Hierarchy Of Semidefinite Programs Research Articles

Peak estimation of rational systems using convex optimization

This paper presents algorithms that upper-bound the peak value of a state function along trajectories of a continuous-time system with rational dynamics. The finite-dimensional but nonconvex peak estimation problem is cast as a convex infinite-dimensional linear program in occupation measures. This infinite-dimensional program is then truncated into finite-dimensions using the moment-Sum-of-Squares (SOS) hierarchy of semidefinite programs. Prior work on treating rational dynamics using the moment-SOS approach involves clearing dynamics to common denominators or adding lifting variables to handle reciprocal terms under new equality constraints. Our solution method uses a sum-of-rational method based on absolute continuity of measures. The Moment-SOS truncations of our program possess lower computational complexity and (empirically demonstrated) higher accuracy of upper bounds on example systems as compared to prior approaches.

Mutually unbiased bases: polynomial optimization and symmetry

A set of k orthonormal bases of Cd is called mutually unbiased if |⟨e,f⟩|2=1/d whenever e and f are basis vectors in distinct bases. A natural question is for which pairs (d,k) there exist k mutually unbiased bases in dimension d. The (well-known) upper bound k≤d+1 is attained when d is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain C∗-algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of Sd and Sk.We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the Sd≀Sk-module C([d]×[k])t into irreducible modules. We present numerical results for small d,k and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist d+2 mutually unbiased bases in dimensions d=2,3,4,5,6,7,8. Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than 3 MUBs in dimension 6 requires polynomials of total degree at least 12.

Tighter Bounds on Transient Moments of Stochastic Chemical Systems

The use of approximate solution techniques for the Chemical Master Equation is a common practice for the analysis of stochastic chemical systems. Despite their widespread use, however, many such techniques rely on unverifiable assumptions and only a few provide mechanisms to control the approximation error quantitatively. Addressing this gap, Dowdy and Barton (J Chem Phys 149(7):074103, 2018) proposed an optimization-based technique for the computation of guaranteed bounds on the moment trajectories associated with stochastic chemical systems, thereby providing a general framework for rigorous uncertainty quantification. Here, we present an extension of this method. The key contribution is a new hierarchy of convex necessary moment conditions that build upon partitioning of the time domain. These conditions reflect the temporal causality that is inherent to the moment trajectories associated with stochastic processes described by the Chemical Master Equation and can be strengthened by simple refinement of the time domain partition. Analogous to the original method, these conditions generate a hierarchy of semidefinite programs that furnishes monotonically improving bounds on the trajectories of the moments and related statistics of stochastic chemical systems. Compared to its predecessor, the presented hierarchy produces bounds that are at least as tight and features new bound tightening mechanisms such as refinement of the time domain partition which often enable the computation of dramatically tighter bounds with lower computational cost. We analyze the properties of the presented hierarchy, discuss some aspects of its practical implementation and demonstrate its merits with several examples.

Bootstrap, Markov Chain Monte Carlo, and LP/SDP hierarchy for the lattice Ising model

Bootstrap is an idea that imposing consistency conditions on a physical system may lead to rigorous and nontrivial statements about its physical observables. In this work, we discuss the bootstrap problem for the invariant measure of the stochastic Ising model defined as a Markov chain where probability bounds and invariance equations are imposed. It is described by a linear programming (LP) hierarchy whose asymptotic convergence is shown by explicitly constructing the invariant measure from the convergent sequence of moments. We also discuss the relation between the LP hierarchy for the invariant measure and a recently introduced semidefinite programming (SDP) hierarchy for the Gibbs measure of the statistical Ising model based on reflection positivity and spin-flip equations.

Shape-Constrained Regression Using Sum of Squares Polynomials

Shape-Constrained Regression Using Algebraic Techniques How can one fit a multivariate polynomial to data points in such a way that this polynomial is guaranteed to have certain shape constraints, such as convexity or monotonicity in some of its variables? In “Shape-constrained regression using sum-of-squares polynomials,” M. Curmei and G. Hall propose a hierarchy of semidefinite programs to address this problem. They show that polynomial functions that are optimal to any fixed level of our hierarchy form a consistent estimator of the underlying shape-constrained function, which generates the data points. As a by-product of the proof, they establish that sum-of-squares-convex polynomials are dense in the set of polynomials that are convex over an arbitrary box. They further demonstrate the performance of the regressor for the problem of computing optimal transport maps in a color transfer task and that of estimating the optimal value function of a conic program. A real-time application of the latter problem to inventory management contract negotiation is presented.

Complete Hierarchy for High-Dimensional Steering Certification.

High-dimensional quantum steering can be seen as a test for the dimensionality of entanglement, where the devices at one side are not characterized. As such, it is an important component in quantum informational protocols that make use of high-dimensional entanglement. Although it has been recently observed experimentally, the phenomenon of high-dimensional steering is lacking a general certification procedure. We provide necessary and sufficient conditions to certify the entanglement dimension in a steering scenario. These conditions are stated in terms of a hierarchy of semidefinite programs, which can also be used to quantify the phenomenon using the steering dimension robustness. To demonstrate the practical viability of our method, we characterize the dimensionality of entanglement in steering scenarios prepared with maximally entangled states measured in mutually unbiased bases. Our methods give significantly stronger bounds on the noise robustness necessary to experimentally certify high-dimensional entanglement.

The inflation hierarchy and the polarization hierarchy are complete for the quantum bilocal scenario

It is a fundamental but difficult problem to characterize the set of correlations that can be obtained by performing measurements on quantum mechanical systems. The problem is particularly challenging when the preparation procedure for quantum states is assumed to comply with a given causal structure. Recently, a first completeness result for this quantum causal compatibility problem has been given based on the so-called quantum inflation technique. However, completeness was achieved by imposing additional technical constraints, such as an upper bound on the Schmidt rank of the observables. Here, we show that these complications are unnecessary in the quantum bilocal scenario, a much-studied abstract model of entanglement swapping experiments. We prove that the quantum inflation hierarchy is complete for the bilocal scenario in the commuting observable model of locality. We also give a bilocal version of an observation by Tsirelson, namely, in finite dimensions, the commuting observable model and the tensor product model of locality coincide. These results answer questions recently posed by Renou and Xu [arXiv:2210.09065v2 (2022)]. Finally, we point out that our techniques can be interpreted more generally as giving rise to a semidefinite programming hierarchy that is complete for the problem of optimizing polynomial functions in the states of operator algebras defined by generators and relations. The completeness of this polarization hierarchy follows from a quantum de Finetti theorem for states on maximal C*-tensor products.

Quantum key distribution rates from semidefinite programming

Computing the key rate in quantum key distribution (QKD) protocols is a long standing challenge. Analytical methods are limited to a handful of protocols with highly symmetric measurement bases. Numerical methods can handle arbitrary measurement bases, but either use the min-entropy, which gives a loose lower bound to the von Neumann entropy, or rely on cumbersome dedicated algorithms. Based on a recently discovered semidefinite programming (SDP) hierarchy converging to the conditional von Neumann entropy, used for computing the asymptotic key rates in the device independent case, we introduce an SDP hierarchy that converges to the asymptotic secret key rate in the case of characterised devices. The resulting algorithm is efficient, easy to implement and easy to use. We illustrate its performance by recovering known bounds on the key rate and extending high-dimensional QKD protocols to previously intractable cases. We also use it to reanalyse experimental data to demonstrate how higher key rates can be achieved when the full statistics are taken into account.

Superstabilizing Control of Discrete-Time ARX Models under Error in Variables

This paper applies a polynomial optimization-based framework towards the superstabilizing control of an Autoregressive with Exogenous Input (ARX) model given noisy data observations. The recorded input and output values are corrupted with L-infinity-bounded noise where the bounds are known. This is an instance of Error in Variables (EIV) in which true internal state of the ARX system remains unknown. The consistency set of ARX models compatible with noisy data has a bilinearity between unknown plant parameters and unknown noise terms. The requirement for a dynamic compensator to superstabilize all consistent plants is expressed using polynomial nonnegativity constraints, and solved using sum-of-squares (SOS) methods in a converging hierarchy of semidefinite programs in increasing size. The computational complexity of this method may be reduced by applying a Theorem of Alternatives to eliminate the noise terms. The effectiveness of this method is demonstrated on control of example ARX models.

Dimension-Free Entanglement Detection in Multipartite Werner States

Werner states are multipartite quantum states that are invariant under the diagonal conjugate action of the unitary group. This paper gives a complete characterization of their entanglement that is independent of the underlying local Hilbert space: for every entangled Werner state there exists a dimension-free entanglement witness. The construction of such a witness is formulated as an optimization problem. To solve it, two semidefinite programming hierarchies are introduced. The first one is derived using real algebraic geometry applied to positive polynomials in the entries of a Gram matrix, and is complete in the sense that for every entangled Werner state it converges to a witness. The second one is based on a sum-of-squares certificate for the positivity of trace polynomials in noncommuting variables, and is a relaxation that involves smaller semidefinite constraints.

Continuous-Variable Nonlocality and Contextuality

Contextuality is a non-classical behaviour that can be exhibited by quantum systems. It is increasingly studied for its relationship to quantum-over-classical advantages in informatic tasks. To date, it has largely been studied in discrete-variable scenarios, where observables take values in discrete and usually finite sets. Practically, on the other hand, continuous-variable scenarios offer some of the most promising candidates for implementing quantum computations and informatic protocols. Here we set out a framework for treating contextuality in continuous-variable scenarios. It is shown that the Fine–Abramsky–Brandenburger theorem extends to this setting, an important consequence of which is that Bell nonlocality can be viewed as a special case of contextuality, as in the discrete case. The contextual fraction, a quantifiable measure of contextuality that bears a precise relationship to Bell inequality violations and quantum advantages, is also defined in this setting. It is shown to be a non-increasing monotone with respect to classical operations that include binning to discretise data. Finally, we consider how the contextual fraction can be formulated as an infinite linear program. Through Lasserre relaxations, we are able to express this infinite linear program as a hierarchy of semi-definite programs that allow to calculate the contextual fraction with increasing accuracy.

Quantum-Inspired Hierarchy for Rank-Constrained Optimization

Many problems in information theory can be reduced to optimizations over matrices, where the rank of the matrices is constrained. We establish a link between rank-constrained optimization and the theory of quantum entanglement. More precisely, we prove that a large class of rank-constrained semidefinite programs can be written as a convex optimization over separable quantum states and, consequently, we construct a complete hierarchy of semidefinite programs for solving the original problem. This hierarchy not only provides a sequence of certified bounds for the rank-constrained optimization problem, but also gives pretty good and often exact values in practice when the lowest level of the hierarchy is considered. We demonstrate that our approach can be used for relevant problems in quantum information processing, such as the optimization over pure states, the characterization of mixed unitary channels and faithful entanglement, and quantum contextuality, as well as in classical information theory including the maximum cut problem, pseudo-Boolean optimization, and the orthonormal representation of graphs. Finally, we show that our ideas can be extended to rank-constrained quadratic and higher-order programming.

Facial Input Decompositions for Robust Peak Estimation under Polyhedral Uncertainty

This work bounds extreme values of state functions for a class of input-affine continuous-time systems that are affected by polyhedral-bounded uncertainty. Instances of these systems may arise in data-driven peak estimation, in which the state function must be bounded for all systems that are consistent with a set of state-derivative data records corrupted under L-infinity bounded noise. Existing occupation measure-based methods form a convergent sequence of outer approximations to the true peak value, given an initial set, by solving a hierarchy of semidefinite programs in increasing size. These techniques scale combinatorially in the number of state variables and uncertain parameters. We present tractable algorithms for peak estimation that scale linearly in the number of faces of the uncertainty-bounding polytope rather than combinatorially in the number of uncertain parameters by leveraging convex duality and a theorem of alternatives (facial decomposition). The sequence of decomposed semidefinite programs will converge to the true peak value under mild assumptions (convergence and smoothness of dynamics).

Simple and maximally robust processes with no classical common-cause or direct-cause explanation

Guided by the intuition of coherent superposition of causal relations, recent works presented quantum processes without classical common-cause and direct-cause explanation, that is, processes which cannot be written as probabilistic mixtures of quantum common-cause and quantum direct-cause relations (CCDC). In this work, we analyze the minimum requirements for a quantum process to fail to admit a CCDC explanation and present "simple" processes, which we prove to be the most robust ones against general noise. These simple processes can be realized by preparing a maximally entangled state and applying the identity quantum channel, thus not requiring an explicit coherent mixture of common-cause and direct-cause, exploiting the possibility of a process to have both relations simultaneously. We then prove that, although all bipartite direct-cause processes are bipartite separable operators, there exist bipartite separable processes which are not direct-cause. This shows that the problem of deciding weather a process is direct-cause processis notequivalent to entanglement certification and points out the limitations of entanglement methods to detect non-classical CCDC processes. We also present a semi-definite programming hierarchy that can detect and quantify the non-classical CCDC robustnesses of every non-classical CCDC process. Among other results, our numerical methods allow us to show that the simple processes presented here are likely to be also the maximally robust against white noise. Finally, we explore the equivalence between bipartite direct-cause processes and bipartite processes without quantum memory, to present a separable process which cannot be realized as a process without quantum memory.

Quantum marginal problem and incompatibility

One of the basic distinctions between classical and quantum mechanics is the existence of fundamentally incompatible quantities. Such quantities are present on all levels of quantum objects: states, measurements, quantum channels, and even higher order dynamics. In this manuscript, we show that two seemingly different aspects of quantum incompatibility: the quantum marginal problem of states and the incompatibility on the level of quantum channels are in many-to-one correspondence. Importantly, as incompatibility of measurements is a special case of the latter, it also forms an instance of the quantum marginal problem. The generality of the connection is harnessed by solving the marginal problem for Gaussian and Bell diagonal states, as well as for pure states under depolarizing noise. Furthermore, we derive entropic criteria for channel compatibility, and develop a converging hierarchy of semi-definite programs for quantifying the strength of quantum memories.

Analysis of Optimization Algorithms via Sum-of-Squares

We introduce a new framework for unifying and systematizing the performance analysis of first-order black-box optimization algorithms for unconstrained convex minimization. The low-cost iteration complexity enjoyed by first-order algorithms renders them particularly relevant for applications in machine learning and large-scale data analysis. Relying on sum-of-squares (SOS) optimization, we introduce a hierarchy of semidefinite programs that give increasingly better convergence bounds for higher levels of the hierarchy. Alluding to the power of the SOS hierarchy, we show that the (dual of the) first level corresponds to the performance estimation problem (PEP) introduced by Drori and Teboulle (Math Program 145(1):451–482, 2014), a powerful framework for determining convergence rates of first-order optimization algorithms. Consequently, many results obtained within the PEP framework can be reinterpreted as degree-1 SOS proofs, and thus, the SOS framework provides a promising new approach for certifying improved rates of convergence by means of higher-order SOS certificates. To determine analytical rate bounds, in this work, we use the first level of the SOS hierarchy and derive new results for noisy gradient descent with inexact line search methods (Armijo, Wolfe, and Goldstein).

Semidefinite programming hierarchies for constrained bilinear optimization

We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form {mathrm {Tr}}big [H(Dotimes E)big ], maximized with respect to semidefinite constraints on D and E. Applied to the problem of approximate error correction in quantum information theory, this gives hierarchies of efficiently computable outer bounds on the success probability of approximate quantum error correction codes in any dimension. The first level of our hierarchies corresponds to a previously studied relaxation (Leung and Matthews in IEEE Trans Inf Theory 61(8):4486, 2015) and positive partial transpose constraints can be added to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our sum-of-squares hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo’s asymptotic de Finetti theorem for quantum channels. Finally, our proof methods answer a question of Brandão and Harrow (Proceedings of the forty-fourth annual ACM symposium on theory of computing, STOC’12, p 307, 2012) by improving the approximation factor of de Finetti theorems with no symmetry from O(d^{k/2}) to {mathrm {poly}}(d,k), where d denotes local dimension and k the number of copies.

Strengthening convex relaxations of 0/1-sets using Boolean formulas

In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set S of feasible integer points, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets S that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set containing a set S subseteq {0,1}^n by exploiting certain additional information about S. Namely, the required extra information will be in the form of a Boolean formula phi defining the target set S. The new relaxation is obtained by “feeding” the convex set into the formula phi . We analyze various aspects regarding the strength of our procedure. As one application, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems.

Quantum Bounds on Detector Efficiencies for Violating Bell Inequalities Using Semidefinite Programming

Loophole-free violations of Bell inequalities are crucial for fundamental tests of quantum nonlocality. They are also important for future applications in quantum information processing, such as device-independent quantum key distribution. Based on a detector model which includes detector inefficiencies and dark counts, we estimate the minimal requirements on detectors needed for performing loophole-free bipartite and tripartite Bell tests. Our numerical investigation is based on a hierarchy of semidefinite programs for characterizing possible quantum correlations. We find that for bipartite setups with two measurement choices and our detector model, the optimal inequality for a Bell test is equivalent to the Clauser–Horne inequality.

Computation of Lyapunov Functions under State Constraints using Semidefinite Programming Hierarchies

We provide algorithms for computing a Lyapunov function for a class of systems where the state trajectories are constrained to evolve within a closed convex set. The dynamical systems that we consider comprise a differential equation which ensures continuous evolution within the domain, and a normal cone inclusion which ensures that the state trajectory remains within a prespecified set at all times. Finding a Lyapunov function for such a system boils down to finding a function which satisfies certain inequalities on the admissible set of state constraints. It is well-known that this problem, despite being convex, is computationally difficult. For conic constraints, we provide a discretization algorithm based on simplicial partitioning of a simplex, so that the search of desired function is addressed by constructing a hierarchy (associated with the diameter of the cells in the partition) of linear programs. Our second algorithm is tailored to semi-algebraic sets, where a hierarchy of semidefinite programs is constructed to compute Lyapunov functions as a sum-of-squares polynomial.