The Grover quantum search algorithm is generalized to deal with an arbitrary mixed initial state. The probability to measure a marked state as a function of time is calculated, and found to depend strongly on the specific initial state. The form of the function, though, remains as it is in the case of initial pure state. We study the role of the von Neumann entropy of the initial state, and show that the entropy cannot be a measure for the usefulness of the algorithm. We give few examples and show that for some extremely mixed initial states carrying high entropy, the generalized Grover algorithm is considerably faster than any classical algorithm.
The Groverian entanglement measure introduced earlier for pure quantum states [O. Biham, M.A. Nielsen and T. Osborne, Phys. Rev. A 65, 062312 (2002)] is generalized to the case of mixed states, in a way that maintains its operational interpretation. The Groverian measure of a mixed state of n qubits is obtained by a purification procedure into a pure state of 2n qubits, followed by an optimization process based on Uhlmann's theorem, before the resulting state is fed into Grover's search algorithm. The Groverian measure, expressed in terms of the maximal success probability of the algorithm, pr…
A misunderstanding that an arbitrary phase rotation of the marked state together with the inversion about average operation in Grover's search algorithm can be used to construct a (less efficient) quantum search algorithm is cleared. The $π$ rotation of the phase of the marked state is not only the choice for efficiency, but also vital in Grover's quantum search algorithm. The results also show that Grover's quantum search algorithm is robust.
The Groverian entanglement measure of pure quantum states of $n$ qubits is generalized to the case in which the qubits are divided into any $m \le n$ parties and the entanglement between these parties is evaluated. To demonstrate this measure we apply it to general states of three qubits and to symmetric states with any number of qubits such as the Greenberg-Horne-Zeiliner state and the W state.
Grover's algorithm for quantum searching is generalized to deal with arbitrary initial complex amplitude distributions. First order linear difference equations are found for the time evolution of the amplitudes of the marked and unmarked states. These equations are solved exactly. New expressions are derived for the optimal time of measurement and the maximal probability of success. They are found to depend on the averages and variances of the initial amplitude distributions of the marked and unmarked states, but not on higher moments. Our results imply that Grover's algorithm is robust agains…
Grover's algorithm for quantum searching of a database is generalized to deal with arbitrary initial amplitude distributions. First order linear difference equations are found for the time evolution of the amplitudes of the r marked and N-r unmarked states. These equations are solved exactly. An expression for the optimal measurement time T \sim O(\sqrt{N/r}) is derived which is shown to depend only on the initial average amplitudes of the marked and unmarked states. A bound on the probability of measuring a marked state is derived, which depends only on the standard deviation of the initial a…
In this paper we propose an approach to prepare GHZ states of an arbitrary multi-particle system in terms of Grover's fast quantum searching algorithm. This approach can be regarded as an extension of the Grover's algorithm to find one or more items in an unsorted database.
In a previous paper [quant-ph/0408045] we described a quantum algorithm to prepare an arbitrary state of a quantum register with arbitrary fidelity. Here we present an alternative algorithm which uses a small number of quantum oracles encoding the most significant bits of the absolute value of the complex amplitudes, and a small number of oracles encoding the most significant bits of the phases. The algorithm given here is considerably simpler than the one described in [quant-ph/0408045], on the assumption that a sufficient amount of knowledge about the distribution of the absolute values of t…
The Groverian entanglement measure, G(psi), is applied to characterize a variety of pure quantum states |psi> of multiple qubits. The Groverian measure is calculated analytically for certain states of high symmetry, while for arbitrary states it is evaluated using a numerical procedure. In particular, it is calculated for the class of Greenberger-Horne-Zeilinger states, the W states as well as for random pure states of n qubits. The entanglement generated by Grover's algorithm is evaluated by calculating G(psi) for the intermediate states that are obtained after t Grover iterations, for variou…
Grover search algorithm drives a quantum system from an initial state to a desired final state by using selective phase inversions of these two states. In (1), we studied a generalization of Grover algorithm which relaxes the assumption of the efficient implementation of the selective phase inversion of the initial state, also known as diffusion operator. This assumption is known to become a serious handicap in cases of physical interest (2,3,4,5). Our general search algorithm works with almost arbitrary diffusion operator with only restriction of having the initial state as one of its eigenst…
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.
We review the basic theoretical underpinnings of the Grover algorithm, providing a rigorous and well motivated derivation. We then present a generalization of Grover's algorithm that searches an arbitrary subspace of the multi-dimensional Hilbert space using a diffusion operation and an amplitude amplification procedure that has been biased by unitary {\em steering operators}. We also outline a generalized Grover's algorithm that takes into account higher level correlations that could exist between database elements. In the traditional Grover algorithm, the Hadamard gate selects a uniform samp…
We consider Grover's search algorithm on a model quantum computer implemented on a chain of four or five nuclear spins with first and second neighbour Ising interactions. Noise is introduced into the system in terms of random fluctuations of the external fields. By averaging over many repetitions of the algorithm, the output state becomes effectively a mixed state. We study its overlap with the nominal output state of the algorithm, which is called fidelity. We find either an exponential or a Gaussian decay for the fidelity as a function of the strength of the noise, depending on the type of n…
We investigate the performance of Grover's quantum search algorithm on a register which is subject to loss of the particles that carry the qubit information. Under the assumption that the basic steps of the algorithm are applied correctly on the correspondingly shrinking register, we show that the algorithm converges to mixed states with 50% overlap with the target state in the bit positions still present. As an alternative to error correction, we present a procedure that combines the outcome of different trials of the algorithm to determine the solution to the full search problem. The procedu…
The steadily growing research interest in quantum computing - together with the accompanying technological advances in the realization of quantum hardware - fuels the development of meaningful real-world applications, as well as implementations for well-known quantum algorithms. One of the most prominent examples till today is Grover's algorithm, which can be used for efficient search in unstructured databases. Quantum oracles that are frequently masked as black boxes play an important role in Grover's algorithm. Hence, the automatic generation of oracles is of paramount importance. Moreover, …
Grover's quantum search algorithm is analyzed for the case in which the initial state is an arbitrary pure quantum state $|ϕ>$ of $n$ qubits. It is shown that the optimal time to perform the measurement is independent of $| ϕ>$, namely, it is identical to the optimal time in the original algorithm in which $| ϕ> = | 0>$, with the same number of marked states, $r$. The probability of success $P_{\rm s}$ is obtained, in terms of the amplitudes of the state $| ϕ>$, and is shown to be independent of $r$. A class of states, which includes fixed points and cycles of the Grover iteration operator is …
Grover's search algorithm is the optimal quantum algorithm that can search an unstructured database quadratically faster than any known classical algorithm. The role of entanglement and correlations in the search algorithm have been studied in great detail and it is known that entanglement between the qubits is necessary to gain a quadratic speedup, for pure state implementation of the Grover search algorithm. Here, we systematically investigate the behavior of quantum coherence and monogamy of entanglement in the discrete analogue of the $\textit{analog analogue of Grover search algorithm}$. …
A new approach to the classical limit of Grover's algorithm is discussed by assuming a very rapid dephasing of a system between consecutive Grover's unitary operations, which drives pure quantum states to decohered mixed states. One can identify a specific element among $N$ unsorted elements by a probability of the order of unity after $k\sim N$ steps of classical amplification, which is realized by a combination of Grover's unitary operation and rapid dephasing, in contrast to $k\sim π\sqrt{N}/4$ steps in quantum mechanical amplification. The initial two-state system with enormously unbalance…
The recursion equation analysis of Grover's quantum search algorithm presented by Biham et al. [PRA 60, 2742 (1999)] is generalized. It is applied to the large class of Grover's type algorithms in which the Hadamard transform is replaced by any other unitary transformation and the phase inversion is replaced by a rotation by an arbitrary angle. The time evolution of the amplitudes of the marked and unmarked states, for any initial complex amplitude distribution is expressed using first order linear difference equations. These equations are solved exactly. The solution provides the number of it…
Given a parameterized quantum circuit such that a certain setting of these real-valued parameters corresponds to Grover's celebrated search algorithm, can a variational algorithm recover these settings and hence learn Grover's algorithm? We studied several constrained variations of this problem and answered this question in the affirmative, with some caveats. Grover's quantum search algorithm is optimal up to a constant. The success probability of Grover's algorithm goes from unity for two-qubits, decreases for three- and four-qubits and returns near unity for five-qubits then oscillates ever-…
We address the case, when querying to the oracle in Grover's algorithm is exposed to noise including phase distortions. The oracle-box wires can be altered by an opposite party that tries to prevent correct receiving data from the oracle. This situation reflects an experienced truth that any access to prophetic knowledge cannot be common and direct. To study the problem, we introduce a simple model of collective phase distortions on the base of phase damping channel. In the model used, the success probability is not altered via the oracle-box wires {\it per se}. Phase distortions of the consid…
Graph coloring is a computationally difficult problem, and currently the best known classical algorithm for $k$-coloring of graphs on $n$ vertices has runtimes $Ω(2^n)$ for $k\ge 5$. The list coloring problem asks the following more general question: given a list of available colors for each vertex in a graph, does it admit a proper coloring? We propose a quantum algorithm based on Grover search to quadratically speed up exhaustive search. Our algorithm loses in complexity to classical ones in specific restricted cases, but improves exhaustive search for cases where the lists and graphs consid…
We generalize Grover's unstructured quantum search algorithm to enable it to use an arbitrary starting superposition and an arbitrary unitary matrix simultaneously. We derive an exact formula for the probability of the generalized Grover's algorithm succeeding after n iterations. We show that the fully generalized formula reduces to the special cases considered by previous authors. We then use the generalized formula to determine the optimal strategy for using the unstructured quantum search algorithm. On average the optimal strategy is about 12% better than the naive use of Grover's algorithm…
We propose an ion trap implementation of Grover's quantum search algorithm for an unstructured database of arbitrary length N. The experimental implementation is appealingly simple because the linear ion trap allows for a straightforward construction, in a single interaction step and without a multitude of Hadamard transforms, of the reflection operator, which is the engine of the Grover algorithm. Consequently, a dramatic reduction in the number of the required physical steps takes place, to just O(\sqrt{N}), the same as the number of the mathematical steps. The proposed setup allows for demo…
An algebraic analysis of Grover's quantum search algorithm is presented for the case in which the initial state is an arbitrary pure quantum state of n qubits. This approach reveals the geometrical structure of the quantum search process, which turns out to be confined to a four-dimensional subspace of the Hilbert space. This work unifies and generalizes earlier results on the time evolution of the amplitudes during the quantum search, the optimal number of iterations and the success probability. Furthermore, it enables a direct generalization to the case in which the initial state is a mixed …
The quantum teleportation with noisy EPR state is discussed. Using an optimal decomposition technique, we compute the concurrence, entanglement of formation and Groverian measure for various noisy EPR resources. It is shown analytically that all entanglement measures reduce to zero when $\bar{F} \leq 2/3$, where $\bar{F}$ is an average fidelity between Alice and Bob. This fact indicates that the entanglement is a genuine physical resource for the teleportation process. This fact gives valuable clues on the optimal decomposition for higher-qubit mixed states. As an example, the optimal decompos…
Quantum walks exhibit properties without classical analogues. One of those is the phenomenon of asymptotic trapping -- there can be non-zero probability of the quantum walker being localised in a finite part of the underlying graph indefinitely even though locally all directions of movement are assigned non-zero amplitudes at each step. We study quantum walks with the flip-flop shift operator and the Grover coin, where this effect has been identified previously. For the version of the walk further modified by a random dynamical disruption of the graph (percolated quantum walks) we provide a re…
In this paper, we study decoherence on Grover's quantum searching algorithm using a perturbative method. We assume that each two-state system (qubit) suffers σ_{z} error with probability p (0\leq p\leq 1) independently at every step in the algorithm. Considering an n-qubit density operator to which Grover's operation is applied M times, we expand it in powers of 2Mnp and derive its matrix element order by order under the n\to \infty limit. (In this large n limit, we assume p is small enough, so that 2Mnp(\geq 0) can take any real positive value or 0.) This approach gives us an interpretation a…
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…
We analyze the robustness of Grover's quantum search algorithm performed by a quantum register under a possibly time-correlated noise acting locally on the qubits. We model the noise as originating from an arbitrary but fixed unitary evolution, $U$, of some noisy qubits. The noise can occur with some probability in the interval between any pair of consecutive noiseless Grover evolutions. Although each run of the algorithm is a unitary process, the noise model leads to decoherence when all possible runs are considered. We derive a set of unitary $U$'s, called the 'good noises,' for which the su…
Sun Yat-sen University（中山大学）；Tsinghua University（清华大学）；Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area (Guangdong)（粤港澳大湾区量子科学中心（广东））；QUDOOR Co, Ltd.（QUDOOR有限公司）
The short path algorithm gives a super-Grover speedup for various optimization problems under the assumption of a unique ground state and under an assumption on the density of low-energy states. Here, we remove the assumption of a unique ground state; this uses the same algorithm but a slightly different analysis and holds for arbitrary MAX-$D$-LIN-$2$ problems. Then, specializing to the case $D=2$, we show that for certain values of the objective function we can always achieve a super-Grover speedup (albeit a very slight one) without any assumptions on the density of states. Finally, for rand…
The effect of unitary noise on the performance of Grover's quantum search algorithm is studied. This type of noise may result from tiny fluctuations and drift in the parameters of the (quantum) components performing the computation. The resulting operations are still unitary, but not precisely those assumed in the design of the algorithm. Here we focus on the effect of such noise in the Hadamard gate $W$, which is an essential component in each iteration of the quantum search process. To this end $W$ is replaced by a noisy Hadamard gate $U$. The parameters of $U$ at each iteration are taken fr…
In this paper we discuss how we can design Hamiltonians to implement quantum algorithms, in particular we focus in Deutsch and Grover algorithms. As main result of this paper, we show how Hamiltonian inverse quantum engineering method allow us to obtain feasible and time-independent Hamiltonians for implementing such algorithms. From our approach for the Deutsch algorithm, different from others techniques, we can provide an alternative approach for implementing such algorithm where no auxiliary qubit and additional resources are required. In addition, by using a single quantum evolution, the G…
We derive a lower bound for the concurrence of mixed bipartite quantum states, valid in arbitrary dimensions. As a corollary, a weaker, purely algebraic estimate is found, which detects mixed entangled states with positive partial transpose.
We propose a quantum algorithm for closest pattern matching which allows us to search for as many distinct patterns as we wish in a given string (database), requiring a query function per symbol of the pattern alphabet. This represents a significant practical advantage when compared to Grover's search algorithm as well as to other quantum pattern matching methods, which rely on building specific queries for particular patterns. Our method makes arbitrary searches on long static databases much more realistic and implementable. Our algorithm, inspired by Grover's, returns the position of the clo…
In this work, we provide a complete analysis to minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities. For the complete anaysis, the most important work to do is to find the necessary and sufficient conditions for the existence of null measurement operator. From the geometric structure of qubit states, we obtain the analytic condition for deciding the existence of a null operator in minimum-error measurement for mixed four qubit states, which also gives the necessary and sufficient conditions for every optimal POVM to have non-zero elements. Using the condi…
Two methods for creating arbitrary two-photon polarization pure states are introduced. Based on these, four schemes for creating two-photon polarization mixed states are proposed and analyzed. The first two schemes can synthesize completely arbitrary two-qubit mixed states, i.e., control all 15 free parameters: Scheme I requires several sets of crystals, while Scheme II requires only a single set, but relies on decohering the pump beam. Additionally, we describe two further schemes which are much easier to implement. Although the total capability of these is still being studied, we show that t…
In this note we consider the problem of preparing a {\em single} copy of an arbitrary two-qubit mixed state $ρ$ starting from an entangled pure state $ψ$ and using only local operations assisted with classical communication. We present an analytical expression for the minimal amount of pure state entanglement required, and describe the corresponding local strategy. We also examine optimal probabilistic generalizations of the previous process.
A general quantum search algorithm with arbitrary unitary transformations and an arbitrary initial state is considered in this work. To serach a marked state with certainty, we have derived, using an SU(2) representation: (1) the matching condition relating the phase rotations in the algorithm, (2) a concise formula for evaluating the required number of iterations for the search, and (3) the final state after the search, with a phase angle in its amplitude of unity modulus. Moreover, the optimal choices and modifications of the phase angles in the Grover kernel is also studied.
An SO(3) picture of the generalized Grover's quantum searching algorithm,with arbitrary unitary transformation and with arbitrary phase rotations, is constructed. In this picture, any quantum search operation is a rotation in a 3 dimensional space. Exact formulas for the rotation angle and rotational axis are given. The probability of finding the marked state is just $(z+1)/2$, where z is the z-component of the state vector. Exact formulas for this probability is easily obtained. The phase matching requirement and the failure of algorithm when phase mismatches are clearly explained.
The nonlocal properties of arbitrary dimensional bipartite quantum systems are investigated. A complete set of invariants under local unitary transformations is presented. These invariants give rise to both sufficient and necessary conditions for the equivalence of quantum states under local unitary transformations: two density matrices are locally equivalent if and only if all these invariants have equal values.
We present observable upper bounds of squared concurrence, which are the dual inequalities of the observable lower bounds introduced in [F. Mintert and A. Buchleitner, Phys. Rev. Lett. 98, 140505 (2007)] and [L. Aolita, A. Buchleitner and F. Mintert, Phys. Rev. A 78, 022308 (2008)]. These bounds can be used to estimate entanglement for arbitrary experimental unknown finite-dimensional states by few experimental measurements on a twofold copy $ρ\otimesρ$ of the mixed states. Furthermore, the degree of mixing for a mixed state and some properties of the linear entropy also have certain relations…
This work addresses a fundamental problem of controllability of open quantum systems, meaning the ability to steer arbitrary initial system density matrix into any final density matrix. We show that under certain general conditions open quantum systems are completely controllable and propose the first, to the best of our knowledge, deterministic method for a laboratory realization of such controllability which allows for a practical engineering of arbitrary pure and mixed quantum states. The method exploits manipulation by the system with a laser field and a tailored nonequilibrium and time-de…
We describe a quantum algorithm to prepare an arbitrary pure state of a register of a quantum computer with fidelity arbitrarily close to 1. Our algorithm is based on Grover's quantum search algorithm. For sequences of states with suitably bounded amplitudes, the algorithm requires resources that are polynomial in the number of qubits. Such sequences of states occur naturally in the problem of encoding a classical probability distribution in a quantum register.
We present two efficient schemes which can simultaneously accomplish hyperentanglement concentration and purification for two-photon four-qubit systems in an unknown partially hyperentangled mixed states. The first can correct errors in the polarization entanglement and extract maximal hyperentanglement in polarization and spatial mode with additional partial frequency entanglement. The second uses additional maximal frequency entanglement to purify and concentrate hyperentanglement in polarization and spatial mode deterministically. Both of the two schemes are only based on existing optical d…
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…
We present a systematic construction of quantum circuits implementing Grover's database search algorithm for arbitrary number of targets. We introduce a new operator which flips the sign of the targets and evaluate its circuit complexity. We find the condition under which the circuit complexity of the database search algorithm based on this operator is less than that of the conventional one.
We present an inequality that classifies mixed multipartite systems of an arbitrary dimension with respect to separability and positivity of partial transpose properties. This inequality gives a way to experimentally classify the observed state of multipartite systems of an arbitrary dimension. The inequality also implies that a sufficient condition for a density operator to have no positive partial transpose with respect to any subsystem is that the fidelity to a generalized Greenberger-Horne-Zeilinger state {[A. Cabello, Phys. Rev. A {\bf 63}, 022104 (2001)]} is larger than 1/2 for mixed mul…
We give a complete, hierarchic classification for arbitrary multi-qubit mixed states based on the separability properties of certain partitions. We introduce a family of N-qubit states to which any arbitrary state can be depolarized. This family can be viewed as the generalization of Werner states to multi-qubit systems. We fully classify those states with respect to their separability and distillability properties. This provides sufficient conditions for nonseparability and distillability for arbitrary states.
We propose a parallel quantum computing mode for ensemble quantum computer. In this mode, some qubits can be in pure states while other qubits in mixed states. It enables a single ensemble quantum computer to perform $"$single-instruction-multi-data" type of parallel computation. In Grover's algorithm and Shor's algorithm, parallel quantum computing can provide additional speedup. In addition, it also makes a fuller use of qubit resources in an ensemble quantum computer. As a result, some qubits discarded in the preparation of an effective pure state in the Schulman-Varizani, and the Cleve-DiV…
