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

数理解析研究所講究録

An extention of Grover's quantum search algorithm (Foundations of Computer Science)

Keisuke Tanaka

Published Apr 1, 2000• 0 citations
OpenAlex

Abstract

Quantum computers and their algorithm are now widely studied.We consider the prob- lem of making a superposition with an arbitrarily specified amplitudes.By modifying Grover's quantum search algorithm, we show that this can be done in time $O(\sqrt{N})$ with two simple operations, one is the Walsh-Hadamard transformation of $N\cross N$ -matrices where $N=2^{n}$ , the other is phase rotation of a state.If we see this problem in the classical computation model, it corresponds to the problem of making a sample with an arbitrarily specified distribution, where its time complexity is $\Omega(N)$ .By making a measurement on the result of our algorithm, it is possible to obtain a sample according to an arbitrarily specified probability distribution in $O(\sqrt{N})$ time.

Related papers

Research on Quantum Searching Algorithms Based on Phase Shifts

Zhong Pu-Cha, Wan‐Su Bao

One iterative in Grover's original quantum search algorithm consists of two Hadamard–Walsh transformations, a selective amplitude inversion and a diffusion amplitude inversion. We concentrate on the relation among the probability of success of the algorithm, the phase shifts, the number of target items and the number of iterations via replacing the two amplitude inversions by phase shifts of an arbitrary ϕ = ϕ(0 ⩽ ϕ, ϕ ⩽ 2π). Then, according to the relation we find out the optimal phase shifts when the number of iterations is given. We present a new quantum search algorithm based on the optimal phase shifts of 1.018 after 0.5π/ iterations. The new algorithm can obtain either a single target item or multiple target items in the search space with the probability of success at least 93.43%.

Chinese Physics Letters•Jul 29, 2008•7 citations

Analysis of generalized Grover quantum search algorithms using recursion equations

Eli Biham, Ofer Biham, David Biron +3 more

The recursion equation analysis of Grover's quantum search algorithm presented by Biham et al. [Phys. Rev. A 60, 2742 (1999)] is generalized. It is applied to the large class of Grover-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 iterations T after which the probability of finding a marked state upon measurement is the highest, as well as the value of this probability, ${P}_{\mathrm{max}}.$ Both T and ${P}_{\mathrm{max}}$ 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.

arXiv:quant-ph/0010077•Physical Review A•Dec 13, 2000•72 citations

Rapid sampling through quantum computing

Lov K. Grover

This paper extends the quantum search class of algorithms to the multiple solution case. It is shown that, like the basic search algorithm, these too can be represented as a rotation in an appropriately defined two dimensional vector space. This yields new applications - an algorithm is presented that can create an arbitrarily specified quantum superposition on a space of size N in O(sqrt(N)) steps. By making a measurement on this superposition, it is possible to obtain a sample according to an arbitrarily specified classical probability distribution in O(sqrt(N)) steps. A classical algorithm would need O(N) steps.

arXiv:quant-ph/9912001•Dec 1, 1999•23 citations

Rapid sampling though quantum computing

Lov K. Grover

This paper extends the quantum search class of algorithms to the multiple solution case.It is shown that, like the basic search algorithm, these too can be represented as a rotation in an appropriately defined two dimensional vector space.This yields new applicationsan algorithm is presented that can create an arbitrarily specified quantum superposition on a space of size N in o(4r-N) steps.By making a measurement on this superposition, it is possible to obtain a sample according to an arbitrarily specified classical probability distribution in O(,f-N) steps.A classical algorithm would need f2(N) steps.

ACM Symposium on Theory of Computing (STOC)•May 1, 2000•12 citations

A Rigorous and Self--Contained Proof of the Grover--Rudolph State Preparation Algorithm

Antonio Falco, Daniela Falco-Pomares, Hermann G. Matthies

We give a rigorous and self-contained analysis of the Grover--Rudolph quantum state-preparation algorithm, which encodes a probability distribution $\{p_k\}$ as an $n$-qubit amplitude state $\sum_k\sqrt{p_k}\ket{k}$ via a hierarchy of controlled $\RY$ rotations determined by a dyadic refinement of the target. We formalize the dyadic probability tree, derive the trigonometric factorization of conditional masses, and prove by induction that the circuit prepares exactly the desired measurement law. We further prove that perturbing each rotation angle by at most $η$ changes the output distribution by at most $\min(1,nη)$ in total variation, and combine this with a Hoeffding concentration bound to obtain an explicit design rule: $b\ge\log_2(2nπ/\varepsilon)$ bits and $S\ge 2^{n+1}\log(2/δ)/\varepsilon^2$ shots suffice to achieve accuracy $\varepsilon$ with confidence $1-δ$. As a circuit-theoretic complement, we provide an ancilla-free transpilation of each stage into $\{\RY(\cdot),X,\CNOT\}$ via Gray-code ladders and a Walsh--Hadamard angle transform.

quant-pharXiv:2601.17930•Jan 27, 2026•0 citations

Synthesis of Quantum Superpositions by Quantum Computation

Lov K. Grover

The quantum search algorithm can be looked at as a technique for synthesizing a particular kind of superposition-one whose amplitude is concentrated in a single basis state. This basis state is defined by a binary function f(&xmacr;) that is nonzero in this desired basis state and zero everywhere else. This paper extends the quantum search algorithm to an algorithm that can create an arbitrarily specified superposition on a space of size N in O(sqrt[N] ) steps. The superposition is specified by a complex valued function f(&xmacr;) that specifies the desired amplitude of the system in basis state &xmacr;.

Physical Review Letters•Aug 7, 2000•120 citations
Quantum computers and their algorithm are now widely studied.We consider the prob- lem of making a superposition with an arbitrarily specified amplitudes.By modifying Grover's quantum search algorithm, we show that this can be done in time O(N)O(\sqrt{N})O(N​) with two simple operations, one is the Walsh-Hadamard transformation of N\crossNN\cross NN\crossN -matrices where N=2nN=2^{n}N=2n , the other is phase rotation of a state.If we see this problem in the classical computation model, it corresponds to the problem of making a sample with an arbitrarily specified distribution, where its time complexity is Ω(N)\Omega(N)Ω(N) .By making a measurement on the result of our algorithm, it is possible to obtain a sample according to an arbitrarily specified probability distribution in O(N)O(\sqrt{N})O(N​) time.
One iterative in Grover's original quantum search algorithm consists of two Hadamard–Walsh transformations, a selective amplitude inversion and a diffusion amplitude inversion. We concentrate on the relation among the probability of success of the algorithm, the phase shifts, the number of target items and the number of iterations via replacing the two amplitude inversions by phase shifts of an arbitrary ϕ = ϕ(0 ⩽ ϕ, ϕ ⩽ 2π). Then, according to the relation we find out the optimal phase shifts when the number of iterations is given. We present a new quantum search algorithm based on the optimal phase shifts of 1.018 after 0.5π/ iterations. The new algorithm can obtain either a single target item or multiple target items in the search space with the probability of success at least 93.43%.
The recursion equation analysis of Grover's quantum search algorithm presented by Biham et al. [Phys. Rev. A 60, 2742 (1999)] is generalized. It is applied to the large class of Grover-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 iterations T after which the probability of finding a marked state upon measurement is the highest, as well as the value of this probability, Pmax.{P}_{\mathrm{max}}.Pmax​. Both T and Pmax{P}_{\mathrm{max}}Pmax​ 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.
This paper extends the quantum search class of algorithms to the multiple solution case. It is shown that, like the basic search algorithm, these too can be represented as a rotation in an appropriately defined two dimensional vector space. This yields new applications - an algorithm is presented that can create an arbitrarily specified quantum superposition on a space of size N in O(sqrt(N)) steps. By making a measurement on this superposition, it is possible to obtain a sample according to an arbitrarily specified classical probability distribution in O(sqrt(N)) steps. A classical algorithm would need O(N) steps.
This paper extends the quantum search class of algorithms to the multiple solution case.It is shown that, like the basic search algorithm, these too can be represented as a rotation in an appropriately defined two dimensional vector space.This yields new applicationsan algorithm is presented that can create an arbitrarily specified quantum superposition on a space of size N in o(4r-N) steps.By making a measurement on this superposition, it is possible to obtain a sample according to an arbitrarily specified classical probability distribution in O(,f-N) steps.A classical algorithm would need f2(N) steps.
We give a rigorous and self-contained analysis of the Grover--Rudolph quantum state-preparation algorithm, which encodes a probability distribution {pk}\{p_k\}{pk​} as an nnn-qubit amplitude state ∑kpk∣k⟩\sum_k\sqrt{p_k}\ket{k}∑k​pk​​∣k⟩ via a hierarchy of controlled $\RY$ rotations determined by a dyadic refinement of the target. We formalize the dyadic probability tree, derive the trigonometric factorization of conditional masses, and prove by induction that the circuit prepares exactly the desired measurement law. We further prove that perturbing each rotation angle by at most ηηη changes the output distribution by at most min⁡(1,nη)\min(1,nη)min(1,nη) in total variation, and combine this with a Hoeffding concentration bound to obtain an explicit design rule: b≥log⁡2(2nπ/ε)b\ge\log_2(2nπ/\varepsilon)b≥log2​(2nπ/ε) bits and S≥2n+1log⁡(2/δ)/ε2S\ge 2^{n+1}\log(2/δ)/\varepsilon^2S≥2n+1log(2/δ)/ε2 shots suffice to achieve accuracy ε\varepsilonε with confidence 1−δ1-δ1−δ. As a circuit-theoretic complement, we provide an ancilla-free transpilation of each stage into $\{\RY(\cdot),X,\CNOT\}$ via Gray-code ladders and a Walsh--Hadamard angle transform.
The quantum search algorithm can be looked at as a technique for synthesizing a particular kind of superposition-one whose amplitude is concentrated in a single basis state. This basis state is defined by a binary function f(&xmacr;) that is nonzero in this desired basis state and zero everywhere else. This paper extends the quantum search algorithm to an algorithm that can create an arbitrarily specified superposition on a space of size N in O(sqrt[N] ) steps. The superposition is specified by a complex valued function f(&xmacr;) that specifies the desired amplitude of the system in basis state &xmacr;.