Geometric Quantum Gates Research Articles

Noncyclic nonadiabatic geometric quantum gates in a superconducting circuit

Noncyclic nonadiabatic geometric quantum gates in a superconducting circuit

Realization of high-fidelity and robust geometric gates with time-optimal control technique in superconducting quantum circuit

One of the key features required to realize fault-tolerant quantum computation is the robustness of quantum gates against errors. Since geometric quantum gate is naturally insensitivity to noise, it appears to be a promising routine to achieve high-fidelity, robust quantum gates. The implementation of geometric quantum gate however faces some troubles such as its complex interaction among multiple energy levels. Moreover, traditional geometric schemes usually take more time than equivalent dynamical ones. Here, we experimentally demonstrate a geometric gate scheme with the time-optimal control (TOC) technique in a superconducting quantum circuit. With a transmon qubit and operations restricted to two computational levels, we implement a set of geometric gates which exhibit better robustness features against control errors than the dynamical counterparts. The measured fidelities of TOC X gate and X/2 gate are 99.81% and 99.79% respectively. Our work shows a promising routine toward scalable fault-tolerant quantum computation.

Effective implementation of nonadiabatic geometric quantum gates of cat-state qubits using an auxiliary qutrit

We propose an effective protocol for the implementation of nonadiabatic geometric quantum gates of cat-state qubits in Kerr-nonlinear resonators driven by two-photon squeezing drives. Coupling the Kerr-nonlinear resonators with an auxiliary qutrit with proper coupling strengths, the selective transition of the auxiliary qutrit is realized. The selective transition can be exploited in the implementation of a set of useful quantum gates, including the phase gates, the NOT gates, the controlled-phase gates, the controlled NOT gates, and the Toffoli gates. Numerical simulations show the implementations of different types of gates are robust against systematic errors, random noise, and decoherence. Therefore, the protocol may be helpful for robust and scalable quantum computation based on cat-state qubits.

Optimizing nonadiabatic geometric quantum gates against off-resonance error in a silicon-based spin qubit

Geometric quantum gates are performed by using the geometric phase, making them particularly robust to the pulse amplitude error. However, in many systems, such as the silicon-based spin qubits, the off-resonance error is the dominant noise, which can cause dephasing and is always difficult to deal with for a geometric gate. Thus how to combat with the off-resonance error is very significant for the application of the geometric gates. A recent work [S. Li and Z.-Y. Xue, Phys. Rev. Appl. 16, 044005 (2021).] revealed that by inserting two $\ensuremath{\pi}$ pulses into the evolution paths, the holonomic quantum gate is effective to suppress the pulse amplitude error, however, it is still useless for combating the off-resonance error. Inspired by this work, we combine using the techniques of dynamical correction and path design. Surprisingly, we find that, by picking up a specific evolution path inserted by only a $\ensuremath{\pi}$-pulse, the obtained optimized geometric gate is robust to the off-resonance error, assuming the noise is static. Further, by calculating the filter function considering the realistic $1/f$-type noise in silicon, the related results show that the performance of the optimized geometric gate can also surpass both the conventional geometric gate and the naive dynamical gate constructed without using the geometric phase. Our results indicate dynamical correction is a powerful tool to improve the geometric quantum computation to achieve a high-fidelity quantum gate in silicon.

Conditions for Equivalent Noise Sensitivity of Geometric and Dynamical Quantum Gates

Geometric quantum gates are often expected to be more resilient than dynamical gates against certain types of error, which would make them ideal for robust quantum computing. However, this is still up for debate due to seemingly conflicting results in the literature. Here we use dynamical invariant theory in conjunction with filter functions in order to analytically characterize the noise sensitivity of an arbitrary quantum gate. For any control Hamiltonian that produces a geometric gate, we find that under certain common conditions one can construct another control Hamiltonian that produces an equivalent dynamical gate with identical noise sensitivity (as characterized by the filter function). Our result holds for a Hilbert space of arbitrary dimensions, but we illustrate our result by examining experimentally relevant single-qubit scenarios and providing explicit examples of equivalent geometric and dynamical gates.

Parallel-path implementation of nonadiabatic geometric quantum gates in a decoherence-free subspace with nitrogen-vacancy centers

We propose a scheme for nonadiabatic noncyclic geometric quantum computation in a decoherence-free subspace (DFS). The physical system we used is a cavity quantum electrodynamics system consisting of nitrogen-vacancy (NV) centers in diamond and a whispering-gallery-mode (WGM) microresonator. This scheme not only has the feature of DFSs insensitive to decoherence, but also has the advantage of nonadiabatic noncyclic geometric quantum computation which has the built-in noise-resilience geometric feature and further enhances the robustness due to the ability beyond the limitation of the cyclic condition. An unconventional two-qubit phase gate in DFS has been realized in one step with our parallel protocol, which can largely shorten the computation time and the system error caused by decoherence. In addition, a scheme to realize a parallel process of two evolution paths is put forward to achieve the distribution quantum computation of two WGM microresonators containing a NV center in each. Our scheme may provide a promising way towards the practical realization of high-fidelity geometric quantum computation with the features of a DFS, geometric phase, and parallel implementation process.

Super-robust nonadiabatic geometric quantum control

Nonadiabatic geometric quantum computation (NGQC) and nonadiabatic holonomic quantum computation (NHQC) have been proposed to reduce the run time of geometric quantum gates. However, in terms of robustness against experimental control errors, the existing NGQC and NHQC scenarios have no advantage over standard dynamical gates in most cases. Here, we give the reasons why nonadiabatic geometric gates are sensitive to the control errors and, further, we propose a scheme of super-robust nonadiabatic geometric quantum control, in which the super-robust condition can guarantee both high speed and robustness of the geometric gate. To illustrate the working mechanism of super-robust geometric quantum gates, we give two simple examples of SR-NGQC and SR-NHQC for two- and three-level quantum systems, respectively. Theoretical and numerical results with the experimental parameters indicate that our scheme can significantly improve the gate performance compared to the previous NGQC, NHQC, and standard dynamical schemes. Super-robust geometric quantum computation can be applied to various physical platforms such as superconducting qubits, quantum dots, and trapped ions. All of these sufficiently show that our scheme provides a promising way towards robust geometric quantum computation.

Implementation of Geometric Quantum Gates on Microwave‐Driven Semiconductor Charge Qubits

AbstractA semiconductor‐based charge qubit, confined in double quantum dots, can be a platform to implement quantum computing. However, it suffers severely from charge noises. Here, a theoretical framework to implement universal geometric quantum gates in this system is provided. It is found that, while the detuning noise can be suppressed by operating near its corresponding sweet spot, the tunneling noise, on the other hand, is amplified and becomes the dominant source of error for single‐qubit gates, a fact previously insufficiently appreciated. It is demonstrated, through numerical simulation, that the geometric gates outperform the dynamical gates across a wide range of tunneling noise levels, making them particularly suitable to be implemented in conjunction with microwave driving. To obtain a nontrivial two‐qubit gate, a hybrid system is introduced with charge qubits coupled by a superconducting resonator. When each charge qubit is in resonance with the resonator, it is possible to construct an entangling geometric gate with fidelity higher than that of the dynamical gate for experimentally relevant noise levels. Therefore, the results suggest that geometric quantum gates are powerful tools to achieve high‐fidelity manipulation for the charge qubit.

Nonadiabatic geometric quantum gates that are insensitive to qubit-frequency drifts

Quantum manipulation based on geometric phases provides a promising way towards robust quantum gates. However, in the current implementation of nonadiabatic geometric phases, operational and/or random errors tend to destruct the conditions that induce geometric phases, thereby smearing their noise-resilient feature. In a recent experiment [Y. Xu et al., Phys. Rev. Lett. 124, 230503 (2020)], high-fidelity universal geometric quantum gates have been implemented in a superconducting circuit, which are robust to different types of errors under different configurations of the geometric evolution paths. Here, we apply the path-design strategy to explain in detail why both configurations can realize universal quantum gates in a single-loop way. Meanwhile, we purposefully induce our geometric manipulation by selecting the path configuration that is robust against the qubit-frequency-drift-induced error, which is the dominant error source on realistic superconducting circuits and has not been deliberately addressed. Moreover, our proposal can further integrate with the composite scheme to enhance the gate robustness, which is verified by numerical simulations. Therefore, our scheme provides a promising way towards practical realization of high-fidelity and robust nonadiabatic geometric quantum gates.

High-Fidelity and Robust Geometric Quantum Gates that Outperform Dynamical Ones

Geometric phase is a promising element to induce high-fidelity and robust quantum operations due to its built-in noise-resilience feature. Unfortunately, its practical applications are usually circumscribed by requiring complex interactions among multiple levels/qubits and the longer gate-time than the corresponding dynamical ones. Here, we propose a general framework of geometric quantum computation with the integration of the time-optimal control technique, where the shortest smooth geometric path is found to realize accelerated geometric quantum gates, and thus greatly decreases the gate errors induced by both the decoherence effect and operational imperfections. Meanwhile, we faithfully implement our idea on a scalable platform of a two-dimensional superconducting transmon-qubit lattice, with simple and experimental accessible interactions. In addition, numerical simulations show that our implemented geometric gates possess higher fidelities and stronger robustness, which outperform the best performance of the corresponding dynamical ones. Therefore, our scheme provides a promising alternative way towards scalable fault-tolerant solid-state quantum computation.

Digital Quantum Simulation of Nonadiabatic Geometric Gates via Shortcuts to Adiabaticity.

Geometric phases are used to construct quantum gates since it naturally resists local noises, acting as the modularized units of geometric quantum computing. Meanwhile, fast nonadiabatic geometric gates are required for reducing the information loss induced by decoherence. Here, we propose a digital simulation of nonadiabatic geometric quantum gates in terms of shortcuts to adiabaticity (STA). More specifically, we combine the invariant-based inverse engineering with optimal control theory for designing the fast and robust Abelian geometric gates against systematic error, in the context of two-level qubit systems. We exemplify X and T gates, in which the fidelities and robustness are evaluated by simulations in ideal quantum circuits. Our results can also be extended to constructing two-qubit gates, for example, a controlled-PHASE gate, which shares the equivalent effective Hamiltonian with rotation around the Z-axis of a single qubit. These STA-inspired nonadiabatic geometric gates can realize quantum error correction physically, leading to fault-tolerant quantum computing in the Noisy Intermediate-Scale Quantum (NISQ) era.

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.

Flexible scheme for the implementation of nonadiabatic geometric quantum computation

In this paper, a flexible and effective scheme is proposed to realize nonadiabatic geometric quantum gates with the invariant based reverse engineering and the nonadiabatic holonomic quantum computation ($\mathrm{NHQC}+$) presented in recent work [B.-J. Liu et al., Phys. Rev. Lett. 123, 100501 (2019)] for extensible geometric quantum computation. The scheme provides variabilities for most of control parameters, and can build up multiple evolution paths with different geometric phases acquired in a cycling evolution. As the computational basis can be covered by the evolution paths, the realization of the nonadiabatic geometric quantum computation is possible without auxiliary levels. Moreover, multiple types of nonadiabatic geometric quantum gates can also be easily constructed by only adjusting boundary conditions of control parameters with the method. To show the applications of the scheme, we discuss the implementations of nonadiabatic geometric quantum gates of spin qubits in a double-quantum-dot system with numerical simulations. The results indicate that the scheme possesses robustness against the errors and noise. Therefore, the scheme may offer some interesting perspectives for the realization of the nonadiabatic geometric quantum computation.

Geometric Rydberg quantum gate with shortcuts to adiabaticity.

We propose a controlled-PHASE gate for neutral atoms in which one of the qubit state components adiabatically evolves along the multiple-atom eigenstate formed by the chirped laser pulse coupling to Rydberg states and intrinsic dipole-dipole exchange interactions and, consequently, accumulates an interaction-induced geometric phase. The geometric Rydberg gate is not limited by an adiabatic condition, which is sped up by shortcuts to adiabaticity (STA). Analyses show that an STA scheme is more robust than a non-adiabatic case against the variations of control parameters and faster than an adiabatic case, which protects from the decay of Rydberg states. Furthermore, an intermediate value of dipole-dipole interaction strength is enough for our scheme.

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.

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.

Proposal of realizing superadiabatic geometric quantum computation in decoherence-free subspaces

We propose a practical scheme to implement universal superadiabatic geometric quantum gates in decoherence-free subspaces in the trapped-ions system. The logical qubit is only encoded by two neighboring physical qubits, which is the minimal resource for the decoherence-free subspace encoding. Different from the nonadiabatic control in decoherence-free subspace (Liang et al. in Phys Rev A 89:062312, 2014), a new Hamiltonian to implement universal effective interaction between logical qubits is proposed in the scheme. The proposed gates are numerically demonstrated to be robust against both systematic errors and collective dephasing noises, which combine the advantages of superadiabatic geometric quantum control and decoherence-free subspace. Since the Hamiltonian we use relies solely on two-body interactions, our scheme would be promising to be realized experimentally in trapped-ions systems.

Nonadiabatic Geometric Quantum Computation with Parametrically Tunable Coupling

The nonadiabatic geometric quantum computation is promising as it is robust against certain types of local noises. However, its experimental implementation is challenging due to the need of complex control on multi-level and/or multiple quantum systems. Here, we propose to implement it on a two-dimensional square superconducting qubit lattice. In the construction of our geometric quantum gates, we only use the simplest and experimentally accessible control over the qubit states of the involved quantum systems, without introducing any auxiliary state. Specifically, our scheme is achieved by parametrically tunable all-resonant interaction, which leads to high-fidelity quantum gates. Meanwhile, this simple implementation can be conveniently generalized to a composite scenario, which can further suppress the systematic error during the gate operations. In addition, universal nonadiabatic geometric quantum gates in decoherence-free subspace can also be realized based on the tunable coupling between only two transmon qubits, without consulting to multiple qubits and only using two physical qubits to encode a logical qubit. Therefore, our proposal provides a promising way of high-fidelity geometric manipulation for robust solid-state quantum computation.

Degenerate eigensubspace in a triangle-level system and its geometric quantum control

The non-Abelian geometric phases appear only in a degenerate subspace and the most familiar model with such a subspace is the $N$-pod system. Here we propose an alternative system to realize a non-Abelian gauge structure. We demonstrate that a three-level system with a triangular form can have two degenerate eigenstates which can induce non-Abelian geometric phases, and notably they are the lowest eigenstates of the system. As an application of the degenerate subspace, we propose an experimentally feasible scheme to realize a universal set of quantum gates based on the adiabatic non-Abelian geometric phases. To shorten the evolution time constrained by the adiabatic condition, the shortcut to adiabaticity is adopted to speed up the adiabatic geometric quantum gates. The combination of the intrinsic geometric characteristic and shortcut to adiabaticity makes the quantum operations fast and robust against certain control errors.

Universal non-adiabatic geometric manipulation of pseudo-spin charge qubits

Reliable quantum information processing requires high-fidelity universal manipulation of quantum systems within the characteristic coherence times. Non-adiabatic holonomic quantum computation offers a promising approach to implement fast, universal, and robust quantum logic gates particularly useful in nano-fabricated solid-state architectures, which typically have short coherence times. Here, we propose an experimentally feasible scheme to realize high-speed universal geometric quantum gates in nano-engineered pseudo-spin charge qubits. We use a system of three coupled quantum dots containing a single electron, where two computational states of a double quantum dot charge qubit interact through an intermediate quantum dot. The additional degree of freedom introduced into the qubit makes it possible to create a geometric model system, which allows robust and efficient single-qubit rotations through careful control of the inter-dot tunneling parameters. We demonstrate that a capacitive coupling between two charge qubits permits a family of non-adiabatic holonomic controlled two-qubit entangling gates, and thus provides a promising procedure to maintain entanglement in charge qubits and a pathway toward fault-tolerant universal quantum computation. We estimate the feasibility of the proposed structure by analyzing the gate fidelities to some extent.