Quantum Computing Algorithms Research Articles

Distribution of controlled unitary quantum gates towards factoring large numbers on today’s small-register devices

Factoring a 2048-bit number using Shor's algorithm, when accounting for error correction, reportedly requires 400,000 qubits. However, it is well known that there is yet much time before we will have this many qubits in the same local system. This is why we propose a protocol for distributed quantum computation applicable to small register devices, specifically for the distribution of controlled unitary gates, the key element in the construction of every quantum computation algorithm. We leverage quantum sharing of partial results to obtain a parallel processing scheme, allowing for the first time the quantum distribution of very large gates with thousands of inputs using only small register devices with tens of qubits. In this way, we improve all previous controlled unitary gate distribution approaches, obtaining surprising results. The impact is quantified for recent milestone hardware realizations of quantum processors.

Rapid solution of logical equivalence problems by quantum computation algorithm

We present a quantum computation algorithm that enables solving the problem of logical equivalence verification in exponentially less time than the classical deterministic computation. In this novel quantum algorithm, the oracles of the two evaluated functions are executed in series to yield a common target qubit which then interacts with an ancillary qubit. We found that the degree of entanglement (measured by the concurrence) of the target and ancillary qubits is a reliable witness for the logical equivalence property of the two functions. The steps number of the quantum algorithm is inversely proportional to the square of the standard error ϵ2 of the measured concurrence value, with no dependence on the input size ′n′ of each function. This corresponds to a number of evaluations of the two functions: O(ϵ−2) for the quantum algorithm compared with O(2n) for the classical approach. To assess the algorithm performance, two sets of experiments are conducted using the IBM Q Experience simulator for input sizes: 2 and 12 variables per function. While the former verifies that the results of the experiment are in a good match with the theory, the latter showcases the quantum supremacy of the presented algorithm. In particular, The latter shows that the quantum algorithm requires only 200 oracles queries compared with 213 queries for the classical algorithm.

Quantum-Compute Algorithm for Exact Laser-Driven Electron Dynamics in Molecules.

In this work, we investigate the capability of known quantum computing algorithms for fault-tolerant quantum computing to simulate the laser-driven electron dynamics of excitation and ionization processes in small molecules such as lithium hydride, which can be benchmarked against the most accurate time-dependent full configuration interaction (TD-FCI) calculations. The conventional TD-FCI wave packet propagation is reproduced using the Jordan-Wigner transformation for wave function and operators and the Trotter product formula for expressing the propagator. In addition, the time-dependent dipole moment, as an example of a time-dependent expectation value, is calculated using the Hadamard test. To include non-Hermitian operators in the ionization dynamics, a similar approach to the quantum imaginary time evolution (QITE) algorithm is employed to translate the propagator, including a complex absorption potential, into quantum gates. The computations are executed on a quantum computer simulator. By construction, all quantum computer algorithms, except for the QITE algorithm used only for ionization but not for excitation dynamics, would scale polynomially on a quantum computer with fully entangled qubits. In contrast, TD-FCI scales exponentially. Hence, quantum computation holds promises for substantial progress in the understanding of electron dynamics of excitation processes in increasingly large molecular systems, as has already been witnessed in electronic structure theory.

Application of a variational hybrid quantum-classical algorithm to heat conduction equation and analysis of time complexity

The prosperous development of both hardware and algorithms for quantum computing (QC) potentially prompts a paradigm shift in scientific computing in various fields. As an increasingly active topic in QC, the variational quantum algorithm leads a promising tool for solving partial differential equations on noisy intermediate scale quantum devices. Although a clear perspective on the advantages of QC over classical computing techniques for specific mathematical and physical problems exists, applications of QC in computational fluid dynamics to solve practical flow problems, though promising, are still at the early stage of development. To explore QC in practical simulation of flow problems, this work applies a variational hybrid quantum-classical algorithm, namely the variational quantum linear solver (VQLS), to resolve the heat conduction equation through finite difference discretization of the Laplacian operator. Details of the VQLS implementation are discussed by various test instances of linear systems. The effect of the number of shots on the accuracy is studied, which reveals a logarithmic relationship. Furthermore, the heuristic scaling of the VQLS with the precision ε, the number of qubits n and the condition number k validates its time complexity reported in the literature. In addition, the successful state vector simulations of the heat conduction equation in one and two dimensions demonstrate the validity of the present VQLS-based algorithm by proof-of-concept results. Finally, the heuristic scaling for the heat conduction problem indicates that the time complexity of the present approach is logarithmically dependent on the precision ε and linearly dependent on the number of qubits n.

Preface to the Special Issue on “Quantum Computing Algorithms and Computational Complexity”

In 1982, Richard Feynman stated that in order to simulate quantum systems, we would rather go for a sort of brand-new powered quantum processor instead of a classical one [...]

Graph Theoretic Molecular Fragmentation for Multidimensional Potential Energy Surfaces Yield an Adaptive and General Transfer Machine Learning Protocol.

Over a series of publications we have introduced a graph-theoretic description for molecular fragmentation. Here, a system is divided into a set of nodes, or vertices, that are then connected through edges, faces, and higher-order simplexes to represent a collection of spatially overlapping and locally interacting subsystems. Each such subsystem is treated at two levels of electronic structure theory, and the result is used to construct many-body expansions that are then embedded within an ONIOM-scheme. These expansions converge rapidly with many-body order (or graphical rank) of subsystems and have been previously used for ab initio molecular dynamics (AIMD) calculations and for computing multidimensional potential energy surfaces. Specifically, in all these cases we have shown that CCSD and MP2 level AIMD trajectories and potential surfaces may be obtained at density functional theory cost. The approach has been demonstrated for gas-phase studies, for condensed phase electronic structure, and also for basis set extrapolation-based AIMD. Recently, this approach has also been used to derive new quantum-computing algorithms that enormously reduce the quantum circuit depth in a circuit-based computation of correlated electronic structure. In this publication, we introduce (a) a family of neural networks that act in parallel to represent, efficiently, the post-Hartree-Fock electronic structure energy contributions for all simplexes (fragments), and (b) a new k-means-based tessellation strategy to glean training data for high-dimensional molecular spaces and minimize the extent of training needed to construct this family of neural networks. The approach is particularly useful when coupled cluster accuracy is desired and when fragment sizes grow in order to capture nonlocal interactions accurately. The unique multidimensional k-means tessellation/clustering algorithm used to determine our training data for all fragments is shown to be extremely efficient and reduces the needed training to only 10% of data for all fragments to obtain accurate neural networks for each fragment. These fully connected dense neural networks are then used to extrapolate the potential energy surface for all molecular fragments, and these are then combined as per our graph-theoretic procedure to transfer the learning process to a full system energy for the entire AIMD trajectory at less than one-tenth the cost as compared to a regular fragmentation-based AIMD calculation.

Accurate and Efficient Quantum Computations of Molecular Properties Using Daubechies Wavelet Molecular Orbitals: A Benchmark Study against Experimental Data

Although quantum computation (QC) is regarded as a promising numerical method for computational quantum chemistry, current applications of quantum-chemistry calculations on quantum computers are limited to small molecules. This limitation can be ascribed to technical problems in building and manipulating more qubits and the associated complicated operations of quantum gates in a quantum circuit when the size of the molecular system becomes large. As a result, reducing the number of required qubits is necessary to make QC practical. Currently, the minimal STO-3G basis set is commonly used in benchmark studies because it requires the minimum number of spin orbitals. Nonetheless, the accuracy of using STO-3G is generally low and thus cannot provide useful predictions. We propose to adopt Daubechies wavelet functions as an accurate and efficient method for QCs of molecular electronic properties. We demonstrate that a minimal basis set constructed from Daubechies wavelet basis can yield accurate results through a better description of the molecular Hamiltonian, while keeping the number of spin orbitals minimal. With the improved Hamiltonian through Daubechies wavelets, we calculate vibrational frequencies for H$_2$ and LiH using quantum-computing algorithm to show that the results are in excellent agreement with experimental data. As a result, we achieve quantum calculations in which accuracy is comparable with that of the full configuration interaction calculation using the cc-pVDZ basis set, whereas the computational cost is the same as that of a STO-3G calculation. Thus, our work provides a more efficient and accurate representation of the molecular Hamiltonian for efficient QCs of molecular systems, and for the first time demonstrates that predictions in agreement with experimental measurements are possible to be achieved with quantum resources available in near-term quantum computers.

Quantum behaved Intelligent Variant of Gravitational Search Algorithm with Deep Neural Networks for Human Activity Recognition

Human activity recognition (HAR) encompasses the detection of daily routine activities to advance usability in detecting crime and preventing dangerous activities. The recognition of activities from videos and image sequences with higher exactitude is a major challenge due to system complexities. The efficient feature optimization approach can reduce system complexities by removing ineffective features, which also improves the activity recognition performance. This research work presents a novel quantum behaved intelligent gravitational search algorithm to optimize the features for human activity recognition. The proposed intelligent variant is termed as INQGSA, which optimizes the features by using the advantageous attributes of quantum computing (QC) and intelligent gravitational search algorithm (INGSA). In INQGSA, the intelligent factor avoids the trapping of mass agents in later iterations by using the information of the best and worst agents to update the position of agents. The addition of quantum computing based attributes (such as quantum bits, their superposition, and quantum gates, etc.) ensures a better diversity of discrete optimized features. To analyze the superiority of INQGSA, the feature optimization is also conducted with the gravitational search algorithm (GSA) and the quantum-inspired binary gravitational search algorithm (QBGSA). Finally, the optimized selected features are utilized by the deep neural networks (DNN) of ResNet-50V2 and ResNet-101V2 for the classification of activities. The activity recognition experiments are conducted on the UCF101 and HMDB51 datasets. The performance comparison of the proposed HAR system with state-of-the-art techniques signifies that the proposed system is superior and effective in detecting the different activities.

A review on quantum computing and deep learning algorithms and their applications.

In this paper, we describe a review concerning the Quantum Computing (QC) and Deep Learning (DL) areas and their applications in Computational Intelligence (CI). Quantum algorithms (QAs), engage the rules of quantum mechanics to solve problems using quantum information, where the quantum information is concerning the state of a quantum system, which can be manipulated using quantum information algorithms and other processing techniques. Nowadays, many QAs have been proposed, whose general conclusion is that using the effects of quantum mechanics results in a significant speedup (exponential, polynomial, super polynomial) over the traditional algorithms. This implies that some complex problems currently intractable with traditional algorithms can be solved with QA. On the other hand, DL algorithms offer what is known as machine learning techniques. DL is concerned with teaching a computer to filter inputs through layers to learn how to predict and classify information. Observations can be in the form of plain text, images, or sound. The inspiration for deep learning is the way that the human brain filters information. Therefore, in this research, we analyzed these two areas to observe the most relevant works and applications developed by the researchers in the world.

Quantum Linear System Solver Based on Time-optimal Adiabatic Quantum Computing and Quantum Approximate Optimization Algorithm

We demonstrate that with an optimally tuned scheduling function, adiabatic quantum computing (AQC) can readily solve a quantum linear system problem (QLSP) with O (κ poly(log (κ ε))) runtime, where κ is the condition number, and ε is the target accuracy. This is near optimal with respect to both κ and ε, and is achieved without relying on complicated amplitude amplification procedures that are difficult to implement. Our method is applicable to general non-Hermitian matrices, and the cost as well as the number of qubits can be reduced when restricted to Hermitian matrices, and further to Hermitian positive definite matrices. The success of the time-optimal AQC implies that the quantum approximate optimization algorithm (QAOA) with an optimal control protocol can also achieve the same complexity in terms of the runtime. Numerical results indicate that QAOA can yield the lowest runtime compared to the time-optimal AQC, vanilla AQC, and the recently proposed randomization method.

Twist: sound reasoning for purity and entanglement in Quantum programs

Quantum programming languages enable developers to implement algorithms for quantum computers that promise computational breakthroughs in classically intractable tasks. Programming quantum computers requires awareness of entanglement , the phenomenon in which measurement outcomes of qubits are correlated. Entanglement can determine the correctness of algorithms and suitability of programming patterns. In this work, we formalize purity as a central tool for automating reasoning about entanglement in quantum programs. A pure expression is one whose evaluation is unaffected by the measurement outcomes of qubits that it does not own, implying freedom from entanglement with any other expression in the computation. We present Twist, the first language that features a type system for sound reasoning about purity. The type system enables the developer to identify pure expressions using type annotations. Twist also features purity assertion operators that state the absence of entanglement in the output of quantum gates. To soundly check these assertions, Twist uses a combination of static analysis and runtime verification. We evaluate Twist’s type system and analyses on a benchmark suite of quantum programs in simulation, demonstrating that Twist can express quantum algorithms, catch programming errors in them, and support programs that existing languages disallow, while incurring runtime verification overhead of less than 3.5%.

Quantum Firefly Secure Routing for Fog Based Wireless Sensor Networks

Wireless Sensor Networks (WSNs) become an integral part of Internet of Things (IoT) and finds their applicability in several domains. As classical WSN faces several issues in service-based IoT applications, fog computing has been introduced in real-time, enabling local data processing and avoid raw data transmission to cloud servers. The Fog-based WSN generally involves advanced nodes, normal nodes, and some Fog Nodes (FN). Though the Fog-based WSN offers several benefits, there is a need to develop an effective trust-based secure routing protocol for data transmission among Cluster Heads (CHs) and FNs. In this view, this paper presents a Quantum Firefly Optimization based Multi-Objective Secure Routing (QFO-MOSR) protocol for Fog-based WSN. The main intention of the QFO-MOSR technique is to derive an optimal selection of routes between CHs and FNs in the network. The QFO-MOSR technique has incorporated the concepts of quantum computing and Firefly (FF) optimization algorithm inspired by the flashing behaviour of FFs. In addition, a multi-objective fitness function is derived by the QFO-MOSR technique using seven objectives: distance, inter-cluster distance, energy, delay, intra-cluster distance, link lifetime, and trust. The proposed routing technique derives a fitness function including trust factor from ensuring security. The design of the QFO-MOSR technique with a multi-objective fitness function shows the novelty of the work. To validate the performance of the QFO-MOSR technique, a series of experiments were carried out, and the results are investigated in terms of different measures. The experimental analysis ensured that the QFO-MOSR technique is superior to other methods in terms of different measures.

Molecular Quantum Dynamics: A Quantum Computing Perspective.

ConspectusSimulating molecular dynamics (MD) within a comprehensive quantum framework has been a long-standing challenge in computational chemistry. An exponential scaling of computational cost renders solving the time dependent Schrödinger equation (TDSE) of a molecular Hamiltonian, including both electronic and nuclear degrees of freedom (DOFs), as well as their couplings, infeasible for more than a few DOFs. In the Born-Oppenheimer (BO), or adiabatic, picture, electronic and nuclear parts of the wave function are decoupled and treated separately. Within this framework, the nuclear wave function evolves along potential energy surfaces (PESs) computed as solutions to the electronic Schrödinger equation parametrized in the nuclear DOFs. This approximation, together with increasingly elaborate numerical approaches to solve the nuclear time dependent Schrödinger equation (TDSE), enabled the treatment of up to a few dozens of degrees of freedom (DOFs). However, for particular applications, such as photochemistry, the BO approximation breaks down. In this regime of non-adiabatic dynamics, solving the full molecular problem including electron-nuclear couplings becomes essential, further increasing the complexity of the numerical solution. Although valuable methods such as multiconfigurational time-dependent Hartree (MCTDH) have been proposed for the solution of the coupled electron-nuclear dynamics, they remain hampered by an exponential scaling in the number of nuclear DOFs and by the difficulty of finding universal variational forms.In this Account, we present a perspective on novel quantum computational algorithms, aiming to alleviate the exponential scaling inherent to the simulation of many-body quantum dynamics. In particular, we focus on the derivation and application of quantum algorithms for adiabatic and non-adiabatic quantum dynamics, which include efficient approaches for the calculation of the BO potential energy surfaces (PESs). Thereafter, we study the time-evolution of a model system consisting of two coupled PESs in first and second quantization. In a first application, we discuss a recently introduced quantum algorithm for the evolution of a wavepacket in first quantization and exploit the potential quantum advantage of mapping its spatial grid representation to logarithmically many qubits. For the second demonstration, we move to the second quantization framework and review the scaling properties of two alternative time-evolution algorithms, namely, a variational quantum algorithm (VQA) (based on the McLachlan variational principle) and conventional Trotter-type evolution (based on a Lie-Trotter-Suzuki formula). Both methods clearly demonstrate the potential of quantum algorithms and their favorable scaling compared to the available classical approaches. However, a clear demonstration of quantum advantage in the context of molecular quantum dynamics may require the implementation of these algorithms in fault-tolerant quantum computers, while their application in near-term, noisy quantum devices is still unclear and deserves further investigation.

Experimental Quantum State Measurement with Classical Shadows.

A crucial subroutine for various quantum computing and communication algorithms is to efficiently extract different classical properties of quantum states. In a notable recent theoretical work by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)NPAHAX1745-247310.1038/s41567-020-0932-7], a thrifty scheme showed how to project the quantum state into classical shadows and simultaneously predict M different functions of a state with only O(log_{2}M) measurements, independent of the system size and saturating the information-theoretical limit. Here, we experimentally explore the feasibility of the scheme in the realistic scenario with a finite number of measurements and noisy operations. We prepare a four-qubit GHZ state and show how to estimate expectation values of multiple observables and Hamiltonians. We compare the measurement strategies with uniform, biased, and derandomized classical shadows to conventional ones that sequentially measure each state function exploiting either importance sampling or observable grouping. We next demonstrate the estimation of nonlinear functions using classical shadows and analyze the entanglement of the prepared quantum state. Our experiment verifies the efficacy of exploiting (derandomized) classical shadows and sheds light on efficient quantum computing with noisy intermediate-scale quantum hardware.

An Improved Quantum-Inspired Gravitational Search Algorithm to Optimize the Facial Features

The optimization of the features is vital to effectively detecting facial expressions. This research work has optimized the facial features by employing the improved quantum-inspired gravitation search algorithm (IQI-GSA). The improvement to the quantum-inspired gravitational search algorithm (QIGSA) is conducted to handle the local optima trapping. The QIGSA is the amalgamation of the quantum computing and gravitational search algorithm that owns the overall strong global search ability to handle the optimization problems in comparison with the gravitational search algorithm. In spite of global searching ability, the QIGSA can be trapped in local optima in the later iterations. This work has adapted the IQI-GSA approach to handle the local optima, stochastic characteristics and maintaining balance among the exploration and exploitation. The IQI-GSA is utilized for the optimized features selection from the set of extracted features using the LGBP (a hybrid approach of local binary patterns with the Gabor filter) method. The system performance is analyzed for the application of automated facial expressions recognition with the classification technique of deep convolutional neural network (DCNN). The extensive experimentation evaluation is conducted on the benchmark datasets of Japanese Female Facial Expression (JAFFE), Radboud Faces Database (RaFD) and Karolinska Directed Emotional Faces (KDEF). To determine the effectiveness of the proposed facial expression recognition system, the results are also evaluated for the feature optimization with GSA and QIGSA. The evaluation results clearly demonstrate the outperformed performance of the considered system with IQI-GSA in comparison with GSA, QIGSA and existing techniques available for the experimentation on utilized datasets.

Quantum-assisted simulator

Quantum simulation can help us study poorly understood topics such as high-temperature superconductivity and drug design. However, existing quantum simulation algorithms for current quantum computers often have drawbacks that impede their application. Here, we provide a hybrid quantum-classical algorithm for simulating the dynamics of quantum systems. Our approach takes the ansatz wave function as a linear combination of quantum states. The quantum states are fixed, and the combination parameters are variationally adjusted. Unlike existing variational quantum simulation algorithms, our algorithm does not require any classical-quantum feedback loop and by construction bypasses the barren plateau problem. Moreover, our algorithm does not require any complicated measurements such as the Hadamard test. The entire framework is compatible with existing experimental capabilities and thus can be implemented immediately.

Search for efficient formulations for Hamiltonian simulation of non-Abelian lattice gauge theories

Hamiltonian formulation of lattice gauge theories (LGTs) is the most natural framework for the purpose of quantum simulation, an area of research that is growing with advances in quantum-computing algorithms and hardware. It, therefore, remains an important task to identify the most accurate, while computationally economic, Hamiltonian formulation(s) in such theories, considering the necessary truncation imposed on the Hilbert space of gauge bosons with any finite computing resources. This paper is a first step toward addressing this question in the case of non-Abelian LGTs, which further require the imposition of non-Abelian Gauss's laws on the Hilbert space, introducing additional computational complexity. Focusing on the case of SU(2) LGT in 1+1 D coupled to matter, a number of different formulations of the original Kogut-Susskind framework are analyzed with regard to the dependence of the dimension of the physical Hilbert space on boundary conditions, system's size, and the cutoff on the excitations of gauge bosons. The impact of such dependencies on the accuracy of the spectrum and dynamics is examined, and the (classical) computational-resource requirements given these considerations are studied. Besides the well-known angular-momentum formulation of the theory, the cases of purely fermionic and purely bosonic formulations (with open boundary conditions), and the Loop-String-Hadron formulation are analyzed, along with a brief discussion of a Quantum Link Model of the same theory. Clear advantages are found in working with the Loop-String-Hadron framework which implements non-Abelian Gauss's laws a priori using a complete set of gauge-invariant operators. Although small lattices are studied in the numerical analysis of this work, and only the simplest algorithms are considered, a range of conclusions will be applicable to larger systems and potentially to higher dimensions.

Quantum algorithm based on the $$\varepsilon $$-random linear disequations for the continuous hidden shift problem

There have been several research works on the hidden shift problem, quantum algorithms for the problem, and their applications. However, all the results have focused on discrete groups with discrete oracle functions. In this paper, we define the continuous hidden shift problem on $\mathbb{R}^n$ with a continuous oracle function as an extension of the hidden shift problem, and also define the $\varepsilon$-random linear disequations which is a generalization of the random linear disequations. By employing the newly defined concepts, we show that there exists a quantum computational algorithm which solves this problem in time polynomial in $n$.

Quantum walk search algorithm for multi-objective searching with iteration auto-controlling on hypercube

Shenvi et al. have proposed a quantum algorithm based on quantum walking called Shenvi–Kempe–Whaley (SKW) algorithm, but this search algorithm can only search one target state and use a specific search target state vector. Therefore, when there are more than two target nodes in the search space, the algorithm has certain limitations. Even though a multi-objective SKW search algorithm was proposed later, when the number of target nodes is more than two, the SKW search algorithm cannot be mapped to the same quotient graph. In addition, the calculation of the optimal target state depends on the number of target states m. In previous studies, quantum computing and testing algorithms were used to solve this problem. But these solutions require more Oracle calls and cannot get a high accuracy rate. Therefore, to solve the above problems, we improve the multi-target quantum walk search algorithm, and construct a controllable quantum walk search algorithm under the condition of unknown number of target states. By dividing the Hilbert space into multiple subspaces, the accuracy of the search algorithm is improved from p c = (1/2) – O(1/n) to p c = 1 – O(1/n). And by adding detection gate phase, the algorithm can stop when the amplitude of the target state becomes the maximum for the first time, and the algorithm can always maintain the optimal number of iterations, so as to reduce the number of unnecessary iterations in the algorithm process and make the number of iterations reach .

Quantum k-community detection: algorithm proposals and cross-architectural evaluation

Emerging quantum technologies represent a promising alternative for solving hard combinatorial problems in the post-Moore’s law era. For practical purposes, however, the current number of qubits limits the direct applicability to larger real-world instances in the near-term future. Therefore, a promising strategy to overcome this issue is represented by hybrid quantum classical algorithms which leverage classical as well as quantum devices. One prominent example of a hard computational problem is the community detection problem: a partition of a graph into distinct communities such that the ratio between intra-community and inter-community connectivity is maximized. In this paper, we explore the current potential of quantum annealing and gate-based quantum technologies to solve the community detection problem for an arbitrary number of communities. For this purpose, existing algorithms are (re-)implemented and new hybrid algorithms, that can be run on gate-model devices, are proposed. Their performance on standardized benchmark graphs has been evaluated and compared to the one of a state-of-the-art classical heuristic algorithm. Although no quantum speed-up has been achieved, the existing quantum annealing-based methods as well as the novel hybrid algorithms for gate-based quantum computers yield modularity values, which are similar to those of the classical heuristic. However, the modular architecture of the used algorithms allows for fast utilization of more powerful quantum technologies once they become available. Reproducibility: Our code and data are publicly available (Github in Quantum Modularization. https://github.com/jku-win se/quantum_modularization 2021).