我们提出了一种新的几何量子计算（GQC）方案，称为Floquet GQC（FGQC），其中基于周期性驱动的二能级系统的容错几何门可以通过最近研究中提出的一种新的非阿贝尔几何相位来构建[V. Noviĉenko等人，Phys. Rev. A 100, 012127 (2019)]。基于Rydberg原子，我们给出了FGQC的通用单比特门和非平凡双比特门的可能实现。通过数值模拟，我们评估了在退相干和某种系统性控制误差存在的情况下FGQC的Z门和X门的性能。Z门和X门的门保真度为$F_{X,\text{gate}}\approx F_{Z,\text{gate}}\approx 0.9992$。数值结果证明，即使在噪声和控制缺陷存在的情况下，FGQC门也能达到相当高的门保真度。此外，我们发现FGQC对全局控制误差具有鲁棒性，并给出了分析证明和数值证据。因此，本研究向鲁棒的几何量子计算迈出了重要一步。
英文摘要：
We proposed a new geometric quantum computation (GQC) scheme, called Floquet GQC (FGQC), where error-resilient geometric gates based on periodically driven two-level systems can be constructed via a new non-Abelian geometric phase proposed in a recent study [V. Noviĉenko \textit{et al}, Phys. Rev. A 100, 012127 (2019) ]. Based on Rydberg atoms, we gave possible implementations of universal single-qubit gates and a nontrivial two-qubit gate for FGQC. By using numerical simulation, we evaluated the performance of the FGQC Z and X gates in the presence of both decoherence and a certain kind of systematic control error. The gate fidelities of the Z and X gates are $F_{X,\text{gate}}\approx F_{Z,\text{gate}}\approx 0.9992$. The numerical results provide evidence that FGQC gates can achieve fairly high gate fidelities even in the presence of noise and control imperfection. In addition, we found FGQC is robust against global control error, both analytical demonstration and numerical evidence were given. Consequently, this study makes an important step towards robust geometric quantum computation.
Noise-resilient nonadiabatic geometric quantum computation for bosonic binomial codes
Fujian Key Laboratory of Quantum Information and Quantum Optics, College of Physics and Information Engineering, Fuzhou University（福州大学物理与信息工程学院）；School of Physics, Hangzhou Normal University（杭州师范大学物理学院）；Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University（天津大学理学院）；Institute of Quantum Science and Technology, Yanbian University（延边大学量子科学与技术研究院）；Quantum Information Physics Theory Research Team, Center for Quantum Computing, RIKEN（理研量子计算中心）
For a periodic Hamiltonian, periodic dynamical invariants may be used to obtain non-degenerate cyclic states. This observation is generalized to the degenerate cyclic states, and the relation between the periodic dynamical invariants and the Floquet decompositions of the time-evolution operator is elucidated. In particular, a necessary condition for the occurrence of cyclic non-adiabatic non-Abelian geometrical phase is derived. Degenerate cyclic states are obtained for a magnetic dipole interacting with a precessing magnetic field.
Quantum gates induced by geometric phases are intrinsically robust against noise due to their global properties of the evolution paths. Compared to conventional nonadiabatic geometric quantum computation (NGQC), 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 for verifying the noise-resilient and fast feature. By performing unitary operations under imperfect conditions, w…
Measurement protocol for non-adiabatic geometric phases of Floquet states
Quantum Advanced Research Center (QuARC), CSIC（量子先进研究中心（QuARC），西班牙国家研究委员会）；Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC（马德里材料科学研究所（ICMM），西班牙国家研究委员会）
Dynamical quantum phase transitions (DQPTs) are manifested by time-domain nonanalytic behaviors of many-body systems.Introducing a quench is so far understood as a typical scenario to induce DQPTs.In this work, we discover a novel type of DQPTs, termed "Floquet DQPTs", as intrinsic features of systems with periodic time modulation.Floquet DQPTs occur within each period of continuous driving, without the need for any quenches.In particular, in a harmonically driven spin chain model, we find analytically the existence of Floquet DQPTs in and only in a parameter regime hosting a certain nontrivia…
The light propagating in a waveguide array or photonic lattice has become an ideal platform to control light and to mimic quantum behaviors in a classical system. We here investigate the propagation of light in a coupled waveguide array with one of the waveguides periodically modulated in its geometric structure or refractive index. Within the framework of Floquet theory, it is interesting to find that the light shows the localized propagation in the modulated waveguide as long as bound quasistationary modes are formed in the band-gap area of the Floquet eigenvalue spectrum. This mechanism giv…
Certain quantum many-body ground states can be prepared by a large-depth quantum circuit consisting of geometrically local gates. In the presence of noise, local expectation values deviate from the correct value at most by an amount comparable to the noise rate. This happens if the action of the noiseless circuit, restricted to certain subsystems, rapidly mixes local observables up to a small correction. The encoding circuit for the surface code is given as an example.
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 fea…
Tracking real-space quantum state breathing through Floquet-projector geometry
Institute for Theoretical Physics and Astrophysics, University of Würzburg（维尔茨堡大学理论物理与天体物理研究所）；Würzburg-Dresden Cluster of Excellence ctd.qmat（维尔茨堡-德累斯顿卓越集群ctd.qmat）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）
Non-Hermitian Hamiltonians provide a simple picture for inspecting dissipative systems with natural or induced gain and loss. We investigate the Floquet dynamical phase transition in the dissipative periodically time driven XY and extended XY models, where the imaginary terms represent the physical gain and loss during the interacting processes with the environment. The time-independent effective Floquet non-Hermitian Hamiltonians disclose three regions by analyzing the non-Hermitian gap: pure real gap (real eigenvalues), pure imaginary gap, and complex gap. We show that each region of the sys…
We study the scattering of photons from periodically modulated quantum-optical systems. For excitation-number conserving quantum optical systems, we connect the analytic structure of the frequency-domain N-photon scattering matrix of the system to the Floquet decomposition of its effective Hamiltonian. Furthermore, it is shown that the first order contribution to the transmission or equal-time N-photon correlation spectrum with respect to the modulation frequency is completely geometric in nature i.e. it only depends on the Hamiltonian trajectory and not on the precise nature of the modulation…
We consider a periodically driven quantum system described by a Hamiltonian which is the product of a slowly varying Hermitian operator $V\left(\boldsymbolλ\left(t\right)\right)$ and a dimensionless periodic function with zero average. We demonstrate that the adiabatic evolution of the system within a fully degenerate Floquet band is accompanied by non-Abelian (non-commuting) geometric phases appearing when the slowly varying parameter $\boldsymbolλ=\boldsymbolλ\left(t\right)$ completes a closed loop. The geometric phases can have significant values even after completing a single cycle of the …
We propose a realistic scheme to construct anomalous Floquet Chern topological insulators using spin-1/2 particles carrying out a discrete-time quantum walk in a two-dimensional lattice. By Floquet engineering the quantum-walk protocol, an Aharonov-Bohm geometric phase is imprinted onto closed-loop paths in the lattice, thus realizing an abelian gauge field---the analog of a magnetic flux threading a two-dimensional electron gas. We show that in the strong field regime, when the flux per plaquette is a sizable fraction of the flux quantum, magnetic quantum walks give rise to nearly flat energy…
We give the explicit expressions of the pairwise quantum correlations present in superpositions of multipartite coherent states. A special attention is devoted to the evaluation of the geometric quantum discord. The dynamics of quantum correlations under a dephasing channel is analyzed. A comparison of geometric measure of quantum discord with that of concurrence shows that quantum discord in multipartite coherent states is more resilient to dissipative environments than is quantum entanglement. To illustrate our results, we consider some special superpositions of Weyl-Heisenberg, SU(2) and SU…
Quantum information processing with geometric features of quantum states may provide promising noise-resilient schemes for quantum metrology. In this work, we theoretically explore phase-space geometric Sagnac interferometers with trapped atomic clocks for rotation sensing, which could be intrinsically robust to certain decoherence noises and reach high precision. With the wave guide provided by sweeping ring-traps, we give criteria under which the well-known Sagnac phase is a pure or unconventional geometric phase with respect to the phase space. Furthermore, corresponding schemes for geometr…
We design an adiabatic quantum algorithm for the counting problem, i.e., approximating the proportion, $α$, of the marked items in a given database. As the quantum system undergoes a designed cyclic adiabatic evolution, it acquires a Berry phase $2πα$. By estimating the Berry phase, we can approximate $α$, and solve the problem. For an error bound $ε$, the algorithm can solve the problem with cost of order $(\frac{1}ε)^{3/2}$, which is not as good as the optimal algorithm in the quantum circuit model, but better than the classical random algorithm. Moreover, since the Berry phase is a purely g…
We describe in detail a general strategy for implementing a conditional geometric phase between two spins. Combined with single-spin operations, this simple operation is a universal gate for quantum computation, in that any unitary transformation can be implemented with arbitrary precision using only single-spin operations and conditional phase shifts. Thus quantum geometrical phases can form the basis of any quantum computation. Moreover, as the induced conditional phase depends only on the geometry of the paths executed by the spins it is resilient to certain types of errors and offers the p…
The geometric phase has been proposed as a candidate for noise resilient coherent manipulation of fragile quantum systems. Since it is determined only by the path of the quantum state, the presence of noise fluctuations affects the geometric phase in a different way than the dynamical phase. We have experimentally tested the robustness of Berry's geometric phase for spin-1/2 particles in a cyclically varying magnetic field. Using trapped polarized ultra-cold neutrons it is demonstrated that the geometric phase contributions to dephasing due to adiabatic field fluctuations vanish for long evolu…
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 a…
We consider a quantum system periodically driven with a strength which varies slowly on the scale of the driving period. The analysis is based on a general formulation of the Floquet theory relying on the extended Hilbert space. It is shown that the dynamics of the system can be described in terms of a slowly varying effective Floquet Hamiltonian that captures the long-term evolution, as well as rapidly oscillating micromotion operators. We obtain a systematic high-frequency expansion of all these operators. Generalizing the previous studies, the expanded effective Hamiltonian is now time-depe…
Confinement-Tunable Synthetic Gauge Fields and Floquet Topological Phenomena in a Driven Quantum Wire Qubit
University of Abomey-Calavi（阿博梅-卡利维大学）；University of Dschang（德尚大学）；Quantum Materials and Computing Group - QMaCG（量子材料与计算群 - QMaCG）；Instituteur principal de l’enseignement général, Délégation Régionale de l’Ouest, Délégation Départementale de la Menoua, Arrondissement de Nkong-Ni, École Publique de Batsing’la Chefferie, Bafou, Cameroun（喀麦隆巴富，班辛拉酋长国公立学校，孔克尼区，梅努瓦省教育厅，西部大区教育厅，普通教育首席教师）；University of Yaoundé I（雅温得第一大学）
Quantum gates based on geometric phases possess intrinsic noise-resilience features and therefore attract much attention. However, the implementations of previous geometric quantum computation typically require a long pulse time of gates. As a result, their experimental control inevitably suffers from the cumulative disturbances of systematic errors due to excessive time consumption. Here, we experimentally implement a set of noncyclic and nonadiabatic geometric quantum gates in a superconducting circuit, which greatly shortens the gate time. And also, we experimentally verify that our univers…
Geometric phase, associated with holonomy transformation in quantum state space, is an important quantum-mechanical effect. Besides fundamental interest, this effect has practical applications, among which geometric quantum computation is a paradigm, where quantum logic operations are realized through geometric phase manipulation that has some intrinsic noise-resilient advantages and may enable simplified implementation of multiqubit gates compared to the dynamical approach. Here we report observation of a continuous-variable geometric phase and demonstrate a quantum gate protocol based on thi…
The experimental realisation of the basic constituents of quantum information processing devices, namely fault-tolerant quantum logic gates, requires conditional quantum dynamics, in which one subsystem undergoes a coherent evolution that depends on the quantum state of another subsystem. In particular, the subsystem may acquire a conditional phase shift. Here we consider a novel scenario in which this phase is of geometric rather than dynamical origin. As the conditional geometric (Berry) phase depends only on the geometry of the path executed it is resilient to certain types of errors, and o…
We study a generic class of fermionic two-band models under synchronized periodic driving, i.e., with the different terms in a Hamiltonian subject to periodic drives with the same frequency and phase. With all modes initially in a maximally mixed state, the synchronized drive is found to produce nonperiodic patterns of dynamical quantum phase transitions, with their appearance determined by an interplay of the band structure and the frequency of the drive. A case study of the anisotropic XY chain in a transverse magnetic field, transcribed to an effective two-band model, shows that the modes c…
Holonomies, arising from non-Abelian geometric transformations of quantum states in Hilbert space, offer a promising way for quantum computation. These holonomies are not commutable and thus can be used for the realization of a universal set of quantum logic gates, where the global geometric feature may result in some noise-resilient advantages. Here we report the first on-chip realization of a non-Abelian geometric controlled-Not gate in a superconducting circuit, which is a building block for constructing a holonomic quantum computer. The conditional dynamics is achieved in an all-to-all con…
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 pa…
We present a scheme to study non-abelian adiabatic holonomies for open Markovian systems. As an application of our framework, we analyze the robustness of holonomic quantum computation against decoherence. We pinpoint the sources of error that must be corrected to achieve a geometric implementation of quantum computation completely resilient to Markovian decoherence.
This is a brief overview of quantum holonomies in the context of quantum computation. We choose an adequate set of quantum logic gates, namely, a phase gate, the Hadamard gate, and a conditional-phase gate and show how they can be implemented by purely geometric means. Such gates may be more resilient to certain types of errors.
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…
Experimental realization of a universal set of quantum logic gates is the central requirement for implementation of a quantum computer. An all-geometric approach to quantum computation offered a paradigm for implementation where all the quantum gates are achieved based on the Berry phases and their non-abelian extensions, the holonomies, from geometric transformation of quantum states in the Hilbert space. Apart from its fundamental interest and rich mathematical structure, the geometric approach has some built-in noise-resilient features. On the experimental side, geometric phases and holonom…
