Solving linear systems of equations is one of the most common and basic problems in classical identification systems. Given a coefficient matrix A and a vector b, the ultimate task is to find the solution x such that Ax=b. Based on the technique of the singular value estimation, the paper proposes a modified quantum scheme to obtain the quantum state |x⟩ corresponding to the solution of the linear system of equations in O(κ2rpolylog(mn)/ϵ) time for a general m×n dimensional A, which is superior to existing quantum algorithms, where κ is the condition number, r is the rank of matrix A and ϵ is the precision parameter. Meanwhile, we also design a quantum circuit for the homogeneous linear equations and achieve an exponential improvement. The coefficient matrix A in our scheme is a sparsity-independent and non-square matrix, which can be applied in more general situations. Our research provides a universal quantum linear system solver and can enrich the research scope of quantum computation.
Solving linear systems of equations is one of the most common and basic problems in classical identification systems. Given a coefficient matrix A and a vector b, the ultimate task is to find the solution x such that Ax=b. Based on the technique of the singular value estimation, the paper proposes a modified quantum scheme to obtain the quantum state |x⟩ corresponding to the solution of the linear system of equations in O(κ2rpolylog(mn)/ϵ) time for a general m×n dimensional A, which is superior to existing quantum algorithms, where κ is the condition number, r is the rank of matrix A and ϵ is the precision parameter. Meanwhile, we also design a quantum circuit for the homogeneous linear equations and achieve an exponential improvement. The coefficient matrix A in our scheme is a sparsity-independent and non-square matrix, which can be applied in more general situations. Our research provides a universal quantum linear system solver and can enrich the research scope of quantum computation.
Solving linear systems of equations is a frequently encountered problem in machine learning and optimization. Given a matrix A and a vector b the task is to find the vector x such that Ax=b. We describe a quantum algorithm that achieves a sparsity-independent runtime scaling of $\mathcal{O}\mathbf{(}{\ensuremath{\kappa}}^{2}\sqrt{n}\text{polylog}(n)/\ensuremath{\epsilon}\mathbf{)}$ for an $n\ifmmode\times\else\texttimes\fi{}n$ dimensional A with bounded spectral norm, where $\ensuremath{\kappa}$ denotes the condition number of A, and $\ensuremath{\epsilon}$ is the desired precision parameter. This amounts to a polynomial improvement over known quantum linear system algorithms when applied to dense matrices, and poses a new state of the art for solving dense linear systems on a quantum computer. Furthermore, an exponential improvement is achievable if the rank of A is polylogarithmic in the matrix dimension. Our algorithm is built upon a singular value estimation subroutine, which makes use of a memory architecture that allows for efficient preparation of quantum states that correspond to the rows of A and the vector of Euclidean norms of the rows of A.
We introduce a quantum linear system solving algorithm based on the Kaczmarz method, a widely used workhorse for large linear systems and least-squares problems that updates the solution by enforcing one equation at a time. Its simplicity and low memory cost make it a practical choice across data regression, tomographic reconstruction, and optimization. In contrast to many existing quantum linear solvers, our method does not rely on oracle access to query entries, relaxing a key practicality bottleneck. In particular, when the rank of the system of interest is sufficiently small and the rows of the matrix of interest admit an appropriate structure, we achieve circuit complexity O(ε1logm), where m is the number of variables and ε is the target precision, without dependence on the sparsity s, and could possibly be without explicit dependence on condition number κ. This shows a significant improvement over previous quantum linear solvers where the dependence on κ,s is at least linear. At the same time, when the rows have an arbitrary structure and have at most s nonzero entries, we obtain the circuit depth O(ε1logs) using extra O(s) ancilla qubits, so the depth grows only logarithmically with sparsity s. When the sparsity s grows as O(logm), then our method can achieve an exponential improvement with respect to circuit depth compared to existing quantum algorithms, while using (asymptotically) the same amount of qubits.
Abstract Linear regression is one of the most important and common analytical methods in mathematical statistics. The letter studies a general model of linear regression problem based on least squares method, and investigates the impact of quantum algorithms on the time complexity of solving linear regression problem when quantum algorithms can be implemented on quantum computers. With the help of the general model of linear regression problem in terms of least squares method, we propose a novel simplified quantum scheme for solving linear regression equation based on the sparsity-independent quantum singular value estimation algorithm. For a linear regression equation with dimension n, our scheme can reduce the time complexity from O(Nn 2) to O ( N log N ) when both the condition number κ of related matrices and the reciprocal of precision ∈ are small in size O(play logN), where m is the dimension of the input u and N is the number of samples.
Abstract Solving linear systems of equations is of vital importance in various fields of science. With the scale of calculation soaring, classical computers are intractable, and the advantage of quantum computing is reflected. Quantum algorithm has exponential acceleration in solving the linear systems A x ⃗ = b ⃗ for sparse and well-conditioned coefficient matrix A with dimension N × N . In this paper, we present a quantum algorithm to solve matrix equations of the form AX = B , where X = ( x ⃗ 1 , x ⃗ 2 , … , x ⃗ n ) , B = ( b ⃗ 1 , b ⃗ 2 , … , b ⃗ n ) , x ⃗ k and b ⃗ k are unit vectors. The problem can be intuitively treated as solving n linear systems simultaneously, which will needs O ( n log N ) complexity by the famous HHL algorithm. We demonstrate that in our algorithm the scale of complexity is O ( log n N ) . In the error analysis part, phase estimation is discussed in details.
Quantum algorithms for linear systems produce the solution state A − 1 | b ⟩ by querying two oracles: O A that block encodes the coefficient matrix and O b that prepares the initial state. We present a quantum linear system algorithm making Θ ( 1 / p ) queries to O b , which is optimal in the success probability, and O ( κ log ( 1 / p ) ( log log ( 1 / p ) + log ( 1 / ϵ ) ) ) queries to O A , nearly optimal in all parameters including the condition number and accuracy. Notably, our complexity scaling of initial state preparation holds even when p is not known a priori . This contrasts with recent results achieving O ( κ log ( 1 / ϵ ) ) complexity to both oracles, which, while optimal in O A , is highly suboptimal in O b as κ can be arbitrarily larger than 1 / p . In various applications such as solving differential equations, preparing ground states of operators with real spectra, and estimating and transforming eigenvalues of non-normal matrices, we can further improve the dependence on p using a block preconditioning scheme to nearly match or outperform best p
Given a matrix A of dimension M×N and a vector b, the quantum linear system (QLS) problem asks for the preparation of a quantum state ∣y⟩ proportional to the solution of Ay=b. Existing QLS algorithms have runtimes that scale linearly with the condition number κ(A), the sparsity of A, and logarithmically with inverse precision, but often overlook structural properties of b, whose alignment with A's eigenspaces can greatly affect performance. In this work, we present a new QLS algorithm that explicitly leverages the structure of the right-hand side vector b. The runtime of our algorithm depends polynomially on the sparsity of the augmented matrix H=[A,−b], the inverse precision, the ℓ2 norm of the solution y=A+b, and a new instance-dependent parameter ET=∑i=1Mpi2⋅di, where p=(AA⊤)+b, and di denotes the squared ℓ2 norm of the i-th row of H. We also introduce a structure-aware rescaling technique tailored to the solution y=A+b. Unlike left preconditioning methods, which transform the linear system to DAy=Db, our approach applies a right rescaling matrix, reformulating the linear system as ADz=b. As an application of our instance-aware QLS algorithm and new rescaling scheme, we develop a quantum algorithm for solving multivariate polynomial systems in regimes where prior QLS-based methods fail. This yields an end-to-end framework applicable to a broad class of problems. In particular, we apply it to the maximum independent set (MIS) problem, formulated as a special case of a polynomial system, and show through detailed analysis that, under certain conditions, our quantum algorithm for MIS runs in polynomial time.