Realization of fast fault-tolerant quantum gates on a single spin is the core requirement for solid-state quantum-information processing. As polarized light shows geometric interference, spin coherence is also geometrically controlled with light via the spin-orbit interaction. Here, we show that a geometric spin in a degenerate subspace of a spin-1 electronic system under a zero field in a nitrogen vacancy center in diamond allows implementation of optical non-adiabatic holonomic quantum gates. The geometric spin under quasi-resonant light exposure undergoes a cyclic evolution in the spin-orbit space, and acquires a geometric phase or holonomy that results in rotations about an arbitrary axis by any angle defined by the light polarization and detuning. This enables universal holonomic quantum gates with a single operation. We demonstrate a complete set of Pauli quantum gates using the geometric spin preparation and readout techniques. The new scheme opens a path to holonomic quantum computers and repeaters.3 Main textA quantum bit or qubit must be capable of being precisely and quickly manipulated, as well as robust against noise. These criteria pose a dilemma in that the qubit must be open for a driving field but not for a noise field. It has been demonstrated that the degenerate subspace of a spin-1 electronic system under a zero field, which we call a geometric spin, can serve as a promising memory qubit that is robust against environmental noise 1 . The challenge is to manipulate the degenerate qubit with the help of a geometric phase. The concept of the geometric phase was first proposed by Pancharatnam in 1956 2 in reference to light polarization. Since then, two kinds of geometric phase have been discussed. Adiabatic geometric phases were first proposed by Berry in 1984 3 , and nonadiabatic non-Abelian geometric phases were proposed by Anandan in 1988 4 . Holonomic quantum computation (HQC) based on the adiabatic geometric phase was then proposed for fault-tolerant quantum gates by Zanardi and Rasetti in 1999 5 , and generalized to non-adiabatic HQC by Wang and Matsumoto in 2001 6,7 and Zhu and Wang in 2002 8 . The geometric phase has been experimentally demonstrated in molecular ensembles 8,9 , in a superconducting qubit 10 , in trapped ions 11,12 , in a quantum dot 13,14 , and in a single nitrogen-vacancy (NV) center in diamond 15-17 .

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 non-abelian 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 nitrogen-vacancy centre in diamond under a zero-magnetic field at room temperature. We also demonstrate a two-qubit holonomic gate to show universality by manipulating the electron−nucleus entanglement. The universal holonomic gates enable fast and fault-tolerant manipulation for realising quantum repeaters interfacing between universal quantum computers and secure communication networks.

We demonstrate universal non-adiabatic non-abelian holonomic single quantum gates over a geometric electron spin with phase-modulated polarized light and 93% average fidelity. This allows purely geometric rotation around an arbitrary axis by any angle defined by light polarization and phase using a degenerate three-level Λ-type system in a negatively charged nitrogen-vacancy center in diamond. Since the control light is completely resonant to the ancillary excited state, the demonstrated holonomic gate not only is fast with low power, but also is precise without the dynamical phase being subject to control error and environmental noise. It thus allows pulse shaping for further fidelity.

Spin echo is a fundamental tool for quantum registers and biomedical imaging. It is believed that a strong magnetic field is needed for the spin echo to provide long memory and high resolution, since a degenerate spin cannot be controlled or addressed under a zero magnetic field. While a degenerate spin is never subject to dynamic control, it is still subject to geometric control. Here we show the spin echo of a degenerate spin subsystem, which is geometrically controlled via a mediating state split by the crystal field, in a nitrogen vacancy centre in diamond. The demonstration reveals that the degenerate spin is protected by inherent symmetry breaking called zero-field splitting. The geometric spin echo under zero field provides an ideal way to maintain the coherence without any dynamics, thus opening the way to pseudo-static quantum random access memory and non-invasive biosensors.

Quantum teleportation is a key principle for quantum information technology. It permits the transfer of quantum information into an otherwise inaccessible space, while also permitting the transfer of photon information into a quantum memory without revealing or destroying the stored quantum information. Here, we show reliable quantum state transfer of photon polarization into a carbon isotope nuclear spin coupled to a nitrogen-vacancy center in diamond based on photon-electron Bell state measurement by photon absorption. The carbon spin is first entangled with the electron spin, which is then permitted to absorb a photon into a spin-orbit correlated eigenstate. Detection of the electron after relaxation into the spin ground state allows post-selected transfer of arbitrary photon polarization into the carbon memory. The quantum state transfer scheme allows individual addressing of integrated quantum memories to realize scalable quantum repeaters for long-haul quantum communications, and distributed quantum computers for large-scale quantum computation and metrology.

Quantum bits or qubits naturally decohere by becoming entangled with uncontrollable environments. Dynamical decoupling is thereby required to disentangle qubits from an environment by periodically reversing the qubit bases, but this causes rotation error to accumulate. Whereas a conventional qubit is rotated within the SU(2) two-level system, a geometric qubit defined in the degenerate subspace of a V-shaped SU(3) three-level system is geometrically rotated via the third ancillary level to acquire a geometric phase. We here demonstrate that, simply by introducing detuning, the dynamical decoupling of the geometric qubit on a spin triplet electron in a nitrogen-vacancy center in diamond can be made to spontaneously suppress error accumulation. The geometric dynamical decoupling extends the coherence time of the geometric qubit up to 1.9 ms, limited by the relaxation time, with 128 decoupling gates at room temperature. Our technique opens a route to holonomic quantum memory for use in various quantum applications requiring sequential operations. 2 Main textQuantum information technology is becoming a reality in the form of quantum computers, simulators, sensors, as well as the repeaters required for the quantum internet. Long memory time and high-fidelity gates are the key factors that must be scaled up to finally achieve real applications.The widely used dynamical decoupling technique [1][2][3][4][5][6][7][8], which in principle extends the memory time or the coherence time, in practice faces the problem of error accumulation after a large number of decoupling gates, which eventually degrades the state fidelity. The Carr-Purcell-Meiboom-Gill (CPMG) sequence [1] has thus been developed to suppress the accumulation of gate errors, while the initial state is restricted to the eigenstate of the driving field of the decoupling gate. As alternatives, a composite pulse technique for achieving high-fidelity gates [3,4] and a specially designed gate sequence [2,5,6] have been developed to be independent of the initial state.A qubit is typically defined as being in a two-level system with an energy gap, which allows direct transition within the bases to implement dynamic quantum gates. Another type of qubit can also be defined in a two-level system without an energy gap; this type requires an indirect transition via a third ancillary level, and thus constitutes a V-shaped three-level system to implement geometric quantum gates [9][10][11][12]. Geometric quantum gates can be either adiabatic [13][14][15] or non-adiabatic [9][10][11][12][16][17][18][19]. In contrast to the adiabatic geometric gate, the non-adiabatic geometric gate enables faster gate operation to reduce the influence of the environmental noise, thereby resulting in high fidelity.Moreover, the degenerate two-level system is independent of the global phase of the driving field, as seen in the polarization or time-bin encoding of a photon, enabling post selection of successful operations to exclude a population loss from the qubit space spanned by the degenerate...

Geometric nature, which appears in photon polarization, also appears in spin polarization under a zero magnetic field. These two polarized quanta, one travelling in vacuum and the other staying in matter, behave the same as geometric quantum bits or qubits, which are promising for noise resilience compared to the commonly used dynamic qubits. Here we show that geometric photon and spin qubits are entangled upon spontaneous emission with the help of the spin − orbit entanglement inherent in a nitrogen-vacancy center in diamond. The geometric spin qubit is defined in a degenerate subsystem of spin triplet electrons and manipulated with a polarized microwave. An experiment shows an entanglement state fidelity of 86.8%. The demonstrated entangled emission, combined with previously demonstrated entangled absorption, generates purely geometric entanglement between remote matters in a process that is insensitive of time, frequency, and space mode matching, which paves the way for building a noise-resilient quantum repeater network or a quantum internet.

Fault-tolerant quantum memory plays a key role in interfacing quantum computers with quantum networks to construct quantum computer networks. Manipulation of spin quantum memory generally requires a magnetic field, which hinders the integration with superconducting qubits. Completely zero-field operation is desirable for scaling up a quantum computer based on superconducting qubits. Here we demonstrate quantum error correction to protect the nuclear spin of the nitrogen as a quantum memory in a diamond nitrogen-vacancy center with two nuclear spins of the surrounding carbon isotopes under a zero magnetic field. The quantum error correction makes quantum memory resilient against operational or environmental errors without the need for magnetic fields and opens a way toward distributed quantum computation and a quantum internet with memory-based quantum interfaces or quantum repeaters.