Methods¶

This page summarizes the physics and algorithms behind AtomQC. For the full treatment see the reference paper, J. Phys.: Condens. Matter 33, 385501 (2021), doi:10.1088/1361-648X/ac1154.

From a material to a qubit Hamiltonian¶

For a crystalline solid, a Wannier tight-binding Hamiltonian (WTBH) gives a small Hermitian matrix \(H(k)\) at every crystal momentum \(k\). Diagonalizing \(H(k)\) across the Brillouin zone produces the band structure \(E_n(k)\).

JARVIS ships pre-computed Wannier Hamiltonians indexed by material ID, e.g. FCC Aluminum JVASP-816 and diamond Silicon JVASP-1002. The bridge to a quantum computer is the fact that any Hermitian matrix can be written as a weighted sum of Pauli strings,

\[ H=\sum_k c_k\, P_k,\qquad P_k\in\{I,X,Y,Z\}^{\otimes n}, \]

and Pauli strings are exactly what a quantum computer can measure. HermitianSolver handles padding \(H(k)\) to a power-of-two dimension so it maps onto an integer number of qubits (\(N \times N\) needs \(\lceil\log_2 N\rceil\) qubits).

VQE — Variational Quantum Eigensolver¶

VQE prepares a parameterized trial state \(|\psi(\theta)\rangle\) (the ansatz), measures its energy, and lets a classical optimizer push that energy down. The variational principle guarantees

\[\langle\psi(\theta)|H|\psi(\theta)\rangle \;\ge\; E_{\text{ground}},\]

so "lower is always better" — VQE can never undershoot the true ground state. AtomQC uses hardware-efficient ansätze from QuantumCircuitLibrary; the number of layers (reps) trades expressiveness for circuit depth (and noise on real hardware).

ADAPT-VQE — adaptively grown ansatz¶

Fixed-depth VQE guesses the circuit shape up front. ADAPT-VQE instead builds the ansatz one operator at a time:

Start from a reference state (e.g. \(|0\dots0\rangle\)).
From a pool of candidate operators \(\{P_j\}\), compute the energy gradient of adding each.
Append the operator with the largest gradient as \(e^{i\theta_j P_j}\), then re-optimize all parameters with VQE.
Repeat until the largest gradient falls below a threshold (converged).

The payoff is a compact, problem-tailored circuit — often fewer gates than fixed VQE for the same accuracy, which matters on noisy hardware.

VQD — excited states and band structure¶

VQE finds only the lowest eigenvalue. A band structure needs all eigenvalues of \(H(k)\) at every \(k\). VQD (Variational Quantum Deflation) finds excited states by re-running VQE while adding a penalty that pushes each new state to be orthogonal to the ones already found:

\[ E_k(\theta)=\langle\psi|H|\psi\rangle+\sum_{j<k}\beta_j\,|\langle\psi|\psi_j\rangle|^2. \]

get_bandstruct repeats the VQD eigenvalue solve along a standard \(k\)-path and overlays the exact NumPy bands for comparison.

Classical cross-check¶

Every run can be compared against exact diagonalization via HermitianSolver.run_numpy() (numpy.linalg.eigh). This is how AtomQC validates the quantum results and how the benchmark CSVs under atomqc/data/ were generated — see Example scripts.