Chapter 1: Why Tapering?

In this chapter, you’ll understand why encoded Hamiltonians have removable qubits and what that means for quantum simulation.

In This Chapter

• What you’ll learn: How fermion-to-qubit encoding creates symmetries that allow safe qubit removal
• Why this matters: Fewer qubits = smaller circuits = less noise = larger problems solvable on near-term hardware
• Try this next: Move to Chapter 2 — The Diagonal Z₂ Approach to see how to detect and exploit these symmetries.

From Encoding to Symmetry

Recall the encoding pipeline:

Each fermionic ladder operator $a_i^\dagger$ or $a_i$ maps to a sum of Pauli strings. When you assemble a molecular Hamiltonian from these encoded operators, the result is a Pauli sum:

where each $\sigma_\alpha$ is a Pauli string (product of single-qubit Paulis I, X, Y, Z).

The Symmetry Observation

In many practical systems — especially those respecting particle-number conservation — the encoded Hamiltonian exhibits a special structure:

For certain qubit indices $j$, every term in the Hamiltonian has only I or Z on qubit $j$, never X or Y.

This is a diagonal Z₂ symmetry. It means:

• The measurement outcome on qubit $j$ is an eigenvalue of the Z operator ($+1$ or $-1$), not a superposition.
• The physics of the system is independent of whether qubit $j$ is measured to be spin-up or spin-down — you can fix it permanently to one value.
• Fixing that value eliminates the qubit entirely without changing the spectrum.

Concrete Illustration

Consider this toy 4-qubit Hamiltonian:

Look at qubit index 1 (second column):

• In $\mathrm{ZIZI}$: position 1 is I
• In $\mathrm{ZZII}$: position 1 is Z
• In $\mathrm{IIZZ}$: position 1 is I

Every term has I or Z at qubit 1. Qubit 1 is a diagonal Z₂ symmetry.

The same is true for qubits 0, 2, and 3. In fact, all four qubits are diagonal Z₂ symmetric in this Hamiltonian!

What “Fixing” a Qubit Means

If we decide that qubit 1 will have eigenvalue $+1$ (spin-up), then:

• Terms with $\mathrm{Z}$ on qubit 1 pick up a coefficient factor of $+1$
• Terms with $\mathrm{I}$ on qubit 1 are unaffected
• Qubit 1 can be removed from the tensor product

Example: fixing qubits 1 and 3 to the $+1$ sector:

\(\hat{H}' = (0.8 \cdot 1) \,\mathrm{ZI} + (-0.4 \cdot 1) \,\mathrm{ZI} + (0.3) \,\mathrm{II}\) \(= 0.4 \,\mathrm{ZI} + 0.3 \,\mathrm{II}\)

The result is a 2-qubit Hamiltonian (qubits 0 and 2), and it has exactly the same eigenvalues as the original 4-qubit system would have in the $(+1, +1)$ sector on qubits $(1, 3)$.

Why It Matters: Circuit Depth

Quantum circuits for simulating a Hamiltonian scale with:

• Number of qubits: Every additional qubit doubles the Hilbert space dimension
• Pauli weight: Each term with weight $w$ requires $\sim 2(w-1)$ two-qubit gates

Removing even one qubit halves the state space. For a 12-qubit molecular Hamiltonian where 3 qubits are taperable, tapering gives you a 9-qubit system — an 8-fold reduction in Hilbert space.

Scope: v1 Diagonal Z₂

FockMap’s v1 tapering handles the “diagonal Z₂” case: generators of the form $Z_j$ where each term is diagonal on qubit $j$ (only I or Z). This covers:

• Particle-number symmetries (the most common and important)
• Single-qubit-only conservation laws
• Spin-projection symmetries (in spin-adapted encodings)

v2 direction (planned): General Z₂ generators (not just single-qubit), combined with Clifford tapering for multi-qubit stabiliser generators.

Next: Chapter 2 — The Diagonal Z₂ Approach