A new notion of controllability, eigenstate controllability, is defined for finite-dimensional bilinear quantum mechanical systems which are neither strongly completely controllably nor completely controllable. And a quantum control algorithm based on Grover iteration is designed to perform a quantum control task of steering a system, which is eigenstate controllable but may not be (strongly) completely controllable, from an arbitrary state to a target state.
Approximate Counting refers to the problem where we are given query access to a function $f : [N] \to \{0,1\}$, and we wish to estimate $K = #\{x : f(x) = 1\}$ to within a factor of $1+ε$ (with high probability), while minimizing the number of queries. In the quantum setting, Approximate Counting can be done with $O\left(\min\left(\sqrt{N/ε}, \sqrt{N/K}/ε\right)\right)$ queries. It has recently been shown that this can be achieved by a simple algorithm that only uses "Grover iterations"; however the algorithm performs these iterations adaptively. Motivated by concerns of computational simplici…
The success probability of a search of $M$ targets from a database of size $N$, using Grover's search algorithm depends critically on the number of iterations of the composite operation of the oracle followed by Grover's diffusion operation. Although the required number of iterations scales as $\mathcal{O}(\sqrt{N})$ for large $N$, the asymptote is not a good indicator of the optimal number of iterations when $\sqrt{M/N}$ is not small. A scheme for the determination of the exact number of iterations, subject to a threshold set for the success probability of the search (probability of detecting…
Grover算法通常用$Q = - H I_{0} H I_{x_0}$形式的复合算子的迭代来描述。虽然验证迭代的代数是非常简单的，但这对算法的工作原理几乎没有帮助。$Q$的化合物结构有什么意义？为什么有个负号？后来发现$H$基本上可以被任何酉$U$所取代。这里涉及的自由是什么？我们给出了一个描述格罗弗的算法，提供了一些澄清这些问题。
Improvement of performance of Grover's algorithm on three generations of Heron family IBM QPUs without and with topological dynamical decoupling
Georgi Nadjakov Institute of Solid State Physics, Bulgarian Academy of Sciences（保加利亚科学院固体物理格奥尔基·纳贾科夫研究所）；Center for Quantum Technologies, Department of Physics, St. Kliment Ohridski University of Sofia（索非亚圣克利门特·奥赫里德斯基大学物理学院量子技术中心）
Grover's quantum search algorithm provides a quadratic quantum advantage over classical algorithms across a broad class of unstructured search problems. The original protocol is probabilistic, returning the desired result with significant probability on each query, but in general, requiring several iterations of the algorithm. We present a modified version to return the correct result with certainty without having user control over the quantum search oracle. Our deterministic, two-parameter "D2p" protocol utilizes generalized phase rotations replacing the phase inversions after a standard orac…
Linear combination of unitaries (LCU for short) is one of the most important techniques in designing quantum algorithms. In this paper, we propose a new quantum algorithm in three different forms to achieve LCU. Different from previous algorithms [Childs-linear-system,Clader,Long11], the complexity now only depends on the number of the unitaries and the precision. So it will play more important role in the design of quantum algorithms when the number of unitaries is small, such as quantum iteration algorithms. %Since in iteration algorithms, $m$ often refers to the iteration steps, which is sm…
A sequential application of the Grover algorithm to solve the iterated search problem has been improved by Ozhigov by parallelizing the application of the oracle. In this work a representation of the parallel Grover as dynamic system of inversion about the mean and Grover operators is given. Within this representation the parallel Grover for $k = 2$ can be interpreted as rotation in three-dimensional space and it can be shown that the sole application of the parallel Grover operator does not lead to a solution for $k > 2$. We propose a solution for $k = 3$ with a number of approximately $1.51\…
Entanglement is considered to be one of the primary reasons for why quantum algorithms are more efficient than their classical counterparts for certain computational tasks. The global multipartite entanglement of the multiqubit states in Grover's search algorithm can be quantified using the geometric measure of entanglement (GME). Rossi {\em et al.} (Phys. Rev. A \textbf{87}, 022331 (2013)) found that the entanglement dynamics is scale invariant for large $n$. Namely, the GME does not depend on the number $n$ of qubits; rather, it only depends on the ratio of iteration $k$ to the total iterati…
In the Grover-type quantum search process a search operator is iteratively applied, say, k times, on the initial database state. We present an additive decomposition scheme such that the iteration process is expressed, in the computational space, as a linear combination of k operators, each of which consists of a single Grover-search followed by an overall phase-rotation. The value of k and the rotation phase are the same as those determined in the framework of the search with certainty. We further show that the final state can be expressed in terms of a single oracle operator of the Grover-se…
Grover's algorithm achieves a quadratic speedup over classical algorithms, but it is considered necessary to know the value of $λ$ exactly [Phys. Rev. Lett. 95, 150501 (2005); Phys. Rev. Lett. 113, 210501 (2014)], where $λ$ is the fraction of target items in the database. In this paper, we find out that the Grover algorithm can actually apply to the case where one can identify the range that $λ$ belongs to from a given series of disjoint ranges. However, Grover's algorithm still cannot maintain high success probability when there exist multiple target items. For this problem, we proposed a com…
Formal Modeling and Verification of Grover's Algorithm
Capital Normal University（首都师范大学）；Beijing Key Laboratory of Electronic System Reliability Technology, Capital Normal University（首都师范大学电子系统可靠性技术北京市重点实验室）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）；International Science and Technology Cooperation Base of Electronic System Reliability and Mathematical Interdisciplinary, Capital Normal University（首都师范大学电子系统可靠性与数学交叉国际科技合作基地）
We generalize Grover algorithm with two arbitrary phases in a density matrix set up. We give exact analytic expressions for the success probability after arbitrary number of iteration of the generalized Grover operator as a function of number of iterations, two phase angles (α, \{beta}) and parameter ξ introduced in the off diagonal terms of the density matrix in a sense to capture the coherence present in the initial quantum register. We extend Li and Li's idea and show for the phase matching condition α = -\{beta} = 0.35π with two iterations and ξ = 1, we can achieve success probability >= 0…
Coherence plays a very important role in Grover search algorithm (GSA). In this paper, we define the normalization coherence N(C), where C is a coherence measurement. In virtue of the constraint of large N and Shannon's maximum entropy principle, a surprising complementary relationship between the coherence and the success probability of GSA is obtained. Namely, P_s(t)+N(C(t))\simeq 1, where C is in terms of the relative entropy of coherence and l_1 norm of coherence, t is the number of the search iterations in GSA. Moreover, the equation holds no matter in ideal or noisy environments. Conside…
Our aim is to determine conditions for quantum computing technology to give rise to security risks associated with quantum Bitcoin mining. Specifically, we determine the speed and energy efficiency a quantum computer needs to offer an advantage over classical mining. We analyze the setting in which the Bitcoin network is entirely classical except for a single quantum miner who has small hash rate compared to that of the network. We develop a closed-form approximation for the probability that the quantum miner successfully mines a block, with this probability dependent on the number of Grover i…
Grover's algorithm confers on quantum computers a quadratic advantage over classical computers for searching in an arbitrary data set, a scenario that describes Bitcoin mining. It has previously been argued that the only side-effect of quantum mining would be an increased difficulty. In this work, we argue that a crucial argument in the analysis of Bitcoin security breaks down when quantum mining is performed. Classically, a Bitcoin fork occurs rarely, i.e., when two miners find a block almost simultaneously, due to propagation time effects. The situation differs dramatically when quantum mine…
Inspired by a class of algorithms proposed by Farhi et al. (arXiv:1411.4028), namely the quantum approximate optimization algorithm (QAOA), we present a circuit-based quantum algorithm to search for a needle in a haystack, obtaining the same quadratic speedup achieved by Grover's original algorithm. In our algorithm, the problem Hamiltonian (oracle) and a transverse field are applied alternately to the system in a periodic manner. We introduce a technique, based on spin-coherent states, to analyze the composite unitary in a single period. This composite unitary drives a closed transition betwe…
A while loop tests a termination condition on every iteration. On a quantum computer, such measurements perturb the evolution of the algorithm. We define a while loop primitive using weak measurements, offering a trade-off between the perturbation caused and the amount of information gained per iteration. This trade-off is adjusted with a parameter set by the programmer. We provide sufficient conditions that let us determine, with arbitrarily high probability, a worst-case estimate of the number of iterations the loop will run for. As an example, we solve Grover's search problem using a while …
In this work, we show the characterization of quantum iterations that would generally construct quantum amplitude amplification algorithms with a quadratic speedup, namely, quantum amplitude amplification operators (QAAOs). Exact quantum search algorithms that find a target with certainty and with a quadratic speedup can be composed of sequential applications of QAAO: existing quantum amplitude amplification algorithms thus turn out to be sequences of QAAOs. We show that an optimal and exact quantum amplitude amplification algorithm corresponds to the Grover algorithm together with a single it…
Amplitude Amplification -- a key component of Grover's Search algorithm -- uses an iterative approach to systematically increase the probability of one or multiple target states. We present novel strategies to enhance the amplification procedure by partitioning the states into classes, whose probabilities are increased at different levels before or during amplification. The partitioning process is based on the binomial distribution. If the classes to which the search target states belong are known in advance, the number of iterations in the Amplitude Amplification algorithm can be drastically …
A Grover-compatible manifold optimization algorithm for quantum search
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
A simpler quantum counting algorithm based on amplitude amplification is presented. This algorithm is bounded by O(sqrt(N/M)) calls to the controlled-Grover operator where M is the number of marked states and N is the total number of states in the search space. This algorithm terminates within log(sqrt(N/M)) consecutive measurement steps when the probability p1 of measuring the state |1> is at least 0.5, and the result from the final step is used in estimating M by a classical post processing. The purpose of controlled-Grover iteration is to increase the probability p1. This algorithm requires…
In 1998, Brassard, Hoyer, Mosca, and Tapp (BHMT) gave a quantum algorithm for approximate counting. Given a list of $N$ items, $K$ of them marked, their algorithm estimates $K$ to within relative error $\varepsilon$ by making only $O\left( \frac{1}{\varepsilon}\sqrt{\frac{N}{K}}\right) $ queries. Although this speedup is of "Grover" type, the BHMT algorithm has the curious feature of relying on the Quantum Fourier Transform (QFT), more commonly associated with Shor's algorithm. Is this necessary? This paper presents a simplified algorithm, which we prove achieves the same query complexity usin…
