Algorithms · Hamiltonian simulation · product formula
Transverse-field Ising simulation
A compact Hamiltonian-simulation pattern for spin chains: alternate local field rotations with entangling ZZ evolution.
Strong empiricalUpdated
Open on the mapView frozen graph
A small worked example, computed in your browser. It is not a result from this record's papers.
2 Trotter steps of a transverse-field Ising chain
4 qubits starting at |0000⟩, J=1, h=0.5, dt=0.3, 2 first-order Trotter steps.
Step 1 of 2 · Ising Trotter step(4)
Probability of each outcome after this step
- 000091%
- 00012%
- 00102%
- 01002%
- 10002%
- 00110%
- 01010%
- 01100%
- 8 more outcomes0%
| Outcome | Probability | Relative phase | |
|---|---|---|---|
| 0000 | 91.4% | 0 | |
| 0001 | 2.1% | 3π/2 | |
| 0010 | 2.1% | 3π/2 | |
| 0100 | 2.1% | 3π/2 | |
| 1000 | 2.1% | 3π/2 | |
| 0011 | 0.0% | π | |
| 0101 | 0.0% | π | |
| 0110 | 0.0% | π | |
8 more outcomes (0.1% between them)
Phases are measured against the largest amplitude. A global phase is not observable, so only the differences carry meaning.
First Trotter step: RZZ(2*J*dt) couples each neighbouring pair on the line, then RX(2*h*dt) applies the transverse field to every qubit.
Probability spreads from 1 outcome to 16.
Second Trotter step, identical to the first. Together the two steps approximate exp(−i·H·2·dt) for this chain's Hamiltonian.
Probability moves between the same 16 outcomes. Up: 0001 2.1% → 6.4%, 1000 2.1% → 6.4%. Down: 0000 91.4% → 75.2%. This is the step that entangles the qubits: from here on no single qubit has a state of its own, only the register as a whole does.
Readout The state spreads from |0000⟩ across several nearby basis states. The exact split is a regression value from this file's own simulator, not a claim about a real material.
Open in Studio- Steps3
- Wires2
- Qubits2 shown · chain generalizes to n
- Step depth1 RZZ + 2 RX rotations
- ApproximationFirst-order product formula
- Papers1
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.
TakesNothingWhat joins here
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.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
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.
Nothing in the Atlas meets this end.
This record publishes no gate sequence and no register, so there is nothing here to read an interface off. Absent rather than empty. See all 152 →
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.
TakesNothingWhat joins here
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.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
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.
Nothing in the Atlas meets this end.
This record publishes no gate sequence and no register, so there is nothing here to read an interface off. Absent rather than empty. See all 152 →
- Qubits
- 2 shown · chain generalizes to n
- Step depth
- 1 RZZ + 2 RX rotations
- Approximation
- First-order product formula
- Hamiltonian
- H = J Z₀Z₁ + h(X₀ + X₁)
- Time step
- Δt = 0.1 in the sketch
- Main limitation
- Trotter error for non-commuting terms
RZZ(2JΔt)on q[0], q[1]RX(2hΔt)on q[0]RX(2hΔt)on q[1]
Quantum vs classical
Classical baseline
For 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.
Quantum claim
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.
How to compare
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.
Nobody has reviewed this record for gaps yet.
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 = qcWhere this sits
This record is named by the layer graph at:
- Simulate Hamiltonian evolution Slot
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.
Related entries
- Trotter–Suzuki Hamiltonian simulation
Algorithms
- Quantum imaginary-time evolution
Algorithms
- Quantum signal processing
Algorithms
- Quantum phase estimation
Algorithms
References
- Efficient quantum algorithms for simulating sparse Hamiltonians
Dominic W. Berry, Graeme Ahokas, Richard Cleve, Barry C. Sanders · 2005
Places Hamiltonian simulation in a query-complexity framework; the catalog sketch shows a small Pauli-term product formula.