A general scheme for an adiabatic geometric phase gate is proposed which is maximally robust against parameter fluctuations. While in systems with SU(2) symmetry geometric phases are usually accompanied by dynamical phases and are thus not robust, we show that in the more general case of a SU(2) x SU(2) symmetry it is possible to obtain a non-vanishing geometric phase without dynamical contributions. The scheme is illustrated for a phase gate using two systems with dipole-dipole interactions in external laser fields which form an effective four-level system.
Motivated for the fault tolerant quantum computation, quantum gate by adiabatic geometric phase shift is extensively investigated. In this paper, we demonstrate the nonadiabatic scheme for the geometric phase shift and conditional geometric phase shift. Essentially, the new scheme is simply to add an appropriate additional field. With this additional field, the state evolution can be controlled exactly on a dynamical phase free path. Geometric quantum gates for single qubit and the controlled NOT gate for two qubits are given.
We consider stimulated Raman adiabatic passage (STIRAP) processes in tripod systems and show how to generate purely geometric phase changes of the quantum states involved. The geometric phases are controlled by three laser fields where pulse shapes, relative field strength and phases can be controlled. We present a robust set of universal gates for quantum computing based on these geometric phases: a one-qubit phase gate, a Hadamard gate and a two-qubit phase gate.
We show that all geometric quantum gates (GQG's in short), which are quantum gates only with geometric phases, are robust against control field strength errors. As examples of this observation, we show (1) how robust composite rf-pulses in NMR are geometrically constructed and (2) a composite rf-pulse based on Trotter-Suzuki Formulas is a GQG.
High-fidelity manipulation is the key for the physical realization of fault-tolerant quantum computation. Here, we present a protocol to realize universal nonadiabatic geometric gates for silicon-based spin qubits. We find that the advantage of geometric gates over dynamical gates depends crucially on the evolution loop for the construction of the geometric phase. Under appropriate evolution loops, both the geometric single-qubit gates and the CNOT gate can outperform their dynamical counterparts for both systematic and detuning noises. We also perform randomized benchmarking using noise ampli…
Geometric gates that use the global property of the geometric phase is believed to be a powerful tool to realize fault-tolerant quantum computation. However, for singlet-triplet qubits in semiconductor quantum dot, the low Rabi frequency of the microwave control leads to overly long gating time, and thus the constructing geometric gate suffers more from the decoherence effect. Here we investigate the key issue of whether the fast geometric gate can be realized for singlet-triplet qubits without introducing an extra microwave-driven pulse, while maintaining the high-fidelity gate operation at t…
We analyze a scheme for quantum computation where quantum gates can be continuously changed from standard dynamic gates to purely geometric ones. These gates are enacted by controlling a set of parameters that are subject to unwanted stochastic fluctuations. This kind of noise results in a departure from the ideal case that can be quantified by a gate fidelity. We find that the maximum of this fidelity corresponds to quantum gates with a vanishing dynamical phase.
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…
We propose a simple but versatile protocol to engineer time-dependent Hamiltonians inversely for geometric quantum computation. By utilizing SU(2) transformation, a speedup goal on gate operation is achieved with more freedom to design the control parameters. As an application, this protocol enables the conventional and unconventional nonadiabatic geometric quantum gates with desired evolution paths by controlling the microwave pulses in the diamond nitrogen-vacancy center system. We show that the inversely designed Hamiltonian can fulfill the geometric gate with more economical evolution time…
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…
We propose an experimentally feasible scheme to achieve quantum computation based on nonadiabatic geometric phase shifts, in which a cyclic geometric phase is used to realize a set of universal quantum gates. Physical implementation of this set of gates is designed for Josephson junctions and for NMR systems. Interestingly, we find that the nonadiabatic phase shift may be independent of the operation time under appropriate controllable conditions. A remarkable feature of the present nonadiabatic geometric gates is that there is no intrinsic limitation on the operation time, unlike adiabatic ge…
We propose a new class of unconventional geometric gates involving nonzero dynamic phases, and elucidate that geometric quantum computation can be implemented by using these gates. Comparing with the conventional geometric gate operation, in which the dynamic phase shift must be removed or avoided, the gates proposed here may be operated more simply. We illustrate in detail that unconventional nontrivial two-qubit geometric gates with built-in fault tolerant geometric features can be implemented in real physical systems.
We introduce the non-adiabatic, or Aharonov-Anandan, geometric phase as a tool for quantum computation and show how it could be implemented with superconducting charge qubits. While it may circumvent many of the drawbacks related to the adiabatic (Berry) version of geometric gates, we show that the effect of fluctuations of the control parameters on non-adiabatic phase gates is more severe than for the standard dynamic gates. Similarly, fluctuations also affect to a greater extent quantum gates that use the Berry phase instead of the dynamic phase.
Quantum computation has demonstrated advantages over classical computation for special hard problems, where a set of universal quantum gates is essential. Geometric phases, which have built-in resilience to local noise, have been used to construct quantum gates with excellent performance. However, this advantage has been smeared in previous schemes. Here, we propose a state-independent nonadiabatic geometric quantum-gate scheme that is able to realize a more fully geometric gate than previous approaches, allowing for the cancelation of dynamical phases accumulated by an arbitrary state. Numeri…
In a scheme of nonadiabatic purely geometric quantum gates in nuclear magnetic resonance(NMR) systems we propose proper magnitudes of magnetic fields that are suitable for an experiment. We impose a natural condition and reduce the degree of freedom of the magnetic fields to the extent. By varying the magnetic fields with essentially one-dimensional degree of freedom, any spin state can acquire arbitrary purely geometric phase ϕ_{g}=-2π(1-cos{theta}), 0 < cosθ < 1. This is an essential ingredient for constructing universal geometric quantum gates.
Geometric quantum gates are performed by using the geometric phase, making them particularly robust to the pulse amplitude error due to the intrinsic global property. 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 deal with the off-resonance error is very significant for the application of the geometric gates. A recent work in \emph{Phy. Rev. Appl. 16, 044005 (2021)} reveals that by inserting two $π$-pulse dynamically corrected sequen…
Producing and maintaining entanglement reside at the heart of the optimal construction of quan- tum operations and are fundamental issues in the realization of universal quantum computation. We here introduce a setup of spin qubits that allows for geometric implementation of entangling gates between the register qubits with any arbitrary entangling power. We show this by demon- strating a circuit through a spin chain, which performs universal nonadiabatic holonomic two-qubit entanglers. The proposed gates are all electric and geometric, which would help to realize fast and robust entangling ga…
We propose a geometric phase gate of two ion qubits that are encoded in two levels linked by an optical dipole-forbidden transition. Compared to hyperfine geometric phase gates mediated by electric dipole transitions, the gate has many interesting properties, such as very low spontaneous emission rates, applicability to magnetic field insensitive states, and use of a co-propagating laser beam geometry. We estimate that current technology allows for infidelities of around 10$^{-4}$.
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 the effects of certain forms of decoherence applied to both adiabatic and non-adiabatic geometric phase quantum gates. For a single qubit we illustrate path-dependent sensitivity to anisotropic noise and for two qubits we quantify the loss of entanglement as a function of decoherence.
We propose a scheme to implement geometric entangling gates for two logical qubits in a coupled cavity system in decoherence-free subspaces. Each logical qubit is encoded with two atoms trapped in a single cavity and the geometric entangling gates are achieved by cavity coupling and controlling the external classical laser fields. Based on the coupled cavity system, the scheme allows the scalability for quantum computing and relaxes the requirement for individually addressing atoms.
Because of using geometric phases, nonadiabatic geometric gates have the robustness against control errors. On the other hand, decoherence still affects nonadiabatic geometric gates, which is a key factor in reducing their fidelities. In this paper, we show that based on the system Hamiltonian that realizes a nonadiabatic geometric gate, one may construct a new system Hamiltonian, by using which not only the geometric feature of the nonadiabatic geometric gate is preserved, but also the system's coherence is protected. As a result, a coherence-protected nonadiabatic geometric gate is realized …
We propose a feasible scheme to implement a universal set of quantum gates based on geometric phases and superadiabatic quantum control. Consolidating the advantages of both strategies, the proposed quantum gates are robust and fast. The diamond nitrogen-vacancy center system is adopted as a typical example to illustrate the scheme. We show that these gates can be realized in a simple two-level configuration by appropriately controlling the amplitude, phase, and frequency of just one microwave field. The gate's robust and fast features are confirmed by comparing the fidelity of the proposed su…
Nonadiabatic geometric quantum computation is dedicated to the realization of high-fidelity and robust quantum gates, which are necessary for fault-tolerant quantum computation. However, it is limited by cyclic and mutative evolution path, which usually requires longer gate-time and abrupt pulse control, weakening the gate performance. Here, we propose a scheme to realize geometric quantum gates with noncyclic and nonadiabatic evolution via invariant-based shortcuts, where universal quantum gates can be induced in one step without path mutation and the gate time is also effectively shortened. …
A 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, we provide a theoretical framework to implement universal geometric quantum gates in this system. We find 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. We demonstrate, through numerical simulation, that the geom…
In this work Controlled phase shift gates are implemented on a qaudrupolar system, by using non-adiabatic geometric phases. A general procedure is given, for implementing controlled phase shift gates in an 'N' level system. The utility of such controlled phase shift gates, is demonstrated here by implementing 3-qubit Deutsch-Jozsa algorithm on a 7/2 quadrupolar nucleus oriented in a liquid crystal matrix.
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-orbi…
Following a key idea of unconventional geometric quantum computation developed earlier [Phys. Rev. Lett. 91, 197902 (2003)], here we propose a more general scheme in such an intriguing way: $γ_{d}=α_{g}+ηγ_{g}$, where $γ_{d}$ and $γ_{g}$ are respectively the dynamic and geometric phases accumulated in the quantum gate operation, with $η$ as a constant and $α_{g}$ being dependent only on the geometric feature of the operation. More arrestingly, we demonstrate the first experiment to implement a universal set of such kind of generalized unconventional geometric quantum gates with high fidelity i…
We study a 2-qubit nuclear spin system for realizing an arbitrary geometric quantum phase gate by means of non-adiabatic operation. A single magnetic pulse with multi harmonic frequencies is applied to manipulate the quantum states of 2-qubit instantly. Using resonant transition approximation, the time dependent Hamiltonian of two nuclear spins can be solved analytically. The time evolution of the wave function is obtained without adiabatic approximation. The parameters of magnetic pulse, such as the frequency, amplitude, phase of each harmonic part as well as the time duration of the pulse, a…
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 describe a hybrid laser-microwave scheme to implement two-qubit geometric phase gates in crystals of trapped ions. The proposed gates can attain errors below the fault-tolerance threshold in the presence of thermal, dephasing, laser-phase, and microwave-intensity noise. Moreover, our proposal is technically less demanding than previous schemes, since it does not require a laser arrangement with interferometric stability. The laser beams are tuned close to a single vibrational sideband to entangle the qubits, while strong microwave drivings provide the geometric character to the gate, and th…
Suppression of Universal Errors in DFS-Encoded Superconducting Geometric Logical \emph{T} Gate
School of Physics and Astronomy, Anqing Normal University（安庆师范大学物理与天文学院）；School of Electronic Information and Integrated Circuits, Anqing Normal University（安庆师范大学电子信息与集成电路学院）；School of Electrical and Photoelectronic Engineering, West Anhui University（皖西大学电气与光电工程学院）
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 NHQCþ, for maintaining both flexibility and robustness against certain types of noises. Consequently, this approach makes it possible to incorporate most of the existing optimal con…
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 er…
Geometric phases are robust against certain types of local noises, and thus provide a promising way towards high-fidelity quantum gates. However, comparing with the dynamical ones, previous implementations of nonadiabatic geometric quantum gates usually require longer evolution time, due to the needed longer evolution path. Here, we propose a scheme to realize nonadiabatic geometric quantum gates with short paths based on simple pulse control techniques, instead of deliberated pulse control in previous investigations, which can thus further suppress the influence from the environment induced n…
We investigate the effects of systematic errors of the control parameters on single-qubit gates based on non-adiabatic non-Abelian geometric holonomies and those relying on purely dynamical evolution. It is explicitly shown that the systematic error in the Rabi frequency of the control fields affects these two kinds of gates in different ways. In the presence of this systematic error, the transformation produced by the non-adiabatic non-Abelian geometric gate is not unitary in the computational space, and the resulting gate infidelity is larger than that with the dynamical method. Our results …
Quantum computing in terms of geometric phases, i.e. Berry or Aharonov-Anandan phases, is fault-tolerant to a certain degree. We examine its implementation based on Zeeman coupling with a rotating field and isotropic Heisenberg interaction, which describe NMR and can also be realized in quantum dots and cold atoms. Using a novel physical representation of the qubit basis states, we construct $π/8$ and Hadamard gates based on Berry and Aharonov-Anandan phases. For two interacting qubits in a rotating field, we find that it is always impossible to construct a two-qubit gate based on Berry phases…
A large-scalable quantum computer model, whose qubits are represented by the subspace subtended by the ground state and the single exciton state on semiconductor quantum dots, is proposed. A universal set of quantum gates in this system may be achieved by a mixed approach, composed of dynamic evolution and nonadibatic geometric phase.
We show that some composite pulses widely employed in NMR experiments are regarded as non-adiabatic geometric quantum gates with Aharanov-Anandan phases. Thus, we reveal the presence of a fundamental issue on quantum mechanics behind the traditional technique of the composite pulses. To examine the robustness of such composite pulses against fluctuations, we present a practical noise model in a two-level system. Then, we find that the composite pulses possesses purely geometrical nature even under a specific type of fluctuations.
We present an implementation of quantum logic gates via internal state dependent displacements of ions in a linear Paul trap caused by optical dipole forces. Based on a general quantum analysis of the system dynamics we consider specific implementations with alkaline earth ions. For experimentally realistic parameters gate infidelities as low as $10^{-4}$ can be obtained.
We present a study of the performance of the trapped-ion driven geometric phase gates introduced in [New J. Phys. 15, 083001 (2013)] when realized in a stimulated Raman transition. We show that the gate can achieve errors below the fault-tolerance threshold in the presence of laser intensity fluctuations. We also find that, in order to reduce the errors due to photon scattering below the fault-tolerance threshold, very intense laser beams are required to allow for large detunings in the Raman configuration without compromising the gate speed.
In the quantum-computation scenario, geometric phase-gates are becoming increasingly attractive for their intrinsic fault tolerance to disturbance. With an adiabatic cyclic evolution, Berry phase appears to realize a geometric transformation. Performing the quantum gates as many as possible within the timescale of coherence, however, remains an inconvenient bottleneck due to the systematic errors. Here we propose an accelerated adiabatic quantum gate based on the Berry phase, the transitionless driving, and the dynamical decoupling. It reconciles a high fidelity with a high speed in the presen…
We propose a scheme for realizing a two-qubit controlled phase gate via an unconventional geometric phase with two nonresonant quantum dots trapped in a photonic crystal cavity. In this system, the quantum dots simultaneously interact with a large detuned cavity mode and strong driving classical light fields. During the gate operation, the quantum dots undergo no transitions, while the cavity mode is displaced along a closed path in the phase space. In this way, the system can acquire geometric phases conditional upon the states of the quantum dots. After implementing single-qubit operations, …
We propose a novel symmetrization procedure to beat decoherence for oscillator-assisted quantum gate operations. The enacted symmetry is related to the global geometric features of qubits transformation based on ancillary oscillator modes, e.g. phonons in an ion-trap system. It is shown that the devised multi-circuit symmetrized evolution endows the system with a two-fold resilience against decoherence: insensitivity to thermal fluctuations and quantum dissipation.
A practical quantum computer must be capable of performing high fidelity quantum gates on a set of quantum bits (qubits). In the presence of noise, the realization of such gates poses daunting challenges. Geometric phases, which possess intrinsic noise-tolerant features, hold the promise for performing robust quantum computation. In particular, quantum holonomies, i.e., non-Abelian geometric phases, naturally lead to universal quantum computation due to their non-commutativity. Although quantum gates based on adiabatic holonomies have already been proposed, the slow evolution eventually compro…
The ability to perform gates in multiqubit systems that are robust to noise is of crucial importance for the advancement of quantum information technologies. However, finding control pulses that cancel noise while performing a gate is made difficult by the intractability of the time-dependent Schrodinger equation, especially in multiqubit systems. Here, we show that this issue can be sidestepped by using a formalism in which the cumulative error during a gate is represented geometrically as a curve in a multi-dimensional Euclidean space. Cancellation of noise errors to leading order correspond…
We propose an implementation scheme for holonomic, i.e., geometrical, quantum information processing based on semiconductor nanostructures. Our quantum hardware consists of coupled semiconductor macroatoms addressed/controlled by ultrafast multicolor laser-pulse sequences. More specifically, logical qubits are encoded in excitonic states with different spin polarizations and manipulated by adiabatic time-control of the laser amplitudes . The two-qubit gate is realized in a geometric fashion by exploiting dipole-dipole coupling between excitons in neighboring quantum dots.
A proposal for applying non-adiabatic geometric phases to quantum computing, called the double-loop method [S.-L. Zhu and Z. D. Wang, Phys. Rev. A {\bf 67}, 022319 (2003)], is demonstrated in a liquid state NMR quantum computer. Using a spin echo technique, the original method is modified so that quantum gates are implemented in a standard high-precision NMR system for chemical analysis. We show that the dynamical phase is successfully eliminated and a one-qubit quantum gate is realized, although the fidelity is not so high.
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…
Fast and robust quantum gates is the cornerstone of fault-tolerance quantum computation. In this paper, we propose to achieve quantum gates based on non-cyclic geometric evolution. Dynamical phase during the evolution is cancelled by spin-echo process and the adiabatic control can be sped up through shortcut to adiabatic manner. Different from geometric gates based on cyclic evolution, the superiority of non-cyclic scheme is that the operation time is proportional to the rotation angle (but not the geometric phase) of quantum state. Therefore, the non-cyclic scheme becomes fairly fast in the c…
We propose a new strategy to physically implement a universal set of quantum gates based on geometric phases accumulated in the nondegenerate eigenstates of a designated invariant operator in a periodic physical system. The system is driven to evolve in such a way that the dynamical phase shifts of the invariant operator eigenstates are the same (or {\it mod} $2π$) while the corresponding geometric phases are nontrivial. We illustrate how this strategy to work in a simple but typical NMR-type qubit system.
