unitarylab/quantum-practices/hamiltonian-simulation

Hamiltonian Simulation

Approximate the time-evolution operator $e^{-iHt}$ for a given Hermitian matrix $H$.

Available Methods

1. Trotter-Suzuki Decomposition

See reference: ./trotter/SKILL.md

Deterministic product formula. Decomposes $e^{-iHt}$ into a sequence of simpler exponentials.

• Key parameters: order (Suzuki order), steps (number of Trotter steps).

2. QDrift (Randomized)

See reference: ./qdrift/SKILL.md

Stochastic channel simulation via $\lambda$-weighted random sampling of Pauli terms.

• Key parameter: steps (number of random samples).

3. Cartan Decomposition

See reference: ./cartan/SKILL.md

Structural decomposition via Lie algebra splitting $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{m}$. Iterates a Lax flow to build a circuit of the form $K \cdot e^{-i\eta} \cdot K^\dagger$.

• Key parameters: lr (learning rate), max_steps (hard cap on Lax update steps), reps (number of repetitions).

4. QSP-HS (Quantum Signal Processing for Hamiltonian Simulation)

See reference: ./qsp/SKILL.md

Use this route only when the user asks for Hamiltonian simulation or time evolution with QSP. This method block-encodes the Hamiltonian and applies polynomial spectral transformations via interleaved signal-processing rotations. It approximates $\cos(tH)$ and $\sin(tH)$ as Chebyshev series and merges them via LCU.

• Key parameters: degree (upper bound on polynomial degree per time slice), beta (block-encoding scaling factor).

For a basic single-qubit QSP demo, scalar signal variable x, phase-factor optimization, or approximating cos(t*x) at a point, route to ../linear-systems/quantum-signal-processing/SKILL.md instead.

5. Taylor Series (LCU)

See reference: ./taylor/SKILL.md

Truncates the Taylor series of $e^{-iHt}$ and implements the result as a Linear Combination of Unitaries over Pauli string products. Applies adaptive time-slicing to keep the per-slice spectral weight small.

• Key parameters: degree (Taylor truncation order, capped at 15).

Method Selection Guidance

This top-level skill is an index and routing guide. For executable examples, open the corresponding method-specific SKILL.md.

Conceptual dispatch logic:

if method == "trotter":
    use ./trotter/SKILL.md
elif method == "qdrift":
    use ./qdrift/SKILL.md
elif method == "cartan-lax":
    use ./cartan/SKILL.md
elif method == "qsp" or method == "qsp-hs":
    use ./qsp/SKILL.md
elif method == "taylor":
    use ./taylor/SKILL.md

The returned object exposes lazy properties: circuit, evolution_result, total_error.

Prerequisites

1. Basic linear algebra (Hermitian matrices, matrix exponential).
2. High-level understanding of Hamiltonian simulation goals.
3. NumPy for matrix construction.