Quantum Computing for Real-World Applications

Distinguish global phase from relative phase in a qubit state.

Calculate computational-basis measurement probabilities using the Born rule.

Apply standard single-qubit gates to a state vector and predict the resulting state.

Represent a pure qubit state on the Bloch sphere.

Compose a sequence of quantum gates in the correct matrix order.

Simulate repeated single-qubit measurements with NumPy.

Reproduce a NumPy state-vector calculation in a Qiskit circuit.

Construct multi-qubit states using tensor products and a stated basis-ordering convention.

Apply CNOT, controlled-Z, and SWAP gates to computational-basis states.

Determine whether a given two-qubit pure state is separable or entangled.

Prepare and verify a Bell state on a simulator.

Calculate joint, marginal, and conditional probabilities from multi-qubit measurement results.

Encode classical numerical features as qubit rotation angles.

Build a parameterized quantum circuit whose parameters can be updated programmatically.

Estimate the expectation value of an X, Y, or Z Pauli observable from sampled measurements using appropriate basis rotations.

Run the same circuit with state-vector and finite-shot simulators and compare their outputs.

Quantify sampling uncertainty in an expectation-value estimate.

Transpile a circuit for a specified native gate set and qubit-connectivity map.

Measure a transpiled circuit's qubit count, gate count, depth, and two-qubit-gate count.

Use density-matrix evolution to predict the effects of depolarizing and amplitude-damping channels and distinguish them from readout noise.

Construct and run a configurable noise model for a quantum circuit.

Calibrate and apply readout-error mitigation to sampled measurement counts.

Compare ideal, noisy, and mitigated results for the same observable.

Construct a phase oracle for a small Boolean objective.

Implement Grover search for a small unstructured-search problem.

Implement the quantum Fourier transform and its inverse.

Use quantum phase estimation on a unitary with a known eigenphase.

Decompose a small Hamiltonian into weighted Pauli strings.

Estimate the expectation value of a weighted Pauli Hamiltonian by grouping compatible measurement terms.

Approximate Hamiltonian time evolution with a first-order Trotter circuit.

Implement amplitude estimation for a small discrete probability model.

Express a binary optimization objective as a QUBO matrix.

Encode an equality constraint with a quadratic penalty term.

Convert an inequality constraint into QUBO form using binary slack variables.

Transform a QUBO objective into an equivalent Ising Hamiltonian.

Formulate Max-Cut as both a QUBO and an Ising model.

Formulate a constrained knapsack instance as a QUBO.

Validate a QUBO formulation by exhaustively enumerating a small instance.

Establish a reproducible classical baseline for a binary optimization instance.

Build a depth-one QAOA circuit, including cost and mixer unitaries, for a supplied Ising Hamiltonian.

Calculate QAOA objective values from sampled bitstrings.

Optimize QAOA circuit parameters with a classical optimizer.

Evaluate how QAOA depth changes solution quality and circuit cost.

Compare classical optimizers under finite-shot objective noise.

Warm-start QAOA from a classical relaxation and compare it with standard initialization.

Solve a QUBO with a free simulated quantum-annealing workflow.

Compare QAOA, simulated annealing, and an exact classical solver on matched instances.

Formulate a small traveling-salesperson problem as a QUBO.

Reduce redundant solutions in a QUBO by fixing a symmetry.

Decode sampled bitstrings and repair infeasible candidate solutions.

Partition an oversized QUBO into smaller subproblems for hybrid solution.

Transfer QAOA parameters between related optimization instances.

Construct a benchmark that records feasibility, objective quality, runtime, and resource use.

Select an appropriate classical, annealing, or gate-model solver for a constrained instance.

Package formulation, solution, decoding, and validation as a reusable optimization pipeline.

State the approximations that reduce a molecular system to an electronic-structure Hamiltonian.

Represent one- and two-electron terms with fermionic creation and annihilation operators.

Select an active space for a small molecule and identify the electrons and orbitals retained.

Generate the active-space electronic Hamiltonian of a small molecule with a free quantum-chemistry library.

Map a fermionic Hamiltonian to qubit Pauli operators with the Jordan-Wigner transformation.

Prepare the Hartree-Fock reference state for a mapped molecular Hamiltonian.

Compute an exact ground-state energy for a small Hamiltonian as a validation target.

Construct a qubit Hamiltonian for a small spin-lattice model.

Implement VQE for the ground-state energy of molecular hydrogen.

Compare the accuracy and circuit depth of hardware-efficient and chemistry-inspired ansatzes.

Generate a molecular bond-dissociation energy curve with VQE.

Measure how active-space selection changes VQE accuracy and qubit requirements.

Apply readout-error mitigation to a noisy VQE energy estimate and quantify its effect.

Use VQE to approximate the ground state of a small Heisenberg spin model.

Estimate magnetization and two-point correlations from a variationally prepared state.

Produce a resource estimate and classical-validation plan for a candidate materials simulation.

Distinguish ground-based quantum acceleration from onboard quantum processing under space-system constraints.

Generate a small mission-planning dataset containing tasks, visibility windows, priorities, power, and storage limits.

Define binary decision variables and an objective for satellite task scheduling.

Encode visibility, exclusivity, power, and storage constraints in a scheduling QUBO.

Formulate a small constellation assignment or routing problem as a QUBO.

Estimate the qubits and couplers required by a space-mission QUBO.

Solve a space-mission instance with an exact or heuristic classical baseline.

Validate a decoded mission schedule against every operational constraint.

Solve a small satellite-scheduling QUBO with QAOA.

Solve the same satellite-scheduling instance with a simulated annealing workflow.

Repair infeasible quantum samples into valid mission schedules.

Apply hybrid decomposition to a mission instance that exceeds a chosen qubit budget.

Measure the sensitivity of schedule quality to penalty weights, shot count, and noise.

Encode a small uncertain mission-risk calculation for amplitude estimation.

Compare amplitude estimation with Monte Carlo on the same mission-risk model.

Produce a deployment recommendation from benchmark results and quantum-resource estimates.

Formulate a cardinality-constrained portfolio-selection problem as a QUBO.

Solve a small portfolio QUBO with the existing QAOA workflow.

Compare a quantum-derived portfolio with exact and heuristic classical baselines.

Encode a discrete return distribution and payoff function in a quantum circuit.

Estimate an expected payoff with amplitude estimation.

Evaluate the sensitivity of portfolio selection to perturbations in expected returns and covariance estimates.

Assess whether a proposed finance use case has a plausible quantum advantage under stated hardware assumptions.