QOC — QCPOP & polynomial optimization

Composition and tensor train structure in polynomial optimization

2026-04-19T00:00:00Z

We study polynomial optimization problems whose objective has a composition or tensor train structure. These polynomials can be evaluated as a sequence of maps, giving rise to intermediate variables (``states'') of dimension lower than the ambient dimension. Structures like these arise naturally in dynamical systems, Markov chains, and neural networks. We develop two moment-SOS (sums of squares) hierarchies that exploit this composition structure in different ways. The first one, termed state-lifting chordal, is based on the correlative sparsity graph of the problem. The second one, termed state-lifting push-forward, encodes the structure at the level of the measures directly. Numerical experiments demonstrate that the proposed methods can compute certified bounds for problems with hundreds or even a thousand variables. To illustrate the versatility of the hierarchies we apply them to Markov chain optimization, quantum optimal control, and neural networks.

Optimization Techniques in Quantum Information

2025-12-17T00:00:00Z

This thesis focuses on the intersection of mathematical and computational optimization and quantum information. Main contributions are open-source software code: A hybrid approach mixing "traditional" nonconvex and convex methods can make difficult problems more accessible. A demonstration of how to efficiently implement such an algorithm, avoiding interfacial bottlenecks, is provided, finding optimal protocols to establish entanglement through a lossy channel. The central software package developed addresses polynomial optimization problems. Many problems naturally involve only a polynomial objective and constraint polynomials. Such problems can automatically be cast into semidefinite programs that provide a hierarchy of outer approximations. The resulting problems are often so large and scale so unfavorably with respect to the variable number and degree involved that the boundary of the doable is reached quickly. However, technical progress both in hardware and algorithms has pushed this boundary - but software frameworks for polynomial optimization have not followed in the same manner, often now making them the bottleneck that before was the solver. The package PolynomialOptimization.jl developed during this thesis aims to fill the gap and provide a very resource-efficient intermediate layer together with a wide number of algorithms to reduce the problem size, and naturally supporting complex numbers and semidefinite constraints ubiquitous in quantum information problems. Its application on an entanglement distribution problem is demonstrated, showing that even relaxations with semidefinite matrices of three- and four-digit size can be solved conveniently. Finally, a new way to calculate interior-point barriers for the cone of sums-of-squares matrices in a nearly time-optimal way is developed, whose efficient implementation has the potential of further reducing resource consumption.

On the convergence of the variational quantum eigensolver and quantum optimal control

2025-09-05T00:00:00Z

When do variational quantum algorithms converge to a globally optimal solution? Despite extensive work, this question remains largely open. We develop a convergence theory for the variational quantum eigensolver (VQE), using quantum control landscape terminology to prove a sufficient criterion guaranteeing convergence to a Hamiltonian's ground state for almost all initial parameters. Specifically, we show that if (i) the parameterized unitary transformation allows movement in all tangent-space directions in a bounded manner (local surjectivity), and (ii) the gradient descent terminates, then VQE converges almost surely to a ground state. Under similar assumptions, we also guarantee almost-sure convergence to a global optimum on a smaller, closed unitary Lie subgroup, with representation theory providing a criterion for when this optimum is a ground state. We construct examples satisfying condition (i) and analyze two common circuit ansatz families, then discuss regularization techniques that ensure gradient descent terminates and connect to the halting problem.

Unitary Gate Synthesis via Polynomial Optimization

2025-08-02T00:00:00Z

Quantum optimal control plays a crucial role in the development of quantum technologies, particularly in the design and implementation of fast and accurate gates for quantum computing. Here, we present a method to synthesize gates using the Magnus expansion. In particular, we formulate a polynomial optimization problem that allows us to find the global solution without resorting to approximations of the exponential. The global method we use provides a certificate of globality and lets us do single-shot optimization, which implies it is generally faster than local methods. By optimizing over Hermitian matrices generating the unitaries, instead of the unitaries themselves, we can reduce the size of the polynomial to optimize, leading to fast convergence and scalability. Numerical experiments comparing our results with CRAB and GRAPE show that we maintain high accuracy while providing globality certificates.

Globally optimal control of quantum dynamics

2022-09-13T00:00:00Z

Optimization of constrained quantum control problems powers quantum technologies. This task becomes very difficult when these control problems are nonconvex and plagued with dense local extrema. For such problems current optimization methods must be repeated many times to find good solutions, each time requiring many simulations of the system. Here, we present quantum control via polynomial optimization (QCPOP), a method that eliminates this problem by directly finding globally optimal solutions. The resulting increase in speed, which can be a thousandfold or more, makes it possible to solve problems that were previously intractable. This remarkable advance is due to global optimization methods recently developed for polynomial functions. We demonstrate the power of this method by showing that it obtains an optimal solution in a single run for a problem in which local extrema are so dense that gradient methods require thousands of runs to reach a similar fidelity. Since QCPOP is able to find the global optimum for quantum control, we expect that it will not only enhance the utility of quantum control by making it much easier to find the necessary protocols, but also provide a key tool for understanding the precise limits of quantum technologies. Finally, we note that the ability to cast quantum control as polynomial optimization resolves an open question regarding the computability of exact solutions to quantum control problems.

Quantum Optimal Control via Magnus Expansion and Non-Commutative Polynomial Optimization

2020-01-17T00:00:00Z

Quantum optimal control has numerous important applications ranging from pulse shaping in magnetic-resonance imagining to laser control of chemical reactions and quantum computing. Our objective is to address two major challenges that have limited the success of applications of quantum optimal control so far: non-commutativity inherent in quantum systems and non-convexity of quantum optimal control problems involving more than three quantum levels. Methodologically, we address the non-commutativity of the control Hamiltonian at different times by the use of Magnus expansion. To tackle the non-convexity, we employ non-commutative polynomial optimisation and non-commutative geometry. As a result, we present the first globally convergent methods for quantum optimal control.

Global optimization for quantum dynamics of few-fermion systems

2018-01-29T00:00:00Z

Quantum state preparation is vital to quantum computation and quantum information processing tasks. In adiabatic state preparation, the target state is theoretically obtained with nearly perfect fidelity if the control parameter is tuned slowly enough. As this, however, leads to slow dynamics, it is often desirable to be able to do processes faster. In this work, we employ two global optimization methods to estimate the quantum speed limit for few-fermion systems confined in a one-dimensional harmonic trap. Such systems can be produced experimentally in a well controlled manner. We determine the optimized control fields and achieve a reduction in the ramping time of more than a factor of four compared to linear ramping. We also investigate how robust the fidelity is to small variations of the control fields away from the optimized shapes.

What is the optimal way to prepare a Bell state using measurement and feedback?

2017-04-02T00:00:00Z

Recent work has shown that use of quantum feedback can significantly enhance both the speed and success rate of measurement-based remote entanglement generation, but it is generally unknown what feedback protocols are optimal for these tasks. Here we consider two common measurements that are capable of projecting into pairwise entangled states, namely half- and full-parity measurements of two qubits, and determine in each case a globally optimal protocol for generation of entanglement. For the half-parity measurement, we rederive a previously described protocol using more general methods and prove that it is globally optimal for several figures of merit, including maximal concurrence or fidelity, and minimal time to reach a specified concurrence or fidelity. For the full-parity measurement, we derive a protocol for rapid entanglement generation related to that of (C. Hill, J. Ralph, Phys. Rev. A 77, 014305 (2008)), and then map the dynamics of the concurrence of the state to the Bloch vector length of an effective qubit. This mapping allows us to prove several optimality results for feedback protocols with full-parity measurements. We further show that our full-parity protocol transfers entanglement optimally from one qubit to the other amongst all measurement-based schemes. The methods developed here will be useful for deriving feedback protocols and determining their optimality properties in many other quantum systems subject to measurement and unitary operations.

Role of control constraints in quantum optimal control

2015-07-31T00:00:00Z

The problems of optimizing the value of an arbitrary observable of the two-level system at both a fixed time and the shortest possible time is theoretically explored. Complete identification and classification along with comprehensive analysis of globally optimal control policies and traps (i.e. policies which are locally but not globally optimal) is presented. The central question addressed is whether the control landscape remains trap-free if control constraints of the inequality type are imposed. The answer is astonishingly controversial, namely, although formally it is always negative, in practice it is positive provided that the control time is fixed and chosen long enough.

Searching for quantum optimal controls under severe constraints

2015-02-03T00:00:00Z

The success of quantum optimal control for both experimental and theoretical objectives is connected to the topology of the corresponding control landscapes, which are free from local traps if three conditions are met: (1) the quantum system is controllable, (2) the Jacobian of the map from the control field to the evolution operator is of full rank, and (3) there are no constraints on the control field. This paper investigates how the violation of assumption (3) affects gradient searches for globally optimal control fields. The satisfaction of assumptions (1) and (2) ensures that the control landscape lacks fundamental traps, but certain control constraints can still introduce artificial traps. Proper management of these constraints is an issue of great practical importance for numerical simulations as well as optimization in the laboratory. Using optimal control simulations, we show that constraints on quantities such as the number of control variables, the control duration, and the field strength are potentially severe enough to prevent successful optimization of the objective. For each such constraint, we show that exceeding quantifiable limits can prevent gradient searches from reaching a globally optimal solution. These results demonstrate that careful choice of relevant control parameters helps to eliminate artificial traps and facilitate successful optimization.

Coherent control of a qubit is trap-free

2014-07-19T00:00:00Z

There is a strong interest in optimal manipulating of quantum systems by external controls. Traps are controls which are optimal only locally but not globally. If they exist, they can be serious obstacles to the search of globally optimal controls in numerical and laboratory experiments, and for this reason the analysis of traps attracts considerable attention. In this paper we prove that for a wide range of control problems for two-level quantum systems all locally optimal controls are also globally optimal. Hence we conclude that two-level systems in general are trap-free. In particular, manipulating qubits---two-level quantum systems forming a basic building block for quantum computation---is free of traps for fundamental problems such as the state preparation and gate generation.

Global versus local optimality in feedback-controlled qubit purification: new insights from minimizing Rényi entropies

2013-12-04T00:00:00Z

It was first shown by Jacobs, in 2003, that the process of qubit state purification by continuous measurement of one observable can be enhanced, on average, by unitary feedback control. Here, we quantify this by the reduction in any one of the family of Rényi entropies $S_α$, with $0< α< \infty$, at some terminal time, revealing the rich structure of stochastic quantum control even for this simple problem. We generalize Jacobs' original argument, which was for (the unique) impurity measure with a linear evolution map under his protocol, by replacing linearity with convexity, thereby making it applicable to Rényi entropies $S_α$ for $α$ in a finite interval about $1$. Even with this generalization, Jacobs' argument fails to identify the surprising fact, which we prove by Bellman's principle of dynamic programming, that his protocol is globally optimal for all Rényi entropies whose decrease is locally maximized by that protocol. Also surprisingly, even though there is a range of Rényi entropies whose decrease is always locally maximized by the null-control protocol, that null-control protocol cannot be shown to be globally optimal in any instance. These results highlight the non-intuitive relation between local and global optimality in stochastic quantum control.

Semidefinite Programming and Reachable Sets of Dissipative Bilinear Control Systems

2005-04-14T00:00:00Z

In this manuscript, we investigate optimal control problems which arise in connection with manipulation of dissipative quantum dynamics. These problems motivate the study of a class of dissipative bilinear control systems. For these systems it is shown that the optimal solution and the reachable set can be found by solving a semidefinite program. In practice, solutions to these problems generate optimal methods for control of quantum mechanical phenomena in presence of dissipation. In the area of coherent spectroscopy, this translates into the maximum signal to noise ratio that can be obtained in a spectroscopy experiment.