Quantum computer based on color centers in diamond
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This document summarizes research on developing a quantum computer based on color centers in diamond. Key points include: - Nitrogen vacancy (NV) centers in diamond can function as qubits at room temperature and have demonstrated quantum properties needed for a quantum computer. - While progress has been made in areas like quantum sensing, scaling up NV centers to build a full quantum computer remains challenging due to fabrication and control limitations. - The document reviews various proposals for controlling and entangling NV center spins, performing readout, and fabrication methods, but acknowledges that most proposals are still in early stages and scaling up faces significant hurdles. It provides an overview of the status and challenges of developing a diamond-based quantum computer.
![change of state in the sense of quantum mechanical rules. Here, a con-
nection with the classical world takes place, and the system can no lon-
ger be regarded as isolated. If possible, the measurement can be made
as a so-called “single-shot readout.”26
This is not a single measure-
ment, but further measurements should not change the result. For
example, it is possible to read out a nuclear spin using an NV center
several times without changing the state of the nuclear spin.26
Only two defect centers in diamond exhibit properties that make
controlling their spin states at room temperature possible: the NV and
the ST1 centers.27,28
At low temperature, the spin state of SiV centers
can be coherently controlled as well, but their T2 times are very
short.29
With regard to the state of the art, the NV center remains the
main candidate for coherently controlling single spin. It is composed
of a nitrogen atom with a nuclear spin of 1 (14
N) or 1=2 (15
N) and a
first-neighbor carbon vacancy. In the negatively charged state of the
NV, the electrons build a triplet state with a zero-field splitting of
2.87 GHz.30
The main advantage is that the m ¼ 0 spin state exhibits
different behavior during optical cycles than the m ¼ 61 spin state.
Thus, an initialization as well as a readout of the spin state is possible.
In a two-qubit scheme, the m ¼ 0 state can be defined as a j0i state
and m ¼ –1 as a j1i qubit state.
Figure 1 shows the scheme of the energy levels of the NV-
center.
The center has been well described in the literature.17,30,31
Compared
with quantum dots, the NV center is a kind of artificial atom with sim-
ilar, well-reproducible properties that can be created via ion implanta-
tion or during diamond growth.
B. Quantum properties of single-color centers
The quantum properties are defined by the Hamiltonian of the
NV center,30
H ¼ D Sz þ cS
~
S ~
B þ~
S
A~
B þ cI
~
I ~
B: (1)
Typically, charged color centers possess a singlet ground state
and are transformed into a triplet by excitation.32
An example is the
ST1 (or L2 center).27,33
Spin polarization is obtained when a state of
the triplet is preferentially occupied. For the NV center, the situation is
the opposite. In the ground state, it is in a triplet state, and the singlet
is reached via intersystem crossing from the excited state (Fig. 1). The
spin polarization then takes place exactly via the singlet state toward
which the transition rates for m ¼ 1 and m ¼ –1 are significantly
stronger than for m ¼ 0.34,35
Because the singlet decays back into the
triplet states with similar transition rates, the spin polarization likely
occurs due only to the first decay from the excited triplet to the sin-
glet.36
Therefore, after only a few excitation cycles, a spin polarization
with a probability of 98% is achieved. As is shown in Eq. (1), even
when no external magnetic field is applied, the ground state is also split
by a spin orbital interaction with an energy of 2.87GHz h. The polari-
zation of the ground state is also given by the spin orbital interaction.
The spin state can be controlled by applying microwave pulses. The NV
center is characterized by a long coherence time because phonon interac-
tion is suppressed. In an isotopically pure diamond, coherence times
(T2) of several milliseconds are achievable at room temperature. Figure 2
shows Rabi oscillations driven on a single NV center. A p-pulse leads to
a conversion of the state m¼ 0 (e.g., qubit j0i state) to m¼ –1 (e.g.,
qubit j1i) and can be realized in 7ns.37,38
Other centers, like the SiV
or ST1, do not exhibit such a long coherence time.27
The properties of the NV can be improved by antenna structures
or even a F€
orster transfer.39
However, it is not possible at present to
electrically excite and initialize NVs. This is an obstacle to the produc-
tion of sensors and limits the design of NV-based quantum computers.
Thus, only all NV centers can be initialized in a register at the same
time. While initialization is performed by using a laser pulse, gate oper-
ations are achieved by microwave pulses. The simplest variant involves
determining the Rabi frequency (Fig. 2) and defining different gate
operations via the microwave pulse length in combination with the
defined waiting times.40
Two-gate operations are extended in a similar
way, where a conditional gate operation is achieved via a small band-
width of the microwave pulses depending on the state of the second
qubit. These operations are also performed in combination with the
defined waiting times.41,42
The times at which the gate operations can
be performed depend on the intensity of the microwaves, the coupling
with the NV, and the interference due to the bandwidth. Typical gate
times for a p-pulse are approximately 10 ns.43,44
With a coherence time
of a few milliseconds, 104
gate operations can be carried out. Although
the number of operations appears to be large, many gate operations are
required, e.g., for error correction or for SWAP gates on the nuclear
spin (see Sec. III). The number of gate operations can be improved by
extending the T2 times, actively decoupling the environment and paral-
lelizing the gate operations. The corresponding concepts are presented
in Sec. III. Figure 3 shows the detected magnetic resonance spectra of
an NV center with hyperfine coupling to 14
N in the ground state. This
coupling allows researchers to swap an electron’s spin state with the
nuclear spin state to prolong coherence times.
C. Defects and Fermi level adjustment in diamond to
optimize properties of NV centers
To use the color centers as qubits, it is necessary to control their
crystal environment. The quality of the crystal structure, and the pres-
ence of foreign atoms or defects with unsaturated electron spins and
1.945 eV
3A2
1A1
ms=0
ms
+1
+1
–1
–1
0
0
3E
FIG. 1. Energy level scheme of the NV-
center. The ground and excited states are
triplet states. There is a 30% mixing by the interband coupling of m ¼ 61 with the
singlet state and the m ¼ 0 in the ground state. This behavior leads to an easy and
efficient polarization of the spin state to m ¼ 0 by optical pumping [Weber et al.,
“Quantum computing with defects,” Proceedings of the National Academy of
Sciences of the United States of America 107(19), 8513–8518 (2010), licensed
under a Creative Commons Attribution (CC BY) license].4
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![in the coherence time because, in addition to the NV, defect centers
are activated and free spin states develop. An improvement by means
of electrical excitation would be advantageous. A first attempt in this
direction has recently been successful with a Forster excitation.39
To
achieve a universal set of gates, it is necessary to realize a NOT gate as
a single-qubit gate operation and a controlled-NOT (CNOT) as a two-
qubit universal gate.2
A coherent spin state switching of the NV can
easily be realized by a microwave pulse of a certain length and inten-
sity, the so-called p-pulse. This p-pulse represents the NOT gate. The
resonant frequency is defined by the spin state of nearby centers or
nuclear spins, because of which a narrow-band microwave p-pulse
allows for the realization of a controlled rotation (CROT) or, in combi-
nation with an additional rotation, a CNOT gate. The CROT gate is,
for NV centers, equivalent to the CNOT gate. Only if the controlled
qubit is in the j1i state is the target gate resonant with the microwave
field and the target qubit flips. Of course, the gate requires a linewidth
that is small enough. Figure 5(a) shows the scheme of an interaction
between NV centers with different orientations by means of magnetic
dipole–dipole interaction. The different orientations lead to different
splittings of the magnetic resonances as detected by optically detected
magnetic resonance (ODMR) [Fig. 5(b)]. Thus, each NV can be
manipulated individually. The magnetic dipole field interaction is
related to the quantum state m. A control qubit can only be flipped if
the target qubit is in the correct state; otherwise, the microwave fre-
quency for the p-pulse does not match the transmission frequency.
The single-shot procedure has been carried out using nuclear
spins.26,67
In this method, the spin state of the NV center is exchanged
with the state of the nucleus. The spin state can be transferred by a
SWAP gate process. The SWAP process can be achieved, e.g., by three
CNOT gate operations. Figure 6 shows a scheme of qubit teleporta-
tion. In this case, the nuclear singlet spin is transformed into a triplet
state jþi or j–i using double quantum transitions, which requires the
use of five CNOT gates. This enables the reading of the spin state sev-
eral thousand times without changing the nuclear spin state. This is
the only way to successfully perform an optical or an electrical
readout.
The best coherence times are currently possible in slightly n-
type-doped 13
C free diamonds,10,68,69
and can reach 2 ms at room tem-
perature. Using nuclear spins, this time can be extended to the order
of seconds.70
Assuming a process time of the gate operation of 10 ns,
several 107
gate operations can be performed, including nuclear spin.
However, error correction requires several 1000 gate operations.
Possible “flying qubits” are realizable by photons,71
phonons,72
or elec-
trons.73,74
None of the previously proposed methods can ensure reli-
able operation. An alternative is to use a small number of qubit NVs
as a register and build a chain of NVs. The transfer operations, how-
ever, strongly limit the number of qubits. The realization of flying
qubits is one of the biggest hurdles to building a fully functioning
quantum computer with more than 1000 qubits.
B. Performance
The qubit systems realized in diamond or SiC are characterized
by excellent quantum control, and single qubit gates with a fidelity of
99.99% and double qubit gates with that of 99.2% have already been
demonstrated.75
Entanglement and high-quality quantum algorithms
in chips with up to 10 quantum bits have also been demonstrated,
with coherent control for up to 30 spins76
(see Sec. IIII). Thus, solid-
state spin systems are already comparable to superconducting quan-
tum bits and better than competing systems, as shown in Table I.
Spins of dopants in diamond or SiC are among the most control-
lable quantum bits in solids.77
They can be manufactured deterministi-
cally within a spatial accuracy of a few nanometers.64
Electron or
nuclear spins are available as qubits. Their coherence times are in the
range of a few seconds depending on the type of qubit.75
One- and
two-qubit gate operations with a precision of 99.95% have been dem-
onstrated.41
The initialization or readouts of qubits with qualities
Fluorescence
(kcounts
s
–1
)
2.75
45
2.80 2.85 2.90 2.95 3.00
MW frequency (GHz)
(a) (b)
FIG. 5. (a) Interaction of the electron spins of two NV centers. (b) Optically detected magnetic resonance ODMR spectra of two coupled NV centers. The outer pairs of transi-
tions correspond to NV1 and the inner pairs to NV2. The splitting of 3 MHz within a pair occurs owing to hyperfine coupling with the 15
N nucleus. Spin transitions of distant NV
centers are separated by 30 MHz. Reproduced with permission from F. Dolde et al., “High-fidelity spin entanglement using optimal control,” Nat. Communications 5, 3371
(2014).41
electron
p
p
p
p
p
0
+
–
+1/2
–1/2
nuclear
FIG. 6. Swap scheme to transfer the spin state from electron spin to nuclear spin.
Reproduced with permission from F. Dolde et al., “High-fidelity spin entanglement
using optimal control,” Nat. Communications 5, 3371 (2014).41
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![higher than 95% have also been demonstrated,26
and the entanglement
of several electron quantum bits, or “entanglement swapping” on adja-
cent quantum bits, has been proven.42
NV centers are one of a few
quantum systems that allow a repetitive error correction.78
Other
quantum algorithms, such as the quantum Fourier transformation or
algorithms for the simulation of simple molecular compounds, have
been proposed79
and have yielded pulse sequences for quantum gates
with qualities above the fault tolerance threshold.80
The strong cou-
pling of defects is the essential basis for scalability.81
C. Error correction
To operate a quantum computer, error correction is neces-
sary.88,89
Different error correction methods have been developed and
can all be applied to NV qubits. The simplest methods are based on
five or seven qubits,90
and complex ones are based on up to 100 qubits.
The choice of error correction protocol depends on the quality of the
qubits. With a fidelity of 99.99%, only a few qubits are sufficient. If
the fidelity drops, a large number of qubits is needed to achieve suffi-
cient correction.91
The main task is therefore to achieve a fidelity as
high as possible. For the NV, an average gate fidelity of a single qubit
has been measured to be 0.999 952(6), and that of a two-qubit-CNOT
gate reaches 0.992(1) at room temperature. NV centers are one of the
few systems where the concept of error correction has been success-
fully demonstrated. Three-qubit systems were tested at room tempera-
ture91
and exhibited a very high fidelity at 4 K.92
D. Magnetic dipole coupling of electronic NV–NV and
nuclear NV–13
C spins
To couple qubits, an interaction must exist or have existed. For
NV spin systems, the coupling is achieved by magnetic dipole–dipole
interaction. The field strengths of single spins are extremely high in
the near field. Magnetic field strengths of 1 T can be achieved at a dis-
tance of 2 nm. The dipole–dipole interaction is given by
Hdip ¼
l0c2
e
4pr3
AB
SASB 3 SA þ nAB
ð Þ nABSB
ð Þ
½ : (2)
Because the dipole–dipole interaction is dependent on r3
, the lateral
resolution in which an optimal structure of a quantum computer
needs to be achieved is very strongly limited. The NV–NV coupling
has an energy of h 10 kHz for a 30 nm separation. For the NV–13
C
coupling (hyperfine interaction), this is about h 12 MHz if the 13
C is
located within the first shell. The couplings of it with 14
N and 15
N can
be carried out at 2 and 5 MHz, respectively, in the ground state. A too
weak coupling strength leads to long switching times for two-qubit
gates, such as a CNOT, while a too high coupling strength reduces the
coherence times of the qubits.55
Relaxation times due to spin flip-flop
depend on r6
. A trade-off therefore needs to be found between these
extremes. An optimal value for the NV–NV coupling is likely reached
at a distance of 10 to 20 nm. This corresponds to a coupling strength
of a few MHz and can be used to realize short switching times. This is
the only way to achieve a sufficient number of switching times within
the coherence time. A disadvantage of solid-state systems is that the
coupling strength cannot be adjusted and is determined only by the
distance between the NVs. By clever selection of the spins, like double
quantum transition (DQT), the value can be quadrupled with the dis-
advantage, however, that the coherence time is halved.93
However, as
discussed in Sec. III E, one can extend the coherence times T2
to T2,
for example, from a value of T2
of 22 ls to that of T2 of 500 ls,94
by
dressing modes and refocusing procedures. Another possibility is the
inclusion of nuclear spins (Fig. 7). Nuclear spins have a magnetic field
three orders of magnitude lower, and the coupling is correspondingly
smaller. However, this lower coupling also leads to a greater decou-
pling to the environment, which results in a long coherence time. The
T2 times can reach up to seconds at room temperature. Nuclear spins
are therefore ideal quantum memories.91
Figure 7 illustrates this kind
of coupling between one NV and three nuclear spins.
E. Active decoupling methods of NV or nuclear qubits
from the environment
The importance of the coherence time T2 has already been noted
several times. The QC can be used only within T2. The width of the
lines and, thus, the design of the CNOT gates also depends on the
coherence time. The coherence time is limited by disturbances, which
in most cases are owing to unsaturated spin systems such as 13
C
nuclear spins, but can also be caused by disturbances outside the sam-
ple. Because the coupling to the spin system is mainly affected by the
magnetic components of the electromagnetic field, only disturbances
caused by magnetic fields are to be considered. These magnetic distur-
bances from outside can have certain defined frequencies but also a
wide noise spectrum, accordingly.95
In the first case, refocusing is pos-
sible using known pulse sequences, such as the XY8 pulse sequence.
The limited coherence time caused by noise only is called T2, and the
limited coherence time caused by defined disturbances is called T2
.
T2
is therefore always shorter than T2. To prevent interference from
direct spin couplings, such as 13
C nuclear spins, double frequency exci-
tations are possible. This method is also called spin state dressing and
allows for the decoupling of nuclear spins from the NV center,9
but
can also be used to prevent electron spin crossover during gate opera-
tions of nuclear spin.70
In the latter case, a CROT gate has been built
to achieve the coupling of an NV with nine nuclear spins.75
Another
possibility is to polarize the spin bath to achieve a defined constant
magnetic field. The polarization of the spin bath is also called hyperpo-
larization, where this method is known in the nuclear magnetic reso-
nance (NMR) technique.96
The spin of an NV center is used to
polarize the surrounding dark spins. To achieve polarization, an opti-
mal coupling from the NV center to the nuclear spins is necessary for
a certain time. This is achieved by a Hartmann–Hahn condition, or by
means of the P1 centers (substitutional neutral nitrogen atoms).97
TABLE I. Comparison of benchmarks for different solid-state systems.
Qubit
(host) T2 [ms]
Benchmarking
one-qubit gate
Gate error
two-qubit
gate Ref.
NV (diamond) 60 000 99.995% 99.2% 70,78
P (Si) 30 000 99.9% — 82,83
Superconducting 0.1 99.92% 99.4% 84,85
e
(GaAs) 0.001 98% 90% 86
e
(Si) 0.1 99.9% 94.7% 87
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![F. Direct NV–N–NV coupling
In addition to the direct NV–NV coupling, electronic dark spins
can be included, just like nuclear spins. The coupling is NV–N–NV,55
or even the spin chain NV–N–N–N… N–NV.97
One method involves
energy transitions in the overlap between the NV center and the
P1 center, due to the Zeeman effect, at a magnetic field of 51.2 mT.
Figure 8 shows the distributions of the N and NV centers created by
implantation. Figure 8(c) shows the possible level transitions between
NV and N. This allows for a resonant coupling between the electron
spins of NV and P1 in order to undergo a targeted spin exchange.98
This spin exchange can then be extended to several P1s via dipole–
dipole coupling, as shown in Fig. 9. However, simulations show that
decoherence is a problem for longer spin chains, and these chains are
not controllable. This hardening has applications both in hyperpolari-
zation and methods of electrical readout (see Sec. IV).
G. NV–photon–NV coupling
In this concept, the NV is coupled directly with photons to
achieve the transfer of spin via the photon. The advantage is that a real
NV
Signal
Fluorescence
NV 0
13
C1
13
C1
13
C1
13
C1
13C1
13
C2
13
C2
13C2
14
N
14
N
14
N
14
N
14
N
Hyperfine
NV
=
=
n
2px
px
0
Frequency 0.5 MHz
0.5 s
10 s
Time Probability
0.05 s
0,–1
+1
–½
–½
+½
+½
2
–
(a)
(b) (d)
(e)
(c)
px
2
FIG. 7. (a) Illustration of the quantum register consisting of three nuclear spins (orange circles with black arrows) coupled by hyperfine interaction to the NV electron spin (blue
circles with black arrows), forming a prototype single node of a larger register. 13
C1,2 refers to 13
C nuclei at different lattice positions. (b) Hyperfine splitting (14
N, 2.16 MHz;
13
C1, 413 kHz; 13
C2, 89 kHz) of the transition of mS ¼ 0 to 1 into 12 individual lines. (c) Wire diagram of the projective readout of nuclear spins; the diagram here is that of a
13
C spin. (d) Wire diagram of a CNOT gate between two nuclear spins. (e) Left side: time trace of NV fluorescence photons during repetitive readout of a particular nuclear
spin [see (c)]. The timescales of the dynamics depend on the lifetimes of the nuclear spins and durations of the CNOT gate. Right side: photon-counting histograms of the time
traces. The red line indicates the threshold for nuclear spin state discrimination. Reproduced with permission from G. Waldherr et al., “Quantum error correction in a solid-state
hybrid spin register,” Nature 506(7487), 204 (2014).91
0.10
6 keV
10 keV
E
B (G)
14 keV
0.05
Probability
density
%
Less
than
stated
value
0
5 10 15
Nearest-neighbour separation (nm)
20 25 30 35 0 514
100 % 3 nm
% 2 nm
% 1.5 nm
80
60
40
20
0
0 2 4 6 8
N2 Implant energy (keV)
101214
+1 +½
+1 –½
–1 +½
–1 –½
0 +½
0 –½
B
B*
A*
A
Diamond
N
N2
NV
(a) (b) (c)
FIG. 8. (a) Schematic diagram of molecular implantation leading to the formation of NV–N spin pairs. After entering the diamond, the chemical bond of the N2
þ
molecule is bro-
ken, and the two N atoms penetrate independently into the diamond. Hence, the implantation of single N2
þ
ions leads to the formation of N–N pairs and vacancies (V) in the
diamond substrate. Annealing leads to the conversion of some of the N into NV, leading to closely spaced NV–N pairs. (b) Monte Carlo simulation of the distribution of intrapair
spacings for implantations of 14-, 10-, and 6-keV N2
þ
dimers. The inset shows the calculated fraction of the implanted pairs, with separations between the N defects smaller
than 3, 2, and 1.5 nm as a function of N2 implantation energy. (c) Energy levels of a dipole–dipole-coupled NV–N pair. The separation between the N states in the excited NV
manifold at low B is the dipolar coupling, D ¼ 13 MHz. Reproduced with permission from T. Gaebel et al., “Room-temperature coherent coupling of single spins in diamond,”
Nat. Phys. 2(6), 408–413 (2006).98
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![standardized. There are especially large differences in hydrogen con-
tent. This is a problem that is hindering the industrialization of dia-
mond products. This is a well-known chicken-and-egg problem.
Because no suitably reproducible diamond layer is available, no stan-
dardized product can be developed, yet.
C. Deterministic creation of single NV centers with
high resolution
The diffusion of atoms of almost all elements into diamond is
negligibly small.17
This has the significant advantage that, once intro-
duced, the structures are very stable in time and do not change even
during subsequent healing.17
However, only ion implantation is suit-
able as a method for the production of dopants. Ion implantation tech-
niques have been significantly improved in recent years, and
procedures that allow for a mask-free deterministic implantation of
single ions down to a few nanometers are now available [Fig. 16(a)].
It is therefore particularly advantageous to choose a diamond as
pure as possible (with the lowest level of impurity atoms and crystal-
line defects) to implant individual ions and convert them into color
centers by heat treatment. Although foreign atoms, such as nitrogen,
can be introduced to diamond during growth and later converted into
color centers by generating additional vacancies (e.g., by laser writing),
the unconverted atoms (e.g., N atoms) interfere and lead to shorter
coherence times. Likewise, it is not possible to create NVs with this
method with a precision of a few nanometers, contrary to the case of
ion implantation. The generation of color centers by ion implantation
also has some disadvantages, however. For example, the ions lose their
energy during penetration into the diamond by collisions not only
with the electrons but also with the nuclei of the crystal atoms. This
leads to larger deviations from the initial trajectory and the creation of
defects. The resulting deviation from the desired path is called strag-
gling. It is a well-known phenomenon, and high-resolution precision
can be ensured by adapting the implantation energy to limit the strag-
gling to a few nanometers or less. We recently achieved an N-to-NV
conversion rate of up to 75% for 40 nm deep NVs. Whether the yield
can be further improved, and whether the NV depth can be reduced
to 10–20 nm are questions for actual research. The key was to dope
the diamond with donors, and sulfur gave promising results.47
The
spin coherence time was extended by more than three times (with
respect to intrinsic diamond), and the charge stability of the NV color
centers was increased. A conversion rate of almost 100% seems to be
achievable by further optimization. An enhanced creation yield was also
confirmed for other color centers. The significant advantage of color
centers over alternative methods in a solid is that these centers consist of
only two components, and thus always exhibit the same properties as
long as the environment does not change. Because they are surrounded
by the crystal lattice, in contrast to, e.g., quantum dots, a very high sta-
bility in terms of charge and temperature can be ensured.
In addition to the high conversion rate, a deterministic implanta-
tion of countable atoms with a high lateral resolution is necessary to
create exactly one NV deterministically.108
This problem has been
solved by using different approaches: In the post-detection method,
the implanted ions are detected after implantation by using secondary
electrons or electron–hole pairs.109,110
In the pre-detection method,
the implanted ion is detected before implantation using single ion
sources111–113
or measuring the mirror charge.114
Pre-detection meth-
ods have the advantage that they help avoid false-negative implanta-
tion events.115
Thus, it may be the case that no ion is implanted (false
positive), but never that two or more atoms are implanted. An optical
inspection and subsequent repetition of the implantation procedure
allow for the implantation of the missing atom. Figures 16 and 17
show sketches of single-ion implantation setups.
D. Thermal treatment
Now that the deterministic implantation of single ions at a high
spatial resolution have been demonstrated, full scalability relies on the
conversion of each implanted atom into an active qubit with the suit-
able properties. In the case of NV centers, N-to-NV conversion is
achieved by thermal treatment that enables the diffusion of vacancies
toward the implanted nitrogen and heals other implantation-induced
defects. A wide range of temperatures, between 700
C and 1200
C for
several hours, has been used to create NVs depending on the applica-
tion. The excellent thermal stability of NV centers surpasses any other
concurrent qubit. NV centers start to dissociate at about 1300
C. The
re-orientation of the NV, which would be an issue for qubit addressing,
does not occur below 1000
C.17,116
The diffusion of neutral vacancies
has been determined experimentally to be about 1.5nm2
/s at 1000
C
with an activation energy of 2.12eV117
or 1.8 nm2
/s at 850
C.118
(a)
pulsed
532 nm
laser
photo-
ionization
ion gun
Pr area
ionization
Wien filter Paul trap
866 nm
x/y deflection
electrodes
Sdet=428 mm
Pr
x
z
y
Pr
A
B
A
B
S1=30 mm
U(1)(t) U(2)(t)
~650 mm
Einzel
lens 3D piezo detector
854 nm
CCD
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Quantum computer based on color centers in diamond
- 1. Appl. Phys. Rev. 8, 011308 (2021); https://doi.org/10.1063/5.0007444 8, 011308 © 2021 Author(s). Quantum computer based on color centers in diamond Cite as: Appl. Phys. Rev. 8, 011308 (2021); https://doi.org/10.1063/5.0007444 Submitted: 22 May 2020 . Accepted: 28 December 2020 . Published Online: 10 February 2021 Sébastien Pezzagna, and Jan Meijer COLLECTIONS Paper published as part of the special topic on Quantum Computing This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Passivation of III–V surfaces with crystalline oxidation Applied Physics Reviews 8, 011309 (2021); https://doi.org/10.1063/1.5126629 Material platforms for defect qubits and single-photon emitters Applied Physics Reviews 7, 031308 (2020); https://doi.org/10.1063/5.0006075 Introduction to quantum optimal control for quantum sensing with nitrogen-vacancy centers in diamond AVS Quantum Science 2, 024701 (2020); https://doi.org/10.1116/5.0006785
- 2. Quantum computer based on color centers in diamond Cite as: Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 Submitted: 22 May 2020 . Accepted: 28 December 2020 . Published Online: 10 February 2021 S ebastien Pezzagna and Jan Meijera) AFFILIATIONS Applied Quantum Systems, Felix-Bloch Institute for Solid State Physics, University Leipzig, Linn estrasse 5, Leipzig, Germany Note: This paper is part of the Special Topic on Quantum Computing. a) Author to whom correspondence should be addressed: jan.meijer@uni-leipzig.de ABSTRACT Artificial atoms like the nitrogen vacancy (NV) centers in diamond enable the realization of fully functional qubits in a solid at room temperature. The functionalities of all the parts needed to create a quantum computer, such as quantum error correction, couplings, quantum teleportation, and a quantum repeater, have already been experimentally demonstrated. These achievements are expected to influence the industrial development of quantum information technology as well as quantum sensing. Whereas quantum sensing has been established and a large number of organizations are working on new developments in this area, a quantum computer itself remains elusive due to technical reasons and limitations of the available materials. For example, only in recent months has it become possible to electrically readout the NV spin state at the level of a single center and significantly improve the scalability of NV center production. A number of ideas have been proposed to overcome the above-mentioned limitations. This paper summarizes the status of research in the area, details the most promising concepts for development, and discusses factors limiting progress as well as the most recent developments in the field. V C 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http:// creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0007444 TABLE OF CONTENTS I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. MAIN PROPERTIES OF SINGLE, COHERENTLY CONTROLLABLE SPINS IN DIAMOND . . . . . . . . . . . . 2 A. Diamond as a host for quantum objects. . . . . . . . . 2 B. Quantum properties of single-color centers . . . . . . 3 C. Defects and Fermi level adjustment in diamond to optimize properties of NV centers . . . . . . . . . . . 3 III. BASIC CONCEPTS AND PROPOSALS TO CONTROL AND ENTANGLE SPIN SYSTEMS . . . . . . 5 A. Main requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 B. Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 C. Error correction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 D. Magnetic dipole coupling of electronic NV–NV and nuclear NV–13 C spins . . . . . . . . . . . . . . . . . . . . 7 E. Active decoupling methods of NV or nuclear qubits from the environment . . . . . . . . . . . . . . . . . . 7 F. Direct NV–N–NV coupling. . . . . . . . . . . . . . . . . . . . 8 G. NV–photon–NV coupling. . . . . . . . . . . . . . . . . . . . . 8 H. NV–15 N–NV–15 N by charge state control . . . . . . . 9 I. NV–13 C multi coupling. . . . . . . . . . . . . . . . . . . . . . . . 9 J. Mechanical control of NV–NV using AFM tips. . . 10 K. Electron bus system: NV–P1–e–P1–NV . . . . . . . . . 11 L. Hybrid concepts: NV–superconducting qubits. . . . 11 IV. FROM CLASSICAL QUANTUM INTERFACE TO QUANTUM DEVICES: READOUT OF QUBITS . . . . . 11 A. Optical spin readout. . . . . . . . . . . . . . . . . . . . . . . . . . 11 B. Electrical spin readout . . . . . . . . . . . . . . . . . . . . . . . . 11 V. TECHNICAL REQUIREMENTS AND FABRICATION METHODS FOR SCALABLE DESIGN. . . . . . . . . . . . . . . 12 A. Diamond in CMOS technology . . . . . . . . . . . . . . . . 12 B. Growth techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 C. Deterministic creation of single NV centers with high resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 D. Thermal treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 VI. SUMMARIZING PERFORMANCE OF QUBITS BASED ON NV CENTERS . . . . . . . . . . . . . . . . . . . . . . . . 14 VII. CONCLUSIONS AND OUTLOOK. . . . . . . . . . . . . . . . . 15 AUTHOR CONTRIBUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . 15 I. INTRODUCTION The discovery of the extraordinary properties of the NV center in 19971 raised hopes that a solid-state quantum computer would be Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-1 V C Author(s) 2021 Applied Physics Reviews REVIEW scitation.org/journal/are
- 3. developed soon after.2,3 Compared with phosphorous qubits in silicon,4 NV centers in a diamond can operate at room temperature. Their crea- tion and control appeared relatively easy,5 and their long coherent times offered promise for an easy means of coupling NV centers.6–10 The first promising experiments and the possibility of quantum error correction led to hopes that a scalable quantum register with hundreds of qubits could be built within only five to ten years.4 Many companies had already attempted to sell single-photon sources based on NV cen- ters, and had shown that quantum information can provide unbreak- able and secure data transport.11 On the basis of this technology, schemes to set up a quantum repeater were developed and a quantum Internet was proposed.12,13 But 20years later, the quantum computer remains elusive. After a large investment in research leading to thou- sands of published papers and dissertations as well as hundreds of pat- ents, it has become clear that the challenge at hand is more daunting than was previously thought. Nevertheless, experiments on a single NV center have yielded a number of exciting results and new sensing appli- cations.14,15 In addition to expensive products intended for researchers, the first sensors were developed for the high-value, low-price market, and products for everyday use are now being developed. Moreover, the understanding of ways to couple the electron spin of the NV center with surrounding nuclear spins has led to the development of hyperpo- larization schemes that have the potential to revolutionize magnetic resonance imaging (MRI) for medical applications.16 The research on NV centers has accelerated the transition from nanotechnology to next-generation quantum technology. However, this low-hanging fruit has also motivated many research groups to shift their focus from quantum information to quantum sensing, with only very few concerns still investing time and money in the original goal. Furthermore, owing to past disappointments, the major organizations that fund such research are not willing to spend more money on work on NV centers aimed at building a quantum computer.17 The development of the quantum information system is compa- rable to the development of the digital computer. In a digital computer (e.g., CMOS 1.8 V), each measured value of voltage higher than 1.2 V can be interpreted as a “1,” and each value lower than 0.7V as a “0.” In contrast to the analog computer, exact values are needed in this case.18 This computer model deals with real numbers, and fault toler- ance has to be kept very small. This is the main reason why analog computers are not adequate. The concept of the quantum computer combines an analog and a digital computer system in a sophisticated way. During quantum computation, the state of the qubit can be represented on the Bloch sphere, and every instance of noise due to the environment has to be considered (like in an analog computer). This means that the quantum processor has to be isolated from the environment. On the contrary, during the measurement procedure, the phase disappears, and only the digital number (e.g., up- or down-spin) is visible. Fortunately, error correction can be carried out to bring the system back into a well- defined state without a measurement procedure. Diamond enables the realization of these concepts at room temperature, and the principles of such an operation in a diamond system at room temperature have already been demonstrated. In this paper, we review the proposals for and challenges to attaining the goal of a diamond-based quantum computer. Most of the relevant ideas are in their initial stages and are more akin to sugges- tions than concrete proposals. Their success depends on unknown or very complex properties of materials, which renders realistic modeling or calculation nearly impossible. A large number of these ideas also lack appropriate methods of fabrication. However, the prospect of building a quantum computer based on a solid-state system is impor- tant and a review of work in the area is thus justified. II. MAIN PROPERTIES OF SINGLE, COHERENTLY CONTROLLABLE SPINS IN DIAMOND A. Diamond as a host for quantum objects Building a quantum computer optimally involves isolating a sys- tem from the environment while ensuring the greatest possible control over it.19 Both requirements are impossible to fulfill according to quantum mechanical laws. The solution lies in a compromise such that the best possible separation from the environment can be achieved together with individual qubit control. The optimal condi- tions for this are found in vacuum. In this way, any disturbance can be shielded, and the qubits can be controlled by means of electromagnetic waves. It is therefore not surprising that ion traps at present have achieved the best results, setting a record in terms of the possibility of controlling qubits.20,21 The disadvantage of this method is that the qubits have to be placed in spatial coordinates to control their position. When using ion traps, this is possible only with a large amount of effort. The vacuum must also be very good (better than 1012 mbar), which implies stringent technical constraints. Such systems are there- fore unsuitable for everyday use, and their scalability is technically complex. Much better scalability can be achieved with a superconduct- ing qubit scheme, as shown recently by Google.22 However, supercon- ducting qubits are large and work only at a very low temperature. This also limits their scalability to a few hundred. A solid-state solution is therefore desirable and can be connected more easily with current CMOS technology.23 Quantum objects are automatically fixed in the solid, and the qubits maintain their position even in the case of the failure of technical devices. The defect-free iso- topic 12 C diamond crystal, with its large bandgap of 5.4 eV, has prop- erties similar to those of a vacuum; thus, a point defect center can be compared to an atom in a trap.10,24 The large bandgap ensures that even at room temperature, the conduction band remains unoccupied and no free electrons can unin- tentionally interact with the qubits. The strong covalent bonds provide a high Debye temperature of over 1800 K so that few phonons disturb the system.25 The low diffusion constants of almost all elements enable the atoms to retain their positions even at temperatures of several hun- dred Kelvin. Because of the well-defined properties of defects based on single atoms, all quantum devices have similar electronic structures, spectral lines, and spin properties. This is their main advantage over solid-state quantum devices based on quantum dots, for which the number of atoms and structure differ. However, this is the ideal case that is valid only inside the bulk of a diamond. In practice, the crystal is imperfect, and magnetic atoms such as 13 C, which has a nuclear spin, or impurity atoms like nitro- gen, which has an electron spin, disturb the quantum phase. To manipulate and read out a specific qubit, an interaction with a classical medium is used such that this can be carried out in two ways. The first is as a gate operation in which the state is changed in a defined way but not read out. The quantum system remains a self- contained isolated system. The second way involves actual measure- ment where the quantum state is read out. This is also connected to a Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-2 V C Author(s) 2021
- 4. change of state in the sense of quantum mechanical rules. Here, a con- nection with the classical world takes place, and the system can no lon- ger be regarded as isolated. If possible, the measurement can be made as a so-called “single-shot readout.”26 This is not a single measure- ment, but further measurements should not change the result. For example, it is possible to read out a nuclear spin using an NV center several times without changing the state of the nuclear spin.26 Only two defect centers in diamond exhibit properties that make controlling their spin states at room temperature possible: the NV and the ST1 centers.27,28 At low temperature, the spin state of SiV centers can be coherently controlled as well, but their T2 times are very short.29 With regard to the state of the art, the NV center remains the main candidate for coherently controlling single spin. It is composed of a nitrogen atom with a nuclear spin of 1 (14 N) or 1=2 (15 N) and a first-neighbor carbon vacancy. In the negatively charged state of the NV, the electrons build a triplet state with a zero-field splitting of 2.87 GHz.30 The main advantage is that the m ¼ 0 spin state exhibits different behavior during optical cycles than the m ¼ 61 spin state. Thus, an initialization as well as a readout of the spin state is possible. In a two-qubit scheme, the m ¼ 0 state can be defined as a j0i state and m ¼ –1 as a j1i qubit state. Figure 1 shows the scheme of the energy levels of the NV- center. The center has been well described in the literature.17,30,31 Compared with quantum dots, the NV center is a kind of artificial atom with sim- ilar, well-reproducible properties that can be created via ion implanta- tion or during diamond growth. B. Quantum properties of single-color centers The quantum properties are defined by the Hamiltonian of the NV center,30 H ¼ D Sz þ cS ~ S ~ B þ~ S A~ B þ cI ~ I ~ B: (1) Typically, charged color centers possess a singlet ground state and are transformed into a triplet by excitation.32 An example is the ST1 (or L2 center).27,33 Spin polarization is obtained when a state of the triplet is preferentially occupied. For the NV center, the situation is the opposite. In the ground state, it is in a triplet state, and the singlet is reached via intersystem crossing from the excited state (Fig. 1). The spin polarization then takes place exactly via the singlet state toward which the transition rates for m ¼ 1 and m ¼ –1 are significantly stronger than for m ¼ 0.34,35 Because the singlet decays back into the triplet states with similar transition rates, the spin polarization likely occurs due only to the first decay from the excited triplet to the sin- glet.36 Therefore, after only a few excitation cycles, a spin polarization with a probability of 98% is achieved. As is shown in Eq. (1), even when no external magnetic field is applied, the ground state is also split by a spin orbital interaction with an energy of 2.87GHz h. The polari- zation of the ground state is also given by the spin orbital interaction. The spin state can be controlled by applying microwave pulses. The NV center is characterized by a long coherence time because phonon interac- tion is suppressed. In an isotopically pure diamond, coherence times (T2) of several milliseconds are achievable at room temperature. Figure 2 shows Rabi oscillations driven on a single NV center. A p-pulse leads to a conversion of the state m¼ 0 (e.g., qubit j0i state) to m¼ –1 (e.g., qubit j1i) and can be realized in 7ns.37,38 Other centers, like the SiV or ST1, do not exhibit such a long coherence time.27 The properties of the NV can be improved by antenna structures or even a F€ orster transfer.39 However, it is not possible at present to electrically excite and initialize NVs. This is an obstacle to the produc- tion of sensors and limits the design of NV-based quantum computers. Thus, only all NV centers can be initialized in a register at the same time. While initialization is performed by using a laser pulse, gate oper- ations are achieved by microwave pulses. The simplest variant involves determining the Rabi frequency (Fig. 2) and defining different gate operations via the microwave pulse length in combination with the defined waiting times.40 Two-gate operations are extended in a similar way, where a conditional gate operation is achieved via a small band- width of the microwave pulses depending on the state of the second qubit. These operations are also performed in combination with the defined waiting times.41,42 The times at which the gate operations can be performed depend on the intensity of the microwaves, the coupling with the NV, and the interference due to the bandwidth. Typical gate times for a p-pulse are approximately 10 ns.43,44 With a coherence time of a few milliseconds, 104 gate operations can be carried out. Although the number of operations appears to be large, many gate operations are required, e.g., for error correction or for SWAP gates on the nuclear spin (see Sec. III). The number of gate operations can be improved by extending the T2 times, actively decoupling the environment and paral- lelizing the gate operations. The corresponding concepts are presented in Sec. III. Figure 3 shows the detected magnetic resonance spectra of an NV center with hyperfine coupling to 14 N in the ground state. This coupling allows researchers to swap an electron’s spin state with the nuclear spin state to prolong coherence times. C. Defects and Fermi level adjustment in diamond to optimize properties of NV centers To use the color centers as qubits, it is necessary to control their crystal environment. The quality of the crystal structure, and the pres- ence of foreign atoms or defects with unsaturated electron spins and 1.945 eV 3A2 1A1 ms=0 ms +1 +1 –1 –1 0 0 3E FIG. 1. Energy level scheme of the NV- center. The ground and excited states are triplet states. There is a 30% mixing by the interband coupling of m ¼ 61 with the singlet state and the m ¼ 0 in the ground state. This behavior leads to an easy and efficient polarization of the spin state to m ¼ 0 by optical pumping [Weber et al., “Quantum computing with defects,” Proceedings of the National Academy of Sciences of the United States of America 107(19), 8513–8518 (2010), licensed under a Creative Commons Attribution (CC BY) license].4 Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-3 V C Author(s) 2021
- 5. nuclear spins has to be considered. The quality of the surrounding crystal structure is determined by simple point defects, extended defects, or impurity atoms. The defects lead to stress and unwanted electrical properties, such as a local shift of the Fermi level as well as unsaturated electron and nuclear spins. The electron spins are particu- larly critical for color centers because their range of interaction is of the order of several nanometers and strongly influences the coherence times. The defects can also be ionized by a laser pulse that can promote free electrons in the conduction band. These free Bloch electrons have a very long range (up to several lm). Point defects can be annealed rel- atively easily by means of a temperature treatment. Extended defects are typically more stable and cannot be easily destroyed. In the case of implanted NV centers, the main challenge is to remove isolated vacan- cies (which mainly act as electron traps and perturb the negative charge state stability of the NV) and di-vacancies (paramagnetic defects that reduce NV coherence) or larger vacancy complexes. V0 and V0 2 start to anneal at 600 C and 850 C, respectively, and an optimal T2 is obtained for annealing at 1100 C.45 However, the kinet- ics of the defects and impurities during thermal annealing may be strongly modified if they possess different charge states. This has been the focus of recent work: considering the charging of vacancies (either to Vþ or V ) by diamond doping (p-type46 or n-type doping47 ) to reduce V2 and larger complexes formed by Coulomb repulsion. This led to a more stable NV charge state and an improvement in T2. The NV creation yield was enhanced by one order of magnitude and was up to 75%. Furthermore, the role of hydrogen in the passivation of NV through the formation of NVH was also shown to be strongly depen- dent on diamond doping. Note that the band-bending close to the sur- face of diamond likely modifies the kinetics of defect formation with respect to those of bulk diamond. While the p-type doping of acceptors in diamond by boron ion implantation has been carried out for a while and yielded good results,48 the doping of donors by phosphor, nitrogen, or sulfur ion implantation is considered impossible.49 However, the relevant studies Fluorescence (a.u.) MW frequency (MHz) Allowed ESR transitions (a) (b) 2900 2905 2910 2915 0 0 ms = ±1 2.8 MHz 4.6 MHz 5.1 MHz ms = 0 mI 14 N N–V + ±1 ±1 ±1 FIG. 3. (a) Optically detected magnetic resonance spectrum from a single NV color center. (b) Energy level scheme for the NV center showing hyperfine coupling in the ground state spin substructure. Reproduced with permission from J. Wrachtrup and F. Jelezko, “Processing quantum information in diamond,” J. Phys.-Condens. Matter 18(21), S807–S824 (2006).32 2π rotation – 7 ns 4μm 0 5 10 15 20 25 30 35 40 MW pulse length (ns) NV centre strip-line wires (a) (b) Fluorescence (a.u.) FIG. 2. (a) Rabi oscillation of a single NV center. (b) Confocal fluorescence image of a sample showing microwave strip line wires and scarce NV centers. Reproduced with permission from J. Wrachtrup and F. Jelezko, “Processing quantum information in diamond,” J. Phys.-Condens. Matter 18(21), S807–S824 (2006).32 Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-4 V C Author(s) 2021
- 6. have been based on measurements of the Hall effect, and whether the doping failed overall or the electrons were captured only by the defects generated during the implantation was not determined. Even if n-type conductivity was not obtained, this is not important here because it is precisely the capture of electrons that needs to be exploited to charge the vacancies. A simple method to determine the charge state of iso- lated vacancies is through optical spectroscopy.17 The PL emission line at 741 nm corresponds to the neutral charge state V0 , whereas nega- tively charged vacancies V (ND1) exhibit absorption in the ultravio- let band, and Vþ is not known to be optically active. PL spectra have been used to prove vacancy charging when carbon was implanted into an n-type-like diamond previously doped with phosphorous donors. As a consequence, the formation and properties of the NVs signifi- cantly improved, not only through phosphorous doping but also through doping by oxygen and sulfur. Further investigation (both theoretical and experimental) is needed, however, on the choice of the best donor atom for the given coherence time of the NV. In addition to charge, the presence of hydrogen is an important parameter. Hydrogen is a mobile species in diamond at relatively high tempera- tures and can bind to an NV center to form NVH.50–52 Typical values for the diffusion coefficient of hydrogen are 0.06 nm2 /s at 550 C and 1000 nm2 /s at 850 C (activation energy of about 2.5 eV), but are strongly dependent on its charge state.17,53 NVH is optically inactive, and the NV center can no longer be used as a qubit. To prevent this capture, either a diamond with the lowest possi- ble H content must be used, or hydrogen must be prevented from dif- fusing or forced to bind to another defect. The high conversion rate of the implanted N into NV in donor-doped diamond likely takes advan- tage of such an effect, together with the larger availability of isolated vacancies enabled by Coulomb repulsion. Note that the same passivation-related issues as for NVs occur also for dopants with the formation of BH or PH.54 Only negatively charged NV centers allow for a simple and coherent control of the spin. It is therefore necessary to pin the local Fermi level above the NV(0/–) transition level using the presence of donors. The NV centers are charged by a tunnel effect.55 It has also been shown that the donor atom “giving” its electron to the NV can be the same over several months.56 By applying voltage, the charge of NV centers can be controlled. That this is possible from NV to NV0 has been shown by many research groups and has also been observed in bulk.57,58 The switching from NV0 to NVþ was achieved only for near-surface NVs. In this case, the surface of the diamond was terminated with hydrogen, which produced significant band bending on the surface.59 Figure 4(e) shows the band deflection and the principle of switching the charge of an NV center. The dark state represents NVþ . The switching of the charge state in the NVþ state has no electron spin, which allows for switching off the qubit. This in turns allows for the design of a quantum com- puter according to the concept described in Sec. III. III. BASIC CONCEPTS AND PROPOSALS TO CONTROL AND ENTANGLE SPIN SYSTEMS A. Main requirements All prerequisites for producing a fully functional quantum com- puter (QC) based on NV centers have been met. What would such a chip look like? The essential features of a quantum system for building a quantum computer were initially summarized in the DiVincenzo conditions.61–63 NVs have well-defined properties as color centers in diamond and are well understood.25 In the past few months, break- throughs have been made in the scalability of an NV-based system in terms of both the deterministic implantation of countable single ions at a high resolution64 (see Sec. V) and the efficient conversion of implanted nitrogen atoms into NV centers47 (see Sec. II C), and this has been improved further in terms of optical and spin properties. NVs can currently be generated with a probability of up to 75% by using charge-assisted defect engineering. The initialization stage of qubits is easy to fulfill, as was shown in 2001.2,65,66 This is, however, not possible electronically but only by means of a light pulse. As shown in Fig. 1, the m¼ 61 state has a 30% chance of undergoing a non- radiative intersystem crossing to the singlet state and decaying to the m¼ 0 state, whereas the state m¼ 0 does not couple to the singlet. This means that after about 10 cycles, the NV center is polarized to 98% at room temperature. This is of course not sufficient for a quantum com- puter to achieve a meaningful fidelity of 99.99%. One can change to low temperatures or to a 13 C transferring spin state procedure. A disadvantage of the optical readout occurs because of a tenfold hyperfine interaction that is stronger in the excited state than in the ground state. This disturbs both the surrounding NV and the nuclear spin states. A partial readout is therefore currently not possible for many applications, but is also not necessary. This leads to a reduction Count rate (kcps) Wavelength (nm) 1 1 2 2 EF EF EF EVBM z EVBM E z eUGS Voltage (V) Intensity (arb.u.) Probability U +10 V –10 V 50 +5 V –5 V NV– NV0/– NV0/– NV0 –0.5 V 100 600 C:H C:H E C:O 700 800 –4 –2 0 2 4 150 dark X (a) (b) (d) (c) (e) FIG. 4. Applied voltage of p-n-p junction fabricated by oxygen, hydrogen, and oxy- gen surface termination. (a) The NVþ center at an applied voltage of þ10 V. (b) NV- obtained using an applied voltage of –10 V. (c) Spectra of the NV centers at dif- ferent voltages. (d) Charge state of the NV center at different voltages. (e) Level scheme for different surface terminations. Reproduced with permission from M. V. Hauf et al., “Addressing single nitrogen-vacancy centers in diamond with transpar- ent in-plane gate structures,” Nano Lett. 14(5), 2359–2364 (2014).60 Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-5 V C Author(s) 2021
- 7. in the coherence time because, in addition to the NV, defect centers are activated and free spin states develop. An improvement by means of electrical excitation would be advantageous. A first attempt in this direction has recently been successful with a Forster excitation.39 To achieve a universal set of gates, it is necessary to realize a NOT gate as a single-qubit gate operation and a controlled-NOT (CNOT) as a two- qubit universal gate.2 A coherent spin state switching of the NV can easily be realized by a microwave pulse of a certain length and inten- sity, the so-called p-pulse. This p-pulse represents the NOT gate. The resonant frequency is defined by the spin state of nearby centers or nuclear spins, because of which a narrow-band microwave p-pulse allows for the realization of a controlled rotation (CROT) or, in combi- nation with an additional rotation, a CNOT gate. The CROT gate is, for NV centers, equivalent to the CNOT gate. Only if the controlled qubit is in the j1i state is the target gate resonant with the microwave field and the target qubit flips. Of course, the gate requires a linewidth that is small enough. Figure 5(a) shows the scheme of an interaction between NV centers with different orientations by means of magnetic dipole–dipole interaction. The different orientations lead to different splittings of the magnetic resonances as detected by optically detected magnetic resonance (ODMR) [Fig. 5(b)]. Thus, each NV can be manipulated individually. The magnetic dipole field interaction is related to the quantum state m. A control qubit can only be flipped if the target qubit is in the correct state; otherwise, the microwave fre- quency for the p-pulse does not match the transmission frequency. The single-shot procedure has been carried out using nuclear spins.26,67 In this method, the spin state of the NV center is exchanged with the state of the nucleus. The spin state can be transferred by a SWAP gate process. The SWAP process can be achieved, e.g., by three CNOT gate operations. Figure 6 shows a scheme of qubit teleporta- tion. In this case, the nuclear singlet spin is transformed into a triplet state jþi or j–i using double quantum transitions, which requires the use of five CNOT gates. This enables the reading of the spin state sev- eral thousand times without changing the nuclear spin state. This is the only way to successfully perform an optical or an electrical readout. The best coherence times are currently possible in slightly n- type-doped 13 C free diamonds,10,68,69 and can reach 2 ms at room tem- perature. Using nuclear spins, this time can be extended to the order of seconds.70 Assuming a process time of the gate operation of 10 ns, several 107 gate operations can be performed, including nuclear spin. However, error correction requires several 1000 gate operations. Possible “flying qubits” are realizable by photons,71 phonons,72 or elec- trons.73,74 None of the previously proposed methods can ensure reli- able operation. An alternative is to use a small number of qubit NVs as a register and build a chain of NVs. The transfer operations, how- ever, strongly limit the number of qubits. The realization of flying qubits is one of the biggest hurdles to building a fully functioning quantum computer with more than 1000 qubits. B. Performance The qubit systems realized in diamond or SiC are characterized by excellent quantum control, and single qubit gates with a fidelity of 99.99% and double qubit gates with that of 99.2% have already been demonstrated.75 Entanglement and high-quality quantum algorithms in chips with up to 10 quantum bits have also been demonstrated, with coherent control for up to 30 spins76 (see Sec. IIII). Thus, solid- state spin systems are already comparable to superconducting quan- tum bits and better than competing systems, as shown in Table I. Spins of dopants in diamond or SiC are among the most control- lable quantum bits in solids.77 They can be manufactured deterministi- cally within a spatial accuracy of a few nanometers.64 Electron or nuclear spins are available as qubits. Their coherence times are in the range of a few seconds depending on the type of qubit.75 One- and two-qubit gate operations with a precision of 99.95% have been dem- onstrated.41 The initialization or readouts of qubits with qualities Fluorescence (kcounts s –1 ) 2.75 45 2.80 2.85 2.90 2.95 3.00 MW frequency (GHz) (a) (b) FIG. 5. (a) Interaction of the electron spins of two NV centers. (b) Optically detected magnetic resonance ODMR spectra of two coupled NV centers. The outer pairs of transi- tions correspond to NV1 and the inner pairs to NV2. The splitting of 3 MHz within a pair occurs owing to hyperfine coupling with the 15 N nucleus. Spin transitions of distant NV centers are separated by 30 MHz. Reproduced with permission from F. Dolde et al., “High-fidelity spin entanglement using optimal control,” Nat. Communications 5, 3371 (2014).41 electron p p p p p 0 + – +1/2 –1/2 nuclear FIG. 6. Swap scheme to transfer the spin state from electron spin to nuclear spin. Reproduced with permission from F. Dolde et al., “High-fidelity spin entanglement using optimal control,” Nat. Communications 5, 3371 (2014).41 Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-6 V C Author(s) 2021
- 8. higher than 95% have also been demonstrated,26 and the entanglement of several electron quantum bits, or “entanglement swapping” on adja- cent quantum bits, has been proven.42 NV centers are one of a few quantum systems that allow a repetitive error correction.78 Other quantum algorithms, such as the quantum Fourier transformation or algorithms for the simulation of simple molecular compounds, have been proposed79 and have yielded pulse sequences for quantum gates with qualities above the fault tolerance threshold.80 The strong cou- pling of defects is the essential basis for scalability.81 C. Error correction To operate a quantum computer, error correction is neces- sary.88,89 Different error correction methods have been developed and can all be applied to NV qubits. The simplest methods are based on five or seven qubits,90 and complex ones are based on up to 100 qubits. The choice of error correction protocol depends on the quality of the qubits. With a fidelity of 99.99%, only a few qubits are sufficient. If the fidelity drops, a large number of qubits is needed to achieve suffi- cient correction.91 The main task is therefore to achieve a fidelity as high as possible. For the NV, an average gate fidelity of a single qubit has been measured to be 0.999 952(6), and that of a two-qubit-CNOT gate reaches 0.992(1) at room temperature. NV centers are one of the few systems where the concept of error correction has been success- fully demonstrated. Three-qubit systems were tested at room tempera- ture91 and exhibited a very high fidelity at 4 K.92 D. Magnetic dipole coupling of electronic NV–NV and nuclear NV–13 C spins To couple qubits, an interaction must exist or have existed. For NV spin systems, the coupling is achieved by magnetic dipole–dipole interaction. The field strengths of single spins are extremely high in the near field. Magnetic field strengths of 1 T can be achieved at a dis- tance of 2 nm. The dipole–dipole interaction is given by Hdip ¼ l0c2 e 4pr3 AB SASB 3 SA þ nAB ð Þ nABSB ð Þ ½ : (2) Because the dipole–dipole interaction is dependent on r3 , the lateral resolution in which an optimal structure of a quantum computer needs to be achieved is very strongly limited. The NV–NV coupling has an energy of h 10 kHz for a 30 nm separation. For the NV–13 C coupling (hyperfine interaction), this is about h 12 MHz if the 13 C is located within the first shell. The couplings of it with 14 N and 15 N can be carried out at 2 and 5 MHz, respectively, in the ground state. A too weak coupling strength leads to long switching times for two-qubit gates, such as a CNOT, while a too high coupling strength reduces the coherence times of the qubits.55 Relaxation times due to spin flip-flop depend on r6 . A trade-off therefore needs to be found between these extremes. An optimal value for the NV–NV coupling is likely reached at a distance of 10 to 20 nm. This corresponds to a coupling strength of a few MHz and can be used to realize short switching times. This is the only way to achieve a sufficient number of switching times within the coherence time. A disadvantage of solid-state systems is that the coupling strength cannot be adjusted and is determined only by the distance between the NVs. By clever selection of the spins, like double quantum transition (DQT), the value can be quadrupled with the dis- advantage, however, that the coherence time is halved.93 However, as discussed in Sec. III E, one can extend the coherence times T2 to T2, for example, from a value of T2 of 22 ls to that of T2 of 500 ls,94 by dressing modes and refocusing procedures. Another possibility is the inclusion of nuclear spins (Fig. 7). Nuclear spins have a magnetic field three orders of magnitude lower, and the coupling is correspondingly smaller. However, this lower coupling also leads to a greater decou- pling to the environment, which results in a long coherence time. The T2 times can reach up to seconds at room temperature. Nuclear spins are therefore ideal quantum memories.91 Figure 7 illustrates this kind of coupling between one NV and three nuclear spins. E. Active decoupling methods of NV or nuclear qubits from the environment The importance of the coherence time T2 has already been noted several times. The QC can be used only within T2. The width of the lines and, thus, the design of the CNOT gates also depends on the coherence time. The coherence time is limited by disturbances, which in most cases are owing to unsaturated spin systems such as 13 C nuclear spins, but can also be caused by disturbances outside the sam- ple. Because the coupling to the spin system is mainly affected by the magnetic components of the electromagnetic field, only disturbances caused by magnetic fields are to be considered. These magnetic distur- bances from outside can have certain defined frequencies but also a wide noise spectrum, accordingly.95 In the first case, refocusing is pos- sible using known pulse sequences, such as the XY8 pulse sequence. The limited coherence time caused by noise only is called T2, and the limited coherence time caused by defined disturbances is called T2 . T2 is therefore always shorter than T2. To prevent interference from direct spin couplings, such as 13 C nuclear spins, double frequency exci- tations are possible. This method is also called spin state dressing and allows for the decoupling of nuclear spins from the NV center,9 but can also be used to prevent electron spin crossover during gate opera- tions of nuclear spin.70 In the latter case, a CROT gate has been built to achieve the coupling of an NV with nine nuclear spins.75 Another possibility is to polarize the spin bath to achieve a defined constant magnetic field. The polarization of the spin bath is also called hyperpo- larization, where this method is known in the nuclear magnetic reso- nance (NMR) technique.96 The spin of an NV center is used to polarize the surrounding dark spins. To achieve polarization, an opti- mal coupling from the NV center to the nuclear spins is necessary for a certain time. This is achieved by a Hartmann–Hahn condition, or by means of the P1 centers (substitutional neutral nitrogen atoms).97 TABLE I. Comparison of benchmarks for different solid-state systems. Qubit (host) T2 [ms] Benchmarking one-qubit gate Gate error two-qubit gate Ref. NV (diamond) 60 000 99.995% 99.2% 70,78 P (Si) 30 000 99.9% — 82,83 Superconducting 0.1 99.92% 99.4% 84,85 e (GaAs) 0.001 98% 90% 86 e (Si) 0.1 99.9% 94.7% 87 Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-7 V C Author(s) 2021
- 9. F. Direct NV–N–NV coupling In addition to the direct NV–NV coupling, electronic dark spins can be included, just like nuclear spins. The coupling is NV–N–NV,55 or even the spin chain NV–N–N–N… N–NV.97 One method involves energy transitions in the overlap between the NV center and the P1 center, due to the Zeeman effect, at a magnetic field of 51.2 mT. Figure 8 shows the distributions of the N and NV centers created by implantation. Figure 8(c) shows the possible level transitions between NV and N. This allows for a resonant coupling between the electron spins of NV and P1 in order to undergo a targeted spin exchange.98 This spin exchange can then be extended to several P1s via dipole– dipole coupling, as shown in Fig. 9. However, simulations show that decoherence is a problem for longer spin chains, and these chains are not controllable. This hardening has applications both in hyperpolari- zation and methods of electrical readout (see Sec. IV). G. NV–photon–NV coupling In this concept, the NV is coupled directly with photons to achieve the transfer of spin via the photon. The advantage is that a real NV Signal Fluorescence NV 0 13 C1 13 C1 13 C1 13 C1 13C1 13 C2 13 C2 13C2 14 N 14 N 14 N 14 N 14 N Hyperfine NV = = n 2px px 0 Frequency 0.5 MHz 0.5 s 10 s Time Probability 0.05 s 0,–1 +1 –½ –½ +½ +½ 2 – (a) (b) (d) (e) (c) px 2 FIG. 7. (a) Illustration of the quantum register consisting of three nuclear spins (orange circles with black arrows) coupled by hyperfine interaction to the NV electron spin (blue circles with black arrows), forming a prototype single node of a larger register. 13 C1,2 refers to 13 C nuclei at different lattice positions. (b) Hyperfine splitting (14 N, 2.16 MHz; 13 C1, 413 kHz; 13 C2, 89 kHz) of the transition of mS ¼ 0 to 1 into 12 individual lines. (c) Wire diagram of the projective readout of nuclear spins; the diagram here is that of a 13 C spin. (d) Wire diagram of a CNOT gate between two nuclear spins. (e) Left side: time trace of NV fluorescence photons during repetitive readout of a particular nuclear spin [see (c)]. The timescales of the dynamics depend on the lifetimes of the nuclear spins and durations of the CNOT gate. Right side: photon-counting histograms of the time traces. The red line indicates the threshold for nuclear spin state discrimination. Reproduced with permission from G. Waldherr et al., “Quantum error correction in a solid-state hybrid spin register,” Nature 506(7487), 204 (2014).91 0.10 6 keV 10 keV E B (G) 14 keV 0.05 Probability density % Less than stated value 0 5 10 15 Nearest-neighbour separation (nm) 20 25 30 35 0 514 100 % 3 nm % 2 nm % 1.5 nm 80 60 40 20 0 0 2 4 6 8 N2 Implant energy (keV) 101214 +1 +½ +1 –½ –1 +½ –1 –½ 0 +½ 0 –½ B B* A* A Diamond N N2 NV (a) (b) (c) FIG. 8. (a) Schematic diagram of molecular implantation leading to the formation of NV–N spin pairs. After entering the diamond, the chemical bond of the N2 þ molecule is bro- ken, and the two N atoms penetrate independently into the diamond. Hence, the implantation of single N2 þ ions leads to the formation of N–N pairs and vacancies (V) in the diamond substrate. Annealing leads to the conversion of some of the N into NV, leading to closely spaced NV–N pairs. (b) Monte Carlo simulation of the distribution of intrapair spacings for implantations of 14-, 10-, and 6-keV N2 þ dimers. The inset shows the calculated fraction of the implanted pairs, with separations between the N defects smaller than 3, 2, and 1.5 nm as a function of N2 implantation energy. (c) Energy levels of a dipole–dipole-coupled NV–N pair. The separation between the N states in the excited NV manifold at low B is the dipolar coupling, D ¼ 13 MHz. Reproduced with permission from T. Gaebel et al., “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2(6), 408–413 (2006).98 Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-8 V C Author(s) 2021
- 10. flying qubit is thus obtained. Due to the linewidths, this method can be used only at very low temperatures, and the coupling with the photon field must be very strong. This can be achieved by means of a cavity. Another major advantage is that long stretches can be over- come, and the construction of a quantum network becomes conceiv- able. Figure 10 shows an example of a waveguide based on silica on silicon with a 50:50 beam splitter. This waveguide can be used to cou- ple a resonator with a single NV center. H. NV–15 N–NV–15 N by charge state control In this concept, the charge state of the NV centers is used to switch the NVs. For the NVs, three charge states are known: NV , NV0 , and NVþ . While NV has the well-known triplet, NV0 is a sin- glet state, and NVþ has no luminescence and should have no spin, similar to 12 C (i.e., it is diamond metric). By switching from NV to NVþ , the coupling to 13 C nuclei as well as to adjacent NVs can be switched on and off. This allows researchers to address NV centers and read out single NVs (because the information is transferred to the neighboring 13 C). Figures 11 and 12 show the structure and function- ality of this idea. I. NV–13 C multi coupling Recently, 27 nuclear spins were coupled to an NV center and switched separately by using the fact that, at low temperatures, the linewidths narrow and the coherence time of the nuclear spins increases.76 The hyperfine interaction involves permanently cou- pling the NV with all of the 13 C. A crossover must be avoided, and this can be realized using advanced decoupling techniques.70 The most effective way to achieve this is to directly link the gate with the decoupling sequence. Because the 13 C have different positions with respect to the NV center, each hyperfine coupling is different. If the NV is in the state m ¼ 1 or m ¼ –1, the hyperfine interaction leads to a shift in the RF frequency, which is in resonance with the spin of 13 C. It is therefore possible to address different 13 C spins separately. The decoherence of the NV is first used to check its environment—to determine how many 13 C spins are situated close by as well as the strengths with which they are interacting—by sweeping the RF frequency. Figure 13 shows the NV decoherence as a function of the RF frequency. The different frequencies are Magnetic Field Gradient k W1 W2–W1 W3–W2 W4–W3 W3–W4 W2 W3 W4 W5 k k k k k (a) (b) FIG. 9. (a) Each NV register contains a nuclear spin I ¼ 1/2 (yellow), providing quantum memory, and an electronic spin S ¼ 1 (green). Dark spins (black) repre- sent elements of an optically unaddressable spin chain, which coherently couples spatially separated NV registers. The NV level structure (in a high B field) is shown. A resonant pulse coherently transfers the electronic spin of the register from 0 e to 1 e. Subsequent manipulation of the nuclear spin is accomplished through a radio- frequency (RF) pulse. The far detuned state can be neglected to create an effective two-qubit register. However, the full three-level NV structure is utilized by the coher- ent coupling of the NV registers. Reproduced with permission from N. Y. Yao et al., “Scalable architecture for a room temperature solid-state quantum information pro- cessor,” Nat. Communications 3, 800 (2012).97 f FIG. 10. (a) Quantum computing with photons. An interferometer with controlled phase shift for single-qubit operations and multi-photon entangled state manipula- tion. Reproduced with permission from T. D. Ladd et al., “Quantum computers,” Nature 464(7285), 45–53 (2010).99 Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-9 V C Author(s) 2021
- 11. then used for the individual control of the NV. The method has many advantages. On the one hand, it is not necessary to use isoto- pically pure 12 C samples, or to implant or place the 13 C in the vicinity of the NV center. Diamond samples with a natural abun- dance of 13 C (1.1%) can simply be used. On the other hand, because the T2 times of the nuclei at low temperatures are of the order of several minutes to hours, the state of the NV centers can be reset. Furthermore, in Ref. 100 a single NV center was coupled with up to 28 13 C. This drastically increases the possibility of scalability as only a few coupled NVs are needed to control a large number of qubits. J. Mechanical control of NV–NV using AFM tips In this proposal, the coupling is achieved via an AFM tip in which an NV center is located. The tip is moved over a surface with NV centers. The advantage is that the distance between the NV centers can be very large. The disadvantage of this method is that it takes a (a) (b) (d) (e) (c) 2.0 RF frequency (MHz) Spin flip probability Change state witness (a.u.) 0.4 1.0 500nm 0.5 100 50 –50 –10 –5 0 5 10 0 0.0 –8 –6 –4 –2 Voltage (V) Voltage (V) Current (nA) 0 2 4 6 8 Set volt. Set Set R(q)x p(n) 1n W± W± W0 W0 0.2 2.1 9.0 9.1 9.2 0.0 0.2 0.4 Cq (+) Cq (0) Cq (–) Cq (+) Cq (0) Cq (–) 0.0 0.2 0.4 RF pulse duration (ms) quadrupole splitting (MHz) Rabi maximum Tu (ms) 0.6 0.6 0.4 0.6 0.4 exper. results + fit ab into results + error bars 0.2 0.8 1.0 4.4 4.5 4.6 4.7 4.8 4.9 5.0 0 5 Spin flip probability Spin flip probability 0e 1n 0e 1n 0e 0e 0e 0e FIG. 11. (a) Confocal microscopy image of single NV centers. The left image (light-blue background) was recorded at a voltage corresponding to the NV charge state. In the right image (light-red background), the NV’s fluorescence is quenched by reversing the gate voltage. The blue, gray, and pink colors denote NV , NV0 , and NVþ , respectively, throughout the figure. (b) I V characteristics of the surface gate structure. (c) The two spectra on the left show one of the 15 N nuclear spin transitions for the NV6 and NV0 charge states. Their normalized amplitudes yield the charge state probabilities W6 and W0 , respectively. W6 increases for any charge state exhibiting an electron spin state m ¼ 0, as is expected for NVþ (S ¼ 0, m ¼ 0). The graph on the right shows the charge state probabilities for varying voltages (error bars represent projection noise). The reappearance of the W6 signal at high voltages suggests the presence of NVþ . (d) 15 N nuclear spins with Rabi oscillations between spin state mI ¼ 61/2 for the NV (m ¼ 0) and the tentative NVþ charge state. The right curve shows the maxima of the NVþ Rabi oscillation for varying durations of the initial NVþ voltage application. We deduced an NVþ settling time of 0.54 6 0.08 ms. For the NV charge state, hyperfine interaction increases the Rabi frequency. (e) 14 N quadrupole splittings for the three known charge states (resonances of experimental spectra represented by circles, those of the theory by squares). Any magnetic hyperfine and nuclear Zeeman terms are subtracted. Reproduced with permission from M. Pfender et al., “Protecting a diamond quantum memory by charge state control,” Nano Lett. 17(10), 5931–5937 (2017).58 init (1) (2) (3) (4) ~20nm operation readout/init (a) (b) FIG. 12. (a) Schematic of individual NV nodes addressable via nanoscopic gate electrodes (gray, insulator; gold, electrode; purple, leads). NV centers under H-terminated sur- face are in the NVþ state, and, therefore, do not couple with other nodes because of the lack of an electron spin, their nuclear spin state remains undisturbed, and they do not contribute to fluorescence response when illuminated. Individual electrodes with a positive voltage deplete the hole-conducting layer locally and shift the Fermi level to the sta- ble NV region. Therefore, such NV centers can be coupled with their NV neighbors via magnetic dipole interaction and are optically accessible. Unsuitable NV nodes are not supplied with an electrode, and remain in NVþ . (b) Sketch of modes of operation of the quantum register, with the charge states highlighted by the red and blue boxes, nuclear and electron spins represented by purple and blue arrows, respectively, laser represented in green, and fluorescence by red stars. Initialization: (1 2) Laser initializes all electron spins, (3) swaps to nuclear spins, and (4) switches to NVþ for storage. Operation: (1) Swap two nuclear spins with electron spins in NV , (2) entangle the electron spins, (3) swap back to the nuclei, and (4) switch all NV centers to NVþ . Readout/init: (1) Switch one NV to NV with an initialized electron spin, (2) correlate the electron spin with nuclear spin, (3) read out one electron spin and project the nuclear spins (2 3), single-shot readout, and (4) switch back to NVþ . Reproduced with permission from M. Pfender et al., “Protecting a diamond quantum memory by charge state control,” Nano Lett. 17(10), 5931–5937 (2017).58 Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-10 V C Author(s) 2021
- 12. long time and can only be realized at low temperatures. Disturbances close to the surface also have to be overcome.101 K. Electron bus system: NV–P1–e–P1–NV Manson et al.73 recently proposed using an electron bus system involving substitutional nitrogen (P1 center). This is an adaptation of the method patented by Schenkel et al. for 32 P in silicon.102 The idea is based on the transfer of the NV spin state on a free electron that, used as a flying qubit, enables coupling with a large number of NV centers. Because the electrons can be controlled by electric fields, this method is flexible, and represents a direct link to conventional information technology. However, a direct ionization of the NV centers is challeng- ing because the spin state of a single electron is not defined in the state m ¼ 0 and is instead defined by a combination of one electron up and one down. In the state m ¼ 61, both electrons are in the same state but photoionization is difficult. This problem can be solved if the spin state is transferred to a P1 center that is then ionized. However, the re- ionization (NV0 to NV ) to a defined spin state with a given electron is not easy. Preliminary experiments have thus far been unsuccessful.103 L. Hybrid concepts: NV–superconducting qubits In this approach, the NV centers are employed as storage ele- ments for superconducting qubits. Because both systems operate at the same microwave frequency, the spin state can be exchanged between systems. An attempt was made to use an NV spin bath. The idea is to combine the functionality of superconducting qubits with a long T2 time of NV centers. Due to the difficulty of coupling, the rele- vant experiments have so far been unsuccessful.104,105 IV. FROM CLASSICAL QUANTUM INTERFACE TO QUANTUM DEVICES: READOUT OF QUBITS As described in Sec. III, the readout (measurement) is one of the most important criteria for building a quantum computer. Technically, this can be done with the help of optical methods (i.e., the photon) or in electronic form (with electrons). With photons as well as electrons, the quantum state is determined several times without changing the state of the respective qubit or other qubits. This proce- dure is called the single-shot readout and is illustrated in Fig. 14. A. Optical spin readout The optical readout must be distinguished between those at low temperatures and at room temperature. The aim of the single-shot readout is to determine the spin state without changing the states of the neighboring NV centers. At room temperature, the direct spin state readout within a single shot cannot be achieved. The solution consists in transferring the spin state to an adjacent nuclear spin and then to measure it (through a CNOT gate). If the measurement is per- formed at the electron spin sublevel m ¼ 0, the spin state of the nucleus is preserved and the measurement can be repeated several times.26 At low temperatures, the situation is more favorable owing to the narrow linewidths. In such a case, a direct readout can be per- formed because the excitation of state m ¼ 0 is achieved only at a cer- tain frequency. This can now be repeated several times: If fluorescence occurs, the m ¼ 0 state is confirmed; if it does not occur, it is the m ¼ 61 state. In principle, the problem of addressing emerges with optical readouts. With the optical setups generally used, the resolution is limited to approximately 250–400 nm ( k/2 NA). However, the distance between the coupled NVs must not exceed 100 nm. A readout of individual NVs is therefore not possible. Attempts to eliminate the addressing by means of stimulated emission depletion (STED) micros- copy100 have been unsuccessful: The laser power of the depletion beam that is necessary to achieve the required resolution is too high and disturbs the coherence times of the nuclear spin. B. Electrical spin readout Photoelectrons can be used to readout the NV. The so-called photocurrent detection of magnetic resonance (PDMR) method has been known for a long time,106,107 but recently it has become possible to readout the spin state of single NV centers. This involves a two- photon process to ionize the NV center, and the laser power needs to 14 N 13C e 0.5 1 5 3 8 4 7 2 6 0.4 0.3 0.2 0.1 0 Ry ( ) 0.0 220 400 420 440 RF frequency (kHz) Amplitude 460 480 2 π Rϕ ( ) ρ m 2 π Rx (± ) 2 π (a) (b) FIG. 13. (a) The electron spin of a single NV center in diamond acts as a central qubit and is connected by two-qubit gates to the intrinsic 14 N nuclear spin and a fur- ther eight 13 C nuclear spins surrounding the NV center. (b) Nuclear spin spectros- copy. After preparing the electron in a superposition state, the gate is applied at different RF frequencies x. The electron spin is then measured along a basis in the equatorial plane defined by angle u (see inset). Each data point corresponds to the fitted amplitude A of the function, where u is swept from 0 to 360 deg., and u0 accounts for deterministic phase shifts induced on the electron by the RF field. By fitting the amplitude, we distinguish such deterministic phase shifts from a loss of coherence due to entangling interactions. The signals due to interactions with the eight 13 C spins used in this work are labeled. The dashed gray line indicates the 13 C Larmor frequency xL. Reproduced with permission from C. E. Bradley et al., “A ten-qubit solid-state spin register with quantum memory up to one minute,” Phys. Rev. X 9(3), 031045 (2019), licensed under a Creative Commons Attribution (CC BY) license.75 Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-11 V C Author(s) 2021
- 13. be chosen accordingly. The principle is described in Fig. 15. It has been shown that the use of a second laser beam speeds up NV recharg- ing from NV0 to NV . If this process is repeated several times, the number of photoelectrons comes to depend on the spin state of the NV center and can be determined accordingly. As with optical read- out, a single shot can be performed at room temperature via a nuclear spin. At low temperature, a direct readout is also possible with PDMR. Addressing is carried out by placing an electrical contact over each NV center to which a suction voltage can be applied. Only NVs with the correspondingly high suction voltages are thus read out. The readout lines can, for example, be attached as a gauze and switched accord- ingly. Thus it should be possible to read out 100 NV centers with only 20 lines, one row being read out at a time. It is also important to pre- serve the spin state as long as possible; otherwise, the arithmetic opera- tion has to be repeated a larger number of times. However, even in such a case, the computational effort increases only linearly in time. V. TECHNICAL REQUIREMENTS AND FABRICATION METHODS FOR SCALABLE DESIGN A. Diamond in CMOS technology The production of quantum computers at an industrial process can be achieved only if the structures of CMOS technology are trans- ferred to processes necessary for the machining of diamond. Therefore, the necessary manufacturing processes, such as single-ion implantation and growth, need to be standardized. Comprehensive quality control, especially of the grown diamonds, must be defined and applied. We currently have a technological standard for diamond that is comparable to those in the 1960s and 1970s for silicon technol- ogy. Only in the 1980s, when it became possible to manufacture silicon of the same quality in a controlled manner, did the semiconductor industry develop at a large scale. The main techniques for the produc- tion of diamond qubits are chemical vapor deposition (CVD) growth, single-ion implantation, and thermal annealing. Other processes correspond to the current CMOS standard and can be transferred over directly. B. Growth techniques The growth of suitable diamond layers represents a major hurdle in the production chain.10 Thus far, no standard CVD diamond is really available. Although the E6 company has made an initial subdivi- sion with the introduction of optical, electronic, and quantum grades as well as set values for nitrogen content, other important parameters, such as hydrogen content, defects, and stress, have not been n y repeat MW off MW on MW off MW p pulse Relative fluorescence intensity 1.4 N e 0 B C V 1.0 1.0 0.6 0.6 0.0 0.5 5 0 10 Time (s) 1.0 1.5 (a) (b) FIG. 14. (a) Single-shot readout reveals quantum jumps of a single nuclear spin in real time. (a) Representation of the single-shot readout scheme. (b) Normalized fluores- cence time traces (gray) showing quantum jumps of a single nuclear spin in real time. When the microwave pulses (CNOT gates) are on, a telegraph-like signal appears, revealing the projective nature of this measurement. A low fluorescence intensity represents nuclear spin state 1n and a high intensity indicates the state 0ni or þ 1ni. When the microwave pulses are off (upper trace), the fluorescent intensity remains high because it is not correlated with the nuclear spin state. Each data point was acquired by continuously repeating the readout scheme for 5 ms (2000 repetitions). Reproduced with permission from P. Neumann et al., “Single-shot readout of a single nuclear spin,” Science 329(5991), 542–544 (2010).26 h* 3 E (a) 3 E 2 E 2 E 3 A2 3A2 2 A2 2A2 NV– NV– NV0 e– NV0 II II I I 2.16 eV 1.95 eV CB VB VB CB FIG. 15. Schematic representation of the experiment. (a) Simplified representation of charge carrier generation within one cycle of NV ionization and recovery. First, two photons produce a free electron in the conduction band (CB) and leave defects in the neutral charge state. Second, the two photons convert the neutral charge state back to NV while a free hole persists in the valence band (VB). I and II indi- cate the first and second transitions of the electrons, respectively, and e and h rep- resent the electron and the hole, respectively. Reproduced with permission from P. Siyushev et al., “Photoelectrical imaging and coherent spin-state readout of single nitrogen-vacancy centers in diamond,” Science 363(6428), 728 (2019).107 Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-12 V C Author(s) 2021
- 14. standardized. There are especially large differences in hydrogen con- tent. This is a problem that is hindering the industrialization of dia- mond products. This is a well-known chicken-and-egg problem. Because no suitably reproducible diamond layer is available, no stan- dardized product can be developed, yet. C. Deterministic creation of single NV centers with high resolution The diffusion of atoms of almost all elements into diamond is negligibly small.17 This has the significant advantage that, once intro- duced, the structures are very stable in time and do not change even during subsequent healing.17 However, only ion implantation is suit- able as a method for the production of dopants. Ion implantation tech- niques have been significantly improved in recent years, and procedures that allow for a mask-free deterministic implantation of single ions down to a few nanometers are now available [Fig. 16(a)]. It is therefore particularly advantageous to choose a diamond as pure as possible (with the lowest level of impurity atoms and crystal- line defects) to implant individual ions and convert them into color centers by heat treatment. Although foreign atoms, such as nitrogen, can be introduced to diamond during growth and later converted into color centers by generating additional vacancies (e.g., by laser writing), the unconverted atoms (e.g., N atoms) interfere and lead to shorter coherence times. Likewise, it is not possible to create NVs with this method with a precision of a few nanometers, contrary to the case of ion implantation. The generation of color centers by ion implantation also has some disadvantages, however. For example, the ions lose their energy during penetration into the diamond by collisions not only with the electrons but also with the nuclei of the crystal atoms. This leads to larger deviations from the initial trajectory and the creation of defects. The resulting deviation from the desired path is called strag- gling. It is a well-known phenomenon, and high-resolution precision can be ensured by adapting the implantation energy to limit the strag- gling to a few nanometers or less. We recently achieved an N-to-NV conversion rate of up to 75% for 40 nm deep NVs. Whether the yield can be further improved, and whether the NV depth can be reduced to 10–20 nm are questions for actual research. The key was to dope the diamond with donors, and sulfur gave promising results.47 The spin coherence time was extended by more than three times (with respect to intrinsic diamond), and the charge stability of the NV color centers was increased. A conversion rate of almost 100% seems to be achievable by further optimization. An enhanced creation yield was also confirmed for other color centers. The significant advantage of color centers over alternative methods in a solid is that these centers consist of only two components, and thus always exhibit the same properties as long as the environment does not change. Because they are surrounded by the crystal lattice, in contrast to, e.g., quantum dots, a very high sta- bility in terms of charge and temperature can be ensured. In addition to the high conversion rate, a deterministic implanta- tion of countable atoms with a high lateral resolution is necessary to create exactly one NV deterministically.108 This problem has been solved by using different approaches: In the post-detection method, the implanted ions are detected after implantation by using secondary electrons or electron–hole pairs.109,110 In the pre-detection method, the implanted ion is detected before implantation using single ion sources111–113 or measuring the mirror charge.114 Pre-detection meth- ods have the advantage that they help avoid false-negative implanta- tion events.115 Thus, it may be the case that no ion is implanted (false positive), but never that two or more atoms are implanted. An optical inspection and subsequent repetition of the implantation procedure allow for the implantation of the missing atom. Figures 16 and 17 show sketches of single-ion implantation setups. D. Thermal treatment Now that the deterministic implantation of single ions at a high spatial resolution have been demonstrated, full scalability relies on the conversion of each implanted atom into an active qubit with the suit- able properties. In the case of NV centers, N-to-NV conversion is achieved by thermal treatment that enables the diffusion of vacancies toward the implanted nitrogen and heals other implantation-induced defects. A wide range of temperatures, between 700 C and 1200 C for several hours, has been used to create NVs depending on the applica- tion. The excellent thermal stability of NV centers surpasses any other concurrent qubit. NV centers start to dissociate at about 1300 C. The re-orientation of the NV, which would be an issue for qubit addressing, does not occur below 1000 C.17,116 The diffusion of neutral vacancies has been determined experimentally to be about 1.5nm2 /s at 1000 C with an activation energy of 2.12eV117 or 1.8 nm2 /s at 850 C.118 (a) pulsed 532 nm laser photo- ionization ion gun Pr area ionization Wien filter Paul trap 866 nm x/y deflection electrodes Sdet=428 mm Pr x z y Pr A B A B S1=30 mm U(1)(t) U(2)(t) ~650 mm Einzel lens 3D piezo detector 854 nm CCD camera FIG. 16. (a) Sketch of single-ion implantation setup. (b) Fluorescence of the trapped ions imaged. (top) Pure Ca crystals; the distance between two Ca ions was 9.5 lm. (bottom) Crystals containing an additional Pr ion. Reproduced with permission from K. Groot-Berning et al., “Deterministic single-ion implantation of rare-earth ions for nanometer-resolution color-center generation,” Phys. Rev. Lett. 123(10), 106802 (2019).64 Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-13 V C Author(s) 2021
- 15. Nevertheless, a competing process to NV creation is the formation of V2 or larger complexes, which easily form, during annealing as well as the implantation itself, especially for larger atoms.17 At an ion implan- tation energy suitable for 10–20-nm-deep qubits, the yield of NV crea- tion is only a few percentage points.119 Each unconverted nitrogen, each divacancy, is a source of decoherence for the NVs. Recent work at the single-center level has provided new insights into the birth, disappearance, and re-orientation of the NV with respect to the type of doping of diamond (see Sec. IIC).46,47,116 This instance of charge-assisted defect engineering offers new possibilities for the targeted and enhanced formation of some color centers to the detriment of unwanted ones. Furthermore, the stability and spin prop- erties of the NV charge state are improved. In addition to NV cen- ters, for which the creation was enhanced along an order of magnitude up to 75%, this method has also proved to work with SnV and MgV centers. Following the same idea of charging defects during thermal annealing to modify the formation and diffusion processes, it can also be tried to use light illumination or to apply a voltage to the sample. As well, rapid thermal annealing would add another degree of free- dom. A recent method that uses femtosecond laser has also been shown to achieve local healing and induce NV centers from native nitrogen atoms.108 Whether this method is also suitable for implanted nitrogen is the subject of ongoing research. Finally, the presence and quantity of hydrogen in high purity CVD-grown diamond needs to be taken into account (and better studied) because hydrogen can easily diffuse and passivate the qubits. VI. SUMMARIZING PERFORMANCE OF QUBITS BASED ON NV CENTERS The current status of the NV-based qubits is shown in Table II. At present, 10 qubits can be fully controlled. In the relevant methods, the NV center is used to control the nuclear qubits consisting of 13 C Electron gun Gas inlet Drift tubes Electron collector Electron beam ion source (EBIS) Extraction lens Condenser lens Interferometer piezo stage Objective lens Beam blanker Sample Wien filter Image charge detection Collimator Threshold Trigger signal FIG. 17. Sketch of a deterministic single ion implanter based on mirror charge col- lection. Reproduced with permission from T. Herzig et al., “Creation of quantum centers in silicon using spatial selective ion implantation of high lateral resolution,” in 22nd International Conference on Ion Implantation Technology (IIT), W€ urzburg, Germany, 16–21 Sept. 2018 (IEEE, 2018), pp. 136–139.114 TABLE II. Summary of the current performance status of NV-based qubits (based on input by T. H. Taminiau). Quantum chips: Value Refs. Remarks Number of qubits demonstrated 7 qubits 75 Entangled GHz state 10 qubits 75 Fully connected 29 qubits 76 Rudimentary qubits controlled Chip linked 2 120 Optical interconnection Coherence T2 (e qubit) 2.4 ms 69 Temperature 300 K 0.6 s 121 77 K 1.5 s 70 4 K T2 (n qubits) 2 s 9 300 K 1 min 75 4 K T1 (e qubits) 1 h 70 4 K T1 (n qubits) 6 min 75 4 K Control gates Fidelity: single qubit gate 99.995% 78 300 K Fidelity: two qubit (e-e) 97% 122 300 K Fidelity: (e-n) 99.2% 78 300 K Speed: single qubit gate (e) 10 ns 38 300 K Speed: two qubit gate (e-n) 700 ns 78 300 K Measurement Fidelity (e) 98% 120 Projectiveness 99% 123 Post-measurement state Measurement time 3 ls 120 Fidelity three-qubit parity measurement 0.63 123 Initialization Fidelity 99.9% 124 Excitation of electron Speed 200 ns 88 Excitation of electron Chip to chip interconnection Entanglement fidelity 92% 120 Distance 1 km 120 Sub routines Algorithm: Q-search 125 Temperature 300 K Algorithm: Q simulation 126 300 K Quantum error correction Logical qubits 3-qubit code 91 300 K 92 4 K Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011308 (2021); doi: 10.1063/5.0007444 8, 011308-14 V C Author(s) 2021
- 16. atoms in its immediate vicinity (a maximum of two lattice sites away). This method can easily be enhanced by using entangled NV centers. The NV center has very good coherence times even at room tempera- ture, which allows for the implementation of simple quantum algo- rithms that currently cannot be used in competing cryogenic systems. The fidelity obtained is comparable to that of a superconducting qubit system or even higher. The error correction as well as the first quan- tum algorithm have already been verified. VII. CONCLUSIONS AND OUTLOOK In recent years, a large number of proposals have been made for a quantum computer based on color centers in diamond. However, many of them do not go beyond the idea of such a device. Recently, however, techniques and procedures such as deterministic single-ion implantation have been developed, which, after special treatment of the diamond, can produce NV centers with high yields. Solutions have also been found to electrically read out single NV centers by means of the photocurrent. It has been shown that an NV with many 13 C nuclei can be interlaced. These findings may represent a game-changer, even if the coupling to the 13 C nuclei is possible only at low temperature (otherwise, the addressing of separate 13 C nuclear spins is hindered by the broadening of the microwave resonances). The next steps in future research in this area should involve building a system using the magnetic dipole–dipole coupling of a 30- nm pitch grating of NVs, where each NV center is connected to several (e.g., ten) 13 C qubits. The depth of the NV centers should be in the range of 10 to 20 nm so that their electrical readouts and control by microwave pulses can be implemented by crossed nanowires above each NV. The use of crossed wires enables the generation of circularly polarized microwave pulses and allows for the addressing of individual NV centers. Such a scheme cannot only enable the coupling of 100 NV qubits within a 10 10 grid controlled with 20 nanowires, but can provide a one-kilo qubit system (because each NV can further control up to ten 13 C qubits). The scheme can be modified if the cou- pling is carried out using photons, free electrons, phonons, and so on. Furthermore, this concept allows for the stabilization of the NV qubits by the 13 C spin bath and operation at room temperature. In Europe, a consortium with several large companies has already been formed to investigate this new scheme. But even if many detailed questions are still open, the relevant technical and technological chal- lenges are pushing prevalent technology to new borders that may also open new avenues for applications in other fields. The current race to produce the first fully functional quantum computer is comparable to the race to put the first human on the moon and can similarly inspire people. AUTHOR CONTRIBUTIONS All authors contributed equally to this work. ACKNOWLEDGMENTS The authors acknowledge the funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 820394 (ASTERIQS), from the European Union project MICROSENS (No. BMBF 16KIS0831) and from the Bundesministerium f€ ur Bildung und Forschung (project DiaQuantFab). The authors acknowledge the support from Leipzig University for Open Access Publishing. We thank Professor T. H. Taminiau (Uni. Delft) for providing the content for Table II. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1 A. Gruber et al., “Magnetic-resonance on single molecules in an external mag- netic-field: The Zeeman-Effect of a single-electron spin and determination of the orientation of individual molecules,” Chem. Phys. Lett. 242(4–5), 465–470 (1995). 2 J. Wrachtrup, S. Y. Kilin, and A. P. 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