Transitionless driving on local adiabatic quantum search algorithm

We apply the transitionless driving on the local adiabatic quantum search algorithm to speed up the adiabatic process.By studying quantum dynamics of the adiabatic search algorithm with the equivalent two-level system, we derive the transitionless driving Hamiltonian for the local adiabatic quantum search algorithm. We found that when adding a transitionless quantum driving term H_D(t) on the local adiabatic quantum search algorithm, the success rate is 1 exactly with arbitrary evolution time by solving the time-dependent Schr odinger equation in eigen-picture. Moreover, we show the reason for the drastic decrease of the evolution time is that the driving Hamiltonian increases the lowest eigenvalues to a maximum of ON~(1/2).     

Perturbations Can Enhance Quantum Search

In general, a quantum algorithm wants to avoid decoherence or perturbation, since such factors may cause errors in the algorithm. We show that some perturbations to the generalized quantum search Hamiltonian can reduce the running time and enhance the success probability. We also provide the narrow bound to the perturbation which can be beneficial to quantum search. In addition, we show that the error induced by a perturbation on the Farhi and Gutmann Hamiltonian can be corrected by another perturbation.     

Nonadiabatic geometric quantum computation protected by dynamical decoupling via the XXZ Hamiltonian

Nonadiabatic geometric quantum computation protected by dynamical decoupling combines the robustness of nonadiabatic geometric gates and the decoherence-resilience feature of dynamical decoupling. Solid-state systems provide an appealing candidate for the realization of nonadiabatic geometric quantum computation protected dynamical decoupling since the solid-state qubits are easily embedded in electronic circuits and scaled up to large registers. In this paper, we put forward a scheme of nonadiabatic geometric quantum computation protected by dynamical decoupling via the XXZ Hamiltonian, which not only combines the merits of nonadiabatic geometric gates and dynamical decoupling but also can be realized in a number of solid-state systems, such as superconducting circuits and quantum dots.     

Optimized quantum random-walk search algorithm for multi-solution search

This study investigates the multi-solution search of the optimized quantum random-walk search algorithm on the hypercube. Through generalizing the abstract search algorithm which is a general tool for analyzing the search on the graph to the multi-solution case, it can be applied to analyze the multi-solution case of quantum random-walk search on the graph directly. Thus, the computational complexity of the optimized quantum random-walk search algorithm for the multi-solution search is obtained. Through numerical simulations and analysis, we obtain a critical value of the proportion of solutions q. For a given q, we derive the relationship between the success rate of the algorithm and the number of iterations when q is no longer than the critical value.     

Implementation of a many-qubit Grover search by cavity QED

We explore the possibility of an N-qubit (N>3) Grover search in cavity QED, based on a fast operation of an N-qubit controlled phase-flip with atoms in resonance with the cavity mode. We demonstrate both analytically and numerically that our scheme can be achieved efficiently to find a marked state with high fidelity and high success probability. As an example, a ten-qubit Grover search is simulated specifically under the discussion of experimental feasibility and challenge. We argue that our scheme is applicable to the case involving an arbitrary number of qubits. As cavity decay is involved in our quantum trajectory treatment, we can analytically understand the implementation of a Grover search subject to dissipation, which will be very helpful for relevant experiments.     

Role of coherence in adiabatic search algorithms

We systematically investigate the role of coherence in adiabatic search algorithms by using the relative entropy measure of coherence. Both in the ideal case (adiabatic evolution) and the non-ideal case (nonadiabatic evolution), the success probability increases with the decreases of coherence. In addition, the coherence depletion in global adiabatic search algorithm, local adiabatic search algorithm and an adiabatic search algorithm with constant evolution time was discussed. The results show that the coherence decreases faster in more efficient algorithm and an exponential decaying of coherence is necessary to achieve fast search (constant evolution time) in the adiabatic search algorithm. More importantly, we demonstrate that the efficiency of adiabatic search algorithm can be improved by utilizing appropriate method to speed up the coherence depletion.     

基于相位匹配的量子行走搜索算法及电路实现

量子行走是经典随机行走在量子力学框架下的对应, 理论上可以用来解决一类无序数据库的搜索问题. 因为携带信息的量子态的扩散速度与经典相比有二次方式的增长, 所以量子行走优于经典随机行走, 量子行走的特性值得加以利用. 量子行走作为一种新发现的物理现象的数学描述, 引发了一种新的思维方式, 孕育了一种新的理论计算模型. 最新研究表明, 量子行走本身也是一种通用计算模型, 可被视为设计量子算法的高级工具, 因此受到部分计算机理论科学领域学者的关注和研究. 对于多数问题求解方案的量子算法的设计, 理论上可以只在量子行走模型下进行考虑. 基于Grover算法的相位匹配条件, 本文提出了一个新的基于量子行走的搜索算法. 理论演算表明: 一般情况下本算法的时间复杂度与Grover算法相同, 但是当搜索的目标数目多于总数的1/3时, 本算法搜索成功的概率要大于Grover算法. 本文不但利用Grover算法中相位匹配条件构造了一个新的量子行走搜索算法, 而且在本研究室原有的量子电路设计研究成果的基础上给出了该算法的量子电路表述.     

Demonstration of superadiabatic population transfer in superconducting qubit

We implemented the superadiabatic population transfer within the nonadiabatic regime in a two-level superconducting qubit system. To realize the superadiabatic procedure, we added an additional term in the Hamiltonian, introducing an auxiliary counter-diabatic field to cancel the nonadiabatic contribution in the evolution. Based on the superadiabatic procedure, we further demonstrated quantum Phase and NOT gates. These operations, which possess both of the fast and robust features, are promising for quantum information processing.     

Damping in the Interaction of a Two-Photon Field and a Two-Level Atom Through Quantized Caldirola-Kanai Hamiltonian

Following the lines of the recent papers (Daneshmand and Tavassoly, Int. J. Theor. Phys. 56, 1218 (2017)), we study quantum mechanical treatments of an interaction between a two-level atom with a single-mode field in the two-photon Jaynes-Cummings model, where the Hamiltonian of the field is considered to be the quantized Caldirola-Kanai (CK) Hamiltonian. As a result, we would expect that the quantum dynamics of the two-photon JCM in terms of the CK Hamiltonian is qualitatively different from that of the usual one-photon case. We analytically calculate the explicit form of the atom-field entangled state and numerically evaluate the dynamics of its physical properties. The degree of entanglement, atomic population as well as sub-Poissonian statistics and quadrature squeezing of the field are analyzed. We adjust the latter evolved parameters by appropriately tuning the damping parameter within the CK Hamiltonian and detuning factor. Finally, we report a field detuning asymmetry in the collective statistical behavior.

     

螺旋光纤系统中非绝热条件几何相移

由于绝热条件几何相位量子逻辑门存在非绝热差错与退相干差错这一冲突，因此在拓扑量子计算中需要设计非绝热条件几何相门，以克服这一不足.证明螺旋光纤系统内光子有效哈密顿量恰好是一个Wang Matsumoto型哈密顿量，因此螺旋光纤系统能自动产生非绝热条件几何相移.同时还证明在螺旋光纤方案中，由极化光子与螺旋光纤相互作用哈密顿量所导致的动力学相位为零（这正是拓扑量子计算所要求的），以及在螺旋光纤系统中可以通过控制极化光子初始波矢，从而较容易获得条件初始态.总之，螺旋光纤系统方案能自动满足Wang与Matsumoto的核磁共振方案中为实现非绝热条件几何相移所提出的全部条件与要求. 关键词： 几何相位 螺旋光纤系统 Wang Matsumoto型哈密顿量 拓扑量子计算     

Landau problem with a general time-dependent electric field

The time evolution is studied for the Landau level problem with a general time dependent electric field E(t) in a plane perpendicular to the magnetic field. A general and explicit factorization of the time evolution operator is obtained with each factor having a clear physical interpretation. The factorization consists of a geometric factor (path-ordered magnetic translation), a dynamical factor generated by the usual time-independent Landau Hamiltonian, and a nonadiabatic factor that determines the transition probabilities among the Landau levels. Since the path-ordered magnetic translation and the nonadiabatic factor are, up to completely determined numerical phase factors, just ordinary exponentials whose exponents are explicitly expressible in terms of the canonical variables, all of the factors in the factorization are explicitly constructed. New quantum interference effects are implied by this result. The factorization is unique from the point of view of the quantum adiabatic theorem and provides a seemingly first rigorous demonstration of how the quantum adiabatic theorem (incorporating the Berry phase phenomenon) is realized when infinitely degenerate energy levels are involved. Since the factorization separates the effect caused by the electric field into a geometric factor and a nonadiabatic factor, it makes possible to calculate the nonadiabatic transition probabilities near the adiabatic limit. A formula for matrix elements that determines the mixing of the Landau levels for a general, nonadiabatic evolution is also provided by the factorization.     

Quantum-Classical Correspondence for Adiabatic Shortcut in Two- and Three-Level Atoms

The methods of quickly achieving the adiabatic effect through a non-adiabatic process has recently drawn widely attention both in quantum and classical regime. In this work ,we study the classical adiabatic shortcut for two- and three-Level atoms by transforming the quantum version into classical one via quantum-classical corresponding theory. The results shows that, the additional couplings between the oscillators can be used to speed up the adiabatic evolution of coupled oscillators. Furthermore, we find that the quantum-classical correspondence theory still holds for the couter-adiabatic driving Hamiltonian for the TQD. This means that, we can obtain the counter-adiabatic driving Hamiltonian for a classical system by averaging over its quantum correspondence in a quantum system. This provides a feasible way to study the classical adiabatic shortcut and the simulation for the quantum adiabatic shortcut in a classical system.

     

One‐Step Implementation of N‐Qubit Nonadiabatic Holonomic Quantum Gates with Superconducting Qubits via Inverse Hamiltonian Engineering

A protocol is proposed to realize one‐step implementation of the N‐qubit nonadiabatic holonomic quantum gates with superconducting qubits. The inverse Hamiltonian engineering is applied in designing microwave pulses to drive superconducting qubits. By combining curve fitting, the wave shapes of the designed pulses can be described by simple functions, which are not hard to realize in experiments. To demonstrate the effectiveness of the protocol, a three‐qubit holonomic controlled π‐phase gate is taken as an example in numerical simulations. The results show that the protocol holds robustness against noise and decoherence. Therefore, the protocol may provide an alternative approach for implementing N‐qubit nonadiabatic holonomic quantum gates.     

Decoherence in optimized quantum random-walk search algorithm

This paper investigates the effects of decoherence generated by broken-link-type noise in the hypercube on an optimized quantum random-walk search algorithm. When the hypercube occurs with random broken links, the optimized quantum random-walk search algorithm with decoherence is depicted through defining the shift operator which includes the possibility of broken links. For a given database size, we obtain the maximum success rate of the algorithm and the required number of iterations through numerical simulations and analysis when the algorithm is in the presence of decoherence. Then the computational complexity of the algorithm with decoherence is obtained. The results show that the ultimate effect of broken-link-type decoherence on the optimized quantum random-walk search algorithm is negative.     

非绝热分子动力学的量子路径模拟

基于近期发展的经典-量子混合模拟非绝热分子动力学的量子路径方案,本文对5个典型势能面模型进行了模拟,包括单交叉模型、双交叉模型、拓展耦合模型、哑铃模型以及双弓模型.由于难以在严格意义上得到退相干速率,数值模拟中,我们比较了三个不同的退相干速率公式,包括冻结高斯波包近似退相干速率、能量分辨速率以及力分辨速率.在模拟过程中,我们恰当地处理了势能面跳跃时的能量守恒和力的反向问题.通过与全量子动力学模拟的精确结果进行对比发现,对于结构较简单的势能面模型,三种退相干速率都能得到较好的结果;然而对于较复杂的势能面模型,由于复杂量子干涉的原因,与其他混合经典-量子动力学方案类似,量子路径方案仍然难以得到较准确的结果.如何发展更加有效的混合经典-量子模拟方案,是未来研究的重要课题.     

The Quantum Search Algorithms for All Solutions

Two quantum search algorithms are proposed for known and unknown number of solutions. The first algorithm begins with an arbitrary rotation phase Grover search algorithm by recursive equations, then a sub-algorithm (G _α algorithm) and the corresponding quantum circuits are designed, the probability of success and expected number of iterations of the sub-algorithm to find a solution are analyzed. Based on these results, we design the whole algorithm and estimate the expected number of iterations to search all solutions. The design of the second algorithm follows the same steps. We firstly study a quantum counting algorithm, then design a sub-algorithm (QCG algorithm), and analyze the probability of success and the expected number of iterations to find a solution. Finally the whole algorithm for all solutions is designed and analyzed. The analysis results show that these two algorithms find all solutions in the expected times of $O(t\sqrt{N})$ (t is a number of solutions and N is a total of states).     

A quantum model for the stock market

Beginning with several basic hypotheses of quantum mechanics, we give a new quantum model in econophysics. In this model, we define wave functions and operators of the stock market to establish the Schrödinger equation for stock price. Based on this theoretical framework, an example of a driven infinite quantum well is considered, in which we use a cosine distribution to simulate the state of stock price in equilibrium. After adding an external field into the Hamiltonian to analytically calculate the wave function, the distribution and the average value of the rate of return are shown.     

How behavior of systems with sparse spectrum can be predicted on a quantum computer

Call a spectrum of Hamiltonian H sparse if each eigenvalue can be quickly restored within ε from its rough approximation within ε₁ by means of some classical algorithm. It is shown how the behavior of a system with a sparse spectrum up to time T=(1?ρ)/14ε can be predicted on a quantum computer with the time complexity t=4/(1?ρ)ε₁ plus the time of classical algorithm, where ρ is the fidelity. The quantum knowledge of Hamiltonian eigenvalues is considered as the new Hamiltonian W _H whose action on each eigenvector of H gives the corresponding eigenvalue. Speedup of evolution for systems with a sparse spectrum is possible, because, for such systems, the Hamiltonian W _H can be quickly simulated on the quantum computer. For an arbitrary system (even in the classical case), its behavior cannot be predicted on a quantum computer even for one step ahead. By this method, we can also restore the history with the same efficiency.     

Quantum Dynamics in Noisy Backgrounds: from Sampling to Dissipation and Fluctuations

We investigate the dynamics of a quantum system coupled linearly to Gaussian white noise using functional methods. By performing the integration over the noisy field in the evolution operator, we get an equivalent non-Hermitian Hamiltonian, which evolves the quantum state with a dissipative dynamics. We also show that if the integration over the noisy field is done for the time evolution of the density matrix, a gain contribution from the fluctuations can be accessed in addition to the loss one from the non-hermitian Hamiltonian dynamics. We illustrate our study by computing analytically the effective non-Hermitian Hamiltonian, which we found to be the complex frequency harmonic oscillator, with a known evolution operator. It leads to space and time localisation, a common feature of noisy quantum systems in general applications.