Grover Mixers for QAOA: Shifting Complexity from Mixer Design to State Preparation

We propose GM-QAOA, a variation of the Quantum Alternating Operator Ansatz (QAOA) that uses Grover-like selective phase shift mixing operators. GM-QAOA works on any NP optimization problem for which it is possible to efficiently prepare an equal superposition of all feasible solutions; it is designed to perform particularly well for constraint optimization problems, where not all possible variable assignments are feasible solutions. GM-QAOA has the following features: (i) It is not susceptible to Hamiltonian Simulation error (such as Trotterization errors) as its operators can be implemented exactly using standard gate sets and (ii) Solutions with the same objective value are always sampled with the same amplitude. We illustrate the potential of GM-QAOA on several optimization problem classes: for permutation-based optimization problems such as the Traveling Salesperson Problem, we present an efficient algorithm to prepare a superposition of all possible permutations of $n$ numbers, defined on $O(n^2)$ qubits; for the hard constraint $k$-Vertex-Cover problem, and for an application to Discrete Portfolio Rebalancing, we show that GM-QAOA outperforms existing QAOA approaches

COMMIT: A Sender-Centric Truthful and Energy-Efficient Routing Protocol for Ad Hoc Networks

In this paper, we consider the problem of establishing a route and sending packets between a source/destination pair in ad hoc networks composed of rational selfish nodes,whose purpose is to maximize their own utility. In order to motivate nodes to follow the protocol specification, we use side payments that are made to the forwarding nodes. Our goal is to design a fully distributed algorithm such that: (i)a node is alwais better off participating in the protocol execution (individual rationality), and (ii) a node is always better off behaving according to the protocol specification (truthfulness). Furthemore, we require that messages are routed along the most energy-efficienth path. We introduce the COMMIT protocol for individually rational, truthful, and energy efficient routing in ad hoc networks. To the best of our knowledge, this is the first ad hoc routing protocol with these features. COMMIT exchanges at most O((|M|^2)*d) messages to find the optimal route, where |M|<= n-2, n is the number of network nodes, and d is the maximum node degree in the communication graph. As anaside, our work demonstrates the advantage of using a cross-layer approach to solving problems: leveraging the exitence of an underlyng topology control, and to reduce its message complessity. On the other hand, our investigation of the routing problem in presence of selfish nodes disclosed a new metric under which topology control protocols can be evaluated: the cost of cooperation

A Framework for Incentive Compatible Topology Control in Non-Cooperative Wireless Multi-Hop Networks

In this paper we consider the problem of building and maintaining a network topology with certain desirable features in a wireless multi-hop network where nodes behave like selfish agents. We first provide examples showing that existing topology control approaches are not resilient to strategic node behavior, indicating the need of considering possible selfish node behavior at the design stage. Given this observation, we propose a general framework that can be used as a guideline in the design of incentive compatible topology control protocols. As examples of application of our framework to specific topology control protocols, we present incentive compatible distributed algorithms for building the minimum spanning tree (MST) and the k-closest neighbors graph, which are very well-known topology control approaches. To the best of our knowledge, the ones presented in this paper are the first incentive compatible realizations of topology control presented in the literature

QAOA-based Fair Sampling on NISQ Devices

We study the status of fair sampling on Noisy Intermediate Scale Quantum (NISQ) devices, in particular the IBM Q family of backends. Using the recently introduced Grover Mixer-QAOA algorithm for discrete optimization, we generate fair sampling circuits to solve six problems of varying difficulty, each with several optimal solutions, which we then run on ten different backends available on the IBM Q system. For a given circuit evaluated on a specific set of qubits, we evaluate: how frequently the qubits return an optimal solution to the problem, the fairness with which the qubits sample from all optimal solutions, and the reported hardware error rate of the qubits. To quantify fairness, we define a novel metric based on Pearson's $\chi^2$ test. We find that fairness is relatively high for circuits with small and large error rates, but drops for circuits with medium error rates. This indicates that structured errors dominate in this regime, while unstructured errors, which are random and thus inherently fair, dominate in noisier qubits and longer circuits. Our results provide a simple, intuitive means of quantifying fairness in quantum circuits, and show that reducing structured errors is necessary to improve fair sampling on NISQ hardware

Threshold-Based Quantum Optimization

We propose and study Th-QAOA (pronounced Threshold QAOA), a variation of the Quantum Alternating Operator Ansatz (QAOA) that replaces the standard phase separator operator, which encodes the objective function, with a threshold function that returns a value $1$ for solutions with an objective value above the threshold and a $0$ otherwise. We vary the threshold value to arrive at a quantum optimization algorithm. 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 and its minimum/maximum finding cousin to approximate optimization. Our main findings include: (i) we show semi-formally that the optimum parameter values of GM-Th-QAOA (angles and threshold value) can be found with $O(\log(p) \times \log M)$ iterations of the classical outer loop, where $p$ is the number of QAOA rounds and $M$ is an upper bound on the solution value (often the number of vertices or edges in an input graph), thus eliminating the notorious outer-loop parameter finding issue of other QAOA algorithms; (ii) GM-Th-QAOA can be simulated classically with little effort up to 100 qubits through a set of tricks that cut down memory requirements; (iii) somewhat surprisingly, GM-Th-QAOA outperforms its non-thresholded counterparts in terms of approximation ratios achieved. This third result holds across a range of optimization problems (MaxCut, Max k-VertexCover, Max k-DensestSubgraph, MaxBisection) and various experimental design parameters, such as different input edge densities and constraint sizes

Predicting Expressibility of Parameterized Quantum Circuits using Graph Neural Network

Parameterized Quantum Circuits (PQCs) are essential to quantum machine learning and optimization algorithms. The expressibility of PQCs, which measures their ability to represent a wide range of quantum states, is a critical factor influencing their efficacy in solving quantum problems. However, the existing technique for computing expressibility relies on statistically estimating it through classical simulations, which requires many samples. In this work, we propose a novel method based on Graph Neural Networks (GNNs) for predicting the expressibility of PQCs. By leveraging the graph-based representation of PQCs, our GNN-based model captures intricate relationships between circuit parameters and their resulting expressibility. We train the GNN model on a comprehensive dataset of PQCs annotated with their expressibility values. Experimental evaluation on a four thousand random PQC dataset and IBM Qiskit's hardware efficient ansatz sets demonstrates the superior performance of our approach, achieving a root mean square error (RMSE) of 0.03 and 0.06, respectively