Quantum Phase Estimation for Energy Levels

Phase estimation extracts eigenvalues of a Hamiltonian to arbitrary precision, the fault-tolerant route to chemically accurate fusion-material energies.

Reading energies from phases

Quantum phase estimation (QPE) is the algorithm that turns Hamiltonian time evolution into a number. If |psi> is an eigenstate of U = exp(-iHt) with eigenvalue exp(-i E t), QPE measures the phase E t and therefore the energy E. Unlike variational VQE, its precision improves systematically with more ancilla qubits rather than more optimization.

# QPE circuit sketch
1. prepare ancilla |0>^m and system state |psi> (overlap with eigenstate)
2. apply controlled-U^{2^k} for k = 0..m-1     (U = exp(-iHt))
3. inverse QFT on the ancilla register
4. measure ancilla -> binary fraction ~ (E t)/(2*pi)

# Precision:  eps ~ 2^{-m}  with m ancilla qubits
# Runtime per estimate:  O(1/eps) applications of U

Cost and preconditions

• Needs a good initial state with non-negligible overlap on the target eigenstate; success probability scales with that overlap squared.
• Needs long coherent circuits (many controlled-U calls), so it is inherently a fault-tolerant algorithm.
• Precision cost is O(1/eps), a favorable scaling that motivates the whole materials case.

Chemical accuracy is the bar

For first-wall and blanket energetics the target is chemical accuracy, roughly 1 kcal/mol (about 1.6 milli-Hartree). Reaching that with QPE means m large enough that eps is well below that threshold, driving the logical-qubit and T-gate counts that define the fault-tolerant era.

chemical accuracy: eps ~ 1.6e-3 Hartree
=> m ~ log2( E_range / eps )  ancilla bits
=> controlled-U calls ~ O(1/eps) ~ hundreds to thousands per digit
# only tractable with error-corrected logical qubits

QPE is the destination, not a near-term tool. Kronos treats it as the reference algorithm behind resource estimates for a first-wall defect cluster and validates the full pipeline on tiny systems where classical exact diagonalization gives the ground truth. It relies on Hamiltonian simulation and qubitization as subroutines.