Holonomic Quantum Control by Coherent Optical Excitation in Diamond

Transform-Limited Photon Emission from a Lead-Vacancy Center in Diamond above 10 K

Transform-limited photon emission from quantum emitters is essential for high-fidelity entanglement generation. In this Letter, we report the coherent optical property of a single negatively charged lead-vacancy (PbV) center in diamond. Photoluminescence excitation measurements reveal stable fluorescence with a linewidth of 39 MHz at 6 K, close to the transform limit estimated from the lifetime measurement. We observe 4 orders of magnitude different linewidths of the two zero-phonon lines, and find that the phonon-induced relaxation in the ground state contributes to this huge difference in the linewidth. Because of the suppressed phonon absorption in the PbV center, we observe nearly transform-limited photon emission up to 16 K, demonstrating its high temperature robustness compared to other color centers in diamond.

Evolution Operator Can Always Be Separated into the Product of Holonomy and Dynamic Operators

The geometric phase is a fundamental quantity characterizing the holonomic feature of quantum systems. It is well known that the evolution operator of a quantum system undergoing a cyclic evolution can be simply written as the product of holonomic and dynamical components for the three special cases concerning the Berry phase, adiabatic non-Abelian geometric phase, and nonadiabatic Abelian geometric phase. However, for the most general case concerning the nonadiabatic non-Abelian geometric phase, how to separate the evolution operator into holonomic and dynamical components is a long-standing open problem. In this Letter, we solve this open problem. We show that the evolution operator of a quantum system can always be separated into the product of holonomy and dynamic operators. Based on it, we further derive a matrix representation of this separation formula for cyclic evolution, and give a necessary and sufficient condition for a general evolution being purely holonomic. Our finding is not only of theoretical interest itself, but also of vital importance for the application of quantum holonomy. It unifies the representations of all four types of evolution concerning the adiabatic/nonadiabatic Abelian/non-Abelian geometric phase, and provides a general approach to realizing purely holonomic evolution.

Effective non-adiabatic holonomic quantum computation of cavity modes via invariant-based reverse engineering

In this paper, we propose a protocol to realize non-adiabatic holonomic quantum computation (NHQC) of cavity modes via invariant-based reverse engineering. Coupling cavity modes with an auxiliary atom trapped in a cavity, we derive effective Hamiltonians with the help of laser pulses. Based on the derived Hamiltonians, invariant-based reverse engineering is used to find proper evolution paths for NHQC. Moreover, the systematic-error-sensitivity nullified optimal control method is considered in the parameter selections, making the protocol insensitive to the influence of systematic errors of pulses. We also estimate the imperfections induced by random noise and decoherence. Numerical results show that the protocol holds robustness against these imperfections. Therefore, the protocol may provide useful perspectives to quantum computation with optical qubits in cavity quantum electrodynamics systems. This article is part of the theme issue 'Shortcuts to adiabaticity: theoretical, experimental and interdisciplinary perspectives'.

Demonstration of the Holonomically Controlled Non-Abelian Geometric Phase in a Three-Qubit System of a Nitrogen Vacancy Center

The holonomic approach to controlling (nitrogen-vacancy) NV-center qubits provides an elegant way of theoretically devising universal quantum gates that operate on qubits via calculable microwave pulses. There is, however, a lack of simulated results from the theory of holonomic control of quantum registers with more than two qubits describing the transition between the dark states. Considering this, we have been experimenting with the IBM Quantum Experience technology to determine the capabilities of simulating holonomic control of NV-centers for three qubits describing an eight-level system that produces a non-Abelian geometric phase. The tunability of the geometric phase via the detuning frequency is demonstrated through the high fidelity (~85%) of three-qubit off-resonant holonomic gates over the on-resonant ones. The transition between the dark states shows the alignment of the gate's dark state with the qubit's initial state hence decoherence of the multi-qubit system is well-controlled through a π/3 rotation.

Single-Atom Verification of the Noise-Resilient and Fast Characteristics of Universal Nonadiabatic Noncyclic Geometric Quantum Gates

Quantum gates induced by geometric phases are intrinsically robust against noise due to the global properties of their evolution paths. Compared to conventional nonadiabatic geometric quantum computation, the recently proposed nonadiabatic noncyclic geometric quantum computation (NNGQC) works in a faster fashion while still remaining the robust feature of the geometric operations. Here, we experimentally implement the NNGQC in a single trapped ultracold ^{40}Ca^{+} ion to verify the noise-resilient and fast feature. By performing unitary operations under imperfect conditions, we witness the advantages of the NNGQC with measured fidelities by quantum process tomography in comparison to other two quantum gates by conventional nonadiabatic geometric quantum computation and by straightforward dynamical evolution. Our results provide the first evidence confirming the possibility of accelerated quantum information processing with limited systematic errors even in an imperfect situation.

Distributed geometric quantum computation based on the optimized-control-technique in a cavity-atom system via exchanging virtual photons

We propose a scheme for quantum geometric computation on a fiber-cavity-fiber system, in which two atoms are located in two single-mode cavities, respectively, connected with each other by optical fiber. This scheme not only has the feature of virtual excitation of photons in the cavity quantum electrodynamics (CQED) that can reduce the effect of decay effectively but also has the advantage of geometric phase to withstand noises due to its built-in noise-resilience feature and robust merit. Specifically, our proposal combined with optimized-control-technology (OCT) can reduce gate operation error by adjusting the time-dependent amplitude and phase of the resonant field which further enhances the robustness of the quantum operation. The robustness against decoherence is demonstrated numerically and the scheme may be applied in the remote quantum information processing tasks and quantum computation.

Experimental Implementation of Universal Nonadiabatic Geometric Quantum Gates in a Superconducting Circuit

Using geometric phases to realize noise-resilient quantum computing is an important method to enhance the control fidelity. In this work, we experimentally realize a universal nonadiabatic geometric quantum gate set in a superconducting qubit chain. We characterize the realized single- and two-qubit geometric gates with both quantum process tomography and randomized benchmarking methods. The measured average fidelities for the single-qubit rotation gates and two-qubit controlled-Z gate are 0.9977(1) and 0.977(9), respectively. Besides, we also experimentally demonstrate the noise-resilient feature of the realized single-qubit geometric gates by comparing their performance with the conventional dynamical gates with different types of errors in the control field. Thus, our experiment proves a way to achieve high-fidelity geometric quantum gates for robust quantum computation.

Geometric Phase through Spatial Potential Engineering

We propose a spatial analog of the Berry's phase mechanism for the coherent manipulation of states of nonrelativistic massive particles moving in a two-dimensional landscape. In our construction the temporal modulation of the system Hamiltonian is replaced by a modulation of the confining potential along the transverse direction of the particle propagation. By properly tuning the model parameters the resulting scattering input-output relations exhibit a Wilczek-Zee non-Abelian phase shift contribution that is intrinsically geometrical, hence insensitive to the specific details of the potential landscape. A theoretical derivation of the effect is provided together with practical examples.

Optical Control of a Single Nuclear Spin in the Solid State

We demonstrate a novel method for coherent optical manipulation of individual nuclear spins in the solid state, mediated by the electronic states of a proximal quantum emitter. Specifically, using the nitrogen-vacancy (NV) color center in diamond, we demonstrate control of a proximal ^{14}N nuclear spin via an all-optical Raman technique. We evaluate the extent to which the intrinsic physical properties of the NV center limit the performance of coherent control, and we find that it is ultimately constrained by the relative rates of transverse hyperfine coupling and radiative decay in the NV center's excited state. Possible extensions and applications to other color centers are discussed.

Exploring Non-Abelian Geometric Phases in Spin-1 Ultracold Atoms

The spin vector of a spin-1 system, unlike that of a spin-1/2 system, can lie anywhere on or inside the Bloch sphere representing the phase space. As a consequence, the geometrical and topological properties of the spin-1 phase space of quantum states are richer and require a generalization of Berry's phase. For special trajectories passing through the center of the Bloch sphere (singular loops), the geometric phase has a non-Abelian nature. Here, we experimentally explore this geometric phase for singular loops in a spin-1 quantum system using ultracold ^{87}Rb atoms confined in an optical trap using microwave and rf control fields.

Plug-and-Play Approach to Nonadiabatic Geometric Quantum Gates

Nonadiabatic holonomic quantum computation (NHQC) has been developed to shorten the construction times of geometric quantum gates. However, previous NHQC gates require the driving Hamiltonian to satisfy a set of rather restrictive conditions, reducing the robustness of the resulting geometric gates against control errors. Here we show that nonadiabatic geometric gates can be constructed in an extensible way, called NHQC+, for maintaining both flexibility and robustness against certain types of noises. Consequently, this approach makes it possible to incorporate most of the existing optimal control methods, such as dynamical decoupling, composite pulses, and a shortcut to adiabaticity, into the construction of single-looped geometric gates. Furthermore, this extensible approach of geometric quantum computation can be applied to various physical platforms such as superconducting qubits and nitrogen-vacancy centers. Specifically, we performed numerical simulation to show how the noise robustness in recent experimental implementations [Phys. Rev. Lett. 119, 140503 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.140503; Nat. Photonics 11, 309 (2017)NPAHBY1749-488510.1038/nphoton.2017.40] can be significantly improved by our NHQC+.approach. These results cover a large class of new techniques combing the noise robustness of both geometric phase and optimal control theory.

Optical signatures of Mott-superfluid transition in nitrogen-vacancy centers coupled to photonic crystal cavities

Detecting optical signatures of quantum phase transitions (QPT) in driven-dissipative systems constitutes a new frontier for many-body physics. Here we propose a practical idea to characterize the extensively studied phenomenon of photonic QPT, based on a many-body system composed of nitrogen-vacancy centers embedded individually in photonic crystal cavities, by detecting the critical behaviors of mean photon number, photon fluctuation, photon correlation, and emitted spectrum. Our results bridge these observables to the distinct optical signatures in different quantum phases and serve as good indicators and invaluable tools for studying dynamical properties of dissipative QPT.

Environment-Assisted Holonomic Quantum Maps

Holonomic quantum computation uses non-Abelian geometric phases to realize error resilient quantum gates. Nonadiabatic holonomic gates are particularly suitable to avoid unwanted decoherence effects, as they can be performed at high speed. By letting the computational system interact with a structured environment, we show that the scope of error resilience of nonadiabatic holonomic gates can be widened to include systematic parameter errors. Our scheme maintains the geometric properties of the evolution and results in an environment-assisted holonomic quantum map that can mimic the effect of a holonomic gate. We demonstrate that the sensitivity to systematic errors can be reduced in a proof-of-concept spin-bath model.

Experimental Realization of Nonadiabatic Shortcut to Non-Abelian Geometric Gates

When a quantum system is driven slowly through a parametric cycle in a degenerate Hilbert space, the state would acquire a non-Abelian geometric phase, which is stable and forms the foundation for holonomic quantum computation (HQC). However, in the adiabatic limit, the environmental decoherence becomes a significant source of errors. Recently, various nonadiabatic holonomic quantum computation (NHQC) schemes have been proposed, but all at the price of increased sensitivity to control errors. Alternatively, there exist theoretical proposals for speeding up HQC by the technique of "shortcut to adiabaticity" (STA), but no experimental demonstration has been reported so far, as these proposals involve a complicated control of four energy levels simultaneously. Here, we propose and experimentally demonstrate that HQC via shortcut to adiabaticity can be constructed with only three energy levels, using a superconducting qubit in a scalable architecture. With this scheme, all holonomic single-qubit operations can be realized nonadiabatically through a single cycle of state evolution. As a result, we are able to experimentally benchmark the stability of STA+HQC against NHQC in the same platform. The flexibility and simplicity of our scheme makes it also implementable on other systems, such as nitrogen-vacancy center, quantum dots, and nuclear magnetic resonance. Finally, our scheme can be extended to construct two-qubit holonomic entangling gates, leading to a universal set of STAHQC gates.

Experimental Realization of Robust Geometric Quantum Gates with Solid-State Spins

We experimentally realize a universal set of single-bit and two-bit geometric quantum gates by adiabatically controlling solid-state spins in a diamond defect. Compared with the nonadiabatic approach, the adiabatic scheme for geometric quantum computation offers a unique advantage of inherent robustness to parameter variations, which is explicitly demonstrated in our experiment by showing that the single-bit gates remain unchanged when the driving field amplitude varies by a factor of 2 or the detuning fluctuates in a range comparable to the inverse of the gate time. The reported adiabatic control technique and its convenient implementation offer a paradigm for achieving quantum computation through robust geometric quantum gates, which is important for quantum information systems with parameter-fluctuation noise such as those from the inhomogeneous coupling or the spectral diffusion.

Single-Loop Realization of Arbitrary Nonadiabatic Holonomic Single-Qubit Quantum Gates in a Superconducting Circuit

Geometric phases are noise resilient, and thus provide a robust way towards high-fidelity quantum manipulation. Here we experimentally demonstrate arbitrary nonadiabatic holonomic single-qubit quantum gates for both a superconducting transmon qubit and a microwave cavity in a single-loop way. In both cases, an auxiliary state is utilized, and two resonant microwave drives are simultaneously applied with well-controlled but varying amplitudes and phases for the arbitrariness of the gate. The resulting gates on the transmon qubit achieve a fidelity of 0.996 characterized by randomized benchmarking and the ones on the cavity show an averaged fidelity of 0.978 based on a full quantum process tomography. In principle, a nontrivial two-qubit holonomic gate between the qubit and the cavity can also be realized based on our presented experimental scheme. Our experiment thus paves the way towards practical nonadiabatic holonomic quantum manipulation with both qubits and cavities in a superconducting circuit.

Universal holonomic quantum gates over geometric spin qubits with polarised microwaves

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.

Holonomic quantum gates represent a promising route to noise-tolerant quantum operations. Here, the authors use polarised microwaves to implement nonadiabatic holonomic quantum gates at room temperature and zero magnetic field on NV centers, both on single-qubit and between electron and nuclear spins.

Universal holonomic single quantum gates over a geometric spin with phase-modulated polarized light

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.