Quantum Legendre-Fenchel Transform
- URL: http://arxiv.org/abs/2006.04823v3
- Date: Wed, 17 Mar 2021 07:07:52 GMT
- Title: Quantum Legendre-Fenchel Transform
- Authors: David Sutter, Giacomo Nannicini, Tobias Sutter, Stefan Woerner
- Abstract summary: We present a quantum algorithm to compute the discrete Legendre-Fenchel transform.
We show that our quantum algorithm is optimal up to polylogarithmic factors.
- Score: 6.643082745560234
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We present a quantum algorithm to compute the discrete Legendre-Fenchel
transform. Given access to a convex function evaluated at $N$ points, the
algorithm outputs a quantum-mechanical representation of its corresponding
discrete Legendre-Fenchel transform evaluated at $K$ points in the transformed
space. For a fixed regular discretization of the dual space the expected
running time scales as $O(\sqrt{\kappa}\,\mathrm{polylog}(N,K))$, where
$\kappa$ is the condition number of the function. If the discretization of the
dual space is chosen adaptively with $K$ equal to $N$, the running time reduces
to $O(\mathrm{polylog}(N))$. We explain how to extend the presented algorithm
to the multivariate setting and prove lower bounds for the query complexity,
showing that our quantum algorithm is optimal up to polylogarithmic factors.
For multivariate functions with $\kappa=1$, the quantum algorithm computes a
quantum-mechanical representation of the Legendre-Fenchel transform at $K$
points exponentially faster than any classical algorithm can compute it at a
single point.
Related papers
- Quantum algorithm for the gradient of a logarithm-determinant [0.0]
The inverse of a sparse-rank input operator may be determined efficiently.
Measuring an expectation value of the quantum state--instead of all $N2$ elements of the input operator--can be accomplished in $O(ksigma)$ time.
The algorithm is envisioned for fully error-corrected quantum computers but may be implementable on near-term machines.
arXiv Detail & Related papers (2025-01-16T09:39:31Z) - Fast Laplace transforms on quantum computers [0.0]
We introduce the Quantum Laplace Transform (QLT), which enables the implementation of $Ntimes N$ discrete Laplace transforms on quantum states encoded in $lceil log_2(N)rceil$-qubits.
In many cases, the associated quantum circuits have a depth that scales with $N$ as $O(log(N))$ and a size that scales as $O(log(N))$, requiring exponentially fewer operations and double-exponentially less computational time than their classical counterparts.
arXiv Detail & Related papers (2024-12-06T16:44:00Z) - Calculating response functions of coupled oscillators using quantum phase estimation [40.31060267062305]
We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer.
Our proposed quantum algorithm operates in the standard $s-sparse, oracle-based query access model.
We show that a simple adaptation of our algorithm solves the random glued-trees problem in time.
arXiv Detail & Related papers (2024-05-14T15:28:37Z) - Quantum algorithms for Hopcroft's problem [45.45456673484445]
We study quantum algorithms for Hopcroft's problem which is a fundamental problem in computational geometry.
The classical complexity of this problem is well-studied, with the best known algorithm running in $O(n4/3)$ time.
Our results are two different quantum algorithms with time complexity $widetilde O(n5/6)$.
arXiv Detail & Related papers (2024-05-02T10:29:06Z) - Efficient unitary designs and pseudorandom unitaries from permutations [35.66857288673615]
We show that products exponentiated sums of $S(N)$ permutations with random phases match the first $2Omega(n)$ moments of the Haar measure.
The heart of our proof is a conceptual connection between the large dimension (large-$N$) expansion in random matrix theory and the method.
arXiv Detail & Related papers (2024-04-25T17:08:34Z) - Towards large-scale quantum optimization solvers with few qubits [59.63282173947468]
We introduce a variational quantum solver for optimizations over $m=mathcalO(nk)$ binary variables using only $n$ qubits, with tunable $k>1$.
We analytically prove that the specific qubit-efficient encoding brings in a super-polynomial mitigation of barren plateaus as a built-in feature.
arXiv Detail & Related papers (2024-01-17T18:59:38Z) - Quantum Algorithms for the Pathwise Lasso [1.8058773918890538]
We present a novel quantum high-dimensional linear regression algorithm based on the classical LARS (Least Angle Regression) pathwise algorithm.
Our quantum algorithm provides the full regularisation path as the penalty term varies, but quadratically faster per iteration under specific conditions.
We prove, via an approximate version of the KKT conditions and a duality gap, that the LARS algorithm is robust to errors.
arXiv Detail & Related papers (2023-12-21T18:57:54Z) - Quantum mutual information redistribution by Number Partitioning
algorithm [9.818805141128935]
We show how a bipartite unitary transformation $U_AB$ redistributes quantum mutual information with the third party $C$ in a tripartite pure state $|psirangle_ABC$ in a $d_Atimes d_Btimes d_C$ dimensional Hilbert space.
Our approximate algorithm provides a practical protocol to implement redistribution of quantum mutual information for a tripartite quantum state with high dimensions.
arXiv Detail & Related papers (2023-06-17T09:00:11Z) - Quantum Algorithms for Sampling Log-Concave Distributions and Estimating
Normalizing Constants [8.453228628258778]
We develop quantum algorithms for sampling logconcave distributions and for estimating their normalizing constants.
We exploit quantum analogs of the Monte Carlo method and quantum walks.
We also prove a $1/epsilon1-o(1)$ quantum lower bound for estimating normalizing constants.
arXiv Detail & Related papers (2022-10-12T19:10:43Z) - Exponential Separation between Quantum and Classical Ordered Binary
Decision Diagrams, Reordering Method and Hierarchies [68.93512627479197]
We study quantum Ordered Binary Decision Diagrams($OBDD$) model.
We prove lower bounds and upper bounds for OBDD with arbitrary order of input variables.
We extend hierarchy for read$k$-times Ordered Binary Decision Diagrams ($k$-OBDD$) of width.
arXiv Detail & Related papers (2022-04-22T12:37:56Z) - Quantum Gram-Schmidt Processes and Their Application to Efficient State
Read-out for Quantum Algorithms [87.04438831673063]
We present an efficient read-out protocol that yields the classical vector form of the generated state.
Our protocol suits the case that the output state lies in the row space of the input matrix.
One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
arXiv Detail & Related papers (2020-04-14T11:05:26Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.