Abstract: According to an aspect of an embodiment, operations include receiving an input comprising a node-based graph associated with a real-world optimization problem and generating a sparse graph by removing a subset of edges from the node-based graph, The operations further include formulating operators of a quantum circuit on a quantum computer based on the sparse graph and formulating a cost function for the real-world optimization problem. The operations further include executing a set of operations which includes operating the quantum circuit on the quantum computer to generate a result, estimating a value of the cost function using the result, and updating parameters of the operators based on the value. The operations further include generating a final solution of the real-world optimization problem by repeating the execution of the set of operations using the updated parameters, until the estimated value of the cost function approaches a predefined threshold value.

Abstract: A method may include obtaining an optimization problem and a quadratic form corresponding to the optimization problem and identifying vectors that represent the quadratic form. The method may include setting a dimensionality of each vector that indicates a number of terms included in each vector and generating a first set of unit vectors based on the dimensionality of the vectors and based on a coefficient corresponding to each of the vectors. The method may include iteratively performing unitary operations to quantize each respective unit vector included in the first updated set as a respective indexed quantum state. The method may include setting a respective final quantum state corresponding to each respective indexed quantum state and determining one or more feasible solutions to the optimization problem based on the final quantum state corresponding to each respective indexed quantum state.

Abstract: A method may include obtaining a graph dataset that includes a plurality of nodes. The method may include specifying two or more clusters into which each node of the plurality of nodes is to be sorted. The method may include assigning each respective node of the plurality of nodes of the graph dataset into a respective cluster of the two or more clusters according to respective costs that are each associated with each of the respective clusters such that all of the costs are within a threshold value of each other. The respective cost associated with its respective cluster may be determined based on a number of external edges connecting the respective nodes in the respective cluster to the nodes in each other cluster. The method may include analyzing the plurality of nodes with respect to the respective cluster to which each of the respective nodes is assigned.

Abstract: According to an aspect of an embodiment, operations include receiving a Quadratic Integer Programming (QIP) problem including an objective function and a set of constraints on integer variables associated with the objective function. The operations further include obtaining an approximation of the QIP problem by relaxing the QIP problem and generating an approximate solution by solving the obtained approximation. The operations further include generating a Quadratic Unconstrained Binary Optimization (QUBO) formulation of the QIP problem based on the generated approximate solution and the received QIP problem. The operations further include submitting the generated QUBO formulation to an optimization solver machine and receiving a solution of the submitted QUBO formulation from the optimization solver machine. The operations further include publishing an integral solution of the received QIP problem on a user device based on the received solution.

Abstract: A method of solving a large scale combinatorial optimization problem including inputting, via at least one processor, an objective function and an initial solution as a mapping from a plurality of n nodes, randomly clustering the plurality of nodes into k clusters of n/k nodes each, for each cluster of the k clusters, assigning binary variables to denote each possible permutation of a label set within the cluster, determining that there are u=k2 variables if k>2, and u=1 variables if k=2, expressing the objective function in terms of the un/k variables, solving the objective function in terms of the un/k variables using a Quadratic Unconstrained Binary Optimization (QUBO) solver to obtain an updated solution, determining whether a convergence criteria is satisfied for the updated solution, and upon a determination that a convergence criteria is satisfied, outputting the updated solution to the objective function.

Abstract: A method may include obtaining a plurality of first data distributions in which each data distribution corresponds to running a first quantum circuit using a first input at a different noise level of a plurality of noise levels. The method may include simulating the first quantum circuit as a classical circuit and obtaining a noiseless data distribution corresponding to running the classical circuit using the first input. The method may also include determining an error mitigation parameter by performing a data regression analysis between the noiseless data distribution and the plurality of first data distributions. The method may additionally include obtaining a second data distribution that corresponds to running a second quantum circuit using a second input. A modified second data distribution may be obtained by applying the error mitigation parameter to the second data distribution such that noise included in the second data distribution is removed.

Abstract: A method may include obtaining a dataset including one or more data points. The method may include separating the dataset into one or more partitions based on a target number of subjects and a dimensionality of the data points included in the dataset. The method may include obtaining one or more weight vectors, each respective weight vector corresponding to a respective subject. The method may include selecting a first partition of the plurality of partitions to remove from the dataset based on respective relationships between a first weighted centroid of the dataset and first partition weights corresponding to each of the partitions. The method may include obtaining a first subset of the dataset by removing the data points associated with the selected first partition from the dataset. The method may include training a machine learning model based on the first subset of the dataset.

Abstract: According to an aspect of an embodiment, operations include receiving an Integer Linear Programming (ILP) problem including an objective function and a set of constraints on integer variables of the objective function. The operations may further include determining a lower bound vector for the integer variables and determining an upper bound vector for the integer variables. The operations further include obtaining a binary variable representation of each of the integer variables and updating the received ILP problem based on the obtained binary variable representation. The operations further include generating a Quadratic Unconstrained Binary Optimization (QUBO) formulation of the updated ILP problem and submitting the generated QUBO formulation to a first optimization solver machine. The operations further include receiving a solution of the submitted QUBO formulation and determining an integral solution of the received ILP problem.

Abstract: According to an aspect of an embodiment, operations include obtaining a first set of input parameters associated with a modified multi-dimensional knapsack problem. The operations further include determining a lattice representation of the modified multi-dimensional knapsack problem and computing a reduced basis by applying a lattice reduction method on the set of basis vectors of the determined lattice representation. The operations further include determining a first mathematical formulation of the modified multi-dimensional knapsack problem based on the reduced basis and submitting the determined first mathematical formulation as an input to an integer linear programming (ILP) solver. The operations further include receiving a first solution of the submitted mathematical formulation from the ILP solver and determining an integral solution of the modified multi-dimensional knapsack problem. The operations further include controlling a user device to output the determined integral solution.

Abstract: According to an aspect of an embodiment, operations include receiving a layout of a maritime facility and a first input including a vehicle count associated with a fleet of transport vehicles on the maritime facility. The operations further include determining a weighted graph representation of the maritime facility based on the received layout and generating a Quadratic Unconstrained Binary Optimization (QUBO) formulation based on the weighted graph representation and the received first input. The operations further include submitting the generated QUBO formulation to a first optimization solver machine and receiving a first solution of the submitted QUBO formulation. The operations further include determining a set of paths to be traversed by the fleet of transport vehicles on the maritime facility for transporting the plurality of shipping containers to respective destination locations based on the received first solution.

Abstract: According to an aspect of an embodiment, operations include receiving a Quadratic Integer Programming (QIP) problem including an objective function and a set of constraints on integer variables associated with the objective function. The operations further include obtaining an approximation of the QIP problem by relaxing the QIP problem and generating an approximate solution by solving the obtained approximation. The operations further include generating a Quadratic Unconstrained Binary Optimization (QUBO) formulation of the QIP problem based on the generated approximate solution and the received QIP problem. The operations further include submitting the generated QUBO formulation to an optimization solver machine and receiving a solution of the submitted QUBO formulation from the optimization solver machine. The operations further include publishing an integral solution of the received QIP problem on a user device based on the received solution.

Abstract: A method may include obtaining multiple operations configured to be performed in a serial fashion to implement a function. In some embodiments, each operation may be performed with respect to a parameter and an input. The method may also include obtaining an indication of multiple resources configured to perform the operations and a duration for each of the multiple resources to perform each of the multiple operations individually. The method may also include modeling, as a binary optimization, a scheduling of the resources to perform the multiple operations that reduces a total duration to perform the multiple operations based on the duration for each of the multiple resources to perform each of the multiple operations individually. The method may further include solving the binary optimization to determine a schedule of the multiple resources and performing, by the multiple resources, the multiple operations according to the schedule to implement the function.

Abstract: A method may include assigning each node of a first network to a different node cluster such that a number of nodes equals a number of node clusters, selecting multiple nodes of the first network as a set of nodes, and selecting multiple node clusters as a set of node clusters. The method may also include solving a first optimization problem by reassigning one or more of the nodes of the set of nodes to different node clusters of the set of node clusters while maintaining assigned node clusters of the nodes that are not part of the set of nodes and after reassigning one or more of the nodes of the set of nodes to different node clusters, merging the nodes assigned to at least one of the node clusters to form a second network with fewer nodes than the number of nodes of the first network.

Abstract: A method of solving a large scale combinatorial optimization problem including inputting, via at least one processor, an objective function and an initial solution as a mapping from a plurality of n nodes, randomly clustering the plurality of nodes into k clusters of n/k nodes each, for each cluster of the k clusters, assigning binary variables to denote each possible permutation of a label set within the cluster, determining that there are u=k2 variables if k>2, and u=1 variables if k=2, expressing the objective function in terms of the un/k variables, solving the objective function in terms of the un/k variables using a Quadratic Unconstrained Binary Optimization (QUBO) solver to obtain an updated solution, determining whether a convergence criteria is satisfied for the updated solution, and upon a determination that a convergence criteria is satisfied, outputting the updated solution to the objective function.

Abstract: A method of converting a HOBO problem into a QUBO problem. The method may include creating a data structure of key-value pairs by sorting the plurality of indices of the variables of the HOBO problem, the key in each key-value pair corresponding to combinations of quadratic terms appearing in the HOBO and the value corresponding to all terms of at least degree three that contain the associated key. For each key of the data structure, a quadratization process is performed including identifying a key with the largest number of associated values, replacing the identified key with an auxiliary variable, and updating the data structure so as to correspond with the replacement of the auxiliary variable, storing the auxiliary variable and a quadratic term the auxiliary variable replaced as a pair in a data map. The method may also include constructing a quadratic polynomial for each pair in the data map.

Abstract: A method may include obtaining multiple operations configured to be performed in a serial fashion to implement a function. In some embodiments, each operation may be performed with respect to a parameter and an input. The method may also include obtaining an indication of multiple resources configured to perform the operations and a duration for each of the multiple resources to perform each of the multiple operations individually. The method may also include modeling, as a binary optimization, a scheduling of the resources to perform the multiple operations that reduces a total duration to perform the multiple operations based on the duration for each of the multiple resources to perform each of the multiple operations individually. The method may further include solving the binary optimization to determine a schedule of the multiple resources and performing, by the multiple resources, the multiple operations according to the schedule to implement the function.