scholarly journals A Quantum Circuit Design for Grover’s Algorithm

2002 ◽  
Vol 57 (8) ◽  
pp. 701-708 ◽  
Author(s):  
Zijian Diao ◽  
M. Suhail Zubairy ◽  
Goong Chen
Keyword(s):  
Circuit Design ◽  
Cavity Qed ◽  
Unitary Transformation ◽  
Quantum Circuit ◽  
The State ◽  
Quantum Phase ◽  
Grover’S Algorithm ◽  
Mathematical Proofs ◽  
Grover's Algorithm

We present a circuit design realizing Grover’s algorithm based on 1-bit unitary gates and 2-bit quantum phase gates implementable with cavity QED techniques. In the first step, we express the circuit block which performs a key unitary transformation that flips only the sign of the state |11 · · · 11〉 using 1-bit and 2-bit gates. The Grover’s iteration operator can then be constructed using this key unitary transformation twice, plus other operations involving only 1-bit unitary gates on each qubit. Mathematical proofs are given to justify that the cricuiting satisfies the desired operator properties.

scholarly journals A framework for reducing the overhead of the quantum oracle for use with Grover’s algorithm with applications to cryptanalysis of SIKE

2020 ◽  
Vol 15 (1) ◽  
pp. 143-156
Author(s):  
Jean-François Biasse ◽  
Benjamin Pring
Keyword(s):  
Search Algorithm ◽  
Circuit Complexity ◽  
Quantum Circuit ◽  
Grover’S Algorithm ◽  
Search Problems ◽  
Trade Offs ◽  
Single Target ◽  
Grover's Algorithm ◽  
Quantum Oracle ◽  
The Cost

AbstractIn this paper we provide a framework for applying classical search and preprocessing to quantum oracles for use with Grover’s quantum search algorithm in order to lower the quantum circuit-complexity of Grover’s algorithm for single-target search problems. This has the effect (for certain problems) of reducing a portion of the polynomial overhead contributed by the implementation cost of quantum oracles and can be used to provide either strict improvements or advantageous trade-offs in circuit-complexity. Our results indicate that it is possible for quantum oracles for certain single-target preimage search problems to reduce the quantum circuit-size from $O\left(2^{n/2}\cdot mC\right)$ (where C originates from the cost of implementing the quantum oracle) to $O(2^{n/2} \cdot m\sqrt{C})$ without the use of quantum ram, whilst also slightly reducing the number of required qubits.This framework captures a previous optimisation of Grover’s algorithm using preprocessing [21] applied to cryptanalysis, providing new asymptotic analysis. We additionally provide insights and asymptotic improvements on recent cryptanalysis [16] of SIKE [14] via Grover’s algorithm, demonstrating that the speedup applies to this attack and impacting upon quantum security estimates [16] incorporated into the SIKE specification [14].

Transformation of quantum states using uniformly controlled rotations

2005 ◽  
Vol 5 (6) ◽  
pp. 467-473
Author(s):  
M. Mottonen ◽  
J. J. Vartiainen ◽  
V. Bergholm ◽  
M. M. Salomaa
Keyword(s):  
Pure State ◽  
Unitary Transformation ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Circuit ◽  
The State ◽  
Quantum States ◽  
State Transformation ◽  
Quantum Register ◽  
Analytic Expression

We consider a unitary transformation which maps any given pure state of an $n$-qubit quantum register into another one. This transformation has applications in the initialization of a quantum computer, and also in some quantum algorithms. Employing uniformly controlled rotations, we present a quantum circuit of $2^{n+2}-4n-4$ CNOT gates and $2^{n+2}-5$ one-qubit elementary rotations that effects the state transformation. The complexity of the circuit is noticeably lower than the previously published results. Moreover, we present an analytic expression for the rotation angles needed for the transformation.

scholarly journals A Quantum interpretation of separating conjunction for local reasoning of Quantum programs based on separation logic

2022 ◽  
Vol 6 (POPL) ◽  
pp. 1-27
Author(s):  
Xuan-Bach Le ◽  
Shang-Wei Lin ◽  
Jun Sun ◽  
David Sanan
Keyword(s):  
State Space ◽  
The State ◽  
Quantum States ◽  
Separation Logic ◽  
Grover’S Algorithm ◽  
State Space Explosion ◽  
Grover's Algorithm ◽  
Local Reasoning ◽  
Quantum Interpretation ◽  
Exponential Size

It is well-known that quantum programs are not only complicated to design but also challenging to verify because the quantum states can have exponential size and require sophisticated mathematics to encode and manipulate. To tackle the state-space explosion problem for quantum reasoning, we propose a Hoare-style inference framework that supports local reasoning for quantum programs. By providing a quantum interpretation of the separating conjunction, we are able to infuse separation logic into our framework and apply local reasoning using a quantum frame rule that is similar to the classical frame rule. For evaluation, we apply our framework to verify various quantum programs including Deutsch–Jozsa’s algorithm and Grover's algorithm.

scholarly journals Understanding mathematics of Grover’s algorithm

2021 ◽  
Vol 20 (5) ◽  
Author(s):  
Paweł J. Szabłowski
Keyword(s):  
Phase Change ◽  
Unitary Operator ◽  
Mathematical Structure ◽  
Complex Numbers ◽  
Unitary Operators ◽  
Grover’S Algorithm ◽  
Great Probability ◽  
Grover's Algorithm ◽  
One Step ◽  
The One

AbstractWe analyze the mathematical structure of the classical Grover’s algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one ‘chosen’ element (sometimes called a ‘solution’) of the dataset, but a set of m such ‘chosen’ elements (out of $$n>m)$$ n > m ) . Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that ‘marks,’ by a suitable phase change $$\varphi $$ φ , all these ‘chosen’ elements. In the first part of the paper, we construct a unique unitary operator that selects all ‘chosen’ elements in one step. The constructed operator is uniquely defined by the numbers $$\varphi $$ φ and $$\alpha $$ α which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on $$\alpha $$ α . In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, ‘convenient’ phase change $$\varphi ,$$ φ , and by sequentially applying the so-constructed operator, we find the number of steps to find these ‘chosen’ elements with great probability. We apply this knowledge to study the generalizations of Grover’s algorithm ($$m=1,\phi =\pi $$ m = 1 , ϕ = π ), which are of the form, the found previously, unitary operators.

Global Optimization With Quantum Walk Enhanced Grover Search

2014 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Global Optimization ◽  
Optimization Problems ◽  
Early Stage ◽  
Hybrid Approach ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Tunneling Effect ◽  
Grover’S Algorithm ◽  
Grover's Algorithm ◽  
Grover Search

One of the significant breakthroughs in quantum computation is Grover’s algorithm for unsorted database search. Recently, the applications of Grover’s algorithm to solve global optimization problems have been demonstrated, where unknown optimum solutions are found by iteratively improving the threshold value for the selective phase shift operator in Grover rotation. In this paper, a hybrid approach that combines continuous-time quantum walks with Grover search is proposed. By taking advantage of quantum tunneling effect, local barriers are overcome and better threshold values can be found at the early stage of search process. The new algorithm based on the formalism is demonstrated with benchmark examples of global optimization. The results between the new algorithm and the Grover search method are also compared.

Cavity-QED-based quantum phase gates for a single longitudinal mode

2005 ◽  
Author(s):  
R. Garcia-Maraver ◽  
R. Corbalan ◽  
J. Mompart ◽  
K. Eckert ◽  
S. Rebic ◽  
...  
Keyword(s):  
Cavity Qed ◽  
Longitudinal Mode ◽  
Single Longitudinal Mode ◽  
Quantum Phase
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