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Computational Physics

The computational wing — where physics meets algorithms. Simulators, solvers, and data techniques that turn equations into experiments.

Code is how theory learns to walk.
QParadox

Quantum Simulators • software for quantum systems

Modern stacks let you describe qubits, circuits, and Hamiltonians in code and either emulate them on a CPU/GPU or ship them to hardware. You can prototype algorithms, run noisy models, and verify results with exact diagonalization for small systems.

Libraries

Qiskit, Cirq, PennyLane

Build circuits as Python objects, transpile for specific backends, and simulate with statevector or density-matrix engines. Parameterized gates enable variational algorithms; built-ins handle noise channels and measurement sampling.

Go deeper • Statevector vs. stabilizer vs. tensor networks

Statevectors scale as $2^n$ amplitudes; stabilizer sims excel for Clifford circuits; tensor networks trade memory for structure, contracting low-entanglement circuits efficiently.

Hamiltonians

Analog & digital simulation

Trotterization splits $e^{-iHt}$ into exponentials of local terms; qubitization uses block encodings; analog simulators emulate models directly (e.g., Bose–Hubbard) with controllable couplings.

noisy modelspulse-leveltensor contraction

Numerical Methods • ODEs, PDEs, eigenproblems

Physics problems become differential equations and linear algebra. Good numerics preserve stability, conservation, and symmetry while staying fast.

Time stepping

Explicit, implicit, symplectic

Stiff systems call for implicit schemes (BDF, implicit Runge–Kutta). Hamiltonian dynamics prefer symplectic integrators (leapfrog) to conserve structure over long times.

Try it • Energy drift test

Simulate a Kepler orbit with Euler vs. leapfrog. Only the symplectic method keeps the orbit closed without artificial heating.

Discretization

Finite difference, finite volume, spectral

Upwind and Riemann solvers capture shocks; spectral methods give exponential accuracy for smooth fields. Multigrid and Krylov methods crush large sparse systems.

CFL stabilitypreconditioningAMR

Quantum Algorithms • speedups with structure

Some problems reveal patterns that quantum states exploit: periodicity, amplitude amplification, and variational optimization.

Exact

Shor & Grover

Period finding through the Quantum Fourier Transform factors integers; Grover’s algorithm gives quadratic speedup for unstructured search and is optimal for that model.

Variational

VQE, QAOA, and beyond

Parameterized circuits plus classical optimizers search for ground states or combinatorial optima; expressivity vs. trainability trades off with depth and ansatz design.

Go deeper • Barren plateaus in a sentence

Random deep ansätze make gradients vanish exponentially with width, demanding structure or problem-aware initialization.

Tensor Networks • compressed many-body physics

Low-entanglement states admit efficient factorizations. Matrix Product States (MPS) and Projected Entangled Pair States (PEPS) turn exponential objects into tractable ones.

1D

MPS & DMRG

Ground states of gapped 1D systems obey area laws. DMRG variationally optimizes an MPS, reaching near-exact energies with modest bond dimensions.

Higher-D

PEPS & contraction

2D networks are powerful but expensive to contract. Approximate schemes (corner transfer, tensor renormalization) make observables practical.

area lawentanglement spectrumcanonical forms

Data-Driven Physics • learning with constraints

Machine learning meets physics when models are steered by symmetries, conservation laws, or known operators — adding structure to reduce data hunger and hallucinations.

Differential

PINNs & neural ODEs

Penalize PDE residuals in the loss to fit fields that satisfy physics; integrate dynamics with neural vector fields trained on sparse trajectories.

Symbolic

Discovering equations

Sparse regression and differentiable libraries recover governing equations from data; equivariant nets build symmetry into the architecture.

Go deeper • Uncertainty matters

Bayesian layers or ensembles quantify epistemic uncertainty, crucial when learned surrogates replace expensive simulators.

Computational Cosmology • universe in a computer

From inflationary perturbations to the cosmic web and CMB anisotropies, simulations connect parameters to observables across 10 orders of magnitude.

Linear

Boltzmann solvers & spectra

Einstein–Boltzmann codes evolve multipoles of photons and neutrinos to predict $C_\ell$ and linear matter power; tight-coupling and recombination demand careful numerics.

Nonlinear

N-body & hydro

TreePM and mesh solvers grow structure; hydrodynamics with feedback models shapes baryons. Light-cone ray tracing adds lensing and secondary CMB signals.

symplecticFFTemulators

Monte Carlo Methods • sampling as a calculator

Random draws estimate integrals, simulate transport, and sample posteriors. Engineering the randomness — proposals, variance reduction, parallelism — makes MC powerful.

Integration

Importance & control variates

Align proposals with integrands; subtract correlated functions to shrink variance; quasi–Monte Carlo uses low-discrepancy sequences for smooth problems.

MCMC

Metropolis, Gibbs, HMC

Construct chains with stationary targets; Hamiltonian dynamics propose distant, high-acceptance moves in smooth, high-dimensional spaces.

Go deeper • Effective sample size

Autocorrelation reduces information; track ESS and $\hat{R}$ across chains to certify convergence and mixing.

Quantum Complexity • limits of computational power

BQP problems are efficiently solvable on quantum machines; QMA problems are efficiently verifiable with quantum proofs. Physics meets theory via Local Hamiltonians, adiabatic gaps, and sampling hardness.

Classes

BQP, QMA, & friends

Speedups live where structure does (periods, amplitudes). Postselection inflates power (PostBQP = PP). Oracle separations hint at sampling tasks beyond classical hierarchies.

Physics link

Ground states & gaps

The Local Hamiltonian problem is QMA-complete; adiabatic algorithms hinge on spectral gaps; error-correction overhead turns asymptotics into real costs.

verificationsamplingfault tolerance

Crossroads • where to go next

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Frameworks and equations that shape physical law.

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Instruments, measurements, and noise made honest.

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