Chapter 6: What Comes Next

In this chapter, you’ll connect the Hamiltonian to VQE/QPE workflows and practical scaling decisions.

In This Chapter

What you’ll learn: How the encoded Hamiltonian is used in VQE and QPE, and why encoding choice affects scaling.
Why this matters: Good encoding choices can be the difference between feasible and impractical circuits.
Try this next: Jump into the Compare Encodings lab and test scaling behavior yourself.

The 15-term qubit Hamiltonian from Chapter 4 is the input to quantum algorithms. Two families of algorithms can extract the ground-state energy:

Variational Quantum Eigensolver (VQE)

VQE prepares a parameterised quantum state $\lvert\psi(\boldsymbol{\theta})\rangle$, measures $\langle\psi \mid\hat{H}\mid \psi\rangle$ by separately measuring each Pauli term, and uses a classical optimiser to minimise the energy over $\boldsymbol{\theta}$.

VQE is designed for near-term noisy quantum hardware: the circuits are short and the measurement overhead is manageable for small molecules. Experiments have already demonstrated VQE for H₂ and small molecules on real quantum computers.

Quantum Phase Estimation (QPE)

QPE applies the time-evolution operator $e^{-i\hat{H}t}$ controlled on an ancilla register to extract eigenvalues directly. QPE requires fault-tolerant quantum hardware but provides exponential speedup over classical exact diagonalisation for large systems.

Why Encoding Choice Matters at Scale

For H₂ with 4 qubits and 15 Pauli terms, both algorithms are trivially executable on current hardware. The challenge is scaling to chemically interesting molecules: LiH (12 spin-orbitals), H₂O (14), and the nitrogen fixation catalyst FeMo-co (~100 active spin-orbitals — the “poster child” of quantum chemistry on quantum computers).

The choice of encoding directly affects the scaling:

More Pauli terms → more shots. Each term must be measured separately.
Higher Pauli weight → deeper CNOT ladders → more gate errors.
The ternary tree encoding’s $O(\log_3 n)$ weight scaling means that for 100 modes, the deepest circuits are roughly 5 CNOTs instead of Jordan–Wigner’s 100 — a difference that may determine whether the simulation is feasible on early fault-tolerant hardware.

System	Spin-orbitals	JW max weight	Ternary tree max weight
H₂	4	4	2
LiH	12	12	4
H₂O	14	14	4
N₂	20	20	5
FeMo-co	~100	~100	~5

What You’ve Learned

If you’ve followed this tutorial from Chapter 1, you can now:

Understand why fermions and qubits are algebraically different and why encoding is necessary
Construct the Jordan–Wigner encoding by hand for any number of modes
Compute the complete 15-term qubit Hamiltonian for H₂ and verify it by diagonalisation
Compare five different encodings and explain why tree-based approaches matter for larger molecules