Trapped ions are among the most promising candidates for performing quantum information processing tasks. Recently, it was demonstrated how the properties of geometric phases can be used to implement an entangling two qubit phase gate with significantly reduced operation time while having a built-in resistance against certain types of errors (Palmero et. al., Phys. Rev. A 95, 022328 (2017)). In this article, we investigate the influence of both quantum and thermal fluctuations on the geometric phase in the Markov regime. We show that additional environmentally induced phases as well as a loss of coherence result from the non-unitary evolution, even at zero temperature. We connect these effects to the associated dynamical and geometrical phases. This suggests a strategy to compensate the detrimental environmental influences and restore some of the properties of the ideal implementation. Our main result is a strategy for zero temperature to construct forces for the geometric phase gate which compensate the dissipative effects and leave the produced phase as well as the final motional state identical to the isolated case. We show that the same strategy helps also at finite temperatures. Furthermore, we examine the effects of dissipation on the fidelity and the robustness of a two qubit phase gate against certain error types.
We describe the optimal realization of entangling quantum gates for trapped ions mediated by a dissipative bus mode. With suitably shaped control pulses one can substantially decrease ion-phonon entanglement while maintaining the mediated interaction.
Heralded near-deterministic multi-qubit controlled phase gates with integrated error detection have recently been proposed by Borregaard et al. [Phys. Rev. Lett. 114, 110502 (2015)]. This protocol is based on a single four-level atom (a heralding quartit) and $N$ three-level atoms (operational qutrits) coupled to a single-resonator mode acting as a cavity bus. Here we generalize this method for two distant resonators without the cavity bus between the heralding and operational atoms. Specifically, we analyze the two-qubit controlled-Z gate and its multi-qubit-controlled generalization (i.e., a…
Optimal control methods for implementing quantum modules with least amount of relaxative loss are devised to give best approximations to unitary gates under relaxation. The potential gain by optimal control using relaxation parameters against time-optimal control is explored and exemplified in numerical and in algebraic terms: it is the method of choice to govern quantum systems within subspaces of weak relaxation whenever the drift Hamiltonian would otherwise drive the system through fast decaying modes. In a standard model system generalising decoherence-free subspaces to more realistic scen…
Using the methods of quantum trajectories we investigate the effects of dissipative decoherence in a quantum computer algorithm simulating dynamics in various regimes of quantum chaos including dynamical localization, quantum ergodic regime and quasi-integrable motion. As an example we use the quantum sawtooth algorithm which can be implemented in a polynomial number of quantum gates. It is shown that the fidelity of quantum computation decays exponentially with time and that the decay rate is proportional to the number of qubits, number of quantum gates and per gate dissipation rate induced b…
Dissipative and stochastic effects in the geometric phase of a qubit are taken into account using a geometrical description of the corresponding open--system dynamics within a canonical Langevin framework based on a Caldeira--Leggett like Hamiltonian. By extending the Hopf fibration $S^{3}\to S^{2}$ to include such effects, the exact geometric phase for a dissipative qubit is obtained, whereas numerical calculations are used to include finite temperature effects on it.
Current typical methods to realize nuclear-nuclear quantum gates require a sequence of electronnuclear quantum gates by using dynamical decoupling techniques, which are implemented at low temperature because of short decoherence and relaxation time of the NV spin at room temperature. This limitation could be overcome by using periodical resets of an NV spin as a mediator of interaction between two nuclear spins [Chen, Schwarz, and Plenio, 119, 010801 (2017)]. However, this method works under stringent coupling strengths condition, which makes it not applicable to heteronuclear quantum gate ope…
Error accumulation in dissipative quantum circuits
Universidad Autónoma de Madrid（马德里自治大学）；Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高级理工学院）；University of Stuttgart（斯图加特大学）；Jagiellonian University（雅盖隆大学）；Polish Academy of Sciences（波兰科学院）；OsloMet – Oslo Metropolitan University（奥斯陆都会大学）
We follow a generalized kinematic approach to compute the geometric phases acquired in both unitary and dissipative Jaynes-Cummings models, which provide a fully quantum description for a two-level system interacting with a single mode of the (cavity) electromagnetic field, in a perfect or dissipative cavity respectively. In the dissipative model, the non-unitary effects arise from the outflow of photons through the cavity walls and the incoherent pumping of the two-level system. Our approach allows to compare the geometric phases acquired in these models, leading to an exhaustive characteriza…
The effect of feedback on a two-level dissipative system is studied in this paper. The results show that it is possible to control the phase in the open system even if its state can not be manipulated from an arbitrary initial one to an arbitrary final one. The dependence of the geometric phase on the control parameters is calculated and discussed.
We show that optimizing a quantum gate for an open quantum system requires the time evolution of only three states irrespective of the dimension of Hilbert space. This represents a significant reduction in computational resources compared to the complete basis of Liouville space that is commonly believed necessary for this task. The reduction is based on two observations: The target is not a general dynamical map but a unitary operation; and the time evolution of two properly chosen states is sufficient to distinguish any two unitaries. We illustrate gate optimization employing a reduced set o…
We analyze the performance of holonomic quantum gates in semi-conductor quantum dots, driven by ultrafast lasers, under the effect of a dissipative environment. In agreement with the standard practice, the environment is modeled as a thermal bath of oscillators linearly coupled with the excitonic states of the quantum dot. Standard techniques of quantum dissipation make the problem amenable to a numerical treatment and allow to determine the fidelity (the common gate-performance estimator), as a function of all the relevant physical parameters. As a consequence of our analysis, we show that, b…
Open quantum walks (OQW) are formulated as quantum Markov chains on graphs. It is shown that OQWs are a very useful tool for the formulation of dissipative quantum computing algorithms and for dissipative quantum state preparation. In particular, single qubit gates and the CNOT-gate are implemented as OQWs on fully connected graphs. Also, dissipative quantum state preparation of arbitrary single qubit states and of all two-qubit Bell-states is demonstrated. Finally, the discrete time version of dissipative quantum computing is shown to be more efficient if formulated in the language of OQWs.
We describe an array of quantum gates implementing Shor's algorithm for prime factorization in a quantum computer. The array includes a circuit for modular exponentiation with several subcomponents (such as controlled multipliers, adders, etc) which are described in terms of elementary Toffoli gates. We present a simple analysis of the impact of losses and decoherence on the performance of this quantum factoring circuit. For that purpose, we simulate a quantum computer which is running the program to factor N = 15 while interacting with a dissipative environment. As a consequence of this inter…
Dissipative collective effects are ubiquitous in quantum physics, and their relevance ranges from the study of entanglement in biological systems to noise mitigation in quantum computers. Here, we put forward the first fully quantum simulation of dissipative collective phenomena on a real quantum computer, based on the recently introduced multipartite collision model. First, we theoretically study the accuracy of this algorithm on near-term quantum computers with noisy gates, and we derive some rigorous error bounds that depend on the timestep of the collision model and on the gate errors. The…
We investigate the geometric phase of a two-level atom (qubit) coupled to a bosonic reservoir with Lorentzian spectral density, and find that for the non-Markovian dynamics in which rotating-wave approximation (RWA) is performed, geometric phase has a $π$-phase jump at the nodal point. However, the exact result without RWA given by hierarchy equation of motion method shows that there is no such a phase jump or nodal structure in geometric phase. Thus our results demonstrate that the counter-rotating terms significantly contribute to the geometric phase in multi-mode Hamiltonian under certain c…
We study the dynamics of geometric measure of quantum discord (GMQD) under the influences of two local phase damping noises. Consider the two qubits initially in arbitrary X-states, we find the necessary and sufficient conditions for which GMQD is unaffected for a finite period. It is further shown that such results also hold for the non-Markovian dephasing process.
The geometric (Berry) phase of a two-level system in a dissipative environment is analyzed by using the second-quantized formulation, which provides a unified and gauge-invariant treatment of adiabatic and nonadiabatic phases and is thus applicable to a quantitative analysis of transitional regions away from ideal adiabaticity. In view of the recent experimental observation of the Berry phase in a superconducting qubit, we illustrate our formulation for a concrete adiabatic case in the Ohmic dissipation. The correction to the total phase together with the geometry-dependent dephasing time is g…
The Geometric Phase as a Diagnostic for Driven-Dissipative Oscillators
National University of Singapore（新加坡国立大学）；MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit（中法联合实验室（CNRS-UNS-NUS-NTU））；Nanyang Technological University（南洋理工大学）
We study the geometric phase of a two-level atom coupled to an environment with Lorentzian spectral density. The non-Markovian effect on the geometric phase is explored analytically and numerically. In the weak coupling limit the lowest-order correction to the geometric phase is derived analytically and the general case is calculated numerically. It is found that the correction to the geometric phase is significantly large if the spectral width is small and in this case the non-Markovian dynamics has a significant impact to the geometric phase. When the spectral width increases, the correction…
We investigate the steady-state phase diagram of the dissipative spin-1/2 XYZ model on a two-dimensional triangular lattice, in which each site is coupled to a local environment. By means of cluster mean-field approximation, we find that the steady-state phases of the system are rather rich, in particular there exist various types of nonuniform antiferromagnetic phases due to the geometrical frustration. As the short-range correlations included in the analysis, the numerical results show that the oscillatory phase disappears while the triantiferromagnetic and biantiferromagnetic phases remain …
Dissipative electromagnetic response and scattering geometry are potential sources for the appearance of a negative Casimir entropy. We show that the dissipative contribution familiar from the plane-plane geometry appears also in the plane-sphere and the sphere-sphere geometries and adds to the negative Casimir entropy known to exist in these geometries even for perfectly reflecting objects. Taking the sphere-sphere geometry as an example, we carry out a scattering-channel analysis which allows to distinguish between the contributions of different polarizations. We demonstrate that dissipation…
In a prevous paper (Phys. Rev. Lett. 96, 150403 (2006)) we have proposed a new way to generate an observable geometric phase on a quantum system by means of a completely incoherent phenomenon. The basic idea was to force the ground state of the system to evolve ciclically by "adiabatically" manipulating the environment with which it interacts. The specific scheme we have previously analyzed, consisting of a multilevel atom interacting with a broad-band squeezed vacuum bosonic bath whose squeezing parameters are smoothly changed in time along a closed loop, is here solved in a more direct way. …
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.
An open quantum walk formalism for dissipative quantum computing is presented. The approach is illustrated with the examples of the Toffoli gate and the Quantum Fourier Transform for 3 and 4 qubits. It is shown that the algorithms based on the open quantum walk formalism are more efficient than the canonical dissipative quantum computing approach. In particular, the open quantum walks can be designed to converge faster to the desired steady state and to increase the probability of detection of the outcome of the computation.
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.
The identification of parameters in the Hamiltonian that describes complex many-body quantum systems is generally a very hard task. Recent attention has focused on such problems of Hamiltonian tomography for networks constructed with two-level systems. For open quantum systems, the fact that injected signals are likely to decay before they accumulate sufficient information for parameter estimation poses additional challenges. In this paper, we consider use of the gateway approach to Hamiltonian tomography \cite{Burgarth2009,Burgarth2009a} to complex quantum systems with a limited set of state …
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.
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.
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…
