Simulate Hamiltonian evolution

Why this is a layer

The families here consume genuinely different inputs and pay in different currencies. Product formulas need only a term decomposition and build no block-encoding, but carry a polynomial dependence on $1/\varepsilon$. LCU and qubitization need a block-encoding and its ancillas and reach a logarithmic dependence on $1/\varepsilon$. Which is cheaper depends on the precision regime and on the structure of $H$, so a cost model that names "Hamiltonian simulation" without naming the family has not costed anything. The norm parameter is inherited from the block-encoding layer below, which is where the constant that dominates real resource estimates is actually fixed.

Records for what it handles

The catalogue's own entries for the objects on either side of this contract.

Ways to do this

3 methods recorded

• Product-formula (Trotter-Suzuki) simulation

  Split $H$ into terms that can each be exponentiated directly and alternate short evolutions of them — the Lie-Trotter formula and its higher-order generalizations. No block-encoding is built and there is no all-zeros flag to amplify.

• Truncated-Taylor LCU simulation

  Truncate the Taylor series of $e^{-iHt}$ over short segments and implement the truncated sum as a linear combination of unitaries — PREPARE loads the coefficients, SELECT applies the terms, PREPARE$†$ unprepares — with the all-zeros ancilla flag amplified.

• Qubitization walk simulation

  From a block-encoding pair $(U, |G⟩)$ with $H = (⟨G|⊗I)U(|G⟩⊗I)$, build a walk operator $W$ that splits the Hilbert space into invariant two-dimensional $SU(2)$ subspaces, one per eigenvalue of $H$, with eigenvalues $e^{±i·arccos(H/α)}$. Quantum signal processing phases applied to $W$ then produce $e^{-iHt}$.

Routes that skip this layer

These do not fill the slot. They replace the span it belongs to, so this layer is not on their path at all.

• Chebyshev LCU inversion

  Expand an approximation of $1/x$ in Chebyshev polynomials and implement that expansion directly through the quantum walk associated with a sparse $A$, rather than through Hamiltonian simulation. Phase estimation is never used, which is what removes the $\mathrm{poly}(1/\varepsilon)$ bottleneck.

This is a step inside

• Quantum simulation of the KvN representation

  Because the Koopman-von Neumann generator is Hermitian and its propagator unitary, the lifted evolution can be run by Hamiltonian simulation directly. No linear system is assembled and no linear solver is called.

• LCHS — linear combination of Hamiltonian simulation

  Express a general non-unitary evolution operator as a linear combination of unitary evolution operators, each of which solves a Hamiltonian simulation problem, rather than converting the problem into a dilated linear system. An, Liu and Lin state that the method can achieve optimal cost in terms of state preparation.

• LCHS with the improved kernel a narrower version of LCHS — linear combination of Hamiltonian simulation

  A family of identities expressing non-unitary evolution as a linear combination of unitary evolutions, built on the kernel $f(z) = 1/(C_β e^{(1+iz)^β})$ with $β ∈ (0,1)$ and $C_β = 2π e^{-2^β}$. The kernel decays at a near-exponential rate $e^{-c|k|^β}$, replacing the original Cauchy kernel's quadratic decay and exponentially enhancing accuracy.

• Schrödingerisation (linear PDEs as Schrödinger equations)

  A simple change of variable — the warped phase transformation, which introduces one extra variable — recasts any linear PDE or ODE system into a system of Schrödinger equations in real time, which ordinary Hamiltonian simulation then runs. The original solution is recovered from the auxiliary dimension.

• HHL: eigenvalue inversion by phase estimation

  Prepare $|b>$, run phase estimation against $e^{-iAt}$ to write eigenvalue estimates into an ancilla register, apply a controlled rotation with amplitude proportional to $1/λ̃$, uncompute the estimation and post-select on the rotation ancilla. The success amplitude is about $1/κ$, so the procedure is amplified $O(κ)$ times.

• Ground-state energy by phase estimation

  Prepare a state that already overlaps the ground state, evolve it under the molecular Hamiltonian, and read the energy off the accumulated phase. Nothing is optimized and nothing is varied — the answer is a measured eigenvalue, and the whole difficulty moves into the starting state.

Routes that run this slot many times

For these routes this slot is inside a loop, so its cost is multiplied rather than paid once. That multiplier is usually the largest single term in what the route costs.

• Quantum simulation of the KvN representation runs 4K ∼ O(1/ε) KvN simulations — "the KvN simulation must be repeated 4K ∼ O(1/ε) times"

  The loop stays coherent: nothing is measured between turns. The preparation may still be reapplied every turn — what the loop never pays is a readout and a restart from classical data. The price is depth, and a success probability that multiplies down the chain.

• HHL: eigenvalue inversion by phase estimation runs O(κ) times — once per amplification round

  The loop stays coherent: nothing is measured between turns. The preparation may still be reapplied every turn — what the loop never pays is a readout and a restart from classical data. The price is depth, and a success probability that multiplies down the chain.

No multiplicity recorded

These routes take this step and no source we have read says how often. That is an absence, not a claim that they take it once.

• LCHS — linear combination of Hamiltonian simulation

  Express a general non-unitary evolution operator as a linear combination of unitary evolution operators, each of which solves a Hamiltonian simulation problem, rather than converting the problem into a dilated linear system. An, Liu and Lin state that the method can achieve optimal cost in terms of state preparation.

• LCHS with the improved kernel a narrower version of LCHS — linear combination of Hamiltonian simulation

  A family of identities expressing non-unitary evolution as a linear combination of unitary evolutions, built on the kernel $f(z) = 1/(C_β e^{(1+iz)^β})$ with $β ∈ (0,1)$ and $C_β = 2π e^{-2^β}$. The kernel decays at a near-exponential rate $e^{-c|k|^β}$, replacing the original Cauchy kernel's quadratic decay and exponentially enhancing accuracy.

• Schrödingerisation (linear PDEs as Schrödinger equations)

  A simple change of variable — the warped phase transformation, which introduces one extra variable — recasts any linear PDE or ODE system into a system of Schrödinger equations in real time, which ordinary Hamiltonian simulation then runs. The original solution is recovered from the auxiliary dimension.

• Ground-state energy by phase estimation

  Prepare a state that already overlaps the ground state, evolve it under the molecular Hamiltonian, and read the energy off the accumulated phase. Nothing is optimized and nothing is varied — the answer is a measured eigenvalue, and the whole difficulty moves into the starting state.

In the Atlas

• Trotter–Suzuki Hamiltonian simulation

  Approximates the time evolution of a Hamiltonian with non-commuting terms by alternating their individual exponentials.

• Transverse-field Ising simulation

  A compact Hamiltonian-simulation pattern for spin chains: alternate local field rotations with entangling ZZ evolution.

• Quantum algorithms to simulate many-body physics of correlated fermions

  Simulate strongly correlated fermionic systems — notoriously hard for classical computers — on a quantum computer with 2D or linear (1D) nearest-neighbor qubit-qubit couplings, of the kind typical of superconducting transmon qubit arrays, including preparing the relevant quantum states and evolving the system in time, with the Fermi-Hubbard model as a worked example.

• Quantum simulation of the kicked rotator model

  Simulate the quantum kicked rotator model — used to study quantum chaos, localization and the Anderson transition — with a quantum algorithm that scales better than classical simulation of the same model.