The standard quantum search lacks a feature, enjoyed by many classical algorithms, of having a fixed point, i.e. monotonic convergence towards the solution. Recently a fixed point quantum search algorithm has been discovered, referred to as the Phase- $π/3$ search algorithm, which gets around this limitation. While searching a database for a target state, this algorithm reduces the error probability from $ε$ to $ε^{2q+1}$ using $q$ oracle queries, which has since been proved to be asymptotically optimal. A different algorithm is presented here, which has the same worst-case behavior as the Phase- $π/3$ search algorithm but much better average-case behavior. Furthermore the new algorithm gives $ε^{2q+1}$ convergence for all integral $q$ , whereas the Phase- $π/3$ search algorithm requires $q$ to be $(3^{n}-1)/2$ with $n$ a positive integer. In the new algorithm, the operations are controlled by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations applied to these ancillas. It is an example of how measurement can allow us to bypass some restrictions imposed by unitarity on quantum computing.The standard quantum search lacks a feature, enjoyed by many classical algorithms, of having a fixed point, i.e. monotonic convergence towards the solution. Recently a fixed point quantum search algorithm has been discovered, referred to as the Phase-$π/3$ search algorithm, which gets around this limitation. While searching a database for a target state, this algorithm reduces the error probability from $ε$ to $ε^{2q+1}$ using $q$ oracle queries, which has since been proved to be asymptotically optimal. A different algorithm is presented here, which has the same worst-case behavior as the Phase-$π/3$ search algorithm but much better average-case behavior. Furthermore the new algorithm gives $ε^{2q+1}$ convergence for all integral $q$, whereas the Phase-$π/3$ search algorithm requires $q$ to be $(3^{n}-1)/2$ with $n$ a positive integer. In the new algorithm, the operations are controlled by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations applied to these ancillas. It is an example of how measurement can allow us to bypass some restrictions imposed by unitarity on quantum computing.

arXiv:quant-ph/0505007May 2, 20050 citations

A new algorithm for directed quantum search

Tathagat Tulsi, Lov K. Grover, Apoorva D. Patel

Quantum searching requires precise knowledge of problem parameters (such as the fraction of target states) for efficient operation. Recently an algorithm has been discovered, referred to as the Phase- $\\pi/3$ search algorithm, which gets around this limitation. This algorithm can search a database with the fraction of target states equal to $1-\\epsilon$ so that in $q$ queries it produces a probability of error equal to $\\epsilon^{2q+1}$ which has since been proved to be optimal. This paper gives a different algorithm which has the same worst-case behavior as the Phase- $\\pi/3$ search algorithm but much better average-case behavior. Furthermore the new algorithm gives $\\epsilon^{2q+1}$ convergence for all integral $q$ , the Phase- $\\pi/3$ search algorithm, requires $q$ to be $(3^{n}-1)/2$ , with $n$ a positive integer. In the new algorithm, the operations are controlled in a special way by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations.Quantum searching requires precise knowledge of problem parameters (such as the fraction of target states) for efficient operation. Recently an algorithm has been discovered, referred to as the Phase-$\\pi/3$ search algorithm, which gets around this limitation. This algorithm can search a database with the fraction of target states equal to $1-\\epsilon$ so that in $q$ queries it produces a probability of error equal to $\\epsilon^{2q+1}$ which has since been proved to be optimal. This paper gives a different algorithm which has the same worst-case behavior as the Phase-$\\pi/3$ search algorithm but much better average-case behavior. Furthermore the new algorithm gives $\\epsilon^{2q+1}$ convergence for all integral $q$, the Phase-$\\pi/3$ search algorithm, requires $q$ to be $(3^{n}-1)/2$, with $n$ a positive integer. In the new algorithm, the operations are controlled in a special way by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations.

May 2, 20055 citations

Quantum Algorithms with Fixed Points: The Case of Database Search

Lov K. Grover, Apoorva D. Patel, Tathagat Tulsi

The standard quantum search algorithm lacks a feature, enjoyed by many classical algorithms, of having a fixed-point, i.e. a monotonic convergence towards the solution. Here we present two variations of the quantum search algorithm, which get around this limitation. The first replaces selective inversions in the algorithm by selective phase shifts of $\fracπ{3}$ . The second controls the selective inversion operations using two ancilla qubits, and irreversible measurement operations on the ancilla qubits drive the starting state towards the target state. Using $q$ oracle queries, these variations reduce the probability of finding a non-target state from $ε$ to $ε^{2q+1}$ , which is asymptotically optimal. Similar ideas can lead to robust quantum algorithms, and provide conceptually new schemes for error correction.The standard quantum search algorithm lacks a feature, enjoyed by many classical algorithms, of having a fixed-point, i.e. a monotonic convergence towards the solution. Here we present two variations of the quantum search algorithm, which get around this limitation. The first replaces selective inversions in the algorithm by selective phase shifts of $\fracπ{3}$. The second controls the selective inversion operations using two ancilla qubits, and irreversible measurement operations on the ancilla qubits drive the starting state towards the target state. Using $q$ oracle queries, these variations reduce the probability of finding a non-target state from $ε$ to $ε^{2q+1}$, which is asymptotically optimal. Similar ideas can lead to robust quantum algorithms, and provide conceptually new schemes for error correction.

arXiv:quant-ph/0603132Mar 15, 20063 citations

Ancilla-mediated fixed-point quantum search using Grover iterations

Yash Prabhat, Snigdha Thakur, Ankur Raina

Grover's quantum search algorithm provides a fundamental quadratic speedup for unstructured datasets, reducing query complexity from $\mathcal{O}(N)$ to $\mathcal{O}(\sqrt{N})$ . However, the algorithm's reliance on precise iteration counts leads to the ``soufflé problem,'' where over-rotation results in a sharp decline in success probability. This limitation is particularly restrictive when the number of solution states, $M$ , is unknown. In this work, we present an ancilla-mediated fixed-point quantum search algorithm that achieves robust convergence by mapping the solution amplitude to a dedicated ancilla qubit. Unlike existing phase-matching fixed-point methods, our approach utilizes Grover's real-plane reflections, thereby maintaining the intuitive geometric architecture of the original algorithm. We demonstrate that this method achieves a success probability of at least $92.6\%$ with a query complexity of approximately $\mathcal{O}(\sqrt{N/M})$ , effectively bridging the gap between standard amplitude amplification and robust fixed-point convergence.Grover's quantum search algorithm provides a fundamental quadratic speedup for unstructured datasets, reducing query complexity from $\mathcal{O}(N)$ to $\mathcal{O}(\sqrt{N})$. However, the algorithm's reliance on precise iteration counts leads to the ``soufflé problem,'' where over-rotation results in a sharp decline in success probability. This limitation is particularly restrictive when the number of solution states, $M$, is unknown. In this work, we present an ancilla-mediated fixed-point quantum search algorithm that achieves robust convergence by mapping the solution amplitude to a dedicated ancilla qubit. Unlike existing phase-matching fixed-point methods, our approach utilizes Grover's real-plane reflections, thereby maintaining the intuitive geometric architecture of the original algorithm. We demonstrate that this method achieves a success probability of at least $92.6\%$ with a query complexity of approximately $\mathcal{O}(\sqrt{N/M})$, effectively bridging the gap between standard amplitude amplification and robust fixed-point convergence.

quant-pharXiv:2608.29839Sep 1, 20260 citations

Fixed-Point Quantum Search

Lov K. Grover

The quantum search algorithm consists of an iterative sequence of selective inversions and diffusion type operations, as a result of which it is able to find a state with desired properties (target state) in an unsorted database of size N in only sqrt[N] queries. This is achieved by designing the iterative transformations in a way that each iteration results in a small rotation of the state vector in a two-dimensional Hilbert space that includes the target state; if we choose the right number of iterative steps, we will stop just at the target state. This Letter shows that by replacing the selective inversions by selective phase shifts of pi/3, the algorithm preferentially converges to the target state irrespective of the step size or number of iterations. This feature leads to robust search algorithms and also to new schemes for quantum control and error correction.The quantum search algorithm consists of an iterative sequence of selective inversions and diffusion type operations, as a result of which it is able to find a state with desired properties (target state) in an unsorted database of size N in only sqrt[N] queries. This is achieved by designing the iterative transformations in a way that each iteration results in a small rotation of the state vector in a two-dimensional Hilbert space that includes the target state; if we choose the right number of iterative steps, we will stop just at the target state. This Letter shows that by replacing the selective inversions by selective phase shifts of pi/3, the algorithm preferentially converges to the target state irrespective of the step size or number of iterations. This feature leads to robust search algorithms and also to new schemes for quantum control and error correction.

arXiv:quant-ph/0503205Physical Review LettersOct 3, 2005209 citations

PHASE MATCHING IN FIXED-POINT QUANTUM SEARCH ALGORITHM

Hong‐Fu Wang, Shou Zhang, Yongfang Zhao

We investigate the phase matching problem in the fixed-point quantum search algorithm proposed by Grover [Phys. Rev. Lett.95 (2005) 150501]. We show that the optimal phase shift is π/3.61, which replaces the π/3 phase shift in fixed-point quantum search algorithm. The π/3.61-phase algorithm can be achieved in [Formula: see text] with the success probability of at least 94.11%, which offsets disadvantage that the success probability of getting correct results usually decreases with the increase of marked items when original Grover quantum search algorithm is applied to search an unordered database. In the meantime, this work also indicates that Grover quantum search algorithm is considerably robust to certain kinds of perturbations and is robust against modest noise in the initialization procedure.We investigate the phase matching problem in the fixed-point quantum search algorithm proposed by Grover [Phys. Rev. Lett.95 (2005) 150501]. We show that the optimal phase shift is π/3.61, which replaces the π/3 phase shift in fixed-point quantum search algorithm. The π/3.61-phase algorithm can be achieved in [Formula: see text] with the success probability of at least 94.11%, which offsets disadvantage that the success probability of getting correct results usually decreases with the increase of marked items when original Grover quantum search algorithm is applied to search an unordered database. In the meantime, this work also indicates that Grover quantum search algorithm is considerably robust to certain kinds of perturbations and is robust against modest noise in the initialization procedure.

International Journal of Quantum InformationSep 1, 20092 citations

π/3

search algorithm, which gets around this limitation. While searching a database for a target state, this algorithm reduces the error probability from

ε

ε^{2q+1}

using

q

oracle queries, which has since been proved to be asymptotically optimal. A different algorithm is presented here, which has the same worst-case behavior as the Phase-

π/3

search algorithm but much better average-case behavior. Furthermore the new algorithm gives

ε^{2q+1}

convergence for all integral

q

, whereas the Phase-

π/3

search algorithm requires

q

to be

(3^{n}-1)/2

with

n

a positive integer. In the new algorithm, the operations are controlled by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations applied to these ancillas. It is an example of how measurement can allow us to bypass some restrictions imposed by unitarity on quantum computing.

\\pi/3

search algorithm, which gets around this limitation. This algorithm can search a database with the fraction of target states equal to

1-\\epsilon

so that in

q

queries it produces a probability of error equal to

\\epsilon^{2q+1}

which has since been proved to be optimal. This paper gives a different algorithm which has the same worst-case behavior as the Phase-

\\pi/3

search algorithm but much better average-case behavior. Furthermore the new algorithm gives

\\epsilon^{2q+1}

convergence for all integral

q

, the Phase-

\\pi/3

search algorithm, requires

q

to be

(3^{n}-1)/2

, with

n

a positive integer. In the new algorithm, the operations are controlled in a special way by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations.

\fracπ{3}

. The second controls the selective inversion operations using two ancilla qubits, and irreversible measurement operations on the ancilla qubits drive the starting state towards the target state. Using

q

oracle queries, these variations reduce the probability of finding a non-target state from

ε

ε^{2q+1}

, which is asymptotically optimal. Similar ideas can lead to robust quantum algorithms, and provide conceptually new schemes for error correction.

Grover's quantum search algorithm provides a fundamental quadratic speedup for unstructured datasets, reducing query complexity from

\mathcal{O}(N)

\mathcal{O}(\sqrt{N})

. However, the algorithm's reliance on precise iteration counts leads to the ``soufflé problem,'' where over-rotation results in a sharp decline in success probability. This limitation is particularly restrictive when the number of solution states,

M

, is unknown. In this work, we present an ancilla-mediated fixed-point quantum search algorithm that achieves robust convergence by mapping the solution amplitude to a dedicated ancilla qubit. Unlike existing phase-matching fixed-point methods, our approach utilizes Grover's real-plane reflections, thereby maintaining the intuitive geometric architecture of the original algorithm. We demonstrate that this method achieves a success probability of at least

92.6\%

with a query complexity of approximately

\mathcal{O}(\sqrt{N/M})

, effectively bridging the gap between standard amplitude amplification and robust fixed-point convergence.