Complete Bell state measurement of diamond nuclear spins under a complete spatial symmetry at zero magnetic field

Complete Bell state measurement (BSM),¹ which enables the deterministic projection of two prepared qubits into one of four Bell states with only a single measurement, is a core element of any quantum processing such as quantum computation,^2–9 quantum communication based on quantum teleportation,^{10–13,25,52} entanglement swapping,¹⁴ super-dense coding,¹⁵ quantum repeater,^16–19 and MDI-QKD (measurement-device-independent quantum key distribution).^20,21 The BSM has been performed by disentangling the Bell state into product states of the individual qubits and measuring the states with the help of a dynamical decoupling technique²² or by measuring the parity of the qubits with the help of an ancillary qubit.²³ Both methods have been demonstrated in nitrogen-vacancy (NV) centers in diamond [Fig. 1(a)] under a magnetic field more than 100 times stronger than the geomagnetic field.^24–26 Under such circumstances, however, the measurement fidelity is not increased independently of carbon sites due to the competition among quantization axes between internal and external magnetic fields [Fig. 1(b)]. Moreover, applying a magnetic field also limits the use of hybrid integration of quantum systems such as superconducting qubits,^27–29 which become unstable in a strong magnetic field due to the penetration of magnetic flux into the superconductor or the Josephson junction. An accessible magnetic field is around 100 Gauss even in the plane of a superconducting loop^30,31 unless an exotic junction is used, such as semiconductor nanowires.

In this work, we demonstrate complete BSM on carbon nuclear spins in the vicinity of an NV center with entanglement manipulations based on the geometric phase with an arbitrary polarized microwave under a zero magnetic field. In contrast to a high magnetic field regime, arbitrary oriented carbons can be equally manipulated under a zero magnetic field because the quantization axis is uniquely defined only by the internal hyperfine field.^32–34 Moreover, the capability of entanglement manipulation under a zero magnetic field opens the possibility of quantum transduction between a diamond spin qubit and a superconducting qubit, which we believe are the most promising candidate processor qubits for building quantum computers. To extend the scale of quantum computers and the distance of quantum communications, it is desirable to use spin memories in diamond, which serve as a quantum interface with long memory time^35–38 and excellent optical accessibility to connect quantum computers to optical quantum networks.^39–44 

The orbital ground state electrons in a negatively charged NV center constitute a spin-1 triplet system represented by $ms=0,±1$ and form a V-shaped three-level system under a zero magnetic field.^45–47 We treat the degenerate $mS=±1S$ states as a processor qubit, called a geometric qubit,^48–58 and the $mS=0S$ state as an ancilla, which is separated from $mS=±1S$ by a zero-field splitting. Two carbon isotope spins (⁠$C13$⁠) coupled to the electron spin via hyperfine interactions are used as memory qubits [Fig. 1(c)]. The Hamiltonian describing the electron spin and two carbon spins under a zero magnetic field is given by

where $Sz$ is the $z$ component of the spin-1 operator of the electron spin and $IzC1,IzC2$ are the $z$ components of the spin-1/2 operator of the carbon spins labeled by the site. $D0/2π=2.87$ GHz is the axial zero-field splitting of the electron spin, and $AC1/2π=−1.14$ MHz, $AC2/2π=−0.33$ MHz are the hyperfine interactions between the electron spin and the corresponding carbon spin [Fig. 1(d)]. As the equation shows, the eigenstates of the carbon spins $↓ (mI=−$1/2) and $↑ (mI=$ 1/2) are determined only by the quantization axis along the hyperfine field created by the electron, resulting in high spatial symmetry that is independent of the carbon sites.

The key challenge of this work is to perform two-qubit operations enabling the BSM between two carbon spins, whose direct interaction is negligibly weak. Instead of using this weak interaction, we make use of the hyperfine interaction of carbons with the electron spin to apply the controlled phase gate (Controlled Z; CZ gate) between carbons, which is a fundamental element of two-qubit gates, based on holonomic quantum manipulation yielding a geometric phase π to the electron spin as $|+1S|↑C1|↑C2→eiπ|+1S|↑C1|↑C2$ [Fig. 1(e)]. A cyclic rotation in the geometric space spanned by $|+1S$ and $0S$ induces an arbitrary geometric phase. Since the carbon spins do not have their own quantization axis and depend strongly on only the electron spin state, we must choose the electron spin state $|+1S$ out of the degenerate states $|±1S$⁠. The $|+1S$ state is selected with only a circularly polarized microwave even under a zero magnetic field by using two orthogonally oriented wires placed on the diamond. In addition, we access the target carbon with a weak microwave resonant to the energy splitting between $|+1S|↑C1|↑C2$ and $|0S|↑C1|↑C2$⁠. It was demonstrated previously that a carbon spin can be selectively manipulated by using a weak enough square wave.^59–61 CPMG (Carr-Purcell-Meiboom-Gill)⁶²-optimized microwaves have also been used to mitigate the limited coherence time of the electron spin. In the present demonstration, the GRAPE (gradient ascent pulse engineering) algorithm^63,64 is used to optimize the waveform to obtain high fidelity with a limited pulse width by adjusting the microwave amplitude to minimize the infidelity of the time evolution operator U as $1−F=1−Trρtarget†UρinitU†$⁠, which compares the state after the operation on the initial state (⁠$ρinit$⁠) with the target state (⁠$ρtarget$⁠). Based on this method, we optimize the CNOT (Controlled Not)-like gate used for the single-shot measurement of carbon spins and the CZ gate used for the complete BSM.

The complete BSM of two spins is composed of two operations: one is the disentanglement operation of the entangled state into product states, which is just the reverse of the entanglement generation explained later, and the other is the projective single-shot measurement of the individual spin state.

The single-shot measurement^56–59 (SSM), which enables projective measurement to provide a deterministic outcome with only a single preparation of a state, is required for the complete BSM to deterministically specify the state from among four Bell states. Figure 2(a) shows the sequence of the SSM. The electron spin is first initialized into $|0S$ by irradiating a laser resonantly exciting the orbital ground state to the $|A1$ orbital excited state, which relaxes to the ground state with the $|0S$ spin state, and then a CNOT-like gate is applied to the electron spin by irradiating a microwave $π$ pulse to excite the electron spin from $|0S$ to $|+1S$ conditioned on the carbon spin state of $|↑C$ or $|↓C$⁠. The remaining $|0S$ state is examined by detecting a photon after irradiating a laser resonantly exciting the orbital ground state with the $|0S$ spin state to the $|Ey$ orbital excited state, which relaxes to the orbital ground state with the remaining $|0S$ spin state. Although the number of photons acquired by a single sequence of the measurement is less than 1, enough photons to distinguish $|↑C$ or $|↓C$ are acquired by repeating the sequence. The number of repetitions is limited to 100 in the experiment because the spins are depolarized probabilistically during the repetitive optical excitations. Figures 2(b) and 2(c) show that the probability of photon detection decreases exponentially with repetition. Since a quantum measurement projects systems onto its eigenstates, the projective SSM is applicable not only to the state readout but also to the state initialization. Figures 2(d) and 2(e) show histograms of photon counts acquired by the SSM. Since the state readout has to deterministically distinguish $|↑C$ or $|↓C$⁠, only one threshold is set at seven photons for the C₁ and seven photons for the C₂. In contrast, since the state initialization does not necessarily have to be deterministic, two thresholds are set at zero photon to distinguish $|↑C$ and 20 photons to distinguish $|↓C$ for both carbons. In addition to the spin state, the charge state is detected at the end of the sequence to post-select the negatively charged state (NV⁻) to eliminate the neutrally charged state (NV⁰) [Fig. 2(f)]. The fidelities of a state readout are each defined as the average of $F↑=N↑↑N↑↑+N↑↓, F↓=N↓↓N↓↓+N↓↑$⁠, where $N↑↓$ means the number of events when the spin state is initialized to be $|↑C$ and measured to be $|↓C$⁠. Experimentally achieved fidelities are 99.7% for C₁ [Fig. 2(d)] and 99.5% for C₂ [Fig. 2(e)]. These results indicate that the maximum achievable fidelity of the BSM is 99.2%. The fidelity can be dramatically improved by increasing the photon extraction efficiency through the use of a solid immersion lens or an optical cavity because it would allow the two distributions to be more clearly distinguished.

The fidelity of the BSM is now evaluated by the combination of the disentanglement operation and the projective SSM for the arbitrary prepared specific entangled state (one of four Bell states) of carbon spins [Fig. 3(a)]. After the individual spins are initialized by the projective SSM, a specific entangled state is generated by applying Hadamard gates with a $π/2$ pulse of radio waves and the holonomic CZ gate shown above. The phase of the second radio-wave pulse applied to C₁ is $π$-phase shifted against the first pulse. First, we evaluate the fidelities of the four Bell states by quantum state tomography. To reconstruct the density matrices corresponding to those four states, we measure the individual carbon spin state on various bases by applying a $π/2$ pulse of radio waves. Since this measurement does not need to be deterministic, we set two thresholds to discriminate the carbon spin states $|↑C$ and $|↓C$⁠. The method of constructing density matrices from experimental results is shown in the supplementary material. The fidelity of the prepared Bell state, defined as $Tr[ρideal†ρ exp]$⁠, where $ρideal$ and $ρ exp$⁠, respectively, denote the ideal and experimental density matrices, is first evaluated by the quantum state tomography measurements to be 82.2% on average [Fig. 3(b)]. The fidelity of the BSM, defined as the probability of distinguishing the Bell state correctly as prepared, is then evaluated by the SSM of the prepared Bell state to be 74.3% on average [Fig. 3(c)]. Note that the intrinsic fidelity of the measurement only should be much more than 74.3% since the obtained BSM fidelity includes the state preparation and measurement (SPAM) errors.

The fidelity degradation is caused mainly by errors in the Hadamard gates for C₁ and C₂ and in the CZ gate. The errors of the Hadamard gates are estimated by measuring the state fidelities after applying radio-wave $π$ pulses to flip the carbon spins, which are 94.9% for C₁ and 93.5% for C₂. Since the Hadamard gate is made of a radio-wave $π/2$ pulse, the estimated fidelities of the Hadamard gates as the square root of the flipping fidelities results in 97.4% for C₁ and 96.7% for C₂. Since the CZ gate is optimized only for carbon spins detectable by the ODMR measurements and not for environmental carbon spins that are undetectable due to inhomogeneous broadening, the gate fidelity is degraded by an imperfect electron spin manipulation such as noncyclic rotation or dark state generation. The construction of fault-tolerant holonomic quantum gates that are robust against the environmental spins is a significant challenge in dealing with a zero magnetic field. Another factor of CZ gate degradation is that an unwanted phase is added to the states other than the $|↑↑$ state [Fig. 4(a)]. This result can be explained by the fact that a severe constraint, in which the time evolution process of the electron spin states is independent of the nitrogen spin state, is required. In this paper, we do not initialize the nitrogen spin state. The microwave frequency is set at $D0−AC1/2−AC2/2$ for the GRAPE optimization to align the energy gap between the $|+1, 0, ↑,↑S,N,C1,C2$ state and the $|0S$ state. The fidelity of the ideal CZ gate, which is averaged over the nitrogen spin states without environmental spins, is estimated from the simulation to be 97.8%. The fidelity of the CZ gate will degrade to about 90% due to hyperfine interactions by farther carbon spins hidden within inhomogeneous broadening. Provided that the nitrogen spin state is initialized into the eigenstate $|0N$⁠, $|+1N$⁠, or $|−1N$⁠, we can easily improve the fidelity of the CZ gate by employing a pulse whose Rabi frequency is too weak to induce the cyclic rotation of states other than $|↑↑$⁠.

To further explain the superiority of a zero magnetic field, we here consider the limitation of CZ gate fidelity. Under a zero magnetic field, the quantization axis of carbon spin is independent of the electron spin. Therefore, we can construct the CZ gate, which consists of the cyclic rotation without any changes in carbon spin eigenstates. By reducing the microwave power to get a lower Rabi frequency, the spin state of the carbon can be finely selected, and a high-fidelity gate operation can be achieved. On the other hand, under a magnetic field, the Hamiltonian describing the electron spin and one of the carbon spins is given by¹⁹ 

where $μe,(C)$ is the magnetic moment of the electron (carbon), $Bz$ is the external magnetic field, and $A⊥$ and $A∥$ represent the hyperfine interactions, respectively, normal and parallel to an external magnetic field [Fig. 1(c)]. We here assume a single carbon that has a hyperfine coupling strength of $A=A⊥2+A∥2=1$ MHz. The quantization axis of carbon spin is determined by an external magnetic field. Due to the existence of a hyperfine field normal to the external magnetic field (⁠$A⊥$⁠), the eigenstate of the carbon spin depends on the electron spin state. We thus apply cyclic rotation between $|0,↑S,C$ and $|+1,↗S,C$⁠, where $|↗C$ is the eigenstate of carbon spin corresponding to a superposition of $|↑C$ and $|↓C$ derived from the Hamiltonian in Eq. (2). The discrepancy in the eigenstates of carbon spins during the cyclic evolution causes a deterioration of CZ gate fidelity even if we consider the small Rabi frequency regime. When the magnetic field is strong, ranging from 1000 Gauss to 1 T,²³ relatively high fidelity can be achieved when the angle between the external field and the hyperfine field is less than 45°, while the fidelity degrades drastically for a large angle close to 90° [Fig. 4(b)].

In summary, we have demonstrated the complete BSM in a spatially symmetric environment where the quantization axis of carbon spins is governed only by the hyperfine field of the electron spin without the influence of an external magnetic field. The projective SSM was realized with the help of a geometric phase for individual carbon spins. The demonstrated scheme is applicable to any single carbon. In principle, it should be possible to realize two-qubit operations with extremely high fidelity, although the fidelity of the obtained two-qubit operations and BSM could be degraded due to the environmental carbon spins, which could be decoupled by introducing optimized composite pulses⁶⁵ robust against environmental noise designed for a spatially symmetric system under a zero magnetic field.

See the supplementary material for details regarding the experimental setups, readout, and initialization of electron spin and the implementation of two-qubit gates.

We thank Hiromitsu Kato, Toshiharu Makino, Tokuyuki Teraji, and Yuichiro Matsuzaki for their discussions and experimental help. This work was supported by the Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research (Nos. 16H0632619, 20H05661, 19H0551929, and 20K2044120), the Japan Science and Technology Agency (JST) CREST (Core Research for Evolutional Science and Technology) (No. JPMJCR1773), and the JST Moonshot R&D (JPMJMS2062). We also acknowledge the Ministry of Internal Affairs and Communications, Research and Development for the construction of a global quantum cryptography network (No. JPMI00316).