Abstract
A microwave shares a nonintuitive phase called the geometric phase with an interacting electron spin after an elastic scattering. The geometric phase, generally discarded as a global phase, allows universal holonomic gating of an ideal logical qubit, which we call a geometric spin qubit, defined in the degenerate subspace of the triplet spin qutrit. We here experimentally demonstrate nonadiabatic and nonabelian holonomic quantum gates over the geometric spin qubit on an electron or nitrogen nucleus. We manipulate purely the geometric phase with a polarised microwave in a nitrogenvacancy centre in diamond under a zeromagnetic field at room temperature. We also demonstrate a twoqubit holonomic gate to show universality by manipulating the electron−nucleus entanglement. The universal holonomic gates enable fast and faulttolerant manipulation for realising quantum repeaters interfacing between universal quantum computers and secure communication networks.
Introduction
The geometric phase^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}, which is closely related to the topological phase, is currently a central issue not only in quantum physics but also in various scientific fields including optics, electronics, nanotechnology, and materials science. In quantum physics, the holonomic quantum gate manipulating purely the geometric phase in the degenerate ground state system is believed to be an ideal way to build a faulttolerant universal quantum computer. The concept of the geometric phase was first proposed by Pancharatnam^{1} for light polarisation, which is a typical degenerate ground state system with inner degree of freedom. Since then, an adiabatic geometric phase was formalised by Berry^{2} and then was generalised into a nonabelian geometric phase by Wilczek and Zee^{3} as well as into nonadiabatic geometric phases by Anandan^{4}. Meanwhile, their application to faulttolerant holonomic quantum computing^{5,6,7,9,10} was proposed, and the geometric phase gate or holonomic quantum gate has been experimentally demonstrated in a superconducting qubit^{12,13}, in trapped ions^{14,15} in a quantum dot^{16}, in molecular ensembles^{17,18}, and recently in a nitrogenvacancy (NV) centre in diamond^{19,20,21,22,23,24}.
However, these experiments required microwaves^{19,20} or light waves^{21,24} in two frequencies differing by the energy gap for manipulating the nondegenerate subspace, leading to the degradation of gate fidelity due to the unwanted interference of the dynamic phase. To avoid such interference, we used a degenerate subspace of the triplet spin qutrit in spin1 system to form a logical qubit called a geometric spin qubit^{11,22,23}, where the geometric phase was manipulated with a light resonant to the optical excited state as an ancillary level^{22,23,24}. This method allowed fast and precise geometric gates, but could only control a single qubit made of an electron spin at a temperature below 10 K and the gate fidelity was limited by radiative relaxation.
The advantage of a polarised microwave^{25,26} for the manipulation of an electron spin has been demonstrated with the use of a circularly polarised microwave generated by two crossed wires (Fig. 1a, b) to overcome the limit of the rotationwave approximation^{26}.
We here demonstrate nonadiabatic and nonabelian holonomic quantum gates over the geometric spin qubit on an electron or nitrogen nucleus with a polarised microwave in a nitrogenvacancy centre in diamond under a zeromagnetic field at room temperature. We also demonstrate a twoqubit holonomic gate to show universality by manipulating the electron−nucleus entanglement.
Results
Principle of geometric spin manipulation by polarised microwaves
We start by showing the analogy between microwave polarisation and geometric spin polarisation (Fig. 1c). The microwave polarisation is represented as a state vector in the geometric phase space known as the Poincaré sphere based on the degenerate ±1〉_{p} states corresponding to the right R〉_{p} and left L〉_{p} circular polarisations, which are two components of the spin1 angular momentum projected along the travel direction excluding the 0〉_{p} state corresponding to the vacuum. The superposition of R〉_{p} and L〉_{p} represents any polarisations, including horizontal H〉_{p} and vertical V〉_{p} polarisations. Similarly, the geometric spin polarisation of an electron (or a nucleus of nitrogen ^{14}N) in an NV centre in diamond is also represented as a state vector in the geometric phase space known as the Bloch sphere. This sphere is based on the degenerate m_{S}(m_{I}) = ±1 (hereafter denoted ±1〉_{e(N)}) states, which correspond to the degenerate T^{±} subspace of the triplet spin states \(T^ + = \left { \uparrow \uparrow } \right\rangle ,T^  = \left { \downarrow \downarrow } \right\rangle ,T^0 = \frac{1}{{\sqrt 2 }}\left( {\left { \uparrow \downarrow } \right\rangle + \left { \downarrow \uparrow } \right\rangle } \right)\) with spin1 angular momentum projected along the NV axis, excluding the 0〉_{e(N)} state corresponding to the T^{0} state. The ±1〉 states are naturally split from the 0〉 state even under a zeromagnetic field to constitute a degenerate Vtype qutrit system^{27} as light polarisation does. In contrast that the polarised electromagnetic wave becomes a flying qubit with keeping the quantum interference between the R〉_{p} and L〉_{p} states, the spinpolarised electron or nucleus becomes a stationary qubit with keeping the quantum interference between the ±1〉 states.
Next, we explain the mechanism underlying the holonomic gate based on the interaction Hamiltonian. Although the direct manipulation of the geometric spin is not possible because the transition between the T^{±} states is magnetically forbidden, indirect manipulation of the geometric spin is possible with the geometric phase^{1,2,3,4,5,6,7,8,9,10} acquired via a cyclic evolution in the bright space driven by a correspondingly polarised microwave (radiowave) tuned to the zerofield splitting ~2870 MHz (the nuclear quadrupole splitting ~4.945 MHz). The magnetic coupling of the polarised microwave to the spinpolarised electron is described by the interaction Hamiltonian \(H = ({{\mathrm{\Omega }}}/{2})\left( {{\mathrm{e}}^{  {\mathrm{i}}\phi _ + }\left B \right\rangle _{\mathrm{e}}\left\langle 0 \right + {\mathrm{e}}^{{\mathrm{i}}\phi _ + }\left 0 \right\rangle _{\mathrm{e}}\left\langle B \right} \right)\), where Ω denotes the Rabi frequency and B〉_{e} denotes the geometric electron spin in the bright state \(\left B \right\rangle _{\mathrm{e}} = {\mathrm{cos}}\left( {{\theta }/{2}} \right)\left { + 1} \right\rangle _{\mathrm{e}} + {\mathrm{e}}^{{\mathrm{i}}\phi }{\mathrm{sin}}\left( {{\theta }/{2}} \right)\left {  1} \right\rangle _{\mathrm{e}}\) corresponding to the microwave polarisation \(\left \psi \right\rangle _{\mathrm{p}} = {\mathrm{cos}}\left( {{\theta }/{2}} \right)\left R \right\rangle _{\mathrm{p}} + {\mathrm{e}}^{{\mathrm{i}}\phi }{\mathrm{sin}}\left( {{\theta }/{2}} \right)\left L \right\rangle _{\mathrm{p}}\) (Fig. 1c). ϕ_{+} indicates the global phase defined by the phase of the driving microwave and determines rotation axis on the bright space (Methods). This indicates that the transition between the B〉_{e} and the 0〉_{e} is induced while leaving the dark state \(\left D \right\rangle _{\mathrm{e}} = {\mathrm{sin}}\left( {{\theta }/{2}} \right)\left { + 1} \right\rangle _{\mathrm{e}}  {\mathrm{e}}^{{\mathrm{i}}\phi }{\mathrm{cos}}\left( {{\theta }/{2}} \right)\left {  1} \right\rangle _{\mathrm{e}}\) unchanged (Methods). The time evolution of the geometric electron spin after a cyclic transition in the bright space is represented as the holonomy matrix^{3} \(U_{\mathrm{V}} = {\mathrm{e}}^{  {\mathrm{i}}\gamma }\left B \right\rangle _{\mathrm{e}}\left\langle B \right + \left D \right\rangle _{\mathrm{e}}\left\langle D \right = {\mathrm{e}}^{  {\mathrm{i}}\gamma {\mathbf{n}} \cdot {\mathbf{\sigma }}}\), where n indicates the unit gyration vector pointing to the B〉_{e} and σ indicates the Pauli vector. This indicates that the bright state acquires a geometric phase or holonomy γ, which corresponds to half of the solid angle enclosed by the trajectory in the bright space.
Geometric spin state preparation and measurement
We first calibrate the amplitude and phase of the microwaves generated by two crossed wires to precisely prepare and measure the geometric electron or nuclear spin in the basis states \( + {1} \rangle ,  {1} \rangle , + \rangle = ( {1}/ {{\sqrt 2}}) ( + {1} \rangle +   {1} \rangle ), {} \rangle = ({1}/{{\sqrt 2 }}) (  + {1} \rangle   {1} \rangle),  + {{i}} \rangle = \)\(({1}/{{\sqrt 2 }}) (  + {1} \rangle +{\mathrm{i}}   {1} \rangle)\), and \(\left {  i} \right\rangle = ({1}/{{\sqrt 2 }})\left( {\left { + 1} \right\rangle  {\mathrm{i}}\left {  1} \right\rangle } \right.\) (Fig. 1d) by the bright state preparation and projection method^{28}. The achieved fidelity is 96.2 ± 1.6% on average, as evaluated by the quantum state tomography method^{29,30} (Methods). Using the same method, the geometric entangled states between the electron and nitrogen nuclear spins are also prepared and measured in the Bell states \(\left {{\mathrm{\Phi }}^ \pm } \right\rangle _{{\mathrm{e}},{\mathrm{N}}} \,=\, ({1}/{{\sqrt 2 }})\left( {\left { + 1, + 1} \right\rangle _{{\mathrm{e}},{\mathrm{N}}} \pm \left {  1,  1} \right\rangle _{{\mathrm{e}},{\mathrm{N}}}} \right)\) and \(\left {{\mathrm{\Psi }}^ \pm } \right\rangle _{{\mathrm{e}},{\mathrm{N}}} = ({1}/{{\sqrt 2 }})\left( {\left { + 1,  1} \right\rangle _{{\mathrm{e}},{\mathrm{N}}} \pm \left {  1, + 1} \right\rangle _{{\mathrm{e}},{\mathrm{N}}}} \right)\) with fidelity of 92.4 ± 3.1% on average (Methods).
Holonomic gate process tomography
Assuming the state preparation and measurement error is negligible, we evaluate the fidelity of the holonomic singlequbit gates over a geometric electron (nitrogen nuclear) spin. A microwave (radiowave) polarised in the \(\left H \right\rangle _{\mathrm{p}},\left D \right\rangle _{\mathrm{p}},\left R \right\rangle _{\mathrm{p}},({1}/{{\sqrt 2 }})\left( {\left H \right\rangle _{\mathrm{p}} + \left R \right\rangle _{\mathrm{p}}} \right)\) is applied to drive the corresponding bright geometric spin state in the X, Y, Z, and \(({1}/{{\sqrt 2 }})({\mathrm{X}} + {\mathrm{Z}})\) into the 0〉_{e(N)} state and back to the original state with a geometric phase π, which results in the holonomic X, Y, Z, and Hadamard (H) quantum gates. The χ matrices, which represent the gate transformation based on the Pauli matrices, are estimated by the quantum process tomography method^{29,31} (Fig. 2a) (Methods). The fidelities of the X, Y, Z, and H gates over the geometric electron spin are respectively 99, 93, 93, and 93%, while those over the geometric nuclear spin are 97, 93, 98, and 95% (Fig. 2b).
In the same way, the holonomic twoqubit gates can also be performed by adding a geometric phase to the corresponding joint state (Fig. 3a). We demonstrate the holonomic controlledZ (CZ) gate by driving the joint state −1,−1〉_{e,N} into 0,−1〉_{e,N} and back to the original state with a circularly polarised microwave resonant to one of hyperfinesplit frequencies. The fidelity degradation of the Bell states after the holonomic CZ gate is evaluated by the twoqubit quantum state tomography method (Fig. 3b). The \(\left {{\mathrm{\Phi }}^ + } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\) state prepared with 98% fidelity is transformed into the \(\left {{\mathrm{\Phi }}^  } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\) state with 86% fidelity, while the \(\left {{\mathrm{\Psi }}^ + } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\) state prepared with 91% fidelity remained in the same state with 90% fidelity (Fig. 3c). These results indicate that we can achieve twoqubit gate fidelity of about 90%. Any arbitrary holonomic twoqubit gates can also be constructed by combining the holonomic CZ gate with holonomic singlequbit gates. For example, the holonomic controlledNOT (CNOT) gate can be constructed with the CZ gate and the Hadamard gate.
Discussion
We showed that the interaction of an arbitrary polarised microwave with a geometric electron spin qubit allows direct holonomic gate via the spin ground state as an ancillary level without the mediation of the spin−orbit interaction as in the previous optical holonomic gate^{22,23} to avoid the unwanted interference and relaxation, thus enabling fast and arbitrarily precise gating at an ambient temperature. Note that polarised microwaves serve as noncommutable basis operators known as Pauli operators to satisfy the universality of the singlequbit holonomic gates^{20}. This also allows the holonomic gate of a geometric nuclear spin, which was impossible by the optical gate. We also demonstrated a nontrivial twoqubit holonomic gate, which manipulates the entanglement between the electron and nuclear geometric spin, to assure the universality of the holonomic gate to perform arbitrary quantum computation tasks.
The significance of the demonstration is that the geometric spin qubit is manipulated under a completely zero field^{11,22,23} to provide spatial inversion symmetry for the external field, leading to time inversion symmetry or commutability of Hamiltonians for the environmental nuclear spins^{11}. The overall symmetry results in ideal manipulation of the qubit. The conventional frequency freedom should not be used since it inevitably lacks the spatial inversion symmetry by applying a magnetic field^{19,20,21,24,25}. In contrast, polarisation freedom enables purely geometric timereversal manipulation of the geometric spin qubit under a zero magnetic field^{22,23}.
The degradation of gate fidelity can be attributed to dephasing due to hyperfine interaction with the environmental nuclear spins or to mutual interference of the two wires. The former can be improved by reducing the gating time while increasing the driving power. The Rabi frequency used for the geometric electron spin manipulation is about 15 MHz, that is not strong enough compared to hyperfine interaction from the ^{13}C or ^{14}N nuclear spins. Significant improvement in gate fidelity is expected with larger Rabi frequency^{32}. This problem can be also improved by the geometric spin echo^{11}, and by optimising microwave pulse with the GRAPE (gradient ascent pulse engineering) algorithm^{33} or composite pulses. The latter can be improved by precisely adjusting the angle so that the two wires are at a right angle to each other.
Our scheme relies only on the polarisation of a microwave instead of the frequency in the twofrequency scheme^{19,20,21,24} to define the rotation axis. In future implementations, this difference should provide a significant advantage in gate fidelity, as shown in Fig. 4a, for the holonomic X gate on the geometric electron spin qubit under the effect of environmental nuclear spins as functions of the Zeeman splitting of the ±1〉_{e} states and the Rabi frequency in the bright space. In the twofrequency scheme, the fidelity is expected to saturate and oscillate to make the error higher than −32 dB as the Rabi frequency increases due to the interference between the two frequencies^{24}, which induces a deviation of the trajectory from a true circle in the bright space to acquire the expected holonomy (Fig. 4b). Although pulse shaping technique can be used to compensate the deviation, the Rabi frequency Ω is restricted by the energy difference of two frequency Δ to satisfy Ω « Δ. The resulting longer manipulation time induces degradation of the fidelity by the effect of environmental nuclear spins. In contrast, at the degeneracy point, i.e., where the Zeeman splitting diminishes and the polarisationbased holonomic gate scheme applies, the gate fidelity is expected to monotonically increase with a simple square pulse as the Rabi frequency increases to reduce the error to below −45 dB (Fig. 4c).
On the other hand, the twoqubit gate can be affected by interference even under a zeromagnetic field, since the gate relies on the hyperfine splitting. This requires a Rabi frequency smaller than the hyperfine splitting to avoid unexpected excitation, resulting in a longer gate time to deviate from the expected holonomy due to unresolvable hyperfine splitting by the environmental nuclear spins. However, the fidelity of the nuclear spin gate conditioned by the electron spin can be recovered by the geometric spin echo^{11} (Methods). The twoqubit gate is also applicable to the carbon isotope (^{13}C) around the NV centre.
Although our scheme utilises the 0〉 state as an ancilla for the holonomic quantum gate, it can also be used to detect errors caused by the gate operation or spin relaxation to make the qubit a digitallike qubit with error detection. In the experiment, the initialisation error is excluded by heralding the 0〉 state even though the initialisation rate of the ^{14}N nuclear spin is limited to 85%. Using this scheme, not only the controlledphase or SWAP gates between a microwave photon and a geometric spin qubit but also the quantum state transfer from a microwave photon to a geometric spin qubit can be implemented, the same as in the case of an optical photon to a geometric spin qubit at low temperature^{34}. The combination of these transfers will also enable the quantum media conversion between the superconducting qubit and a photon qubit via the geometric spin qubit.
We have demonstrated universal holonomic quantum gates consisting of singlequbit X, Y, Z, and H gates and a twoqubit CZ gate over a geometric electron and nitrogen nuclear spin qubits with polarised microwaves and radiowaves. The holonomic gates with a geometric phase other than π, such as the phase gate (S) or the \(({{\mathrm{\pi }}}/{8})\) gate (T), are also achievable with this scheme. Although the H, S, T, and CZ gates offer the elementary discrete set required for the universal quantum gates to construct arbitrary unitary quantum gates, the availability of arbitrary phase gates would be beneficial for fast gating in many practical applications, such as quantum Fourier transformation and blind quantum computing^{35}. The scheme allows a purely holonomic gate without requiring an energy gap, which would have induced dynamic phase interference to degrade the gate fidelity, and thus enables precise and fast control over longlived quantum memories for realising quantum repeaters interfacing between universal quantum computers and secure communication networks.
Methods
Theory of geometric spin manipulation by polarised microwaves
The Hamiltonian of an electron spin in an NV centre under the irradiation of microwaves emitted from two wires is described as follows.
where D_{0} is the zerofield splitting of the electron spin, γ_{e} is the gyromagnetic ratio of the electron spin, B_{i} is the magnetic field vector, ω_{i} is the frequency, and ϕ_{i} is the phase of the microwave generated by the wire i = 1, 2 and S is the spin1 operator described as
The zaxis is defined as being in the direction of the zerofield splitting of the electron spin. Note that ħ is omitted in this formula.
We estimate that the effect of the z component of the microwave on the electron spin dynamics is 0 on average. The effective microwave fields are represented as B_{1} = B_{1}(1, 0, 0) and \({\mathbf{B}}_2 = B_2({\mathrm{cos}}\theta _{{\mathrm{eff}}},\sin \theta _{{\mathrm{eff}}},0)\), where θ_{eff} is defined as the effective angle between wires 1 and 2 as shown in Fig. 5a. The Hamiltonian can then be rewritten as
where Ω_{i} = γ_{e}B_{i} represents the effective intensity of the microwave with respect to the electron spin and H.c. represents the Hermitian conjugate.
By tuning the microwave frequencies of the two wires as ω_{1} = ω_{2} = D_{0}, the Hamiltonian is reduced with the interaction picture and the rotating wave approximation to
where \({\mathrm{\Omega }}_ \pm = \left {{\mathrm{\Omega }}_1{\mathrm{e}}^{{\mathrm{i}}\phi _1} + {\mathrm{\Omega }}_2{\mathrm{e}}^{{\mathrm{i}}\phi _2 \pm \theta _{{\mathrm{eff}}}}} \right\) and \(\phi _ \pm = {\mathrm{arg}}\left( {{\mathrm{\Omega }}_1{\mathrm{e}}^{{\mathrm{i}}\phi _1} + {\mathrm{\Omega }}_2{\mathrm{e}}^{{\mathrm{i}}(\phi _2 \pm \theta _{{\mathrm{eff}}})}} \right)\). Further calculations result in the following form:
where \({\mathrm{\Omega }}_{{\mathrm{eff}}} = \sqrt {{\mathrm{\Omega }}_ + ^2 + {\mathrm{\Omega }}_  ^2}\) is the effective Rabi frequency, ϕ_{+} is the global phase, \(\theta = 2{\mathrm{arctan}}\left( {{{{\mathrm{\Omega }}_  }}/{{{\mathrm{\Omega }}_ + }}} \right)\) is the relative amplitude, and ϕ = ϕ_{−} − ϕ_{+} is the relative phase. θ and ϕ correspond respectively to the polar angle and azimuth angle of the bright state vector B〉 in the geometric spin space. The phase of the rotation axis in the bright space is specified by ϕ_{+}.
From Eq. (5), the π rotation in the bright space around the axis with phase ϕ_{+} for the state preparation and projection is represented as
and the 2π rotation in the bright space around the axis with phase ϕ_{+} for the holonomic quantum gate is represented as
where
The holonomic quantum gate^{8,9,10} U(γ) can also be achieved by sequentially applying two π rotations with the global phase \(\phi _ + ^{(1)} = 0\) and \(\phi _ + ^{(2)} = {\mathrm{\gamma }}  {\mathrm{\pi }}\) for each pulse as
By tuning γ, the geometric phase given for the geometric spin can be arbitrarily determined between [−π, π].
The θ_{eff} can be estimated by the effective Rabi oscillation. In the experiment, the relative amplitudes of microwaves by the two wires \(\theta _{\mathrm{w}} = 2{\mathrm{arctan}}\left( ({{{{\mathrm{\Omega }}_1  {\mathrm{\Omega }}_2}})/({{{\mathrm{\Omega }}_1 + {\mathrm{\Omega }}_2}}}) \right)\) were adjusted to be 0 (Ω_{1} = Ω_{2} = Ω), and the relative phase ϕ_{w} = ϕ_{2}−ϕ_{1} of the microwaves by the two wires was swept to show the phase dependence. The Ω_{eff} should become
The θ_{eff} in the experiment was estimated from the ϕ_{w} dependence of the Ω_{eff} to be 44.3° (Fig. 5b).
The θ and ϕ can be determined by controlling the θ_{w} and ϕ_{w} as long as the two wires are not completely parallel (Fig. 5c).
The same calculation can be applied to the ^{14}N nuclear spin. The Hamiltonian of ^{14}N is described as follows.
where Q is the nuclear quadrupole splitting and γ_{N} is the gyromagnetic ratio of the nuclear spin. Only the coefficient is different from the Hamiltonian of the electron spin.
Dynamic CNOT between geometric electron and nuclear spin qubits
“Dynamic CNOT” (DCNOT) is one of the entangling manipulations in the spin1 system and is a conceptual extension of general (in a spinhalf system) CNOT. In the presence of Isingtype interaction between the two spin1 systems S_{z}S_{z}, the degeneracy is lifted by the hyperfine interaction even under a zeromagnetic field, as shown in Fig. 6. The degenerate +1, +1〉_{e,N} and −1,−1〉_{e,N} states corresponding to the evenparity Bell states \(\left {{\mathrm{\Phi }}^ \pm } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\) and the degenerate +1, −1〉_{e,N} and −1, + 1〉_{e,N} states corresponding to the oddparity Bell states \(\left {{\mathrm{\Psi }}^ \pm } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\) split with each other by the hyperfine interaction, thus allowing the direct generation of the entangled state. A microwave resonant to the \(\left {{\mathrm{\Phi }}^ \pm } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\)or \(\left {{\mathrm{\Psi }}^ \pm } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\) states induces the transition of the electron spin (target) between the ±1〉_{e} and 0〉_{e} states depending on the nitrogen nuclear spin (control) state ±1〉_{N}. In the case of πrotation, the DCNOT makes direct transitions from the initial state 0, ±〉_{e,N} to the Bell state \(\left {{\mathrm{\Phi }}^ \pm } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\) or \(\left {{\mathrm{\Psi }}^ \pm } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\). The phase of the initial state is transferred to that of the entangled state. Note that all of the levels relating to the transition are degenerated to allow the DCNOT gate without any phase evolution unless any additional hyperfine interactions exist.
Experimental setup
The experimental setup is shown in Fig. 7. We used a native NV centre in a highpurity typeIIa bulk diamond grown by chemical vapour deposition and having a <100> crystal orientation (electronic grade from Element Six) without any dosing or annealing. A negatively charged NV centre located about 22 μm below the surface was found using a confocal laser microscope with a 0.8 NA ×100 objective lens. This confocal microscope was used to detect the phonon sideband (PSB) emission of the NV centre by excitation with a green laser (532 nm) for the active locking of the position and the spin state measurement. An external magnetic field was applied to carefully compensate for the geomagnetic field of about 0.045 mT using a coil magnet during the monitoring of the optically detected magnetic resonance (ODMR) spectrum. From the ODMR spectrum, the NV centre used in the experiment showed hyperfine splitting caused by a ^{14}N nuclear spin at 2.168 MHz, and the inhomogeneous broadening of the spectrum caused by the environmental nuclear spin bath (^{13}C) was 0.38 MHz. All experiments were performed at room temperature.
We placed two crossed copper wires 25 μm in diameter on the surface of a bulk diamond to generate arbitrary polarised microwaves and radiowaves (Fig. 1a). The arbitrary control of the microwave properties (intensity, frequency, and phase) was possible with IQ modulation using the arbitrary waveform generator (AWG). Radiowaves were generated directly from the same type of AWG. We symmetrically duplicated these arbitrary waveform control units to perform independent and synchronised control over microwaves (radiowaves) flowing in the two crossed wires. To avoid unwanted inductive coupling between the wires, we carefully adjusted their angle to be as orthogonal as possible.
Initialisation and readout of the spin state
The electron spin state is first initialised into the 0〉_{e} state^{36} by optical pumping with a green laser for 60 μs. During the initialisation process, ^{14}N nuclear spin is driven into the completely mixed state. The 0〉_{e} state is then transferred to the ^{14}N nuclear spin to initialise into the 0〉_{N} state with two consecutive DCNOT gates controlled by each other (Fig. 8). After that, the electron spin is initialised again by optical pumping with a green laser for 3 μs.
The photoluminescence in the PSB indicates the population of the 0〉_{e} state to allow the projective measurement of the geometric spin state. We can also indirectly measure the population of the 0〉_{N} state by the photoluminescence after applying the DCNOT gate by using the joint state 0, 0〉_{e,N}.
Preparation and measurement of the geometric spin state
We show that the arbitrary single geometric electron or nuclear spin state is prepared and measured (projected) by the pulse sequence as shown in Fig. 8. After the electron and nuclear spin initialisation into the 0〉 state, we apply two π pulses with different polarisations. From Eq. (6), the unitary operator contributing to the state transition is written as
where B_{pre}〉 is the bright state of the first π pulse, B_{proj}〉 and D_{proj}〉 are the bright and dark states of the second π pulse. The first term indicates that the bright state is projected into the 0〉 state to be measured, and the second term indicates that the dark state remains dark since we can measure only the 0〉 state by photoluminescence. We can reconstruct the density matrix ρ = B_{pre}〉〈B_{pre} by the quantum state tomography method^{29,30} with the mutually unbiased bases {±1〉, ±〉, ±i〉}. In the case of electron spin, to minimise the state preparation and measurement error (SPAM error) inherent in the state tomography, the π pulses for the preparation and projection were individually optimised by the GRAPE (gradient ascent pulse engineering) algorithm^{33}.
Preparation and measurement of the geometric entangled state
The DCNOT gate enables the preparation and measurement of the entangled states needed for the twoqubit state tomography based on the energy levels shown in Fig. 9a and the pulse sequence shown in Fig. 9b. The real parts of the twoqubit density matrices and the fidelities compared with the expected Bell states are shown in Fig. 9c. The selection of the resonant frequency enables parity measurement, whether even \(\left {{\mathrm{\Phi }}^ \pm } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\) or odd \(\left {{\mathrm{\Psi }}^ \pm } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\), and the selection of radiowave polarisation enables phase measurement, whether the inphase \(\left {{\mathrm{\Phi }}^ + } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\) \(\left {{\mathrm{\Psi }}^ + } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\) or the outphase \(\left {{\mathrm{\Phi }}^  } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\) \(\left {{\mathrm{\Psi }}^  } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\). The unbalance of the fidelities between even \(\left {{\mathrm{\Phi }}^ \pm } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\) and odd \(\left {{\mathrm{\Psi }}^ \pm } \right\rangle _{{\mathrm{e}},{\mathrm{N}}}\) states is considered to be due to the fact that the flipflop term of the hyperfine interaction affects only the odd states.
Consideration of controlledZ gate
The CZ gate rotates the electron spin with the circularly polarised microwave controlled by the nuclear spin. The microwave needs to be weak enough not to excite the other states (Fig. 10). However, the gate fidelity becomes degraded due to the dephasing of the geometric spin qubit induced by the hyperfine interaction between the electron and environmental nuclear spins (blue line). The gate fidelity is partially recovered by the geometric spin echo^{11}, although the recovery is not as expected (pink line) since the dephasing effect on the targeted state is not the same as on the other state. In contrast, the gate fidelity for the nuclear spin rotation with the circularly polarised radiowave controlled by the electron spin is recovered as expected (red line) since the drive Hamiltonian is commutable with the interaction.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Change history
05 September 2018
This Article was originally published without the accompanying Peer Review File. This file is now available in the HTML version of the Article; the PDF was correct from the time of publication.
References
 1.
Pancharatnam, S. Generalized theory of interference, and its applications. Proc. Indian Acad. Sci. Sect. A 44, 247–262 (1956).
 2.
Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984).
 3.
Wilczek, F. & Zee, A. Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett. 52, 2111 (1984).
 4.
Anandan, J. Nonadiabatic nonabelian geometric phase. Phys. Lett. A 133, 171–175 (1988).
 5.
Zanardi, P. & Rasetti, M. Holonomic quantum computation. Phys. Lett. A 264, 94–99 (1999).
 6.
XiangBin, W. & Keiji, M. Nonadiabatic conditional geometric phase shift with NMR. Phys. Rev. Lett. 87, 097901 (2001).
 7.
Zhu, S.L. & Wang, Z. Implementation of universal quantum gates based on nonadiabatic geometric phases. Phys. Rev. Lett. 89, 097902 (2002).
 8.
Rippe, L., Julsgaard, B., Walther, A., Ying, Y. & Kröll, S. Experimental quantumstate tomography of a solidstate qubit. Phys. Rev. A 77, 022307 (2008).
 9.
Sjöqvist, E. et al. Nonadiabatic holonomic quantum computation. New J. Phys. 14, 103035 (2012).
 10.
Herterich, E. & Sjöqvist, E. Singleloop multiplepulse nonadiabatic holonomic quantum gates. Phys. Rev. A 94, 052310 (2016).
 11.
Sekiguchi, Y. et al. Geometric spin echo under zero field. Nat. Commun. 7, 11668 (2016).
 12.
Falci, G., Fazio, R., Palma, G. M., Siewert, J. & Vedral, V. Detection of geometric phases in superconducting nanocircuits. Nature 407, 355–358 (2000).
 13.
Abdumalikov, A. Jr. et al. Experimental realization of nonAbelian nonadiabatic geometric gates. Nature 496, 482–485 (2013).
 14.
Leibfried, D., Blatt, R., Monroe, C. & Wineland, D. Experimental demonstration of a robust, highfidelity geometric two ionqubit phase gate. Nature 422, 412–415 (2003).
 15.
Toyoda, K., Uchida, K., Noguchi, A., Haze, S. & Urabe, S. Realization of holonomic singlequbit operations. Phys. Rev. A 87, 052307 (2013).
 16.
Economou, S. E. & Reinecke, T. Theory of fast optical spin rotation in a quantum dot based on geometric phases and trapped states. Phys. Rev. Lett. 99, 217401 (2007).
 17.
Feng, G., Xu, G. & Long, G. Experimental realization of nonadiabatic holonomic quantum computation. Phys. Rev. Lett. 110, 190501 (2013).
 18.
Li, H., Liu, Y. & Long, G. Experimental realization of singleshot nonadiabatic holonomic gates in nuclear spins. Sci. China Phys. Mech. Astron. 60, 080311 (2017).
 19.
ArroyoCamejo, S., Lazariev, A., Hell, S. W. & Balasubramanian, G. Room temperature highfidelity holonomic singlequbit gate on a solidstate spin. Nat. Commun. 5, 4870 (2014).
 20.
Zu, C. et al. Experimental realization of universal geometric quantum gates with solidstate spins. Nature 514, 72–75 (2014).
 21.
Yale, C. G. et al. Optical manipulation of the Berry phase in a solidstate spin qubit. Nat. Photon. 10, 184–189 (2016).
 22.
Sekiguchi, Y., Niikura, N., Kuroiwa, R., Kano, H. & Kosaka, H. Optical holonomic single quantum gates with a geometric spin under a zero field. Nat. Photon. 11, 309–314 (2017).
 23.
Ishida, N. et al. Universal holonomic single quantum gates over a geometric spin with phasemodulated polarized light. Opt. Lett. 43, 2380–2383 (2018).
 24.
Zhou, B. B. et al. Holonomic quantum control by coherent optical excitation in diamond. Phys. Rev. Lett. 119, 140503 (2017).
 25.
Alegre, T. P. M., Santori, C., MedeirosRibeiro, G. & Beausoleil, R. G. Polarizationselective excitation of nitrogen vacancy centers in diamond. Phys. Rev. B 76, 165205 (2007).
 26.
London, P., Balasubramanian, P., Naydenov, B., McGuinness, L. & Jelezko, F. Strong driving of a single spin using arbitrarily polarized fields. Phys. Rev. A 90, 012302 (2014).
 27.
Maze, J. et al. Properties of nitrogenvacancy centers in diamond: the group theoretic approach. New J. Phys. 13, 025025 (2011).
 28.
Kosaka, H. et al. Coherent transfer of light polarization to electron spins in a semiconductor. Phys. Rev. Lett. 100, 096602 (2008).
 29.
Nielsen, M. A. & Chuang, I. Quantum Computation and Quantum Information. (Cambridge University Press, Cambridge, UK, 2000).
 30.
Kosaka, H. et al. Spin state tomography of optically injected electrons in a semiconductor. Nature 457, 702–705 (2009).
 31.
Howard, M. et al. Quantum process tomography and Linblad estimation of a solidstate qubit. New J. Phys. 8, 33 (2006).
 32.
Toyli, D. M., Weis, C. D., Fuchs, G. D., Schenkel, T. & Awschalom, D. D. Chipscale nanofabrication of single spins and spin arrays in diamond. Nano Lett. 10, 3168–3172 (2010).
 33.
Khaneja, N., Reiss, T., Kehlet, C., SchulteHerbrüggen, T. & Glaser, S. J. Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. J. Magn. Reson. 172, 296–305 (2005).
 34.
Yang, S. et al. Highfidelity transfer and storage of photon states in a single nuclear spin. Nat. Photon. 10, 507–511 (2016).
 35.
Broadbent, A., Fitzsimons, J. & Kashefi, E. Universal blind quantum computation. In Proc. 50th Annual Symp. Foundations of Computer Science 517–526 (IEEE Computer society, Los Alamitos, USA, 2009).
 36.
Doherty, M. W. et al. The nitrogenvacancy colour centre in diamond. Phys. Rep. 528, 1–45 (2013).
Acknowledgements
We thank Yuichiro Matsuzaki, Burkhard Scharfenberger, Kae Nemoto, William Munro, Norikazu Mizuochi, Nobuyuki Yokoshi, Fedor Jelezko and Joerg Wrachtrup for their discussions and experimental help. This work was supported by the National Institute of Information and Communications Technology (NICT) Quantum Repeater Project; by Japan Society for the Promotion of Science (JSPS) GrantsinAid for Scientific Research (24244044, 16H06326, 16H01052); by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) as an “Exploratory Challenge on PostK computer” (Frontiers of Basic Science: Challenging the Limits); by the Research Foundation for OptoScience and Technology; and by a Japan Science and Technology Agency (JST) CREST Grant Number JPMJCR1773, Japan.
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K.N. carried out the experiment. K.K. supported the experiment and provided theoretical framework. K.N. and K.K. analysed the data and wrote the manuscript. Y.S. provided theoretical support. H.K. supervised the project. All authors discussed the results and commented on the manuscript.
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Nagata, K., Kuramitani, K., Sekiguchi, Y. et al. Universal holonomic quantum gates over geometric spin qubits with polarised microwaves. Nat Commun 9, 3227 (2018). https://doi.org/10.1038/s4146701805664w
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