[PDF] A near-term quantum algorithm for solving linear systems of equations based on the Woodbury identity | Semantic Scholar (2024)

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6 Citations

Near term algorithms for linear systems of equations
    Aidan Pellow-JarmanI. SinayskiyA. PillayFrancesco Petruccione

    Computer Science, Physics

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This paper focuses on the Variational Quantum Linear Solvers (VQLS), and other closely related methods and adaptions, and implements and contrasts the first application of the Evolutionary Ansatz to the VQLS, the first implementation of the Logical Ansatz VZLS, based on the Classical Combination of Quantum States (CQS) method.

Identifying Bottlenecks of NISQ-friendly HHL algorithms
    Marc Andreu MarfanyAlona SakhnenkoJeanette Miriam Lorenz

    Physics, Computer Science

  • 2024

This work performs an empirical study to test scaling properties and directly related noise resilience of the the most resources-intense component of the HHL algorithm, namely QPE and its NISQ-adaptation Iterative QPE and deduces an approximate bottleneck for algorithms that consider a similar time evolution as QPE.

Quantum algorithms for geologic fracture networks
    Jessie M. HendersonM. Podzorova Daniel O’Malley

    Physics, Computer Science

    Scientific Reports

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Two quantum algorithms for fractured flow are introduced, designed for future quantum computers which operate without error and designed to be noise resilient, which already performs well for problems of small to medium size.

Addressing Quantum's"Fine Print": State Preparation and Information Extraction for Quantum Algorithms and Geologic Fracture Networks
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    Computer Science, Physics

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This work addresses two further requirements for solving geologic fracture flow systems with quantum algorithms: efficient system state preparation and efficient information extraction that are consistent with an overall exponential speed-up.

Novel real number representations in Ising machines and performance evaluation: Combinatorial random number sum and constant division
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This work proposes three new types of real number representation and compared these representations under the problem of solving linear equations, finding experimentally that the accuracy of the solution varies significantly depending on how the real numbers are represented.

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Near-term quantum algorithms for linear systems of equations with regression loss functions

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Variational Quantum Linear Solver
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It is proved that C⩾ϵ2/κ2, where C is the VQLS cost function and κ is the condition number of A, is the operationally meaningful termination condition for VZLS that allows one to guarantee that a desired solution precision ϵ is achieved.

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This work exhibits a quantum algorithm for estimating x(-->)(dagger) Mx(-->) whose runtime is a polynomial of log(N) and kappa, and proves that any classical algorithm for this problem generically requires exponentially more time than this quantum algorithm.

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Two quantum algorithms based on evolution randomization, a simple variant of adiabatic quantum computing, to prepare a quantum state |x⟩ that is proportional to the solution of the system of linear equations Ax[over →]=b[ over →], yielding an exponential quantum speed-up under some assumptions.

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This paper generalizes quantum amplitude amplification to the case when parts of the algorithm that is being amplified stop at different times, and applies it to give two new quantum algorithms for linear algebra problems.

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Operator Sampling for Shot-frugal Optimization in Variational Algorithms
    A. ArrasmithL. CincioR. SommaPatrick J. Coles

    Chemistry, Computer Science

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This work introduces a strategy for reducing the number of measurements by randomly sampling operators from the overall Hamiltonian, and implements an improved optimizer called Rosalin (Random Operator Sampling for Adaptive Learning with Individual Number of shots), which outperforms other optimizers in most cases.

Quantum state verification in the quantum linear systems problem
    R. SommaY. Subaşı

    Physics, Computer Science

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The complexity of quantum state verification in the context of solving systems of linear equations of the form $A \vec x = \vec b$ is analyzed, where state preparation, gate, and measurement errors will need to decrease rapidly with $\kappa$ for worst-case and typical instances if error correction is not used, and present some open problems.



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    [PDF] A near-term quantum algorithm for solving linear systems of equations based on the Woodbury identity | Semantic Scholar (2024)


    What is the quantum algorithm for solving linear systems? ›

    The Harrow–Hassidim–Lloyd algorithm or HHL algorithm is a quantum algorithm for numerically solving a system of linear equations, designed by Aram Harrow, Avinatan Hassidim, and Seth Lloyd. The algorithm estimates the result of a scalar measurement on the solution vector to a given linear system of equations.

    What is the most famous quantum algorithm? ›

    The best-known algorithms are Shor's algorithm for factoring and Grover's algorithm for searching an unstructured database or an unordered list. Shor's algorithm runs much (almost exponentially) faster than the best-known classical algorithm for factoring, the general number field sieve.

    What is the full form of HHL in computer? ›

    Harrow-Hassidim-Lloyd (HHL) quantum algorithm, which can solve linear system problems with exponential speed-up over the classical method and is the basis of many important quantum computing algorithms, is used to serve this purpose.

    What is the HHL algorithm for quantum computing? ›

    What is HHL in Quantum Computing? The HHL Algorithm, named after its creators Harrow, Hassidim, and Lloyd, is a quantum algorithm designed to solve linear systems of equations. Given a system of equations represented by Ax = b , where A is a matrix and b is a vector, the HHL Algorithm finds the solution vector \( x \).

    What math is used in quantum? ›

    The main tools include: linear algebra: complex numbers, eigenvectors, eigenvalues. functional analysis: Hilbert spaces, linear operators, spectral theory. differential equations: partial differential equations, separation of variables, ordinary differential equations, Sturm–Liouville theory, eigenfunctions.

    What is an example of a simple quantum algorithm? ›

    The best-known examples are Shor's algorithm and Grover's algorithm. Shor's algorithm is a quantum algorithm for integer factorization. Simply put, when given an integer N, it will find its prime factors. It can solve this problem exponentially faster than the best-known classical algorithm can.

    What language is used in quantum algorithms? ›

    Q# Developed by Microsoft, Q# (pronounced as 'Q sharp') is a domain-specific programming language used for expressing quantum algorithms. It is integrated with the . NET framework and can work with classical languages like C# and Python.

    Who is the king of quantum physics? ›

    If physicists wrote history, we would now be in the second century of our era, specifically the year 116 of Planck, the German physicist who changed our view of the world when he laid the cornerstone of quantum theory in the year 1900 (of the Christian era.)

    Why is designing quantum algorithms difficult? ›

    difficult to go about finding a quantum algorithm compared to classical algorithms because quantum computers are very different than classical computers, so the approach to an algorithm is very different too. speed-up cannot arise from problems that have polynomial-time classical algorithms, like P AND NP).

    What does Hartron stand for? ›

    Haryana State Electronics Development Corporation Ltd. ( HARTRON)

    What does HPCs stand for? ›

    High-Performance Computers (HPCs) form the computing backbone of the software-defined vehicle. They provide the necessary computing power for the software functions in the vehicle enabling them to be further developed throughout their life cycle.

    What does MBL stand for in computer? ›

    Microcomputer-Based Laboratories (MBL)

    How long would it take a quantum computer to crack AES 256? ›

    If a classical computer needs 2^256 operations to brute force a 256 bit key, a quantum computer would need 2^128 operations. That's still a huge number, dude. Even if you had a quantum computer with millions of qubits (which we don't have yet), it would still take years or decades to crack 256 bit encryption.

    What code do quantum computers use? ›

    QCL. Quantum Computation Language (QCL) is one of the first implemented quantum programming languages. The most important feature of QCL is the support for user-defined operators and functions.

    What is the algorithm for solving systems of linear equations? ›

    In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients.

    What is the quantum linear system theory? ›

    Quantum linear systems are a class of systems whose dynamics, which are described by the laws of quantum mechanics, take the specific form of a set of linear quantum stochastic differential equations (QSDEs).

    What are the algorithms for linear problems? ›

    The simplex algorithm, developed by George Dantzig in 1947, solves LP problems by constructing a feasible solution at a vertex of the polytope and then walking along a path on the edges of the polytope to vertices with non-decreasing values of the objective function until an optimum is reached for sure.

    Which algorithm is used in quantum computing? ›

    One of the first applications of quantum computers discovered was Shor's algorithm for integer factorisation. In the factorisation problem, given an integer N=p×q for some prime numbers p and q, our task is to determine p and q.


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