Almost Optimal Classical Approximation Algorithms for a Quantum Generalization of Max-Cut

Approximation algorithms for constraint satisfaction problems (CSPs) are a central direction of study in theoretical computer science. In this work, we study classical product state approximation algorithms for a physically motivated quantum generalization of Max-Cut, known as the quantum Heisenberg model. This model is notoriously difficult to solve exactly, even on bipartite graphs, in stark contrast to the classical setting of Max-Cut. Here we show, for any interaction graph, how to classically and efficiently obtain approximation ratios 0.649 (anti-feromagnetic XY model) and 0.498 (anti-ferromagnetic Heisenberg XYZ model). These are almost optimal; we show that the best possible ratios achievable by a product state for these models is 2/3 and 1/2, respectively

From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz

The next few years will be exciting as prototype universal quantum processors emerge, enabling implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation, and which have the potential to significantly expand the breadth of quantum computing applications. A leading candidate is Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates between applying a cost-function-based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the Quantum Alternating Operator Ansatz, is the consideration of general parametrized families of unitaries rather than only those corresponding to the time-evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach to a wide variety of approximate optimization, exact optimization, and sampling problems. Here, we introduce the Quantum Alternating Operator Ansatz, lay out design criteria for mixing operators, detail mappings for eight problems, and provide brief descriptions of mappings for diverse problems.Comment: 51 pages, 2 figures. Revised to match journal pape

Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems

The quantum k-Local Hamiltonian problem is a natural generalization of classical constraint satisfaction problems (k-CSP) and is complete for QMA, a quantum analog of NP. Although the complexity of k-Local Hamiltonian problems has been well studied, only a handful of approximation results are known. For Max 2-Local Hamiltonian where each term is a rank 3 projector, a natural quantum generalization of classical Max 2-SAT, the best known approximation algorithm was the trivial random assignment, yielding a 0.75-approximation. We present the first approximation algorithm beating this bound, a classical polynomial-time 0.764-approximation. For strictly quadratic instances, which are maximally entangled instances, we provide a 0.801 approximation algorithm, and numerically demonstrate that our algorithm is likely a 0.821-approximation. We conjecture these are the hardest instances to approximate. We also give improved approximations for quantum generalizations of other related classical 2-CSPs. Finally, we exploit quantum connections to a generalization of the Grothendieck problem to obtain a classical constant-factor approximation for the physically relevant special case of strictly quadratic traceless 2-Local Hamiltonians on bipartite interaction graphs, where a inverse logarithmic approximation was the best previously known (for general interaction graphs). Our work employs recently developed techniques for analyzing classical approximations of CSPs and is intended to be accessible to both quantum information scientists and classical computer scientists

Quantum SDP-Solvers: Better upper and lower bounds

Brand\~ao and Svore very recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension $n$ of the problem and the number $m$ of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure. We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimizations problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with $mn$ when $m\approx n$, which is the same as classical.Comment: v4: 69 pages, small corrections and clarifications. This version will appear in Quantu

4. generációs mobil rendszerek kutatása = Research on 4-th Generation Mobile Systems

A 3G mobil rendszerek szabványosítása a végéhez közeledik, legalábbis a meghatározó képességek tekintetében. Ezért létfontosságú azon technikák, eljárások vizsgálata, melyek a következő, 4G rendszerekben meghatározó szerepet töltenek majd be. Több ilyen kutatási irányvonal is létezik, ezek közül projektünkben a fontosabbakra koncentráltunk. A következőben felsoroljuk a kutatott területeket, és röviden összegezzük az elért eredményeket. Szórt spektrumú rendszerek Kifejlesztettünk egy új, rádiós interfészen alkalmazható hívásengedélyezési eljárást. Szimulációs vizsgálatokkal támasztottuk alá a megoldás hatékonyságát. A projektben kutatóként résztvevő Jeney Gábor sikeresen megvédte Ph.D. disszertációját neurális hálózatokra épülő többfelhasználós detekciós technikák témában. Az elért eredmények Imre Sándor MTA doktori disszertációjába is beépültek. IP alkalmazása mobil rendszerekben Továbbfejlesztettük, teszteltük és általánosítottuk a projekt keretében megalkotott új, gyűrű alapú topológiára épülő, a jelenleginél nagyobb megbízhatóságú IP alapú hozzáférési koncepciót. A témakörben Szalay Máté Ph.D. disszertációja már a nyilvános védésig jutott. Kvantum-informatikai módszerek alkalmazása 3G/4G detekcióra Új, kvantum-informatikai elvekre épülő többfelhasználós detekciós eljárást dolgoztunk ki. Ehhez új kvantum alapú algoritmusokat is kifejlesztettünk. Az eredményeket nemzetközi folyóiratok mellett egy saját könyvben is publikáltuk. | The project consists of three main research directions. Spread spectrum systems: we developed a new call admission control method for 3G air interfaces. Project member Gabor Jeney obtained the Ph.D. degree and project leader Sandor Imre submitted his DSc theses from this area. Application of IP in mobile systems: A ring-based reliable IP mobility mobile access concept and corresponding protocols have been developed. Project member Máté Szalay submitted his Ph.D. theses from this field. Quantum computing based solutions in 3G/4G detection: Quantum computing based multiuser detection algorithm was developed. Based on the results on this field a book was published at Wiley entitled: 'Quantum Computing and Communications - an engineering approach'

Beyond Product State Approximations for a Quantum Analogue of Max Cut

We consider a computational problem where the goal is to approximate the maximum eigenvalue of a two-local Hamiltonian that describes Heisenberg interactions between qubits located at the vertices of a graph. Previous work has shed light on this problem's approximability by product states. For any instance of this problem the maximum energy attained by a product state is lower bounded by the Max Cut of the graph and upper bounded by the standard Goemans-Williamson semidefinite programming relaxation of it. Gharibian and Parekh described an efficient classical approximation algorithm for this problem which outputs a product state with energy at least 0.498 times the maximum eigenvalue in the worst case, and observe that there exist instances where the best product state has energy 1/2 of optimal. We investigate approximation algorithms with performance exceeding this limitation which are based on optimizing over tensor products of few-qubit states and shallow quantum circuits. We provide an efficient classical algorithm which achieves an approximation ratio of at least 0.53 in the worst case. We also show that for any instance defined by a 3- or 4-regular graph, there is an efficiently computable shallow quantum circuit that prepares a state with energy larger than the best product state (larger even than its semidefinite programming relaxation)

The Quantum and Classical Streaming Complexity of Quantum and Classical Max-Cut

We investigate the space complexity of two graph streaming problems: Max-Cut and its quantum analogue, Quantum Max-Cut. Previous work by Kapralov and Krachun [STOC `19] resolved the classical complexity of the \emph{classical} problem, showing that any $(2 - \varepsilon)$-approximation requires $\Omega(n)$ space (a $2$-approximation is trivial with $\textrm{O}(\log n)$ space). We generalize both of these qualifiers, demonstrating $\Omega(n)$ space lower bounds for $(2 - \varepsilon)$-approximating Max-Cut and Quantum Max-Cut, even if the algorithm is allowed to maintain a quantum state. As the trivial approximation algorithm for Quantum Max-Cut only gives a $4$-approximation, we show tightness with an algorithm that returns a $(2 + \varepsilon)$-approximation to the Quantum Max-Cut value of a graph in $\textrm{O}(\log n)$ space. Our work resolves the quantum and classical approximability of quantum and classical Max-Cut using $\textrm{o}(n)$ space. We prove our lower bounds through the techniques of Boolean Fourier analysis. We give the first application of these methods to sequential one-way quantum communication, in which each player receives a quantum message from the previous player, and can then perform arbitrary quantum operations on it before sending it to the next. To this end, we show how Fourier-analytic techniques may be used to understand the application of a quantum channel