In this short notes, we present the formulation of a kind of quantum algorithm, applicable to the polynomial optimization. The present algorithm introduces the quantum effect by means of the coordinate variables, the momentum variables, and the commutators, and the Plank constant, which shall establish the correspondence between the quantum and the classical cases. The formalism is an extension to the polynomial optimization in the ``classical'' sense; the relations among quantum mechanical values are represented by a semi-algebraic set ( by the set of polynomials and sign conditions), and the optimum is searched in this semi-algebraic set. In the context of this formalism, the optimization of the Ising model [QUBO model] will be a special case. However, the computational procedure proposed here is different from the quantum annealing which is now in vogue. We demonstrate the exemplary calculation through quantifier elimination (which is a kind of symbolic computation applicable to optimization and other problems).
In this short notes, we present the formulation of a kind of quantum algorithm, applicable to the polynomial optimization. The present algorithm introduces the quantum effect by means of the coordinate variables, the momentum variables, and the commutators, and the Plank constant, which shall establish the correspondence between the quantum and the classical cases. The formalism is an extension to the polynomial optimization in the ``classical'' sense; the relations among quantum mechanical values are represented by a semi-algebraic set ( by the set of polynomials and sign conditions), and the optimum is searched in this semi-algebraic set. In the context of this formalism, the optimization of the Ising model [QUBO model] will be a special case. However, the computational procedure proposed here is different from the quantum annealing which is now in vogue. We demonstrate the exemplary calculation through quantifier elimination (which is a kind of symbolic computation applicable to optimization and other problems).
Semidefinite programming hierarchies for commutative and non-commutative polynomial optimization represent a powerful computational tool with many applications in quantum information. In such applications, a given variable is typically not either commuting or non-commuting with all other variables, but instead commutes with some variables and does not commute with others, i.e., the variables satisfy some partial commutation relations. While such partial commutation relations can always be incorporated in a fully non-commutative setting through suitable linear constraints in the semidefinite programming relaxations, exploiting their algebraic properties from the onset can result in more compact relaxations. This leads us to introduce partially-commutative polynomial optimization, a framework that encompasses commutative and non-commutative polynomial optimization, allowing for arbitrary commutation relations among the variables. We point out that the underlying algebraic structure is that of a partially-commutative monoid. We present and review several key aspects of such monoids and show how they can be used to build SDP relaxations for partially-commutative polynomial optimization problems in which the partial commutations are natively implemented in the monomial structure, without the need of additional linear constraints.
In this article, we present our version of quantum particle swarm optimization. The objective function is the expectation value of the quantum mechanical Hamiltonian, as the sum of the kinetic term (−h2dx2d2) and the polynomial objective function (V(x)). The quantum mechanical feature in the system is represented by the set of expectation values (ϕ∣xiuj∣ϕ) for the possible combination of the kinetic momentum operator (u=hdxd) and the positional operator (x). These elements should satisfy the semi-definite condition of a matrix which is the representation of the uncertainty principle. The Plank constant h is a control parameter which would switch from the quantum mechanical case to the classical case, and vice versa. In order to get the optimum, these quantum mechanical expectation values are assumed to be the particle coordinates and propagated by the particle swarm optimization with the restriction. We present a simple example of the computational process.
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 quantum annealing devices, which encode the solution to a computational problem in the ground state of a quantum Hamiltonian, are implemented in D-Wave systems with more than 2,000 qubits. However, quantum annealing can solve only a classical combinatorial optimization problem such as an Ising model, or equivalently, a quadratic unconstrained binary optimization (QUBO) problem. In this paper, we formulate the QUBO model to solve elliptic problems with Dirichlet and Neumann boundary conditions using the finite element method. In this formulation, we develop the objective function of quadratic binary variables represented by qubits and the system finds the binary string combination minimizing the objective function globally. Based on the QUBO formulation, we introduce an iterative algorithm to solve the elliptic problems. We discuss the validation of the modeling on the D-Wave quantum annealing system.
This paper introduces an innovative approach to explore the capabilities of Quantum Annealing (QA) for trajectory optimization in dynamic systems. The proposed method involves transforming trajectory optimization problems into equivalent binary optimization problems using Quadratic Unconstrained Binary Optimization (QUBO) representation. The procedure is general and adaptable, making it applicable to a wide range of optimal control problems that entail the satisfaction of dynamic, boundary, and path constraints. Specifically, the trajectory is discretized and approximated using polynomials. In contrast to the conventional approach of determining the polynomial degree (n) solely based on the number of boundary conditions, a specific factor is introduced in our method to augment the polynomial degree. As a result, the ultimate polynomial degree is calculated as a composite of two components: n = l + (m−1), where m denotes the count of boundary conditions and l signifies the number of independent variables. By leveraging inverse dynamics, the control required to follow the approximated trajectory can be determined as a linear function of independent variables l. As a result, the optimization function, which is represented by the integral of the square of the control, can be formulated as a QUBO problem and the QA is employed to find the optimal binary solutions.
Abstract The use of graphical language in quantum computing for the representation of algorithms, although intuitive, is not very useful for different tasks such as the description of quantum circuits in text environments, the calculation of quantum states or the optimization of quantum circuits. While classical circuits can be represented either by circuit graphs or by Boolean expressions, quantum circuits have until now predominantly been illustrated as circuit graphs because no formal language for quantum circuits that allows algebraic manipulations has so far been accepted. This work proposes a means to represent quantum circuits in a convenient and concise manner, similar to the way Boolean expressions are used in classical circuits. The proposed notation allows the consistent and parameterized description of quantum algorithms, as well as the easy handling of the elements that compose it to achieve powerful optimizations in the number of gates of the circuits. To visualize it, a software implementation of an algebraic quantum circuit framework has been made, which allows describing quantum circuits, as well as their respective state vectors, using the proposed algebraic language.