A benchmarking study of grover's algorithm for solving boolean sat with quantum circuits

In order to understand the bounds of utilization of the Grover's search algorithm for the large unstructured data in presence of the quantum computer noise, we undertake a series of simulations by inflicting various types of noise, modelled by the IBM QISKit. We apply three forms of Grover's algorithms: (1) the standard one, with 4-10 qubits, (2) recently published modified Grover's algorithm, set to reduce the circuit depth, and (3) the algorithms in (1) and (2) with multi-control Toffoli's modified by addition of an ancilla qubit. Based on these simulations, we find the upper bound of noise for these cases, establish its dependence on the quantum depth of the circuit and provide comparison among them. By extrapolation of the fitted thresholds, we predict what would be the typical gate error bounds when apply the Grover's algorithms for the search of a data in a data set as large as thirty two thousands.

Over the past few years, several quantum software stacks (QSS) have been developed in response to rapid hardware advances in quantum computing. A QSS includes a quantum programming language, an optimizing compiler that translates a quantum algorithm written in a high-level language into quantum gate instructions, a quantum simulator that emulates these instructions on a classical device, and a software controller that sends analog signals to a very expensive quantum hardware based on quantum circuits. In comparison to traditional compilers and architecture simulators, QSSes are difficult to tests due to the probabilistic nature of results, the lack of clear hardware specifications, and quantum programming complexity.This work devises a novel differential testing approach for QSSes, named QDiff with three major innovations: (1) We generate input programs to be tested via semantics-preserving, source to source transformation to explore program variants. (2) We speed up differential testing by filtering out quantum circuits that are not worthwhile to execute on quantum hardware by analyzing static characteristics such as a circuit depth, 2-gate operations, gate error rates, and T1 relaxation time. (3) We design an extensible equivalence checking mechanism via distribution comparison functions such as Kolmogorov–Smirnov test and cross entropy.We evaluate QDiff with three widely-used open source QSSes: Qiskit from IBM, Cirq from Google, and Pyquil from Rigetti. By running QDiff on both real hardware and quantum simulators, we found several critical bugs revealing potential instabilities in these platforms. QDiff's source transformation is effective in producing semantically equivalent yet not-identical circuits (i.e., 34% of trials), and its filtering mechanism can speed up differential testing by 66%.

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.

A significant amount of study is being done to review the security promises made for the various ciphers now in use as a result of the development of quantum computing technology. A general attack against symmetric key cryptography primitives that can reduce search costs to the square root is Grover's search algorithm. To implement Grover's algorithm, it is necessary that the target cipher be implemented as a quantum circuit. Despite being relatively new, this area of study has received significant attention from the research community. The authors have estimated the cost of Grover's key search attack against the stream cipher Atom, for the first time, under circuit depth restrictions defined in National Institute of Standards and Technology (NIST) PQC standardisation process. The authors implement the quantum circuit of Atom in QISKIT, (open‐source software development kit for working with quantum computers running on IBM Quantum Experience). The results are also compared with other existing literature on LFSR‐based stream ciphers, such as Grain‐v1, Grain‐128‐AEAD, and Lizard. The authors also find that, to the best of their knowledge, in the existing literature on estimating the cost of Grover's attack on symmetric ciphers, Atom is the only 128‐bit key cipher that meets the threshold of ≈2 170 set by NIST for quantum security of 128‐bit key ciphers. The authors also analyse the security of Atom against quantum TMDTO attacks.

Finding the optimal solution to a complex optimisation problem is of great importance in practically all fields of science, technology, technical design and econometrics. We demonstrate that a modified Grover's quantum algorithm can be applied to real problems of finding a global minimum using modest numbers of quantum bits. Calculations of the global minimum of simple test functions and Lennard-Jones clusters have been carried out on a quantum computer simulator using a modified Grover's algorithm. The number of function evaluations N reduced from O(N) in classical simulation to O(N 1/2) in quantum simulation. We also show how the Grover's quantum algorithm can be combined with the classical Pivot method for global optimisation to treat larger systems.

While current quantum computers, referred to as Noisy Intermediate-Scale Quantum (NISQ) computers, are expected to be beneficial for different applications, they are prone to different types of errors. In order to enhance the reliability of quantum systems, noise-aware quantum compilers are used to generate physical quantum circuits to be executed on NISQ computers. The quantum hardware is calibrated very frequently and its error rates are computed accordingly. Based on the hardware error rates, a quantum compiler allocates physical qubits and schedules quantum operations. However, error rates may change post-calibration. To incorporate dynamic error rates into quantum circuit compilation with minimum cost, we propose a Machine Learning (ML)-based scheme to detect the incorrect output of the quantum circuit and predict the Probability of Successful Trials (PST) with high accuracy. Our approach can verify the error rates of the quantum hardware and validate the correctness of the extracted quantum circuit output. We provide a case study of our ML-based reliability models using IBM Q16 Melbourne quantum computer. Our results show that the proposed scheme achieves a very high prediction accuracy.

We propose the concepts of the explicit oracle and the implicit oracle for realizing quantum algorithms. Then, the quantum circuit of the well-known Grover algorithm is constructed with the explicit oracle to solve the Hamiltonian cycle problem for the complete graph. The quantum circuit has a quadratic speedup over the classical unstructured search algorithm for solving the same problem. The IBM quantum computer simulator is used to run the quantum circuit to validate that it can indeed derive the Hamiltonian cycle of the complete graph.

Local Friendliness (LF) inequalities follow from seemingly reasonable assumptions about reality: (i) ``absoluteness of observed events'' (e.g., every observed event happens for all observers) and (ii) ``local agency'' (e.g., free choices can be made uncorrelated with other events outside their future light cone). Extended Wigner's Friend Scenario (EWFS) thought experiments show that textbook quantum mechanics violates these inequalities. Thus, experimental evidence of these violations would make these two assumptions incompatible. In [Nature Physics 16, 1199 (2020)], the authors experimentally implemented an EWFS, using a photonic qubit to play the role of each of the ``friends'' and measured violations of LF. One may question whether a photonic qubit is a physical system that counts as an ``observer'' and thereby question whether the experiment's outcome is significant. Intending to measure increasingly meaningful violations, we propose using a statistical measure called the ``branch factor'' to quantify the ``observerness'' of the system. We then encode the EWFS as a quantum circuit such that the components of the circuit that define the friend are quantum systems of increasing branch factor. We run this circuit on quantum simulators and hardware devices, observing LF violations as the system sizes scale. As errors in quantum computers reduce the significance of the violations, better quantum computers can produce better violations. Our results extend the state of the art in proof-of-concept experimental violations from branch factor 0.0 to branch factor 16.0. This is an initial result in an experimental program for measuring LF violations at increasingly meaningful branch factors using increasingly more powerful quantum processors and networks. We introduce this program as a fundamental science application for near-term and developing quantum technology.

Quantum communication relies on the existence of entanglement between two nodes of a network. However, due to its fragile nature, it is nearly impossible to establish entanglement at large distances through the direct transmission of qubits. Quantum repeaters have been proposed to solve this problem, which split-up the network to create small-scale entangled links and then connect them up to create the large-scale link. As researchers race to establish entanglement over larger and larger distances, it becomes essential to gauge the performance and robustness of the different protocols that have been proposed to design a quantum repeater, before deploying them in real life. Currently available noisy quantum computers are ideal for this task, as they can emulate the noisy environment in a quantum communication channel, and provide a measure for how the protocols will perform on real-life hardware. In this paper, we report the circuit-level implementation of the complete architecture of a quantum repeater, and benchmark this protocol on IBM's cloud quantum computer - IBMQ. Our experiments indicate a 26% fidelity of shared bell-pairs for a complete on-chip quantum repeater with a yield of 49%. We also compare these results with simulation data from IBM Qiskit. The results of our experiments provide a quantitative measure for the fidelity of entanglement that currently available repeaters can establish. In addition, the proposed circuit-implementation provides a robust benchmark for state-of-the-art quantum computing hardware.

Numerous conceptually important quantum algorithms rely on a blackbox device known as an oracle, which is typically difficult to construct without knowing the answer to the problem that the algorithm is intended to solve. A notable example is Grover's search algorithm. Here we propose a Grover search for solutions to a class of NP-complete decision problems known as subset sum problems, including the special case of number partitioning. Each problem instance is encoded in the couplings of a set of qubits to a central spin or boson, which enables a realization of the oracle without knowledge of the solution. The algorithm provides a quantum speedup across a known phase transition in the computational complexity of the partition problem, and we identify signatures of the phase transition in the simulated performance. Whereas the naive implementation of our algorithm requires a spectral resolution that scales exponentially with system size for NP-complete problems, we also present a recursive algorithm that enables scalability. We propose and analyze implementation schemes with cold atoms, including Rydberg-atom and cavity-QED platforms.

Generative modeling is a flavor of machine learning with applications ranging from computer vision to chemical design. It is expected to be one of the techniques most suited to take advantage of the additional resources provided by near-term quantum computers. Here, we implement a data-driven quantum circuit training algorithm on the canonical Bars-and-Stripes dataset using a quantum-classical hybrid machine. The training proceeds by running parameterized circuits on a trapped ion quantum computer and feeding the results to a classical optimizer. We apply two separate strategies, Particle Swarm and Bayesian optimization to this task. We show that the convergence of the quantum circuit to the target distribution depends critically on both the quantum hardware and classical optimization strategy. Our study represents the first successful training of a high-dimensional universal quantum circuit and highlights the promise and challenges associated with hybrid learning schemes.

The development of near-term quantum computers, referred to as Noisy Intermediate-Scale Quantum (NISQ) computers, has progressed rapidly in the past few years resulting in several quantum computers which vary in their underlying technology and physical constraints. The performance of these computers also varies from one quantum algorithm to another. To enable efficient selection of the quantum computer that provides the highest output fidelity for a given application, an accurate noise modeling of each quantum hardware is required. However, noise modeling for a given application is a complex problem because of the unknown interaction between the quantum circuit parameters and the noise parameters of NISQ devices. We propose the use of Machine Learning (ML) to model the performance of different quantum computers at the application level. The ML models predict the output fidelity of the quantum application executed on different quantum computers given their publicly available physical constraints. We use a diverse training dataset to cover the key features for application-level benchmarking of the quantum hardware. Our results obtained from different superconducting quantum devices show that our proposed ML models enable cost-effective quantum computer selection for different quantum applications with different fidelity metrics.

Grover’s quantum search algorithm promises a quadratic speedup for unstructured search over its classical counterpart. But this advantage is affected by noise acting on the search space. Here, we show that a quantum switch can act as a resource to mitigate the effects of noise. In this scenario, the noise is modeled by a depolarizing channel, which coherently acts on the entire quantum register. We show that a quantum switch can significantly reduce the error in Grover’s search algorithm. We consider the success probability of finding the marked item as the sole quantifier of diminishing the effect of noise in the search space in the presence of quantum switch. We propose two frameworks for the application of quantum switches. In the first framework, we apply the superposition of channel’s orders in the form of a switch and do a post-selection at every iteration of the applications of the Grover operator. In the second framework, we delay this measurement and post-selection until the very end. The number of post selections is minimal in the second scenario, and hence the noise reduction can be attributed more to the presence of quantum switch. We illustrate with an example of significant advantage in the success probability of Grover’s algorithm using quantum switch.

Solving electronic structure problems represents a promising field of applications for quantum computers. Currently, much effort is spent in devising and optimizing quantum algorithms for near-term quantum processors, with the aim of outperforming classical counterparts on selected problem instances using limited quantum resources. These methods are still expected to feature a runtime preventing quantum simulations of large scale and bulk systems. In this work, we propose a strategy to extend the scope of quantum computational methods to large scale simulations using a machine learning potential trained on quantum simulation data. The challenge of applying machine learning potentials in today’s quantum setting arises from the several sources of noise affecting the quantum computations of electronic energies and forces. We investigate the trainability of a machine learning potential selecting various sources of noise: statistical, optimization, and hardware noise. Finally, we construct the first machine learning potential from data computed on actual IBM Quantum processors for a hydrogen molecule. This already would allow us to perform arbitrarily long and stable molecular dynamics simulations, outperforming all current quantum approaches to molecular dynamics and structure optimization.

Quantum computing has rapidly advanced in recent years due to substantial development in both hardware and algorithms. These advances are carrying quantum computers closer to their impending commercial utility. Drug discovery is a promising area of application that will find a number of uses for these new machines. As a prominent example, quantum simulation will enable faster and more accurate characterizations of molecular systems than existing quantum chemistry methods. Furthermore, algorithmic developments in quantum machine learning offer interesting alternatives to classical machine learning techniques, which may also be useful for the biochemical efforts involved in early phases of drug discovery. Meanwhile, quantum hardware is scaling up rapidly into a regime where an exact simulation is difficult even using the world’s largest supercomputers. We review how these recent advances can shift the paradigm with which one thinks about drug discovery, focusing on both the promises and caveats associated with each development. In particular, we highlight how hybrid quantum-classical approaches to quantum simulation and quantum machine learning could yield substantial progress using noisy-intermediate scale quantum devices, whereas fault-tolerant, error-corrected quantum computers are still in their development phase.