Documentation
Combinatorial
Optimisation.
The branch of mathematics and computer science concerned with finding the best solution — from an astronomically large discrete set — under hard constraints. It is the language of logistics, finance, scheduling, network design, and drug discovery. NEROX is built entirely around it.
Why it matters
The hardest problems in industry are combinatorial.
Exponential search spaces
Combinatorial problems grow faster than any polynomial. A scheduling problem with 20 jobs and 5 machines has over 60 sextillion possible orderings. Exact enumeration is computationally impossible — intelligent search is the only path.
NP-hardness is the norm
Most real-world decision problems are NP-hard — meaning no known algorithm can solve them in polynomial time. TSP, VRP, QAP, MaxCut, 3-SAT, and hundreds of industrial variants all belong to this class. Approximation and heuristics dominate practice.
Solutions are worth billions
A 1% improvement in a global logistics network saves hundreds of millions annually. Better portfolio construction compounds over years. Tighter job schedules mean more throughput, less waste, and fewer delays. The return on optimisation is among the highest in software engineering.
Universal encoding
QUBO: one matrix
to rule them all.
Quadratic Unconstrained Binary Optimisation is a unifying formulation. Any combinatorial problem can be written as minimising xᵀQx over binary vectors x ∈ {0,1}ⁿ, where Q encodes both the objective and all hard constraints via penalty terms.
This universality is powerful: a single solver that efficiently minimises QUBO can solve TSP, portfolio selection, scheduling, MaxCut, and any other combinatorial problem — simply by changing the Q matrix. QUBO is also the native format of quantum annealers (D-Wave), quantum gate circuits (QAOA), and all quantum-inspired GPU solvers.
TSP → QUBO reduction (sketch)
All three terms combined → a single Q matrix.
Submit to any NEROX solver without modification.
Problem catalogue
Every major NP-hard class. One API.
Submit problems as natural language, QUBO matrices, or structured JSON. NEROX maps them to the appropriate solver and returns results with provenance.
Travelling Salesman
Find the shortest route visiting N cities exactly once. At N=50 cities, brute-force requires more operations than atoms in the observable universe.
Portfolio Optimization
Select and weight assets to maximize risk-adjusted return under capital and sector constraints. A cornerstone of modern quantitative finance.
Job-Shop Scheduling
Assign N jobs to M machines with ordering and resource constraints to minimize makespan. Underpins manufacturing, cloud orchestration, and hospital scheduling.
Maximum Cut
Partition graph vertices into two sets to maximise crossing edges. The canonical NP-hard benchmark for quantum and quantum-inspired hardware.
Graph Colouring
Assign colours to vertices such that no adjacent vertices share a colour, using the minimum number of colours. Models register allocation and frequency assignment.
Bin Packing
Pack items of varying sizes into the fewest possible containers of fixed capacity. Critical to shipping, cloud bin-packing, and container loading.
Vehicle Routing
Route a fleet of vehicles to serve customers under capacity and time-window constraints. A generalization of TSP that governs last-mile delivery worldwide.
Quadratic Assignment
Assign facilities to locations minimising a flow×distance objective. One of the hardest NP-hard instances known — even small instances defeat exact solvers.
Solver portfolio
Quantum-inspired algorithms,
no quantum hardware required.
Quantum optimisation algorithms are mathematically beautiful and practically powerful. NEROX implements them on GPU hardware today — delivering quantum-grade solution quality at classical scale and classical cost.
GPU-Parallel Annealing
HeuristicSimulated annealing — one of the most studied metaheuristics in combinatorial optimisation — is reimplemented to exploit massively parallel GPU thread blocks. Thousands of independent Markov chains explore the energy landscape simultaneously, dramatically increasing solution diversity and convergence speed versus single-chain CPU annealing.
Tabu Search
MetaheuristicA memory-structured local search that maintains a "tabu list" of recently visited configurations to escape local optima. Particularly effective for structured combinatorial problems where neighbourhood moves have clear semantics, such as permutation-based routing and scheduling instances.
Hybrid Decomposition
HybridLarge-scale instances are decomposed into overlapping sub-problems via graph partitioning. Each sub-problem is solved independently then recombined — enabling problems with millions of variables to be addressed with hardware designed for thousands, without sacrificing solution quality.
QAOA
Quantum-InspiredThe Quantum Approximate Optimisation Algorithm encodes combinatorial problems into parameterised quantum circuits. NEROX implements a classical simulation of QAOA using tensor-network contraction on GPU, giving circuit-level fidelity without quantum hardware requirements.
VQE
VariationalThe Variational Quantum Eigensolver minimises a Hamiltonian encoding the cost function via gradient descent over circuit parameters. The ground-state energy corresponds to the optimal solution. Used for MaxCut, portfolio selection, and Ising-model instances.
The quantum horizon
Built for the world that's coming.
Fault-tolerant quantum computers will one day solve certain NP-hard problems faster than any classical machine. The algorithms and problem encodings used today — QAOA, VQE, QUBO — are the same algorithms that will run on those machines. Software written for NEROX today is already in the right format for quantum hardware.
This is not speculative roadmap positioning. QUBO and Ising models are the native problem formats for D-Wave annealers, IBM Quantum gate hardware (via QAOA), and IonQ systems. The ecosystem is converging on these representations.
Reference
Documentation index
Quickstart
Solve your first problem in 5 minutes. Install the SDK, submit a TSP, stream results.
API Reference
Full REST API documentation — endpoints, authentication, request formats, and response schemas.
Python SDK
SDK installation, async client, job management, streaming, and type reference.
QUBO Formulation
How to express any combinatorial problem as a QUBO matrix. Penalty methods, constraint encoding, scaling tips.
Solver Selection
Decision guide for choosing the right solver — GPU Annealing, Tabu, Hybrid, QAOA, or VQE.
GPU Deployment
On-premise Docker setup, GPU requirements, CUDA version compatibility, performance tuning.
Benchmarking
How to measure solver performance, reproduce benchmark results, and compare against baselines.
Production
Rate limits, error handling, retry strategies, monitoring, and best practices for production workloads.
Examples
Runnable notebooks for TSP, portfolio optimization, MaxCut, job scheduling, and more.
Ready to solve something?
Quickstart takes under 5 minutes. No GPU setup required to get a result.
GPU-native quantum-inspired optimization for combinatorial problems at any scale.