Excited states and state geometry

Excited states

Two methods find the lowest few eigenstates of H.

• VQD (variational quantum deflation; Higgott, Wang and Brierley, Quantum 3, 156 (2019)) finds the states one after another. State k minimises ⟨H⟩ + Σ_{j<k} β_j |⟨ψ_j|ψ⟩|². Each penalty β_j must exceed the gap E_k − E_j. Every gap is at most twice the coefficient one-norm of H; the default, three times that norm, always exceeds it.
• SSVQE (subspace-search VQE; Nakanishi, Mitarai and Fujii, Phys. Rev. Research 1, 033062 (2019)) applies one circuit to |0⟩, |1⟩, … and minimises a weighted sum of their energies. The weights must decrease strictly; with equal weights the result is only the subspace sum. It needs a more expressive circuit than VQD, because one unitary must map all inputs to eigenstates.

Both methods use exact adjoint gradients and seeded restarts:

import numpy as np
import metatron_qso as q

h = q.h2_sto3g()["hamiltonian"]                       # two qubits
exact = np.linalg.eigvalsh(h.to_matrix(2))
v = q.vqd(h, q.efficient_su2(2, 2), 4, restarts=6, seed=2)
s = q.ssvqe(h, q.efficient_su2(2, 2), 4, restarts=6, seed=2)
assert np.allclose(v["energies"], exact, atol=1e-6)
assert np.allclose(s["energies"], exact, atol=1e-6)

The tests check both methods against exact diagonalisation: all four levels of H₂ and the three lowest levels of a three-qubit Ising chain.

Quantum Fisher information

For |ψ(θ)⟩ = U(θ)|ψ₀⟩, the quantum Fisher information is

F_ij = 4 Re(⟨∂_iψ|∂_jψ⟩ − ⟨∂_iψ|ψ⟩⟨ψ|∂_jψ⟩).

It is four times the Fubini–Study metric, which measures how fast the state moves when the parameters change. quantum_fisher_information computes it exactly. One forward pass through the circuit carries one derivative vector per parameter. The tests compare F with finite differences of the fidelity |⟨ψ(θ)|ψ(θ + δ)⟩|².

Natural gradient and imaginary time

• quantum_natural_gradient (Stokes, Izaac, Killoran and Carleo, Quantum 4, 269 (2020)) steps along (F/4 + λI)⁻¹ ∇E. This is the steepest descent in the geometry of states, not of parameters. On H₂ with real_amplitudes(2, 2) and step size 0.2, it converges in 64 steps, where plain gradient descent needs 151.
• imaginary_time_evolution (McArdle et al., npj Quantum Inf. 5, 75 (2019)) integrates McLachlan’s equation θ̇ = −2F⁺∇E with fourth-order Runge–Kutta. Redundant parameters make F singular, so F⁺ is a pseudo-inverse. When the ansatz can represent the normalised state e^{−Hτ}|ψ₀⟩, the method follows it: the tests reproduce the exact energy curve to 10⁻⁶.

import numpy as np
import metatron_qso as q

h = q.PauliSum("Z0 + 0.6 X0 X1 - 0.4 Z1 + 0.3 X1")
ansatz = q.real_amplitudes(2, 2)
theta0 = np.full(ansatz.num_parameters, 0.2)
ite = q.imaginary_time_evolution(ansatz, h, theta0, time=3.0, steps=100)
qng = q.quantum_natural_gradient(ansatz, h, theta0, learning_rate=0.2)
print(ite["energies"][-1], qng["energy"], h.ground_state_energy())