Low-depth amplitude estimation without really trying

The two-site Heisenberg model has an extraordinarily simple analytical solution and is traditionally used as a benchmark against numerical methods, such as exact diagonalization and Monte Carlo methods. In the same spirit, we benchmark three quantum algorithms that are implemented in a quantum computer against the analytical solution of this model. In particular, this presentation includes a description of the standard and iterative quantum phase estimation algorithms, as well as the variational quantum eigensolver. These quantum algorithms are introduced in a pedagogical fashion allowing newcomers to the subject, familiar with only the most basic quantum mechanical systems, to easily reproduce the presented results and apply the methods to other problems, thus building a seemingly under-appreciated path towards useful quantum algorithms through the lens of simulating and computing properties of physical quantum systems.

Accurately and efficiently estimating the carbon market risk is paramount for ensuring financial stability, promoting environmental sustainability, and facilitating informed decision‐making. Although classical risk estimation methods are extensively utilized, the implicit pre‐assumptions regarding distribution are predominantly contained and challenging to balance accuracy and computational efficiency. A quantum computing‐based carbon market risk estimation framework is proposed to address this problem with the quantum conditional generative adversarial network‐quantum amplitude estimation (QCGAN‐QAE) algorithm. Specifically, quantum conditional generative adversarial network (QCGAN) is employed to simulate the future distribution of the generated return rate, whereas quantum amplitude estimation (QAE) is employed to measure the distribution. Moreover, the quantum circuit of the QCGAN improved by reordering the data interaction layer and data simulation layer is coupled with the introduction of the quantum fully connected layer. The binary search method is incorporated into the QAE to bolster the computational efficiency. The simulation results based on the European Union Emissions Trading System reveals that the proposed framework markedly enhances the efficiency and precision of value‐at‐risk and conditional value‐at‐risk compared to original methods.

Variational quantum algorithms (VQAs) are promising methods to demonstrate quantum advantage on near-term devices as the required resources are divided between a quantum simulator and a classical optimizer. As such, designing a VQA which is resource-efficient and robust against noise is a key factor to achieve a potential advantage with the existing noisy quantum simulators. It turns out that a fixed VQA circuit design, such as the widely used hardware-efficient ansatz, is not necessarily robust against imperfections. In this work, we propose a genome-length-adjustable evolutionary algorithm to design a robust VQA circuit that is optimized over variations of both circuit ansatz and gate parameters, without any prior assumptions on circuit structure or depth. Remarkably, our method not only generates a noise-effect-minimized circuit with shallow depth, but also accelerates the classical optimization by substantially reducing the number of parameters. In this regard, the optimized circuit is far more resource-efficient with respect to both quantum and classical resources. As applications, based on two typical error models in VQA, we apply our method to calculate the ground energy of the hydrogen and the water molecules as well as the Heisenberg model. Simulations suggest that, compared with conventional hardware-efficient ansatz, our circuit-structure-tunable method can generate circuits apparently more robust against both coherent and incoherent noise and hence is more likely to be implemented on near-term devices.

Topology optimization is a key methodology in engineering design for finding efficient and robust structures. Due to the enormous size of the design space, evaluating all possible configurations is typically infeasible. In this work, we present an end-to-end, fault-tolerant quantum algorithm for topology optimization that operates on an exponentially large Hilbert space representing the design space. We demonstrate the algorithm on the two-dimensional Messerschmitt-Bölkow-Blohm beam problem. By restricting design variables to binary values, we reformulate the compliance minimization task as a combinatorial satisfiability problem, solved using Grover’s algorithm. Within Grover’s oracle, the compliance is computed through the finite-element method using established quantum algorithms, including block-encoding of the stiffness matrix, quantum singular value transformation for matrix inversion, Hadamard test, and quantum amplitude estimation. The complete algorithm is implemented and validated using classical quantum-circuit simulations. A detailed complexity analysis shows that the method evaluates the compliance of exponentially many structures in quantum superposition in polynomial time. In the global search, our approach maintains Grover’s quadratic speedup compared to classical unstructured search. Overall, the proposed quantum workflow demonstrates how quantum algorithms can advance the field of computational science and engineering.

Quantum support vector machine without iteration

This report discusses Shor’s quantum factorization algorithm and ρ–Pollard’s factorization algorithm. Shor’s quantum factorization algorithm consists of classical and quantum parts. In the classical part, it is proposed to use Euclidean algorithm, to find the greatest common divisor (GCD), but now exist large number of modern algorithms for finding GCD. Results of calculations of 8 algorithms were considered, among which algorithm with lowest execution rate of task was identified, which allowed the quantum algorithm as whole to work faster, which in turn provides greater potential for practical application of Shor’s quantum algorithm. Standard quantum Shor’s algorithm was upgraded by replacing the binary algorithm with iterative shift algorithm, canceling random number generation operation, using additive chain algorithm for raising to power. Both Shor’s algorithms (standard and upgraded) are distinguished by their high performance, which proves much faster and insignificant increase in time in implementation of data processing. In addition, it was possible to modernize Shor’s quantum algorithm in such way that its efficiency turned out to be higher than standard algorithm because classical part received an improvement, which allows an increase in speed by 12%.

Quantum computing serves as a revolutionary paradigm in computing, harnessing quantum mechanics principles such as quantum entanglement and superposition for optimization simulation, machine learning, and other problems that are a challenge to classical computer systems. The article will examine different usage opportunities in retail and supply chain operations, healthcare and life sciences domains, insurance operations, and financial service industries. Better computation can be achieved with quantum algorithms in discrete and continuous optimization, logistics network optimization, warehouse management, and product delivery optimized by quantum algorithms. Application domains in healthcare include simulation of molecules for medical breakthroughs, genomic analysis, and medical studies optimized by quantum eigenvalue algorithms and quantum eigensolver algorithms. The insurance industry portfolio targets catastrophic modeling, identification, and risk assessment optimized by quantum amplitude estimation algorithms and quantum machine learning algorithms. Financial service industries include derivative pricing, portfolio optimization, credit risk evaluation, and real-time risk calculation optimized in quantum algorithms. The path towards achieving quantum benefit in quantum technology involves using noisy intermediate quantum technology with hybrid quantum and classical designs and error management schemes, where integration opportunities for different industries arise from their common computation problem schemas in need of further scaling in hardware technology.

A dominating set of a graph G = (V, E) is a subset U of its vertices V, such that any vertex of G is either in U, or has a neighbor in U. The dominating-set problem is to find a minimum dominating set in G. Dominating sets are of critical importance for various types of networks/graphs, and find therefore potential applications in many fields. Particularly, in the area of communication, dominating sets are prominently used in the efficient organization of large-scale wireless ad hoc and sensor networks. However, the dominating set problem is also a hard optimization problem and thus currently is not efficiently solvable on classical computers. Here, we propose a biomolecular and a quantum algorithm for this problem, where the quantum algorithm provides a quadratic speedup over any classical algorithm. We show that the dominating set problem can be solved in O(2^{n/2}) queries by our proposed quantum algorithm, where n is the number of vertices in G. We also demonstrate that our quantum algorithm is the best known procedure to date for this problem. We confirm the correctness of our algorithm by executing it on IBM Quantum’s qasm simulator and the Brooklyn superconducting quantum device. And lastly, we show that molecular solutions obtained from solving the dominating set problem are represented in terms of a unit vector in a finite-dimensional Hilbert space.

An interactive proof of proximity (\(\mathsf{{IPP}}\)) is an interactive protocol in which a prover tries to convince a sublinear-time verifier that \(x \in \mathcal {L}\). Since the verifier runs in sublinear-time, following the property testing literature, the verifier is only required to reject inputs that are far from \(\mathcal {L}\). In a recent work, Rothblum et. al (STOC, 2013) constructed an \(\mathsf{{IPP}}\) for every language computable by a low depth circuit.

Coherent control errors, for which ideal Hamiltonians are perturbed by unknown multiplicative noise terms, are a major obstacle for reliable quantum computing. In this paper we present a framework for analyzing the robustness of quantum algorithms against coherent control errors using Lipschitz bounds. We derive worst-case fidelity bounds which show that the resilience against coherent control errors is mainly influenced by the norms of the Hamiltonians generating the individual gates. These bounds are explicitly computable even for large circuits and they can be used to guarantee fault tolerance via threshold theorems. Moreover, we apply our theoretical framework to derive a guideline for robust quantum algorithm design and transpilation, which amounts to reducing the norms of the Hamiltonians. Using the three-qubit quantum Fourier transform as an example application, we demonstrate that this guideline targets robustness more effectively than existing ones based on circuit depth or gate count. Furthermore, we apply our framework to study the effect of parameter regularization in variational quantum algorithms. The practicality of the theoretical results is demonstrated via implementations in simulation and on a quantum computer.

The recent advent of noisy intermediate-scale quantum devices, especially near-term quantum computers, has sparked extensive research efforts concerned with their possible applications. At the forefront of the considered approaches are variational methods that use parametrized quantum circuits. The classical and quantum Fisher information are firmly rooted in the field of quantum sensing and have proven to be versatile tools to study such parametrized quantum systems. Their utility in the study of other applications of noisy intermediate-scale quantum devices, however, has only been discovered recently. Hoping to stimulate more such applications, this article aims to further popularize classical and quantum Fisher information as useful tools for near-term applications beyond quantum sensing. We start with a tutorial that builds an intuitive understanding of classical and quantum Fisher information and outlines how both quantities can be calculated on near-term devices. We also elucidate their relationship and how they are influenced by noise processes. Next, we give an overview of the core results of the quantum sensing literature and proceed to a comprehensive review of recent applications in variational quantum algorithms and quantum machine learning.

Solving Computational Fluid Dynamics (CFD) problems requires the inversion of a linear system of equations, which can be done using a quantum algorithm for matrix inversion.
 However, the number of shots required to measure the output of the system can be prohibitive and remove any advantage obtained by quantum computing.
 In this work we propose a scheme for measuring the output of a Quantum Singular Value Transform (QSVT) matrix inversion algorithm specifically for the CFD use case.
 We use a Quantum Signal Processing (QSP) based amplitude estimation algorithm and show how it can be combined with the QSVT matrix inversion algorithm.
 We perform a detailed resource estimation of the amount of computational resources required for a single iteration of amplitude estimation, and compare the costs of amplitude estimation with the cost of not doing amplitude estimation and measuring the whole wavefunction.
 We also propose a measurement scheme to reduce the number of amplitudes measured in the CFD example by focussing on large amplitudes only. We simulate the whole CFD loop, finding that thus measuring only a small number of the total amplitudes in the output vector still results in an acceptable level of overall error.

Discrete stochastic processes (DSP) are instrumental for modeling the dynamics of probabilistic systems and have a wide spectrum of applications in science and engineering. DSPs are usually analyzed via Monte-Carlo methods since the number of realizations increases exponentially with the number of time steps, and importance sampling is often required to reduce the variance. We propose a quantum algorithm for calculating the characteristic function of a DSP, which completely defines its probability distribution, using the number of quantum circuit elements that grows only linearly with the number of time steps. The quantum algorithm reduces the Monte-Carlo sampling to a Bernoulli trial while taking all stochastic trajectories into account. This approach guarantees the optimal variance without the need for importance sampling. The algorithm can be further furnished with the quantum amplitude estimation algorithm to provide quadratic speed-up in sampling. The Fourier approximation can be used to estimate an expectation value of any integrable function of the random variable. Applications in finance and correlated random walks are presented. Proof-of-principle experiments are performed using the IBM quantum cloud platform.

Quantum computing introduces an efficient way for unfolding complex problems on computing systems. It uses concept of superposition and entanglement, & hence supports intrinsic parallelism for obtaining the speedup in context of time over classical computing. Quantum algorithms can be efficiently implemented over quantum computing system. Most popular quantum algorithms like Grover's and Shor's provide a great foundation to specifically design the algorithms. These algorithms work on improving the lower runtime bounds of classical algorithms. Grover's algorithm exhibits a quadratic polynomial speedup while searching an element among ‘N’ elements unstructured database. The algorithmic strategy can be applied over various computer science related applications and some important are being considered in this article i.e. pattern matching, data mining and machine learning. This article explores quantum fundamentals; history based evolution followed by Grover's search algorithm in detail with cascading advantages, limitations & applications and also finally concludes with the significance of quantum computing & quantum algorithmic domains.

Quantum computer simulators play an important role when we evaluate quantum algorithms. Quantum computation can be regarded as parallel computation in some sense, and thus, it is suitable to implement a simulator on a hardware, which can process a lot of operations in parallel. In this paper, we propose a processor architecture dedicated to simulating quantum algorithms. The proposed architecture is based on the register reordering method that shifts and swaps registers containing probability amplitudes so that the probability amplitudes of target basis states can be quickly selected. This reduces the number of large multiplexers and improves clock frequency. We implemented the processor on an FPGA. Experimental results show that the proposed processor has scalability in terms of the number of quantum bits, and can simulate quantum algorithms faster than software simulators.