On the quantum version of the shifted power method

time. The method can be potentially used in any eigenvalue related problems and to find minimum/maximum of a quantum state in lieu of Grover's search algorithm. In addition, if the solution space of an optimization problem with

n

parameters can be encoded as the eigenspace of an

2^n

dimensional unitary operator in

O(poly(n))

time, then the solution for such a problem can be found in

O(poly(n))

if the eigengap is polynomial. As an example, using the quantum gates, we show how to generate the solution space of the quadratic unconstrained binary optimization as eigenvectors of a diagonal unitary matrix and find the solution for the problem.

Related papers

The quantum version of the shifted power method and its application in quadratic binary optimization

Ammar Daskin

In this paper, we present a direct quantum adaptation of the classical shifted power method. The method is very similar to the iterative phase estimation algorithm; however, it does not require any initial estimate of an eigenvector, and as in the classical case its convergence and the required number of iterations are directly related to the eigengap. If the amount of the gap is in the order of $1/poly(n)$ , then the algorithm can converge to the dominant eigenvalue in $O(poly(n))$ time. The method can be potentially used for solving any eigenvalue related problem and finding minimum/maximum of a data set in lieu of Grover's search algorithm. In addition, if the solution space of an optimization problem with $n$ parameters is encoded as the eigenspace of an $2^n$ dimensional unitary operator in $O(poly(n))$ time and the eigengap is not too small, then the solution for such a problem can be found in $O(poly(n))$ . As an example, using the quantum gates, we show how to generate the solution space of the quadratic unconstrained binary optimization as the eigenvectors of a diagonal unitary matrix and find the solution for the problem.In this paper, we present a direct quantum adaptation of the classical shifted power method. The method is very similar to the iterative phase estimation algorithm; however, it does not require any initial estimate of an eigenvector, and as in the classical case its convergence and the required number of iterations are directly related to the eigengap. If the amount of the gap is in the order of $1/poly(n)$, then the algorithm can converge to the dominant eigenvalue in $O(poly(n))$ time. The method can be potentially used for solving any eigenvalue related problem and finding minimum/maximum of a data set in lieu of Grover's search algorithm. In addition, if the solution space of an optimization problem with $n$ parameters is encoded as the eigenspace of an $2^n$ dimensional unitary operator in $O(poly(n))$ time and the eigengap is not too small, then the solution for such a problem can be found in $O(poly(n))$. As an example, using the quantum gates, we show how to generate the solution space of the quadratic unconstrained binary optimization as the eigenvectors of a diagonal unitary matrix and find the solution for the problem.

arXiv:1809.01378TURKISH JOURNAL OF ELECTRICAL ENGINEERING & COMPUTER SCIENCESJan 7, 20202 citations

Quantum estimates for classical polynomial optimization

Oleg Evnin

The problem of finding lower and upper bounds on multivariate homogeneous polynomials is both difficult and important given its applications to questions ranging from dynamical stability in complex potential landscapes to data analysis. From the standpoint of tensor eigenvalue theory, the question is equivalent to finding the smallest and the largest eigenvalues of the coefficient tensor corresponding to the given polynomial. Standard approaches outlined in the literature amount to running nonlinear iterations in search for the optimal rays along which the growth of the polynomial is fastest or slowest. Unlike the case of matrices (or their corresponding multivariate quadratic forms) convergence of such algorithms for higher-rank tensors is capricious due to the complex topography of polynomial objective functions. In this essay, a very different strategy, inspired by quantum-mechanical variational methods, is introduced for finding bounds on polynomials. The original polynomial is replaced by an operator acting in a suitably chosen (large) space of states, such that in an appropriate "classical" limit this operator approaches the original polynomial expression made of commutative variables. As a result, approximating the smallest and largest eigenvalues of the coefficient tensor, and thus finding bounds on polynomials, amounts to diagonalizing the resulting quantum operator, represented as a large matrix, and then inspecting the smallest and largest eigenvalues of this matrix. This approach is then successfully applied to standard test examples from tensor eigenvalue literature and other problems of interest in mathematical physics including Strichartz-type inequalities.The problem of finding lower and upper bounds on multivariate homogeneous polynomials is both difficult and important given its applications to questions ranging from dynamical stability in complex potential landscapes to data analysis. From the standpoint of tensor eigenvalue theory, the question is equivalent to finding the smallest and the largest eigenvalues of the coefficient tensor corresponding to the given polynomial. Standard approaches outlined in the literature amount to running nonlinear iterations in search for the optimal rays along which the growth of the polynomial is fastest or slowest. Unlike the case of matrices (or their corresponding multivariate quadratic forms) convergence of such algorithms for higher-rank tensors is capricious due to the complex topography of polynomial objective functions. In this essay, a very different strategy, inspired by quantum-mechanical variational methods, is introduced for finding bounds on polynomials. The original polynomial is replaced by an operator acting in a suitably chosen (large) space of states, such that in an appropriate "classical" limit this operator approaches the original polynomial expression made of commutative variables. As a result, approximating the smallest and largest eigenvalues of the coefficient tensor, and thus finding bounds on polynomials, amounts to diagonalizing the resulting quantum operator, represented as a large matrix, and then inspecting the smallest and largest eigenvalues of this matrix. This approach is then successfully applied to standard test examples from tensor eigenvalue literature and other problems of interest in mathematical physics including Strichartz-type inequalities.

math-phCross-listedarXiv:2607.25445Jul 29, 20260 citations

Quantum gradient descent and Newton’s method for constrained polynomial optimization

Rebentrost, Frank Patrick, Seth Lloyd

Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for high-dimensional problems where a small number of iterations is sufficient.Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for high-dimensional problems where a small number of iterations is sufficient.

Jul 1, 2019127 citations

Quantum gradient descent and Newton’s method for constrained polynomial optimization

Patrick Rebentrost, Maria Schuld, Leonard Wossnig +2 more

Abstract Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton’s method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for high-dimensional problems where a small number of iterations is sufficient.Abstract Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton’s method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for high-dimensional problems where a small number of iterations is sufficient.

arXiv:1612.01789New Journal of PhysicsJun 20, 201915 citations

A quantum analogue of convex optimization

Eunou Lee

Convex optimization is the powerhouse behind the theory and practice of optimization. We introduce a quantum analogue of unconstrained convex optimization: computing the minimum eigenvalue of a Schrödinger operator $h = -Δ+ V $ with convex potential $V:\mathbb R^n \rightarrow \mathbb R_{\ge 0}$ such that $V(x)\rightarrow\infty $ as $\|x\|\rightarrow\infty$ . For this problem, we present an efficient quantum algorithm, called the Fundamental Gap Algorithm (FGA), that computes the minimum eigenvalue of $h$ up to error $ε$ in polynomial time in $n$ , $1/ε$ , and parameters that depend on $V$ . Adiabatic evolution of the ground state is used as a key subroutine, which we analyze with novel techniques that allow us to focus on the low-energy space. We apply the FGA to give the first known polynomial-time algorithm for finding the lowest frequency of an $n$ -dimensional convex drum, or mathematically, the minimum eigenvalue of the Dirichlet Laplacian on an $n$ -dimensional region that is defined by $m$ linear constraints in polynomial time in $n$ , $m$ , $1/ε$ and the radius $R$ of a ball encompassing the region.Convex optimization is the powerhouse behind the theory and practice of optimization. We introduce a quantum analogue of unconstrained convex optimization: computing the minimum eigenvalue of a Schrödinger operator $h = -Δ+ V $ with convex potential $V:\mathbb R^n \rightarrow \mathbb R_{\ge 0}$ such that $V(x)\rightarrow\infty $ as $\|x\|\rightarrow\infty$. For this problem, we present an efficient quantum algorithm, called the Fundamental Gap Algorithm (FGA), that computes the minimum eigenvalue of $h$ up to error $ε$ in polynomial time in $n$, $1/ε$, and parameters that depend on $V$. Adiabatic evolution of the ground state is used as a key subroutine, which we analyze with novel techniques that allow us to focus on the low-energy space. We apply the FGA to give the first known polynomial-time algorithm for finding the lowest frequency of an $n$-dimensional convex drum, or mathematically, the minimum eigenvalue of the Dirichlet Laplacian on an $n$-dimensional region that is defined by $m$ linear constraints in polynomial time in $n$, $m$, $1/ε$ and the radius $R$ of a ball encompassing the region.

arXiv:2510.02151Oct 2, 20250 citations

A Quantum Speed-Up for Approximating the Top Eigenvectors of a Matrix

Yanlin Chen, András Gilyén, Ronald de Wolf

Finding a good approximation of the top eigenvector of a given $d\times d$ matrix $A$ is a basic and important computational problem, with many applications. We give two different quantum algorithms that, given query access to the entries of a Hermitian matrix $A$ and assuming a constant eigenvalue gap, output a classical description of a good approximation of the top eigenvector: one algorithm with time complexity $\mathcal{\tilde{O}}(d^{1.75})$ and one with time complexity $d^{1.5+o(1)}$ (the first algorithm has a slightly better dependence on the $\ell_2$ -error of the approximating vector than the second, and uses different techniques of independent interest). Both of our quantum algorithms provide a polynomial speed-up over the best-possible classical algorithm, which needs $Ω(d^2)$ queries to entries of $A$ , and hence $Ω(d^2)$ time. We extend this to a quantum algorithm that outputs a classical description of the subspace spanned by the top- $q$ eigenvectors in time $qd^{1.5+o(1)}$ . We also prove a nearly-optimal lower bound of $\tildeΩ(d^{1.5})$ on the quantum query complexity of approximating the top eigenvector. Our quantum algorithms run a version of the classical power method that is robust to certain benign kinds of errors, where we implement each matrix-vector multiplication with small and well-behaved error on a quantum computer, in different ways for the two algorithms. Our first algorithm estimates the matrix-vector product one entry at a time, using a new "Gaussian phase estimation" procedure. Our second algorithm uses block-encoding techniques to compute the matrix-vector product as a quantum state, from which we obtain a classical description by a new time-efficient unbiased pure-state tomography procedure.Finding a good approximation of the top eigenvector of a given $d\times d$ matrix $A$ is a basic and important computational problem, with many applications. We give two different quantum algorithms that, given query access to the entries of a Hermitian matrix $A$ and assuming a constant eigenvalue gap, output a classical description of a good approximation of the top eigenvector: one algorithm with time complexity $\mathcal{\tilde{O}}(d^{1.75})$ and one with time complexity $d^{1.5+o(1)}$ (the first algorithm has a slightly better dependence on the $\ell_2$-error of the approximating vector than the second, and uses different techniques of independent interest). Both of our quantum algorithms provide a polynomial speed-up over the best-possible classical algorithm, which needs $Ω(d^2)$ queries to entries of $A$, and hence $Ω(d^2)$ time. We extend this to a quantum algorithm that outputs a classical description of the subspace spanned by the top-$q$ eigenvectors in time $qd^{1.5+o(1)}$. We also prove a nearly-optimal lower bound of $\tildeΩ(d^{1.5})$ on the quantum query complexity of approximating the top eigenvector. Our quantum algorithms run a version of the classical power method that is robust to certain benign kinds of errors, where we implement each matrix-vector multiplication with small and well-behaved error on a quantum computer, in different ways for the two algorithms. Our first algorithm estimates the matrix-vector product one entry at a time, using a new "Gaussian phase estimation" procedure. Our second algorithm uses block-encoding techniques to compute the matrix-vector product as a quantum state, from which we obtain a classical description by a new time-efficient unbiased pure-state tomography procedure.

arXiv:2405.14765May 23, 20240 citations

n

, then the algorithm can converge to the dominant eigenvalue in

O(poly(n))

n

parameters can be encoded as the eigenspace of an

2^n

dimensional unitary operator in

O(poly(n))

time, then the solution for such a problem can be found in

O(poly(n))

1/poly(n)

, then the algorithm can converge to the dominant eigenvalue in

O(poly(n))

time. The method can be potentially used for solving any eigenvalue related problem and finding minimum/maximum of a data set in lieu of Grover's search algorithm. In addition, if the solution space of an optimization problem with

n

parameters is encoded as the eigenspace of an

2^n

dimensional unitary operator in

O(poly(n))

time and the eigengap is not too small, then the solution for such a problem can be found in

O(poly(n))

. As an example, using the quantum gates, we show how to generate the solution space of the quadratic unconstrained binary optimization as the eigenvectors of a diagonal unitary matrix and find the solution for the problem.

V:\mathbb R^n \rightarrow \mathbb R_{\ge 0}

such that $V(x)\rightarrow\infty $ as

\|x\|\rightarrow\infty

. For this problem, we present an efficient quantum algorithm, called the Fundamental Gap Algorithm (FGA), that computes the minimum eigenvalue of

h

up to error

ε

in polynomial time in

n

1/ε

, and parameters that depend on

V

. Adiabatic evolution of the ground state is used as a key subroutine, which we analyze with novel techniques that allow us to focus on the low-energy space. We apply the FGA to give the first known polynomial-time algorithm for finding the lowest frequency of an

n

-dimensional convex drum, or mathematically, the minimum eigenvalue of the Dirichlet Laplacian on an

n

-dimensional region that is defined by

m

linear constraints in polynomial time in

n

m

1/ε

and the radius

R

of a ball encompassing the region.

Finding a good approximation of the top eigenvector of a given

d\times d

matrix

A

is a basic and important computational problem, with many applications. We give two different quantum algorithms that, given query access to the entries of a Hermitian matrix

A

and assuming a constant eigenvalue gap, output a classical description of a good approximation of the top eigenvector: one algorithm with time complexity

\mathcal{\tilde{O}}(d^{1.75})

and one with time complexity

d^{1.5+o(1)}

(the first algorithm has a slightly better dependence on the

\ell_2

-error of the approximating vector than the second, and uses different techniques of independent interest). Both of our quantum algorithms provide a polynomial speed-up over the best-possible classical algorithm, which needs

Ω(d^2)

queries to entries of

A

, and hence

Ω(d^2)

time. We extend this to a quantum algorithm that outputs a classical description of the subspace spanned by the top-

q

eigenvectors in time

qd^{1.5+o(1)}

. We also prove a nearly-optimal lower bound of

\tildeΩ(d^{1.5})

on the quantum query complexity of approximating the top eigenvector. Our quantum algorithms run a version of the classical power method that is robust to certain benign kinds of errors, where we implement each matrix-vector multiplication with small and well-behaved error on a quantum computer, in different ways for the two algorithms. Our first algorithm estimates the matrix-vector product one entry at a time, using a new "Gaussian phase estimation" procedure. Our second algorithm uses block-encoding techniques to compute the matrix-vector product as a quantum state, from which we obtain a classical description by a new time-efficient unbiased pure-state tomography procedure.