Analytical results for the Quantum Alternating Operator Ansatz with Grover Mixer
- URL: http://arxiv.org/abs/2401.11056v2
- Date: Thu, 28 Mar 2024 21:29:09 GMT
- Title: Analytical results for the Quantum Alternating Operator Ansatz with Grover Mixer
- Authors: Guilherme Adamatti Bridi, Franklin de Lima Marquezino,
- Abstract summary: We introduce a statistical approach to analyze QAOA with Grover mixer.
We get an expression for the expectation value independent of the number of layers.
We generalize all the bounds by using a contradiction argument with the optimality of Grover's algorithm.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: An important property of QAOA with Grover mixer is that its expectation value is invariant over any permutation of states. As a consequence, the algorithm is independent of the structure of the problem. If, on the one hand, this characteristic raises serious doubts about the capacity of the algorithm to overcome the bound of the unstructured search problem, on the other hand, it can pave the way to its analytical study. In this sense, a prior work introduced a statistical approach to analyze GM-QAOA that results in an analytical expression for the expectation value depending on the probability distribution associated with the problem Hamiltonian spectrum. Although the method provides surprising simplifications in calculations, the expression depends exponentially on the number of layers, which makes direct analytical treatment unfeasible. In this work, we extend the analysis to the more simple context of Grover Mixer Threshold QAOA (GM-Th-QAOA), a variant that replaces the phase separation operator of GM-QAOA to encode a threshold function. As a result, we obtain an expression for the expectation value independent of the number of layers and, with it, we provide bounds for different performance metrics. Furthermore, we extend the analysis to a more general context of QAOA with Grover mixer, which we called Grover-based QAOA. In that framework, which allows the phase separation operator to encode any compilation of the cost function, we generalize all the bounds by using a contradiction argument with the optimality of Grover's algorithm on the unstructured search problem. As a result, we get the main contribution of this work, an asymptotic lower bound on the quantile achieved by the expectation value that formalizes the notion that the Grover mixer, at most, reflects a quadratic Grover-style speed-up over classical brute force.
Related papers
- Performance Upper Bound of Grover-Mixer Quantum Alternating Operator Ansatz [3.5023108034606256]
The Quantum Alternating Operator Ansatz (QAOA) represents a branch of quantum algorithms for solving optimization problems.
A specific variant, the Grover-Mixer Quantum Alternating Operator Ansatz (GM-QAOA), ensures uniform amplitude across states that share equivalent objective values.
We show that the GM-QAOA provides a quadratic enhancement in sampling probability and requires circuit depth that scales exponentially with problem size to maintain consistent performance.
arXiv Detail & Related papers (2024-05-06T05:47:27Z) - Quantum Alternating Operator Ansatz (QAOA) beyond low depth with
gradually changing unitaries [0.0]
We study the underlying mechanisms governing the behavior of Quantum Alternating Operator Ansatz circuits.
We use the discrete adiabatic theorem, which complements and generalizes the insights obtained from the continuous-time adiabatic theorem.
Our analysis explains some general properties that are conspicuously depicted in the recently introduced QAOA performance diagrams.
arXiv Detail & Related papers (2023-05-08T04:21:42Z) - Quasi-parametric rates for Sparse Multivariate Functional Principal
Components Analysis [0.0]
We show that the eigenelements can be expressed as the solution to an optimization problem.
We establish a minimax lower bound on the mean square reconstruction error of the eigenelement, which proves that the procedure has an optimal variance in the minimax sense.
arXiv Detail & Related papers (2022-12-19T13:17:57Z) - Equivariant Transduction through Invariant Alignment [71.45263447328374]
We introduce a novel group-equivariant architecture that incorporates a group-in hard alignment mechanism.
We find that our network's structure allows it to develop stronger equivariant properties than existing group-equivariant approaches.
We additionally find that it outperforms previous group-equivariant networks empirically on the SCAN task.
arXiv Detail & Related papers (2022-09-22T11:19:45Z) - Clipped Stochastic Methods for Variational Inequalities with
Heavy-Tailed Noise [64.85879194013407]
We prove the first high-probability results with logarithmic dependence on the confidence level for methods for solving monotone and structured non-monotone VIPs.
Our results match the best-known ones in the light-tails case and are novel for structured non-monotone problems.
In addition, we numerically validate that the gradient noise of many practical formulations is heavy-tailed and show that clipping improves the performance of SEG/SGDA.
arXiv Detail & Related papers (2022-06-02T15:21:55Z) - General Hamiltonian Representation of ML Detection Relying on the
Quantum Approximate Optimization Algorithm [74.6114458993128]
The quantum approximate optimization algorithm (QAOA) conceived for solving optimization problems can be run on the existing noisy intermediate-scale quantum (NISQ) devices.
We solve the maximum likelihood (ML) detection problem for general constellations by appropriately adapting the QAOA.
In particular, for an M-ary Gray-mapped quadrature amplitude modulation (MQAM) constellation, we show that the specific qubits encoding the in-phase components and those encoding the quadrature components are independent in the quantum system of interest.
arXiv Detail & Related papers (2022-04-11T14:11:24Z) - Numerical Evidence for Exponential Speed-up of QAOA over Unstructured
Search for Approximate Constrained Optimization [0.0]
We present evidence for an exponential speed-up of QAOA over Grover-style unstructured search.
Our result suggests that maximizing QAOA performance requires a judicious choice of mixer and phase separator.
arXiv Detail & Related papers (2022-02-01T18:39:52Z) - Differentiable Annealed Importance Sampling and the Perils of Gradient
Noise [68.44523807580438]
Annealed importance sampling (AIS) and related algorithms are highly effective tools for marginal likelihood estimation.
Differentiability is a desirable property as it would admit the possibility of optimizing marginal likelihood as an objective.
We propose a differentiable algorithm by abandoning Metropolis-Hastings steps, which further unlocks mini-batch computation.
arXiv Detail & Related papers (2021-07-21T17:10:14Z) - Threshold-Based Quantum Optimization [0.0]
Th-QAOA (pronounced Threshold QAOA) is a variation of the Quantum Alternating Operator Ansatz (QAOA)
We focus on a combination with the Grover Mixer operator; the resulting GM-Th-QAOA can be viewed as a generalization of Grover's quantum search algorithm.
arXiv Detail & Related papers (2021-06-25T19:36:49Z) - VAE Approximation Error: ELBO and Conditional Independence [78.72292013299868]
This paper analyzes VAE approximation errors caused by the combination of the ELBO objective with the choice of the encoder probability family.
We show that the ELBO subset can not be enlarged, and the respective error cannot be decreased, by only considering deeper encoder networks.
arXiv Detail & Related papers (2021-02-18T12:54:42Z) - Total Deep Variation: A Stable Regularizer for Inverse Problems [71.90933869570914]
We introduce the data-driven general-purpose total deep variation regularizer.
In its core, a convolutional neural network extracts local features on multiple scales and in successive blocks.
We achieve state-of-the-art results for numerous imaging tasks.
arXiv Detail & Related papers (2020-06-15T21:54:15Z)
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.