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

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.

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.

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.

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.

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.

This work designs near-term quantum algorithms for linear systems of equations based on the classical combination of variational quantum states (CQS), and exhibits several provable guarantees for these algorithms, supported by the representation of the linear system on a so-called ansatz tree.

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.

A quantum linear solver algorithm combining ideas from adiabatic quantum computing with filtering techniques based on quantum signal processing is introduced, which reduces the cost of quantumlinear solvers over state-of-the-art close to an order of magnitude for early implementations.

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.

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.

The algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation, and allows the quantum phase estimation algorithm, whose dependence on $\epsilon$ is prohibitive, to be bypassed.

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.

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.

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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