Quantum Search Algorithm Research Articles

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

Abstract In 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 2 n / 2 ⋅ m C $O\left(2^{n/2}\cdot mC\right)$ (where C originates from the cost of implementing the quantum oracle) to O ( 2 n / 2 ⋅ m C ) $O(2^{n/2} \cdot m\sqrt{C})$ 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].

Playing Pool with |ψ⟩: from Bouncing Billiards to Quantum Search

In ``Playing Pool with π'' \cite{Galperin}, Galperin invented an extraordinary method to learn the digits of π by counting the collisions of billiard balls. Here I demonstrate an exact isomorphism between Galperin's bouncing billiards and Grover's algorithm for quantum search. This provides an illuminating way to visualize Grover's algorithm.

Quantum search with an ensemble of single-query search processes

Recently we introduced an additive composition formulation of an iterative quantum-search algorithm of Grover type. In this paper, an ensemble-processing scheme suggested by the additive composition is presented. The ensemble-processing consists of single-query search processes and yields the quantum search with certainty. Furthermore, we propose a possible quantum circuit for the ensemble processing.

Implementation of Grover’s Search Algorithm in the QED Circuit for Two Superconducting Qubits

In this work, we discuss the realization of Grover’s quantum search algorithm using a set of given gates which can be implemented in the QED circuit for two superconducting qubits of transmon type. In this realization, we employ the X-rotation gate, instead of Hadamard transformation, and we implement the phase oracle. We show that a probability of success greater than $\frac {1}{2}$ can be reached. We propose also an effective way to implement the two-qubit Grover’s search algorithm based on qubit-qubit interaction via a circuit QED for two different situations. In this sense, we use two transmon qubits that are coupled capacitively to a resonator driven by a strong microwave field. The capacitive coupling in dispersive regime gives a X-rotation gate for two qubits during an appropriate interaction time. This given gate allows us to generate the states of Bell type. The obtained numerical results show that our proposal is feasible with a high fidelity.

Optimality of spatial search via continuous-time quantum walks

One of the most important algorithmic applications of quantum walks is to solve spatial search problems. A widely used quantum algorithm for this problem, introduced by Childs and Goldstone [Phys. Rev. A 70, 022314 (2004)], finds a marked node on a graph of $n$ nodes via a continuous-time quantum walk. This algorithm is said to be optimal if it can find any of the nodes in $O(\sqrt{n})$ time. However, given a graph, no general conditions for the optimality of the algorithm are known and previous works demonstrating optimal quantum search for certain graphs required an instance-specific analysis. In fact, the demonstration of necessary and sufficient conditions a graph must fulfill for quantum search to be optimal has been a long-standing open problem. In this work, we make significant progress towards solving this problem. We derive general expressions, depending on the spectral properties of the Hamiltonian driving the walk, that predict the performance of this quantum search algorithm provided certain spectral conditions are fulfilled. Our predictions are valid, for example, for (normalized) Hamiltonians whose spectral gap is considerably larger than $n^{-1/2}$. This allows us to derive necessary and sufficient conditions for optimal quantum search in this regime, as well as provide new examples of graphs where quantum search is sub-optimal. In addition, by extending this analysis, we are also able to show the optimality of quantum search for certain graphs with very small spectral gaps, such as graphs that can be efficiently partitioned into clusters. Our results imply that, to the best of our knowledge, all prior results analytically demonstrating the optimality of this algorithm for specific graphs can be recovered from our general results.

The Principle and Prospect of Quantum Communication

With the progress of science and technology, as well as the development of the times, people begin to have higher requirements on the security and efficiency of communication, so quantum communication has attracted much more attention. Quantum communication has lots of advantages such as unconditional security, high transmission efficiency, strong anti-interference ability and good concealment performance. In this paper, the author analyzes the advantages and the present situation of quantum communication. This paper briefly introduces the model of quantum communication network, the quantum computing, data decoding based on quantum computation and the quantum search algorithm concepts, which are described in simple terms. Two quantum secure direct communication protocols are also introduced and analyzed, and the prospects of quantum communication are presented.

Optimal quantum search on truncated simplex lattices

First-order truncated simplex lattices have been used as interesting data structures to explore the properties of quantum search such as the effect of connectivity on search speed [D. A. Meyer and T. G. Wong, Phys. Rev. Lett. 114, 110503 (2015)]. In this paper, we further discuss quantum search algorithms for truncated simplex lattices. We first propose a multistage quantum algorithm for an $r\mathrm{th}$-order truncated simplex lattice with $N$ vertices based on the numerical calculation up to the fifth-order lattice, which requires an $(r+1)$-stage search process and consumes a run time $\mathrm{\ensuremath{\Theta}}({N}^{(2r+1)/(2r+2)})$ in general. Furthermore, with edge weights suitably adjusted (which increases the connectivity according to certain definitions), we merge the multistage search process into a single stage and achieve a fast quantum search algorithm with an optimal run time $\mathrm{\ensuremath{\Theta}}(\sqrt{N})$ for first-order truncated simplex lattices, which we conjecture is generally true for any-order lattice. Under small noise of the lattice structure, both our multistage and our single-stage search algorithms are quite robust.

Web-app realization of Shor’s quantum factoring algorithm and Grover’s quantum search algorithm

Quantum algorithms are well-known for their quadratic if not exponential speedup over their classical counterparts. The two widely-known quantum algorithms are Shor’s quantum factoring algorithm and Grover’s quantum search algorithm. Shor’s quantum factoring algorithm could perform integer factorization in O(logN). Grover’s quantum search algorithm could solve the unsorted search problem in O(√N). However, both algorithms are introduced as theoretical concepts in the original papers due to the limitations of quantum technology at that time. In this paper, an improved way is presented to realize the two algorithms into a web application using state-of-the-art quantum technology. The web-app is designed and built considering the uses of a quantum simulator and libraries provided by ProjectQ and Rigetti Forest. The result shows that both algorithms are realizable into web-applications.

Modifying quantum Grover’s algorithm for dynamic multi-pattern search on reconfigurable hardware

Grover’s quantum search algorithm is a highly studied quantum algorithm that has potential applications in unstructured data search, achieving quadratic speedup over existing classical search algorithms. In this work, we propose a modified quantum circuit for multi-pattern quantum Grover’s search algorithm. Our proposed modification simplifies the conventional Grover’s algorithm circuit and makes it capable of processing dynamically changing input patterns. The proposed techniques are demonstrated using reconfigurable hardware architectures that are designed for cost-effective, scalable, high-precision and high-throughput emulation of quantum algorithms. We experimentally evaluate the modified algorithm using Field Programmable Gate Array (FPGA) hardware and provide analysis of experimental results in terms of hardware resource utilization and emulation time. Our results demonstrate successful emulation of multi-pattern Grover’s algorithm using up to 22 quantum bits on a single FPGA, which is the highest and most efficient among existing work. Hardware implementations were performed for up to 32 qubits, and emulation time results of up to 32 qubits were projected using a performance estimation model.

Ironwood meta key agreement and authentication protocol

<p style='text-indent:20px;'>Number theoretic public-key solutions, currently used in many applications worldwide, will be subject to various quantum attacks, making them less attractive for longer-term use. Certain group theoretic constructs are now showing promise in providing quantum-resistant cryptographic primitives, and may provide suitable alternatives for those looking to address known quantum attacks. In this paper, we introduce a new protocol called a <i>Meta Key Agreement and Authentication Protocol</i> (MKAAP) that has some characteristics of a public-key solution and some of a shared-key solution. Specifically, it has the deployment benefits of a public-key system, allowing two entities that have never met before to authenticate without requiring real-time access to a third-party, but does require secure provisioning of key material from a trusted key distribution system (similar to a symmetric system) prior to deployment. We then describe a specific MKAAP instance, the Ironwood MKAAP, discuss its security, and show how it resists certain quantum attacks such as Shor's algorithm or Grover's quantum search algorithm. We also show Ironwood implemented on several "internet of things" (IoT devices), measure its performance, and show how it performs significantly better than ECC using fewer device resources.

Depth optimization of quantum search algorithms beyond Grover's algorithm

Grover's quantum search algorithm provides a quadratic speedup over the classical one. The computational complexity is based on the number of queries to the oracle. However, depth is a more modern metric for noisy intermediate-scale quantum computers. We propose a new depth optimization method for quantum search algorithms. We show that Grover's algorithm is not optimal in depth. We propose a quantum search algorithm, which can be divided into several stages. Each stage has a new initialization, which is a rescaling of the database. This decreases errors. The multistage design is natural for parallel running of the quantum search algorithm.

On quantum methods for machine learning problems part I: Quantum tools

This is a review of quantum methods for machine learning problems that consists of two parts. The first part, “quantum tools”, presents the fundamentals of qubits, quantum registers, and quantum states, introduces important quantum tools based on known quantum search algorithms and SWAP-test, and discusses the basic quantum procedures used for quantum search methods. The second part, “quantum classification algorithms”, introduces several classification problems that can be accelerated by using quantum subroutines and discusses the quantum methods used for classification.

On quantum methods for machine learning problems part II: Quantum classification algorithms

This is a review of quantum methods for machine learning problems that consists of two parts. The first part, “quantum tools”, presented some of the fundamentals and introduced several quantum tools based on known quantum search algorithms. This second part of the review presents several classification problems in machine learning that can be accelerated with quantum subroutines. We have chosen supervised learning tasks as typical classification problems to illustrate the use of quantum methods for classification.

Topological order on the Bloch sphere

A Bloch sphere is the geometrical representation of an arbitrary two-dimensional Hilbert space. Possible classes of entanglement and separability for the pure and mixed states on the Bloch sphere were suggested by [Boyer et al 2017 PRA 95 032 308]. Here we construct a Bloch sphere for the Hilbert space spanned by one of the ground states of Kitaev’s toric code model and one of its closest product states. We prove that this sphere contains only one separable state, thus belonging to the fourth class suggested by the said paper. We furthermore study the topological order of the pure states on its surface and conclude that, according to conventional definitions, only one state (the toric code ground state) seems to present non-trivial topological order. We conjecture that most of the states on this Bloch sphere are neither ‘trivial’ states (namely, they cannot be generated from a product state using a trivial circuit) nor topologically ordered. In addition, we show that the whole setting can be understood in terms of Grover rotations with gauge symmetry, akin to the quantum search algorithm.

Nonergodic Delocalized States for Efficient Population Transfer within a Narrow Band of the Energy Landscape

We analyze the role of coherent tunneling that gives rise to bands of delocalized quantum states providing a coherent pathway for population transfer (PT) between computational states with similar energies. Given an energy function ${\cal E}(z)$ of a binary optimization problem and a bit-string $z_i$ with atypically low energy, our goal is to find other bit-strings with energies within a narrow window around ${\cal E}(z_i)$. We study PT due to quantum evolution under a transverse field $B_\perp$ of an n-qubit system that encodes ${\cal E}(z)$. We focus on a simple yet nontrivial model: $M$ randomly chosen "marked" bit-strings ($2^n \gg M$) are assigned energies in the interval ${\cal E}(z)\in[-n -W/2, n + W/2]$ with $W << B_\perp$, while the rest of the states are assigned energy $0$. The PT starts at a marked state $z_i$ and ends up in a superposition of $\sim \Omega$ marked states inside the PT window. The scaling of a typical runtime for PT with $n$ and $\Omega$ is the same as in the multi-target Grover's algorithm, except for a factor that is equal to $\exp(n \,B_{\perp}^{-2}/2)$ for $n \gg B_{\perp}^{2} \gg 1$. Unlike the Hamiltonians used in analog quantum search algorithms, the model we consider is non-integrable, and the transverse field delocalizes the marked states. PT protocol is not sensitive to the value of B and may be initialized at a marked state. We develop microscopic theory of PT. Under certain conditions, the band of the system eigenstates splits into mini-bands of non-ergodic delocalized states, whose width obeys a heavy-tailed distribution directly related to that of PT runtimes. We find analytical form of this distribution by solving nonlinear cavity equations for the random matrix ensemble. We argue that our approach can be applied to study the PT protocol in other transverse field spin glass models, with a potential quantum advantage over classical algorithms.

Efficient design of quaternary quantum comparator with only a single ancillary input

Circuit design with no energy dissipation is possible in the reversible logic. Also, using the quantum multiple-valued logic in the design of circuits due to the reduction of circuit width leads to the smaller chip area and lower power consumption. Quaternary reversible logic potentially has several advantages in comparison to the equivalent ternary and binary quantum technologies. Hence, the quaternary quantum technology is a promising choice for the future digital system design. Here, the authors proposed an efficient design of 1-qudit and n -qudit quantum comparators. The comparator module is widely used in search conditions, such as Grover's quantum search algorithm in quantum computing. The proposed circuits were better than their existing counterparts in terms of quantum cost, the number of constant inputs, the number of garbage outputs, and hardware complexity. To design these circuits, they have used 1-qudit gates and 2-qudit Muthukrishnan-Stroud gates. The efficiency of their proposed circuits was reported.

A Proof for the Birch Swinnerton-Dyer Conjecture

The Birch Swinnerton-Dyer Conjecture states that the number of rational points on an elliptic curve are infinite. This paper develops the logical formalism for a reinforcement learning system embedded with a quantum search algorithm. The formalism proves the rational points on high dimensional elliptical curves are infinite.

Algorithm for solving unconstrained unitary quantum brachistochrone problems

We introduce an iterative method to search for time-optimal Hamiltonians that drive a quantum system between two arbitrary, and in general mixed, quantum states. The method is based on the idea of progressively improving the efficiency of an initial, randomly chosen, Hamiltonian, by reducing its components that do not actively contribute to driving the system. We show that our method converges rapidly even for large dimensional systems, and that its solutions saturate any attainable bound for the minimal time of evolution. We provide a rigorous geometric interpretation of the iterative method by exploiting an isomorphism between geometric phases acquired by the system along a path and the Hamiltonian that generates it, and discuss resulting similarities with Grover's quantum search algorithm. Our method is directly applicable as a powerful tool for state preparation and gate design problems.

Transition probabilities in generalized quantum search Hamiltonian evolutions

A relevant problem in quantum computing concerns how fast a source state can be driven into a target state according to Schrödinger’s quantum mechanical evolution specified by a suitable driving Hamiltonian. In this paper, we study in detail the computational aspects necessary to calculate the transition probability from a source state to a target state in a continuous time quantum search problem defined by a multiparameter generalized time-independent Hamiltonian. In particular, quantifying the performance of a quantum search in terms of speed (minimum search time) and fidelity (maximum success probability), we consider a variety of special cases that emerge from the generalized Hamiltonian. In the context of optimal quantum search, we find it is possible to outperform, in terms of minimum search time, the well-known Farhi–Gutmann analog quantum search algorithm. In the context of nearly optimal quantum search, instead, we show it is possible to identify sub-optimal search algorithms capable of outperforming optimal search algorithms if only a sufficiently high success probability is sought. Finally, we briefly discuss the relevance of a tradeoff between speed and fidelity with emphasis on issues of both theoretical and practical importance to quantum information processing.

Lackadaisical quantum walk for spatial search

Lackadaisical quantum walk (LQW) has been an efficient technique in searching for a target state in a database which is distributed in a two-dimensional lattice. We numerically study the quantum search algorithm based on the lackadaisical quantum walk in one and two dimensions. It is observed that specific values of the self-loop weight at each vertex of the graph is responsible for such a speedup of the algorithm. Searching for a target state in one-dimensional lattice with periodic boundary conditions is possible using lackadaisical quantum walk, which can find a target state with [Formula: see text] success probability after [Formula: see text] time steps. In two dimensions, our numerical simulation up to [Formula: see text] for specific sets of target states suggests that the lackadaisical quantum walk can search one of the [Formula: see text] target states in [Formula: see text] time steps.