L11: Hamiltonian Simulation (Trotterization & Mapping Matter to Qubits)

Learning Objectives¶

• Explain why simulating quantum matter is the ‘native’ application of quantum computers.

• Decompose $e^{-iHt}$ via Lie-Trotter-Suzuki formulas and bound the error.

• Map spin Hamiltonians (Ising, Heisenberg) directly to circuits.

• Map fermions to qubits (Jordan-Wigner; Bravyi-Kitaev in outline).

Feynman’s Charge¶

Classical cost of an $n$-site quantum system grows as $2^n$; a quantum computer stores the state in $n$ qubits.

Target problems for materials science: ground-state energies, band structures, reaction barriers, dynamics after a quench, correlated phases (Hubbard physics, high-$T_c$).

Eureka! Simulation is the application where quantum advantage needs the least algorithmic magic: the computer and the problem obey the same physics.

The Task¶

Given $H=\sum_{k} h_k H_k$ (each $H_k$ a local/Pauli term), implement $U(t)=e^{-iHt}$ as a circuit.

Problem: the $H_k$ generally do not commute, so $e^{-iHt} \ne \prod_k e^{-ih_kH_kt}$.

Trotterization¶

First-order Lie–Trotter:

Second-order (Suzuki) symmetric splitting reduces error to $\mathcal{O}(t^3/r^2)$.

Eureka! Error is governed by commutators: terms that commute cost nothing to split. Locality of physical Hamiltonians is what makes simulation efficient.

Circuit Primitives for Pauli Evolutions¶

• $e^{-i\theta Z}$: an $R_z(2\theta)$ gate.

• $e^{-i\theta Z\otimes Z}$: CNOT ladder $\to R_z(2\theta) \to$ CNOT ladder.

• Any Pauli string $e^{-i\theta P}$: basis change ($H$ for $X$, $S^\dagger H$ for $Y$), then the $ZZ\cdots Z$ pattern.

Worked example on board: one Trotter step of the transverse-field Ising model $H = -J\sum_j Z_jZ_{j+1} - g\sum_j X_j$.

Fermions to Qubits: Jordan-Wigner¶

Creation/annihilation operators need antisymmetry; qubits do not have it natively.

• Preserves $\{a_i, a_j^\dagger\}=\delta_{ij}$; cost: strings of $Z$’s (nonlocality).

• Bravyi--Kitaev: $\mathcal{O}(\log n)$-weight strings; parity/ternary-tree variants.

• Pipeline: molecule/lattice $\to$ second quantization $\to$ qubit Hamiltonian (openfermion, qiskit-nature).

Eureka! After JW, any electronic-structure problem is a sum of Pauli strings — exactly what our Trotter and VQE (L12) machinery consumes.

Measuring Observables¶

• Energy: $\langle H\rangle = \sum_k h_k \langle P_k\rangle$; measure each Pauli string by basis rotation + $Z$-basis shots.

• Grouping commuting strings reduces the number of measurement settings.

• Dynamics: measure correlators $\langle Z_j(t) Z_k(0)\rangle$ after Trotterized evolution.

Beyond Trotter (One-Slide Horizon)¶

qDRIFT (randomized compiling of terms), linear combination of unitaries (LCU), qubitization / quantum signal processing achieve optimal $t$ and $\varepsilon$ scaling — the fault-tolerant-era toolkit built on the QFT/QPE ideas of L08-L09.

Mini-Lab¶

• Simulate a 4-spin transverse-field Ising quench; compare Trotter steps $r=1,2,4,8$ against exact diagonalization (NumPy).

• Verify the $\mathcal{O}(1/r)$ error scaling on a log-log plot.

• Map H$_2$ (STO-3G, 2 qubits after reduction) to a qubit Hamiltonian and print its Pauli decomposition.

Takeaways¶

• $e^{-iHt}$ is compiled by splitting $H$ into local terms; commutators set the price.

• Pauli-string evolution circuits are the universal subroutine.

• Jordan-Wigner turns electrons into qubits; materials problems become Pauli sums.