Quantum Algorithm For Finding Research Articles

Quantum algorithms for the generalized eigenvalue problem

The generalized eigenvalue (GE) problems are of particular importance in various areas of science engineering and machine learning. We present a variational quantum algorithm for finding the desired generalized eigenvalue of the GE problem, $\mathcal{A}|\psi\rangle=\lambda\mathcal{B}|\psi\rangle$, by choosing suitable loss functions. Our approach imposes the superposition of the trial state and the obtained eigenvectors with respect to the weighting matrix $\mathcal{B}$ on the Rayleigh-quotient. Furthermore, both the values and derivatives of the loss functions can be calculated on near-term quantum devices with shallow quantum circuit. Finally, we propose a full quantum generalized eigensolver (FQGE) to calculate the minimal generalized eigenvalue with quantum gradient descent algorithm. As a demonstration of the principle, we numerically implement our algorithms to conduct a 2-qubit simulation and successfully find the generalized eigenvalues of the matrix pencil $(\mathcal{A},\,\mathcal{B})$. The numerically experimental result indicates that FQGE is robust under Gaussian noise.

Variational quantum algorithm for molecular geometry optimization

Classical algorithms for predicting the equilibrium geometry of strongly correlated molecules require expensive wave function methods that become impractical already for few-atom systems. In this work, we introduce a variational quantum algorithm for finding the most stable structure of a molecule by explicitly considering the parametric dependence of the electronic Hamiltonian on the nuclear coordinates. The equilibrium geometry of the molecule is obtained by minimizing a more general cost function that depends on both the quantum circuit and the Hamiltonian parameters, which are simultaneously optimized at each step. The algorithm is applied to find the equilibrium geometries of the ${\mathrm{H}}_{2},$ ${\mathrm{H}}_{3}{}^{+},$ ${\mathrm{BeH}}_{2},$ and ${\mathrm{H}}_{2}\mathrm{O}$ molecules. The quantum circuits used to prepare the electronic ground state for each molecule were designed using an adaptive algorithm where excitation gates in the form of Givens rotations are selected according to the norm of their gradient. All quantum simulations are performed using the PennyLane library for quantum differentiable programming. The optimized geometrical parameters for the simulated molecules show an excellent agreement with their counterparts computed using classical quantum chemistry methods.

Quantum Phase Estimation Algorithm for Finding Polynomial Roots

AbstractQuantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum algorithm for finding the roots ofnthdegree polynomials where n is any positive integer. In classical algorithm, the resources required for solving this problem increase drastically when n increases and it would be impossible to practically solve the problem when n is large. It was found that any polynomial can be rearranged into a corresponding companion matrix, whose eigenvalues are roots of the polynomial. This leads to a possibility to perform a quantum algorithm where the number of computational resources increase as a polynomial of n. In this study, we construct a quantum circuit representing the companion matrix and use eigenvalue estimation technique to find roots of polynomial.

On the quantum attacks against schemes relying on the hardness of finding a short generator of an ideal in ℚ(𝜁2𝑠 )

Abstract A family of ring-based cryptosystems, including the multilinear maps of Garg, Gentry and Halevi [Candidate multilinear maps from ideal lattices, Advances in Cryptology—EUROCRYPT 2013, Lecture Notes in Comput. Sci. 7881, Springer, Heidelberg 2013, 1–17] and the fully homomorphic encryption scheme of Smart and Vercauteren [Fully homomorphic encryption with relatively small key and ciphertext sizes, Public Key Cryptography—PKC 2010, Lecture Notes in Comput. Sci. 6056, Springer, Berlin 2010, 420–443], are based on the hardness of finding a short generator of a principal ideal (short-PIP) in a number field typically in ℚ ⁢ ( ζ 2 s ) {\mathbb{Q}(\zeta_{2^{s}})} . In this paper, we present a polynomial-time quantum algorithm for recovering a generator of a principal ideal in ℚ ⁢ ( ζ 2 s ) {\mathbb{Q}(\zeta_{2^{s}})} , and we recall how this can be used to attack the schemes relying on the short-PIP in ℚ ⁢ ( ζ 2 s ) {\mathbb{Q}(\zeta_{2^{s}})} by using the work of Cramer et al. [R. Cramer, L. Ducas, C. Peikert and O. Regev, Recovering short generators of principal ideals in cyclotomic rings, IACR Cryptology ePrint Archive 2015, https://eprint.iacr.org/2015/313], which is derived from observations of Campbell, Groves and Shepherd [SOLILOQUY, a cautionary tale]. We put this attack into perspective by reviewing earlier attempts at providing an efficient quantum algorithm for solving the PIP in ℚ ⁢ ( ζ 2 s ) {\mathbb{Q}(\zeta_{2^{s}})} . The assumption that short-PIP is hard was challenged by Campbell, Groves and Shepherd. They proposed an approach for solving short-PIP that proceeds in two steps: first they sketched a quantum algorithm for finding an arbitrary generator (not necessarily short) of the input principal ideal. Then they suggested that it is feasible to compute a short generator efficiently from the generator in step 1. Cramer et al. validated step 2 of the approach by giving a detailed analysis. In this paper, we focus on step 1, and we show that step 1 can run in quantum polynomial time if we use an algorithm for the continuous hidden subgroup problem (HSP) due to Eisenträger et al. [K. Eisenträger, S. Hallgren, A. Kitaev and F. Song, A quantum algorithm for computing the unit group of an arbitrary degree number field, Proceedings of the 2014 ACM Symposium on Theory of Computing—STOC’14, ACM, New York 2014, 293–302].

Using Bernstein\u2013Vazirani algorithm to attack block ciphers

In this paper, we study applications of Bernstein–Vazirani algorithm and present several new methods to attack block ciphers. Specifically, we first present a quantum algorithm for finding the linear structures of a function. Based on it, we propose new quantum distinguishers for the 3-round Feistel scheme and a new quantum algorithm to recover partial key of the Even–Mansour construction. Afterwards, by observing that the linear structures of a encryption function are actually high probability differentials of it, we apply our algorithm to differential analysis and impossible differential cryptanalysis respectively. We also propose a new kind of differential cryptanalysis, called quantum small probability differential cryptanalysis, based on the fact that the linear structures found by our algorithm are also the linear structure of each component function. To our knowledge, no similar method was proposed before. The efficiency and success probability of all attacks are analyzed rigorously. Since our algorithm treats the encryption function as a whole, it avoid the disadvantage of traditional differential cryptanalysis that it is difficult to extending the differential path.

Quantum algorithm for association rules mining

Association rules mining is one of the most important problems in knowledge discovery and data mining. The goal of it is to acquire consumption habits of customers by discovering the relationships between items from a transaction database that has a large number of transactions and items. The most compute intensive process for ARM is to find out the frequent 1-itemsets and 2-itemsets. In this paper, we propose a quantum algorithm for finding out the frequent 1-itemsets and 2-itemsets. In our algorithm, to mine the frequent 1-itemsets efficiently, we use the technique of amplitude amplification. To mine the frequent 2-itemsets efficiently, we propose a new tomography scheme, i.e., pure-state-based quantum state tomography. It is shown that our algorithm is potential to achieve exponential speedup in the number of transactions and polynomial speedup in the number of items over the classical algorithm.

An Efficient Exact Quantum Algorithm for the Integer Square-free Decomposition Problem

Quantum computers are known to be qualitatively more powerful than classical computers, but so far only a small number of different algorithms have been discovered that actually use this potential. It would therefore be highly desirable to develop other types of quantum algorithms that widen the range of possible applications. Here we propose an efficient and exact quantum algorithm for finding the square-free part of a large integer - a problem for which no efficient classical algorithm exists. The algorithm relies on properties of Gauss sums and uses the quantum Fourier transform. We give an explicit quantum network for the algorithm. Our algorithm introduces new concepts and methods that have not been used in quantum information processing so far and may be applicable to a wider class of problems.

Quantum algorithms for Simon's problem over nonabelian groups

Daniel Simon's 1994 discovery of an efficient quantum algorithm for finding “hidden shifts” of Z 2 n provided the first algebraic problem for which quantum computers are exponentially faster than their classical counterparts. In this article, we study the generalization of Simon's problem to arbitrary groups. Fixing a finite group G , this is the problem of recovering an involution m = ( m 1 ,…, m n ) ∈ G n from an oracle f with the property that f ( x ⋅ y ) = f ( x ) ⇔ y ∈ {1, m }. In the current parlance, this is the hidden subgroup problem (HSP) over groups of the form G n , where G is a nonabelian group of constant size, and where the hidden subgroup is either trivial or has order two. Although groups of the form G n have a simple product structure, they share important representation--theoretic properties with the symmetric groups S n , where a solution to the HSP would yield a quantum algorithm for Graph Isomorphism. In particular, solving their HSP with the so-called “standard method” requires highly entangled measurements on the tensor product of many coset states. In this article, we provide quantum algorithms with time complexity 2 O (√ n ) that recover hidden involutions m = ( m 1 ,… m n ) ∈ G n where, as in Simon's problem, each m i is either the identity or the conjugate of a known element m which satisfies κ( m ) = −κ(1) for some κ ∈ Ĝ . Our approach combines the general idea behind Kuperberg's sieve for dihedral groups with the “missing harmonic” approach of Moore and Russell. These are the first nontrivial HSP algorithms for group families that require highly entangled multiregister Fourier sampling.

A Quantum Algorithm for Finding the Modal Value

We present a quantum algorithm for finding the most often occurring (or modal) value of a data set. We thereby supplement other algorithms that can determine the mean value or similar quantities. Our algorithm requires the combined use of quantum counting and extended quantum search.

Realizations of standard quantum computational circuits by adiabatic evolution

We study the adiabatic equivalent of the standard quantum circuit of elementary logic operators. We propose a scheme for constructing time variations of the Hamiltonian. This scheme can be implemented sufficiently simply, for example, on nuclear spins controlled by radio-frequency pulses. As an illustration, we numerically simulate an adiabatic quantum algorithm for finding the permutation order for a system of five spins (qubits).

Quantum lower bounds for the collision and the element distinctness problems

Given a function f as an oracle, the collision problem is to find two distinct indexes i and j such that f ( i ) = f ( j ), under the promise that such indexes exist. Since the security of many fundamental cryptographic primitives depends on the hardness of finding collisions, our lower bounds provide evidence for the existence of cryptographic primitives that are immune to quantum cryptanalysis. We prove that any quantum algorithm for finding a collision in an r -to-one function must evaluate the function Ω(( n / r ) 1/3 ) times, where n is the size of the domain and r | n . This matches an upper bound of Brassard, Høyer, and Tapp. No lower bound better than constant was previously known. Our result also implies a quantum lower bound of Ω( n 2/3 ) queries for the element distinctness problem, which is to determine whether n integers are all distinct. The best previous lower bound was Ω(√ n ) queries.