arXiv:1503.05712 · Physical Review A

Faster quantum searching with almost any diffusion operator

Avatar Tulsi

Published May 7, 2015• 10 citations

Abstract

Grover's search algorithm drives a quantum system from an initial state $|s\ensuremath{\rangle}$ to a desired final state $|t\ensuremath{\rangle}$ by using selective phase inversions of these two states. Earlier, we studied a generalization of Grover's algorithm that relaxes the assumption of the efficient implementation of ${I}_{s}$, the selective phase inversion of the initial state, also known as a diffusion operator. This assumption is known to become a serious handicap in cases of physical interest. Our general search algorithm works with almost any diffusion operator ${D}_{s}$ with the only restriction of having $|s\ensuremath{\rangle}$ as one of its eigenstates. The price that we pay for using any operator is an increase in the number of oracle queries by a factor of $O(B)$, where $B$ is a characteristic of the eigenspectrum of ${D}_{s}$ and can be large in some situations. Here we show that by using a quantum Fourier transform, we can regain the optimal query complexity of Grover's algorithm without losing the freedom of using any diffusion operator for quantum searching. However, the total number of operators required by the algorithm is still $O(B)$ times more than that of Grover's algorithm. So our algorithm offers an advantage only if the oracle operator is computationally more expensive than the diffusion operator, which is true in most search problems.

DOI: 10.1103/physreva.91.052307

Related papers

Postprocessing for speeding up general quantum search algorithms

Avatar Tulsi

arXiv:1504.04787Physical Review AAug 27, 20154 citations

On the class of diffusion operators for fast quantum search

Avatar Tulsi

arXiv:1610.05661Oct 18, 20160 citations

Practical Guide to Quantum Computing: Grover's Algorithm # 4

Mikhail Pavlov

Mar 4, 20260 citations

Generalized Grover’s Algorithm for Multiple Phase Inversion States

Tim Byrnes, Gary Forster, Louis Tessler

arXiv:1801.02809Physical Review LettersFeb 7, 201824 citations

Generalization of Quantum Search Algorithm

Debasis Sadhukhan

Feb 13, 20130 citations

The Quadratic Leap: How Grover's Algorithm is Redefining Core Problems in Computer Science and Engineering

Palaganti Venkata Nagendra Sai Sandeep

Aug 2, 20252 citations

arXiv:1503.05712 · Physical Review A

Faster quantum searching with almost any diffusion operator

Avatar Tulsi

Published May 7, 2015• 10 citations

arXiv HTML PDF

Abstract

Grover's search algorithm drives a quantum system from an initial state

|s\ensuremath{\rangle}

to a desired final state

|t\ensuremath{\rangle}

by using selective phase inversions of these two states. Earlier, we studied a generalization of Grover's algorithm that relaxes the assumption of the efficient implementation of

{I}_{s}

, the selective phase inversion of the initial state, also known as a diffusion operator. This assumption is known to become a serious handicap in cases of physical interest. Our general search algorithm works with almost any diffusion operator

{D}_{s}

with the only restriction of having

|s\ensuremath{\rangle}

as one of its eigenstates. The price that we pay for using any operator is an increase in the number of oracle queries by a factor of

O(B)

, where

B

is a characteristic of the eigenspectrum of

{D}_{s}

and can be large in some situations. Here we show that by using a quantum Fourier transform, we can regain the optimal query complexity of Grover's algorithm without losing the freedom of using any diffusion operator for quantum searching. However, the total number of operators required by the algorithm is still

O(B)

times more than that of Grover's algorithm. So our algorithm offers an advantage only if the oracle operator is computationally more expensive than the diffusion operator, which is true in most search problems.

DOI: 10.1103/physreva.91.052307

Related papers

Postprocessing for speeding up general quantum search algorithms

Avatar Tulsi

A general quantum search algorithm aims to evolve a quantum system from a known source state $|s\ensuremath{\rangle}$ to an unknown target state $|t\ensuremath{\rangle}$. It uses a diffusion operator ${D}_{s}$ that has the source state as one of its eigenstates and ${I}_{t}$ , where ${I}_{\ensuremath{\psi}}$ denotes the selective phase inversion of the $|\ensuremath{\psi}\ensuremath{\rangle}$ state. It evolves $|s\ensuremath{\rangle}$ to a particular state $|w\ensuremath{\rangle}$, which we will call the $w$ state, in $O(B/\ensuremath{\alpha})$ time steps, where $\ensuremath{\alpha}$ is $|\ensuremath{\langle}t|s\ensuremath{\rangle}|$ and $B$ is a characteristic of the diffusion operator. Measuring the $w$ state gives the target state with a success probability of $O(1/{B}^{2})$ , and $O({B}^{2})$ applications of the algorithm can boost it from $O(1/{B}^{2})$ to $O(1)$ , making the total time complexity $O({B}^{3}/\ensuremath{\alpha})$. In the special case of Grover's algorithm, ${D}_{s}$ is ${I}_{s}$ and $B$ is very close to 1. A more efficient way to boost the success probability is quantum amplitude amplification provided we can efficiently implement ${I}_{w}$ . Such an efficient implementation is not known so far. In this paper, we present an efficient algorithm to approximate selective phase inversions of the unknown eigenstates of an operator using a phase estimation algorithm. This algorithm is used to efficiently approximate ${I}_{w}$ , which reduces the time complexity of general algorithm to $O(B/\ensuremath{\alpha})$. Although $O(B/\ensuremath{\alpha})$ algorithms are known to exist, our algorithm offers physical implementation advantages.A general quantum search algorithm aims to evolve a quantum system from a known source state $|s\ensuremath{\rangle}$ to an unknown target state $|t\ensuremath{\rangle}$. It uses a diffusion operator ${D}_{s}$ that has the source state as one of its eigenstates and ${I}_{t}$, where ${I}_{\ensuremath{\psi}}$ denotes the selective phase inversion of the $|\ensuremath{\psi}\ensuremath{\rangle}$ state. It evolves $|s\ensuremath{\rangle}$ to a particular state $|w\ensuremath{\rangle}$, which we will call the $w$ state, in $O(B/\ensuremath{\alpha})$ time steps, where $\ensuremath{\alpha}$ is $|\ensuremath{\langle}t|s\ensuremath{\rangle}|$ and $B$ is a characteristic of the diffusion operator. Measuring the $w$ state gives the target state with a success probability of $O(1/{B}^{2})$, and $O({B}^{2})$ applications of the algorithm can boost it from $O(1/{B}^{2})$ to $O(1)$, making the total time complexity $O({B}^{3}/\ensuremath{\alpha})$. In the special case of Grover's algorithm, ${D}_{s}$ is ${I}_{s}$ and $B$ is very close to 1. A more efficient way to boost the success probability is quantum amplitude amplification provided we can efficiently implement ${I}_{w}$. Such an efficient implementation is not known so far. In this paper, we present an efficient algorithm to approximate selective phase inversions of the unknown eigenstates of an operator using a phase estimation algorithm. This algorithm is used to efficiently approximate ${I}_{w}$, which reduces the time complexity of general algorithm to $O(B/\ensuremath{\alpha})$. Although $O(B/\ensuremath{\alpha})$ algorithms are known to exist, our algorithm offers physical implementation advantages.

arXiv:1504.04787Physical Review AAug 27, 20154 citations

On the class of diffusion operators for fast quantum search

Avatar Tulsi

Grover's quantum search algorithm evolves a quantum system from a known source state $|s\rangle$ to an unknown target state $|t\rangle$ using the selective phase inversions, $I_{s}$ and $I_{t}$ , of these two states. In one of the generalizations of Grover's algorithm, $I_{s}$ is replaced by a general diffusion operator $D_{s}$ having $|s\rangle$ as an eigenstate and $I_{t}$ is replaced by a general selective phase rotation $I_{t}^ϕ$ . A fast quantum search is possible as long as the operator $D_{s}$ and the angle $ϕ$ satisfies certain conditions. These conditions are very restrictive in nature. Specifically, suppose $|\ell\rangle$ denote the eigenstates of $D_{s}$ corresponding to the eigenphases $θ_{\ell}$ . Then the sum of the terms $|\langle \ell|t\rangle|^{2}\cot(θ_{\ell}/2)$ over all $\ell \neq s$ has to be almost equal to $\cot(ϕ/2)$ for a fast quantum search. In this paper, we show that this condition can be significantly relaxed by introducing appropriate modifications of the algorithm. This allows access to a more general class of diffusion operators for fast quantum search.Grover's quantum search algorithm evolves a quantum system from a known source state $|s\rangle$ to an unknown target state $|t\rangle$ using the selective phase inversions, $I_{s}$ and $I_{t}$, of these two states. In one of the generalizations of Grover's algorithm, $I_{s}$ is replaced by a general diffusion operator $D_{s}$ having $|s\rangle$ as an eigenstate and $I_{t}$ is replaced by a general selective phase rotation $I_{t}^ϕ$. A fast quantum search is possible as long as the operator $D_{s}$ and the angle $ϕ$ satisfies certain conditions. These conditions are very restrictive in nature. Specifically, suppose $|\ell\rangle$ denote the eigenstates of $D_{s}$ corresponding to the eigenphases $θ_{\ell}$. Then the sum of the terms $|\langle \ell|t\rangle|^{2}\cot(θ_{\ell}/2)$ over all $\ell \neq s$ has to be almost equal to $\cot(ϕ/2)$ for a fast quantum search. In this paper, we show that this condition can be significantly relaxed by introducing appropriate modifications of the algorithm. This allows access to a more general class of diffusion operators for fast quantum search.

arXiv:1610.05661Oct 18, 20160 citations

Practical Guide to Quantum Computing: Grover's Algorithm # 4

Mikhail Pavlov

Abstract This guide is based on the IBM Quantum course Phase Estimation and Factoring and focuses on Grover’s algorithm, a quantum algorithm for unstructured search problems that provides a quadratic speedup over classical approaches. In an unstructured search problem, we are given a function (often represented as an oracle) that marks one or more “correct” solutions among NNN possible candidates. A classical algorithm requires on the order of O(N)O(N)O(N) evaluations in the worst case. In contrast, Grover’s algorithm finds a solution using only O(N)O(\sqrt{N})O(N) oracle queries. This quadratic improvement implies: The quantum algorithm requires roughly the square root of the number of operations needed classically. Any classical algorithm for unstructured search must have cost at least quadratic relative to Grover’s quantum procedure. Core Idea of the Algorithm Grover’s algorithm relies on: Preparation of a uniform superposition (typically using Hadamard gates). An oracle that marks valid solutions via phase inversion. The Grover diffusion operator, which amplifies the probability amplitude of the marked states through interference. By repeatedly applying the Grover operator, the amplitude of the desired solution grows while others diminish, allowing measurement to reveal the correct answer with high probability. Practical Considerations Despite its theoretical elegance and wide applicability, the quadratic speedup of Grover’s algorithm is unlikely to yield near-term practical advantage. Modern classical hardware operates at extremely high clock rates and is vastly more mature than current quantum devices. For problem sizes accessible to today’s quantum computers, classical systems remain dominant. However, history shows that even sub-quadratic classical improvements — such as: Fast Fourier Transform (FFT) Efficient sorting algorithms like quicksort and mergesort have had transformative practical impact. The crucial distinction is that Grover’s algorithm depends on fundamentally different hardware — quantum processors — which are still in early development. Long-Term Perspective As quantum hardware advances in qubit count, coherence time, and error correction, Grover’s quadratic advantage may become practically significant for: Cryptographic key search Optimization heuristics Combinatorial search problems Amplitude amplification subroutines in broader quantum algorithms Through hands-on experimentation on IBM Q, this lesson demonstrates how theoretical amplitude amplification translates into executable quantum circuits, reinforcing the relationship between interference, probability amplitudes, and computational complexity.Abstract This guide is based on the IBM Quantum course Phase Estimation and Factoring and focuses on Grover’s algorithm, a quantum algorithm for unstructured search problems that provides a quadratic speedup over classical approaches. In an unstructured search problem, we are given a function (often represented as an oracle) that marks one or more “correct” solutions among NNN possible candidates. A classical algorithm requires on the order of O(N)O(N)O(N) evaluations in the worst case. In contrast, Grover’s algorithm finds a solution using only O(N)O(\sqrt{N})O(N) oracle queries. This quadratic improvement implies: The quantum algorithm requires roughly the square root of the number of operations needed classically. Any classical algorithm for unstructured search must have cost at least quadratic relative to Grover’s quantum procedure. Core Idea of the Algorithm Grover’s algorithm relies on: Preparation of a uniform superposition (typically using Hadamard gates). An oracle that marks valid solutions via phase inversion. The Grover diffusion operator, which amplifies the probability amplitude of the marked states through interference. By repeatedly applying the Grover operator, the amplitude of the desired solution grows while others diminish, allowing measurement to reveal the correct answer with high probability. Practical Considerations Despite its theoretical elegance and wide applicability, the quadratic speedup of Grover’s algorithm is unlikely to yield near-term practical advantage. Modern classical hardware operates at extremely high clock rates and is vastly more mature than current quantum devices. For problem sizes accessible to today’s quantum computers, classical systems remain dominant. However, history shows that even sub-quadratic classical improvements — such as: Fast Fourier Transform (FFT) Efficient sorting algorithms like quicksort and mergesort have had transformative practical impact. The crucial distinction is that Grover’s algorithm depends on fundamentally different hardware — quantum processors — which are still in early development. Long-Term Perspective As quantum hardware advances in qubit count, coherence time, and error correction, Grover’s quadratic advantage may become practically significant for: Cryptographic key search Optimization heuristics Combinatorial search problems Amplitude amplification subroutines in broader quantum algorithms Through hands-on experimentation on IBM Q, this lesson demonstrates how theoretical amplitude amplification translates into executable quantum circuits, reinforcing the relationship between interference, probability amplitudes, and computational complexity.

Mar 4, 20260 citations

Generalized Grover’s Algorithm for Multiple Phase Inversion States

Tim Byrnes, Gary Forster, Louis Tessler

Grover's algorithm is a quantum search algorithm that proceeds by repeated applications of the Grover operator and the Oracle until the state evolves to one of the target states. In the standard version of the algorithm, the Grover operator inverts the sign on only one state. Here we provide an exact solution to the problem of performing Grover's search where the Grover operator inverts the sign on N states. We show the underlying structure in terms of the eigenspectrum of the generalized Hamiltonian, and derive an appropriate initial state to perform the Grover evolution. This allows us to use the quantum phase estimation algorithm to solve the search problem in this generalized case, completely bypassing the Grover algorithm altogether. We obtain a time complexity of this case of sqrt[D/M^{α}], where D is the search space dimension, M is the number of target states, and α≈1, which is close to the optimal scaling.Grover's algorithm is a quantum search algorithm that proceeds by repeated applications of the Grover operator and the Oracle until the state evolves to one of the target states. In the standard version of the algorithm, the Grover operator inverts the sign on only one state. Here we provide an exact solution to the problem of performing Grover's search where the Grover operator inverts the sign on N states. We show the underlying structure in terms of the eigenspectrum of the generalized Hamiltonian, and derive an appropriate initial state to perform the Grover evolution. This allows us to use the quantum phase estimation algorithm to solve the search problem in this generalized case, completely bypassing the Grover algorithm altogether. We obtain a time complexity of this case of sqrt[D/M^{α}], where D is the search space dimension, M is the number of target states, and α≈1, which is close to the optimal scaling.

arXiv:1801.02809Physical Review LettersFeb 7, 201824 citations

Generalization of Quantum Search Algorithm

Debasis Sadhukhan

Grover's quantum search algorithm is an important landmark discovery in the area of quantum computing that drives a quantum computer from a known initial state(source state) to an unknown final state(target state) by using selective phase inversions of these states. The algorithm is simply a successive iteration of the Grover's search operator, which is a product of selective inversion operators of source and target states. In its original form, Grover's operator performs the selective inversions only on a unique source state and a unique target state. In the generalized version, we make our search operator to perform selective inversions on more than one source and target states. First, we find that the case of selective inversions of two source and two target states can be easily analysed somewhat similar to the case of Grover's algorithm. But unlike Grover's operator, the selective inversions of three source and three target states is not so easy to study analytically and hence we perform numerical calculations on some specific cases. We find that these generalized search operators also give us a successful quantum search as expected.Grover's quantum search algorithm is an important landmark discovery in the area of quantum computing that drives a quantum computer from a known initial state(source state) to an unknown final state(target state) by using selective phase inversions of these states. The algorithm is simply a successive iteration of the Grover's search operator, which is a product of selective inversion operators of source and target states. In its original form, Grover's operator performs the selective inversions only on a unique source state and a unique target state. In the generalized version, we make our search operator to perform selective inversions on more than one source and target states. First, we find that the case of selective inversions of two source and two target states can be easily analysed somewhat similar to the case of Grover's algorithm. But unlike Grover's operator, the selective inversions of three source and three target states is not so easy to study analytically and hence we perform numerical calculations on some specific cases. We find that these generalized search operators also give us a successful quantum search as expected.

Feb 13, 20130 citations

The Quadratic Leap: How Grover's Algorithm is Redefining Core Problems in Computer Science and Engineering

Palaganti Venkata Nagendra Sai Sandeep

Grover's algorithm stands as a cornerstone of quantum computing, offering a theoretical quadratic speedup for unstructured search problems. This allows for finding a target item in an unsorted database of N entries in approximately the square root of N steps, a significant improvement over classical algorithms which require a number of operations on the order of N in the worst case. This speedup becomes particularly compelling for very large datasets, where classical exhaustive search becomes computationally intractable. The algorithm achieves this by leveraging fundamental quantum mechanical principles, primarily superposition and interference. It operates by preparing qubits in a uniform superposition of all possible states, then iteratively applying a quantum oracle to mark the desired state by flipping its phase, followed by a diffusion operator that amplifies the probability of the marked state. Despite its theoretical promise, real-world implementation faces substantial challenges stemming from Noisy Intermediate-Scale Quantum (NISQ) hardware, which is characterized by high error rates and short coherence times. Consequently, achieving a true, scalable quantum advantage remains a future endeavor, contingent on significant hardware advancements. Nevertheless, Grover's algorithm holds significant potential across diverse fields, including enhancing brute-force attacks on cryptographic systems, accelerating solutions for optimization problems, and speeding up search processes in machine learning and unstructured database queries.Grover's algorithm stands as a cornerstone of quantum computing, offering a theoretical quadratic speedup for unstructured search problems. This allows for finding a target item in an unsorted database of N entries in approximately the square root of N steps, a significant improvement over classical algorithms which require a number of operations on the order of N in the worst case. This speedup becomes particularly compelling for very large datasets, where classical exhaustive search becomes computationally intractable. The algorithm achieves this by leveraging fundamental quantum mechanical principles, primarily superposition and interference. It operates by preparing qubits in a uniform superposition of all possible states, then iteratively applying a quantum oracle to mark the desired state by flipping its phase, followed by a diffusion operator that amplifies the probability of the marked state. Despite its theoretical promise, real-world implementation faces substantial challenges stemming from Noisy Intermediate-Scale Quantum (NISQ) hardware, which is characterized by high error rates and short coherence times. Consequently, achieving a true, scalable quantum advantage remains a future endeavor, contingent on significant hardware advancements. Nevertheless, Grover's algorithm holds significant potential across diverse fields, including enhancing brute-force attacks on cryptographic systems, accelerating solutions for optimization problems, and speeding up search processes in machine learning and unstructured database queries.

Aug 2, 20252 citations

Grover's search algorithm drives a quantum system from an initial state

|s\ensuremath{\rangle}

to a desired final state

|t\ensuremath{\rangle}

by using selective phase inversions of these two states. Earlier, we studied a generalization of Grover's algorithm that relaxes the assumption of the efficient implementation of

{I}_{s}

{D}_{s}

with the only restriction of having

|s\ensuremath{\rangle}

as one of its eigenstates. The price that we pay for using any operator is an increase in the number of oracle queries by a factor of

O(B)

, where

B

is a characteristic of the eigenspectrum of

{D}_{s}

O(B)

A general quantum search algorithm aims to evolve a quantum system from a known source state $|s\ensuremath{\rangle}$ to an unknown target state $|t\ensuremath{\rangle}$. It uses a diffusion operator

{D}_{s}

that has the source state as one of its eigenstates and

{I}_{t}

, where ${I}_{\ensuremath{\psi}}$ denotes the selective phase inversion of the $|\ensuremath{\psi}\ensuremath{\rangle}$ state. It evolves $|s\ensuremath{\rangle}$ to a particular state $|w\ensuremath{\rangle}$, which we will call the

w

state, in $O(B/\ensuremath{\alpha})$ time steps, where $\ensuremath{\alpha}$ is $|\ensuremath{\langle}t|s\ensuremath{\rangle}|$ and

B

is a characteristic of the diffusion operator. Measuring the

w

state gives the target state with a success probability of

O(1/{B}^{2})

, and

O({B}^{2})

applications of the algorithm can boost it from

O(1/{B}^{2})

O(1)

, making the total time complexity $O({B}^{3}/\ensuremath{\alpha})$. In the special case of Grover's algorithm,

{D}_{s}

{I}_{s}

and

B

is very close to 1. A more efficient way to boost the success probability is quantum amplitude amplification provided we can efficiently implement

{I}_{w}

. Such an efficient implementation is not known so far. In this paper, we present an efficient algorithm to approximate selective phase inversions of the unknown eigenstates of an operator using a phase estimation algorithm. This algorithm is used to efficiently approximate

{I}_{w}

, which reduces the time complexity of general algorithm to $O(B/\ensuremath{\alpha})$. Although $O(B/\ensuremath{\alpha})$ algorithms are known to exist, our algorithm offers physical implementation advantages.

Grover's quantum search algorithm evolves a quantum system from a known source state

|s\rangle

to an unknown target state

|t\rangle

using the selective phase inversions,

I_{s}

and

I_{t}

, of these two states. In one of the generalizations of Grover's algorithm,

I_{s}

is replaced by a general diffusion operator

D_{s}

having

|s\rangle

as an eigenstate and

I_{t}

is replaced by a general selective phase rotation

I_{t}^ϕ

. A fast quantum search is possible as long as the operator

D_{s}

and the angle

ϕ

satisfies certain conditions. These conditions are very restrictive in nature. Specifically, suppose

|\ell\rangle

denote the eigenstates of

D_{s}

corresponding to the eigenphases

θ_{\ell}

. Then the sum of the terms

|\langle \ell|t\rangle|^{2}\cot(θ_{\ell}/2)

over all

\ell \neq s

has to be almost equal to

\cot(ϕ/2)

for a fast quantum search. In this paper, we show that this condition can be significantly relaxed by introducing appropriate modifications of the algorithm. This allows access to a more general class of diffusion operators for fast quantum search.