P. Tetienne, T. Hingant, J.-F. Roch, P. Maletinsky, and V. Jacques, “Magnetometry with nitrogen-vacancy defects in diamond,” Rep. Prog. Phys. 77(5), 056503 (2014).
[Crossref]

G. Waldherr, Y. Wang, S. Zaiser, M. Jamali, T. Schulte-Herbrüggen, H. Abe, T. Ohshima, J. Isoya, J. F. Du, P. Neumann, and J. Wrachtrup, “Quantum error correction in a solid-state hybrid spin register,” Nature 506(7487), 204–207 (2014).
[Crossref]

2013 (6)

N. Yu, R. Duan, and M. Ying, “Five two-qubit gates are necessary for implementing the Toffoli gate,” Phys. Rev. A 88(1), 010304 (2013).
[Crossref]

F. Dolde, I. Jakobi, B. Naydenov, N. Zhao, S. Pezzagna, C. Trautmann, J. Meijer, P. Neumann, F. Jelezko, and J. Wrachtrup, “Room-temperature entanglement between single defect spins in diamond,” Nat. Phys. 9(3), 139–143 (2013).
[Crossref]

J. Cai, A. Retzker, F. Jelezko, and M. B. Plenio, “A large-scale quantum simulator on a diamond surface at room temperature,” Nat. Phys. 9(3), 168–173 (2013).
[Crossref]

J. Cai, A. Retzker, F. Jelezko, and M. B. Plenio, “A large-scale quantum simulator on a diamond surface at room temperature,” Nat. Phys. 9(3), 168–173 (2013).
[Crossref]

H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339(6119), 557–560 (2013).
[Crossref]

N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, “Solid-state electronic spin coherence time approaching one second,” Nat. Commun. 4(1), 1743 (2013).
[Crossref]

2012 (4)

P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N. Y. Yao, S. D. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. I. Cirac, and M. D. Lukin, “Room-temperature quantum bit memory exceeding one second,” Science 336(6086), 1283–1286 (2012).
[Crossref]

N. Yao, L. Jiang, A. Gorshkov, P. Maurer, G. Giedke, J. Cirac, and M. Lukin, “Scalable architecture for a room temperature solid-state quantum information processor,” Nat. Commun. 3(1), 800 (2012).
[Crossref]

A. Jarmola, V. M. Acosta, K. Jensen, S. Chemerisov, and D. Budker, “Temperature- and magnetic-field-dependent longitudinal spin relaxation in nitrogen-vacancy ensembles in diamond,” Phys. Rev. Lett. 108(19), 197601 (2012).
[Crossref]

M. W. Doherty, F. Dolde, H. Fedder, F. Jelezko, J. Wrachtrup, N. B. Manson, and L. C. L. Hollenberg, “Theory of the ground-state spin of the NV− center in diamond,” Phys. Rev. B 85(20), 205203 (2012).
[Crossref]

2011 (4)

T. M. Babinec, J. T. Choy, K. J. M. Smith, M. Khan, and M. Lončar, “Design and focused ion beam fabrication of single crystal diamond nanobeam cavities,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 29(1), 010601 (2011).
[Crossref]

S. Machnes, U. Sander, S. J. Glaser, P. de Fouquières, A. Gruslys, S. Schirmer, and T. Schulte-Herbrüggen, “Comparing, optimizing, and benchmarking quantum-control algorithms in a unifying programming framework,” Phys. Rev. A 84(2), 022305 (2011).
[Crossref]

S. Machnes, U. Sander, S. J. Glaser, P. de Fouquières, A. Gruslys, S. Schirmer, and T. Schulte-Herbrüggen, “Comparing, optimizing, and benchmarking quantum-control algorithms in a unifying programming framework,” Phys. Rev. A 84(2), 022305 (2011).
[Crossref]

N. Y. Yao, L. Jiang, A. V. Gorshkov, Z. X. Gong, A. Zhai, L. M. Duan, and M. D. Lukin, “Robust quantum state transfer in random unpolarized spin chains,” Phys. Rev. Lett. 106(4), 040505 (2011).
[Crossref]

2010 (3)

G. de Lange, Z. H. Wang, D. Ristè, V. V. Dobrovitski, and R. Hanson, “Universal dynamical decoupling of a single solid-state spin from a spin bath,” Science 330(6000), 60–63 (2010).
[Crossref]

P. Neumann, J. Beck, M. Steiner, F. Rempp, H. Fedder, P. R. Hemmer, J. Wrachtrup, and F. Jelezko, “Single-shot readout of a single nuclear spin,” Science 329(5991), 542–544 (2010).
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2009 (1)

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2008 (2)

D. Press, T. D. Ladd, B. Zhang, and Y. Yamamoto, “Complete quantum control of a single quantum dot spin using ultrafast optical pulses,” Nature 456(7219), 218–221 (2008).
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P. Tamarat, N. B. Manson, J. P. Harrison, R. L. McMurtrie, C. Nizovtsev, A. Santori, R. Beausoleil, P. Neumann, T. Gaebel, F. Jelezko, P. Hemmer, and J. Wrachtrup, “Spin-flip and spin-conserving optical transitions of the nitrogen-vacancy centre in diamond,” New J. Phys. 10(4), 045004 (2008).
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2006 (3)

R. Hanson, F. M. Mendoza, R. J. Epstein, and D. D. Awschalom, “Polarization and readout of coupled single spins in diamond,” Phys. Rev. Lett. 97(8), 087601 (2006).
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2005 (2)

N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, “Optimal control of coupled spin dynamics: design of nmr pulse sequences by gradient ascent algorithms,” J. Magn. Reson. 172(2), 296–305 (2005).
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2004 (1)

F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett. 92(7), 076401 (2004).
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2002 (2)

M. A. Nielsen, “A simple formula for the average gate fidelity of a quantum dynamical operation,” Phys. Lett. A 303(4), 249–252 (2002).
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Figures (10)

Fig. 1.
Fig. 1. Schematic of quantum computing with high-density diamond NV centers via strain and magnetic encoding. (a). By the contact stress from an AFM tip, an inhomogeneous strain field is applied on the diamond cantilever. Such a gradient strain field can shift the optical transition frequencies of the NV centers depending on their positions. An external gradient magnetic field is applied on the diamond to split the ODMR frequencies of each NV center. Well-designed global composite microwave (MW) pulses with multiple frequencies are applied on the diamond to control the electron spins individually. (b). The gradient strain field and the magnetic field are both along the long axis of the cantilever. (c). A NV center is embedded in a diamond cantilever. The red sphere denotes the nitrogen atom and the blue are for the carbon atoms. The gradient strain field and the magnetic field are applied along z axis in the cantilever coordinates, which is [$\bar 1\bar 1\bar 1$] in Miller indices for diamonds. $z'$ denotes the direction of the NV axis. One possible orientation of the NV axis is [$\bar 1\bar 1\bar 1$] as shown in the figure. The other three are [$11\bar 1$], [$1\bar 11$] and [$\bar 111$]. (d). Energy splits of optical transition frequencies and ODMR frequencies, due to strain field interaction and Zeeman effect, respectively. The frequencies of each NV center are designed to be unmatched with any other spins. Individual control and readout can be realized by this approach.
Fig. 2.
Fig. 2. Transition frequencies between each sublevel of NV centers and nitrogen impurities as a function of the external magnetic field, which is along $\left [\bar 1\bar 1\bar 1\right ]$ axis. The black solid lines are transition frequencies between sublevels for NV centers aligned with the external magnetic field, i.e, the NV axis is along $\left [\bar 1\bar 1\bar 1\right ]$. They represent transitions from $\left | m_s=0\right \rangle$ to $\left | m_s=1\right \rangle$ and from $\left | m_s=0\right \rangle$ to $\left | m_s=-1\right \rangle$, from top to bottom, respectively. The red dashed lines are transition frequencies between sublevels for NV centers aligned at an angle of $109^\circ$ with respect to $\left [\bar 1\bar 1\bar 1\right ]$, i.e, NV-axis along $\left [\bar {1}11\right ]$, $\left [1\bar {1}1\right ]$ or $\left [11\bar {1}\right ]$. From top to bottom, the transitions are from $\left | m_s=0\right \rangle$ to $\left | m_s=-1\right \rangle$, $\left | m_s=0\right \rangle$ to $\left | m_s=1\right \rangle$ and $\left | m_s=1\right \rangle$ to $\left | m_s=-1\right \rangle$, respectively. The blue dotted line stands for the transition from $\left | m_s=-1/2\right \rangle$ to $\left | m_s=1/2\right \rangle$ for nitrogen impurities. Three cyan shaded area stand for the applicable external magnetic field without cross relaxation and can be used to manipulate each NV center spin states individually.
Fig. 3.
Fig. 3. (a) Schematic of the diamond cantilever. The strain is applied along z axis in the cantilever coordinates, which is [$\bar 1\bar 1\bar 1$] in Miller indices for diamonds. A red tunable laser is applied to a cluster of NV centers in the cantilever for individual control and readout. (b)-(e) The simulated distribution of strain and strain gradient of the diamond cantilever when it is bent by an external force. The cantilever is taken to be $5$ $\mu$m in length, $0.5$ $\mu$m in width and $0.25$ $\mu$m in height. The maximum value of the stress in the simulation is $487$ MPa, which is much smaller than its ultimate tensile strength. In such condition, the local strain on the cantilever can reach up to $10^{-4}$. The side view ($z-o-x$ plane) of the simulated distribution of strain is shown in (b) and of strain gradient is shown in (c). The top view ($z-o-y$ plane) of the distribution of strain is shown in (d) and of strain gradient is shown in (e).
Fig. 4.
Fig. 4. Detuning optical transition frequencies for NV centers with different orientations as a function of the applied strain field along $[\bar 1\bar 1\bar 1]$. For NV centers with $\hat {z}$ axis of $[\bar 111]$, $[1\bar 11]$ and $[11\bar 1]$ have identical energy for $E_x$ and $E_y$, shown as the blue dashed-dotted line and the green dashed line, respectively. While for NV centers orientated along $[\bar 1\bar 1\bar 1]$, the detuning frequencies for $E_x$ and $E_y$ are shown in the red solid line and the black dotted line, respectively. (a). For strain field in a range from $0$ to $1\times 10^{-5}$. (b) For a larger range from $0$ to $5\times 10^{-4}$.
Fig. 5.
Fig. 5. Schematic for the quantum gates of single-qubit and double-qubit system in diamond. (a) Arbitrary rotation gate $M$ on a single qubit by applying resonant MW pulse and tuning the duration and phase of the pulse. (b) Hadamard gate on a single qubit by three rotations of $\pi /2$. (c). The controlled-Z (CZ) gate on two qubits by two rotations on single qubit and their ZZ interaction. (d) The controlled-NOT gate by a CZ gate and two rotations on single qubit. (e) Under two simultaneous resonant MW pulses on two qubits, XX interaction between two qubits is turned on. The SWAP gate is realized when the duration of the XX interaction is $2/\nu _{dip}$. (f) The square root of SWAP (denoted by $\sqrt {SWAP}$) can be obtained by changing the duration time to be able $1/\nu _{dip}$. (g) The CNOT gate is obtained by two $\sqrt {SWAP}$ gates and seven single-qubit rotations.
Fig. 6.
Fig. 6. Schematic for the Toffoli gate of three-qubit system in diamond.
Fig. 7.
Fig. 7. Optimal control for the CNOT gate for two interacting NV electron spins, with the spin-dependent splitting of $2600$ MHz and $2700$ MHz, respectively. The dipolar coupling strength between two NVs is taken to be $100$kHz. The sequence consists of 50 rectangular independent pulses with the total duration time of 25 $\mu$s. Each pulse consists of four MW controls, with the frequency of the transition frequency between the $m_s=0\rightarrow m_s=-1$ transitions for two NV electron spins and the direction along $x$ and $y$ axis, respectively. All four control fields are applied to the NV-NV system simultaneously during the whole pulse sequence.
Fig. 8.
Fig. 8. Process matrix of CNOT gate for two interacting NV center electron spins. (a) is the process matrix of the ideal CNOT gate in the computation basis defined by tensor products of Pauli operators $\{I, X, Y, Z\}$. (b) is the process matrix of the CNOT gate realized by optimal control shown above (F=0.999). The bar height and the color show the absolute value and the phase of matrix elements in complex numbers form, respectively.
Fig. 9.
Fig. 9. Optimal control for the Toffoli gate for three mutually interacting NV electron spins, with the spin-dependent splitting of $2500$ MHz, $2600$ MHz and $2700$ MHz, respectively. The dipolar coupling strengths between each two NVs are all identical be $100$kHz. The sequence consists of 50 rectangular independent pulse of 1 $\mu$s. Each pulse consists of six control field components and they correspond to the transition frequencies of NV1, NV2 and NV3 along $x$ and $y$ axis. All six MW controls are applied to the three NVs system simultaneously during the whole pulse sequence.
Fig. 10.
Fig. 10. Process matrix of Toffoli gate for three mutually interacting NV centers. (a) is the process matrix of the ideal Toffoli gate in the computation basis defined by tensor products of Pauli operators $\{I, X, Y, Z\}$, of 64 elements. (b) is the process matrix of the Toffoli gate realized by optimal control shown above (F=0.999). The bar height and the color show the absolute value and the phase of matrix elements in complex numbers form, respectively.

Equations (14)

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H N V = [ D + δ c o s θ δ s i n θ e i ϕ 2 0 δ s i n θ e i ϕ 2 0 δ s i n θ e i ϕ 2 0 δ s i n θ e i ϕ 2 D δ c o s θ ]
E x = f Z P L + g A 1 + ( g E 1 + δ f E 1 ) 2 + ( g E 2 + δ f E 2 ) 2 , E y = f Z P L + g A 1 ( g E 1 + δ f E 1 ) 2 + ( g E 2 + δ f E 2 ) 2 ,
g A 1 = λ A 1 ϵ z z + λ A 1 ( ϵ x x + ϵ y y ) , g E 1 = λ E ( ϵ y y ϵ x x ) + λ E ( ϵ x z + ϵ z x ) , g E 2 = λ E ( ϵ x y + ϵ y x ) + λ E ( ϵ y z + ϵ z y ) ,
ϵ 1 ¯ 11 = [ 2 9 7 9 ν 2 3 9 ( ν + 1 ) 2 9 ( ν + 1 ) 2 3 9 ( ν + 1 ) 2 3 1 3 ν 6 9 ( ν + 1 ) 2 9 ( ν + 1 ) 6 9 ( ν + 1 ) 1 9 8 9 ν ] ϵ .
ϵ 1 ¯ 1 ¯ 1 ¯ = ϵ c
H = H A + H B + H C + H A B d i p + H A C d i p + H B C d i p ,
H A B d i p = μ 0 4 π g e 2 μ B 2 r A B 3 [ S A S B 3 ( S A n A B ) ( S B n A B ) ] ν d i p S z A S z B ,
U C N O T = H 1 U C Z H 1
U C Z = e i π / 4 e i S z 1 S z 2 π / 4 e i S z 1 π / 4 e i S z 2 π / 4 .
H i n t = ν v i p 4 ( S 1 + S 2 + S 1 S 2 + ) + Ω 1 S 1 z + Ω 2 S 2 z ,
U T o f f o l i = H 3 U C N O T 23 R z 3 ( π 4 ) U C N O T 13 R z 3 ( π 4 ) U C N O T 23 R z 3 ( π 4 ) R Z 2 ( π 4 ) H 3 U C N O T 12 R z 2 ( π 4 ) U C N O T 12 R z 2 ( π 2 ) R z 1 ( π 4 ) .
P e r r = P T 1 + P T 2 + P m w + P m a g + P s t r + P d i p = t T 1 + t 3 T 2 3 + δ 1 2 Ω m w 2 + Ω m w 2 Δ m a g 2 + Ω o p t 2 Δ s t r 2 + ν d i p 2 Ω m w 2 ,
F = 1 d ( d + 1 ) ( t r ( M M + ) + | t r ( M ) | 2 ) ,
H c = γ e Σ i c o s [ ω i t + ϕ i ( t ) ] B i ( t ) ( S 1 + S 2 ) ,

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