Chapter 13: Tapering Benchmarks

Numbers, not promises. This chapter measures exactly how much tapering saves on real Hamiltonians.

In This Chapter

• What you’ll learn: Concrete before-and-after measurements of qubit count, term count, Pauli weight, and estimated CNOT cost for tapering on multiple test systems.
• Why this matters: Tapering sounds good in theory. This chapter shows exactly how good — and where the limits are.
• Prerequisites: Chapters 10–12 (you understand both diagonal and Clifford tapering).

Methodology

For each test system, we:

1. Build the encoded Hamiltonian (JW encoding)
2. Count: qubits, terms, max weight, total CNOT cost per Trotter step
3. Apply tapering (all detected symmetries, +1 sector)
4. Re-count the same metrics on the tapered Hamiltonian
5. Report the reduction

CNOT cost per Trotter step is estimated as $\sum_k 2(w_k - 1)$ where $w_k$ is the weight of term $k$.

Benchmark Results

Fully diagonal 6-qubit Hamiltonian

A synthetic Hamiltonian where all terms are I/Z only — the best case for tapering.

let h6 =
    [| PauliRegister("ZIIIII", Complex(0.5, 0.0))
       PauliRegister("IZIIII", Complex(-0.3, 0.0))
       PauliRegister("IIZIII", Complex(0.8, 0.0))
       PauliRegister("IIIZII", Complex(0.2, 0.0))
       PauliRegister("IIIIZI", Complex(-0.4, 0.0))
       PauliRegister("IIIIIZ", Complex(0.7, 0.0))
       PauliRegister("ZZZZII", Complex(0.1, 0.0))
       PauliRegister("IIZZZZ", Complex(-0.2, 0.0)) |]
    |> PauliRegisterSequence

Interpretation: All symmetries were diagonal. The entire Hamiltonian collapsed to a scalar — the energy eigenvalue in the +1 sector. This is the extreme case: a fully classical Hamiltonian with no quantum content.

Mixed Hamiltonian (partial tapering)

A 4-qubit system where qubits 0 and 2 are diagonal but qubits 1 and 3 have X/Y terms.

let hmixed =
    [| PauliRegister("ZIZI", Complex(0.5, 0.0))
       PauliRegister("IXIX", Complex(-0.3, 0.0))
       PauliRegister("ZIIZ", Complex(0.2, 0.0))
       PauliRegister("IYIY", Complex(0.1, 0.0)) |]
    |> PauliRegisterSequence

Interpretation: Half the qubits removed, but the term count and weight stay the same — the remaining terms still have structure. The Hilbert space shrank by 4×, which is the primary gain.

Heisenberg model (Clifford needed)

$\hat{H} = XX + YY + ZZ$ on 2 qubits. No diagonal Z₂ symmetries, but $Z_0Z_1$ is a general Z₂ generator.

Interpretation: Diagonal-only tapering misses the symmetry entirely. Clifford tapering finds and uses it. This is why the general method matters.

The Impact on Circuit Cost

The real payoff of tapering shows in the CNOT staircase (Chapter 15). Each Pauli rotation $e^{-i\theta P}$ with weight $w$ costs $2(w-1)$ CNOTs. Tapering reduces both term count and weight:

These savings multiply across every Trotter step. For a simulation with 1000 Trotter steps, the 50% reduction in the mixed case means 4000 fewer CNOTs total.

Tapering + Encoding: Compounding Savings

Tapering and encoding choice address different aspects of circuit cost, and they compound:

flowchart LR
    H["Original Hamiltonian<br/>n qubits, O(n) weight"]
    H --> T["After tapering<br/>n-k qubits"]
    T --> E["After encoding choice<br/>O(log₃(n-k)) weight"]
    style E fill:#d1fae5,stroke:#059669

For a 14-qubit H₂O system (STO-3G, frozen core):

The combination of tapering (14→11 qubits) and ternary tree encoding (weight 14→4) gives roughly a 5× total reduction in CNOT cost.

Tapering Stage Complete

flowchart LR
    S1["The Molecule<br/>(Ch.1–3)"]
    S2["The Machine<br/>(Ch.4)"]
    S3["Encoding<br/>(Ch.5–9)"]
    S4["Tapering ✓<br/>(Ch.10–13)"]
    S5["Circuits<br/>(Ch.14–17)"]
    S1 --> S2 --> S3 --> S4 --> S5
    style S4 fill:#d1fae5,stroke:#059669
    style S5 fill:#fde68a,stroke:#d97706

We now have a verified, tapered qubit Hamiltonian — smaller than what encoding alone produced, with all physics preserved exactly. The next stage turns this Hamiltonian into a sequence of quantum gates.

How Much Tapering Is Enough?

A natural question: should you always taper everything you can, or is there a point of diminishing returns?

The answer is simple: taper every symmetry you find. Unlike basis-set truncation, where you trade accuracy for cost, tapering has no downside. Every tapered qubit preserves the physics exactly — same eigenvalues, same ground-state energy — while reducing qubit count, Hilbert space dimension, and usually circuit cost. There is no accuracy–cost tradeoff. You never want to leave a symmetry unexploited.

The real question is not how much to taper but whether you found all the symmetries. Diagonal tapering (Chapter 11) catches qubits that are individually frozen — easy to spot. Clifford tapering (Chapter 12) catches symmetries hidden across combinations of qubits — harder to spot, but FockMap finds them automatically. Between the two, you exhaust all Z₂ symmetries (operators that square to the identity and commute with every Hamiltonian term).

Could you do more? In principle, yes. The Hamiltonian has other symmetries beyond Z₂ — particle-number conservation (U(1)), spin symmetry (SU(2)) — that could remove additional qubits. But exploiting them requires different mathematical machinery (symmetry-adapted encodings, qubit-efficient mappings) that goes beyond the stabiliser framework we’ve developed here. This is an active area of research; Chapter 23 touches on it briefly.

In practice, Z₂ tapering removes 2 qubits from H₂ (4 → 2), 3 from H₂O (14 → 11), and typically 3–5 from larger molecules. The absolute savings shrink as a fraction of total qubits — at FeMo-co scale (~100 qubits), tapering saves ~5 qubits. Still worth doing (it’s free), but encoding choice matters more at scale. The complete optimization stack is: taper first (remove every qubit you can for free), then choose the encoding (to minimize the weight of what remains).

Key Takeaways

• Tapering reduces qubit count, Hilbert space size, and often term count and Pauli weight.
• Diagonal tapering handles the easy cases; Clifford tapering catches multi-qubit symmetries that diagonal misses.
• Taper everything — there is no accuracy cost, so every symmetry you find is worth exploiting.
• The savings compound with encoding choice: taper first, then the encoding operates on a smaller system.
• The real metric is CNOTs per Trotter step — tapering can reduce this by 50% or more.
• At large scale, encoding choice dominates; tapering provides a smaller but free additional reduction.

Previous: Chapter 12 — General Clifford Tapering

Next: Chapter 14 — From Hamiltonian to Time Evolution