A compact Hamiltonian-simulation pattern for spin chains: alternate local field rotations with entangling ZZ evolution.
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The transverse-field Ising model is a useful bridge between abstract time evolution and hardware-friendly circuits. Its Pauli terms make the approximation error visible without hiding the physical model.
No input port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
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No output port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
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Takes An access model for — a sum of efficiently exponentiable terms, sparse-access oracles, or a block-encoding — plus an evolution time and a target error . Returns A circuit approximating to within , with a stated query or gate count, an ancilla count, and the norm parameter — sparsity times , or the LCU 1-norm — that the cost is measured against.
For a two-qubit Ising model, first-order Trotterization replaces the exact evolution with a product of exponentials of Pauli terms. The approximation is useful because each factor maps directly to a short hardware circuit, but the product is not exact when the terms do not commute. The catalog's small-instance check should compare this product with the exact 4×4 matrix at the same time step and report the error as the number of steps changes.
from qiskit import QuantumCircuit
J, h, dt = 1.0, 0.5, 0.1
qc = QuantumCircuit(2)
qc.rzz(2 * J * dt, 0, 1)
qc.rx(2 * h * dt, 0)
qc.rx(2 * h * dt, 1)
FINAL_CIRCUIT = qcFor two qubits, a classical matrix exponential gives the reference evolution directly; for larger systems, sparse or tensor-network methods exploit structure that a circuit must also expose.
The circuit implements local Pauli evolution with a number of steps controlled by the desired error, but sampling and gate noise remain part of the end-to-end cost.
Compare the circuit state with exact matrix evolution at the same Δt and report the operator or observable error rather than only showing a plausible circuit.
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Places Hamiltonian simulation in a query-complexity framework; the catalog sketch shows a small Pauli-term product formula.
arxiv.org/abs/quant-ph/0508139 ↗