Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0504149 · Physical Review A

Algebraic analysis of quantum search with pure and mixed states

Daniel Shapira, Yishai Shimoni, Ofer Biham

Published Apr 12, 2005• 11 citations
arXivHTMLPDF

Abstract

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 $\ensuremath{\mid}\ensuremath{\psi}⟩$ 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. It unifies and generalizes earlier results on the time evolution of the amplitudes during the 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 state, providing an exact formula for the success probability.

DOI
10.1103/physreva.71.042320

Related papers

Analysis of Grover’s quantum search algorithm as a dynamical system

Ofer Biham, Daniel Shapira, Yishai Shimoni

Grover's quantum search algorithm is analyzed for the case in which the initial state is an arbitrary pure quantum state $|\ensuremath{\varphi}〉$ of n qubits. It is shown that the optimal time to perform the measurement is independent of $|\ensuremath{\varphi}〉,$ namely, it is identical to the optimal time in the original algorithm in which $|\ensuremath{\varphi}〉=|0〉,$ with the same number of marked states, r. The probability of success ${P}_{\mathrm{s}}$ is obtained in terms of the amplitudes of the state $|\ensuremath{\varphi}〉$ and is shown to be independent of r. A class of states, which includes fixed points and cycles of the Grover iteration operator, is identified. The relevance of these results in the context of using the success probability as an entanglement measure is discussed. In particular, the Groverian entanglement measure, previously limited to a single marked state, is generalized to the case of several marked states.

arXiv:quant-ph/0307141•Physical Review A•Aug 29, 2003•15 citations

Grover Algorithm and an Analysis of Successful Probability of Quantum Mixed State

Duolu Mao

Grover proposed how a quantum computer can find a single marked object in a database of size N by using only O(N) queries of the oracle that identifies the object with the probability of finding the marked object being not exactly 1.The Grover quantum search algorithm is generalized to deal with an arbitrary mixed initial state.The probability of finding the marked object is calculated and it is found that it depends strongly on the specific initial state.Our results contribute to analyze the influence of quantum noise on Grover algorithm based on pure state.Finally,we give few examples to show that for some extremely mixed initial states.

Acta Sinica Quantum Optica•Jan 1, 2009•0 citations

Grover’s quantum search algorithm for an arbitrary initial mixed state

Eli Biham, Dan Kenigsberg

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.

arXiv:quant-ph/0306183•Physical Review A•Dec 4, 2002•49 citations

Grover’s quantum search algorithm for an arbitrary initial amplitude distribution

Eli Biham, Ofer Biham, David Biron +2 more

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 against modest noise in the amplitude initialization procedure.

arXiv:quant-ph/9807027•Physical Review A•Oct 1, 1999•119 citations

Quantum Search in a Four-Complex-Dimensional Subspace

Wenliang Jin

For there to be $M> 1$ target items to be searched in an unsorted database of size $N$, with $M/N\ll 1$ for a sufficiently large $N$, we explore the performance of Grover's search algorithm when considering some possible situations that may arise in a four-complex-dimensional subspace, for which in the case of identical rotation angles $\phi=\theta$, we give the maximum success probabilities of finding a desired state and their corresponding numbers of Grover iterations in an approximate fashion. Our analysis reveals that the case of identical rotation angles $\phi=\theta$ is energetically favorable compared to the case $\left| {\theta - \phi } \right|\gg 0$ for boosting the probability to detect a desired state.

•Jul 1, 2013•0 citations

Generalized quantum search with parallelism

Robert M. Gingrich, Colin P. Williams, Nicolas J. Cerf

We generalize Grover's unstructured quantum search algorithm to enable it to work with arbitrary starting superpositions and arbitrary unitary operators. We show that the generalized quantum search algorithm, when cast in a special orthonormal basis, can be understood as performing an exact rotation of a starting superposition into a target superposition. We derive a formula for the success probability of the generalized quantum search algorithm after n rounds of amplitude amplification. We then use this formula to determine the optimal strategy for a punctuated quantum search algorithm, i.e., one in which the amplitude amplified state is observed before the point of maximum success probability. On average, the optimal strategy is about 12% better than the naive use of Grover's algorithm. The speedup obtained is not dramatic but it illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. In addition, we show that a punctuated quantum algorithm that takes the same average computation time as Grover's standard algorithm only requires half the coherence time. We then extend the analysis to the case of a society of k quantum searches acting in parallel. We derive an analytic formula that connects the degree of parallelism with the expected computation time for k-parallel quantum search. The resulting parallel speedup scales as $O(\sqrt{k}),$ while the minimum number of agents needed to ensure success, k, decreases as the inverse of the square of the achievable coherence time. This result has practical significance for the design of rudimentary quantum computers that are likely to have a limited coherence time.

arXiv:quant-ph/9904049•Physical Review A•Apr 19, 2000•41 citations
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 \ensuremath∣\ensuremathψ⟩\ensuremath{\mid}\ensuremath{\psi}⟩\ensuremath∣\ensuremathψ⟩ of nnn 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. It unifies and generalizes earlier results on the time evolution of the amplitudes during the 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 state, providing an exact formula for the success probability.
Grover's quantum search algorithm is analyzed for the case in which the initial state is an arbitrary pure quantum state $|\ensuremath{\varphi}〉$ of n qubits. It is shown that the optimal time to perform the measurement is independent of $|\ensuremath{\varphi}〉,$ namely, it is identical to the optimal time in the original algorithm in which $|\ensuremath{\varphi}〉=|0〉,$ with the same number of marked states, r. The probability of success Ps{P}_{\mathrm{s}}Ps​ is obtained in terms of the amplitudes of the state $|\ensuremath{\varphi}〉$ and is shown to be independent of r. A class of states, which includes fixed points and cycles of the Grover iteration operator, is identified. The relevance of these results in the context of using the success probability as an entanglement measure is discussed. In particular, the Groverian entanglement measure, previously limited to a single marked state, is generalized to the case of several marked states.
Grover proposed how a quantum computer can find a single marked object in a database of size N by using only O(N) queries of the oracle that identifies the object with the probability of finding the marked object being not exactly 1.The Grover quantum search algorithm is generalized to deal with an arbitrary mixed initial state.The probability of finding the marked object is calculated and it is found that it depends strongly on the specific initial state.Our results contribute to analyze the influence of quantum noise on Grover algorithm based on pure state.Finally,we give few examples to show that for some extremely mixed initial states.
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.
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 against modest noise in the amplitude initialization procedure.
For there to be M>1M> 1M>1 target items to be searched in an unsorted database of size NNN, with M/N≪1M/N\ll 1M/N≪1 for a sufficiently large NNN, we explore the performance of Grover's search algorithm when considering some possible situations that may arise in a four-complex-dimensional subspace, for which in the case of identical rotation angles ϕ=θ\phi=\thetaϕ=θ, we give the maximum success probabilities of finding a desired state and their corresponding numbers of Grover iterations in an approximate fashion. Our analysis reveals that the case of identical rotation angles ϕ=θ\phi=\thetaϕ=θ is energetically favorable compared to the case ∣θ−ϕ∣≫0\left| {\theta - \phi } \right|\gg 0∣θ−ϕ∣≫0 for boosting the probability to detect a desired state.
We generalize Grover's unstructured quantum search algorithm to enable it to work with arbitrary starting superpositions and arbitrary unitary operators. We show that the generalized quantum search algorithm, when cast in a special orthonormal basis, can be understood as performing an exact rotation of a starting superposition into a target superposition. We derive a formula for the success probability of the generalized quantum search algorithm after n rounds of amplitude amplification. We then use this formula to determine the optimal strategy for a punctuated quantum search algorithm, i.e., one in which the amplitude amplified state is observed before the point of maximum success probability. On average, the optimal strategy is about 12% better than the naive use of Grover's algorithm. The speedup obtained is not dramatic but it illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. In addition, we show that a punctuated quantum algorithm that takes the same average computation time as Grover's standard algorithm only requires half the coherence time. We then extend the analysis to the case of a society of k quantum searches acting in parallel. We derive an analytic formula that connects the degree of parallelism with the expected computation time for k-parallel quantum search. The resulting parallel speedup scales as O(k),O(\sqrt{k}),O(k​), while the minimum number of agents needed to ensure success, k, decreases as the inverse of the square of the achievable coherence time. This result has practical significance for the design of rudimentary quantum computers that are likely to have a limited coherence time.