Algorithms · Optimization · time-dependent Hamiltonian
Quantum adiabatic evolution
An optimization pattern that slowly deforms an easy ground state into the ground state of a problem Hamiltonian.
Strong empiricalUpdated
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3 steps across 2 wires. Hover a step, or press play to go through them in order.
Expected outcomes
- Ground-state sector85%
- Excited / diabatic leakage15%
- Steps3
- Wires2
- Qubits2 in the toy interpolation
- Schedules = 0 → 1 over 20 steps
- Problem termZ₀Z₁ coupling
- Papers1
Adiabatic evolution encodes a problem in the final Hamiltonian and relies on remaining near the instantaneous ground state during the interpolation. In a gate model, the continuous path is approximated by a sequence of short product-formula steps.
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.
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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 →
The adiabatic path uses H(s) = (1−s)H₀ + sHₚ. If the schedule is slow compared with the inverse square of the minimum gap, the state can track the ground state; near a small gap, the required runtime can grow sharply. A gate-model implementation approximates each short interval and must still measure final success, so a smooth schedule is only a construction check until the gap and diabatic error are evaluated.
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 in the toy interpolation
- Schedule
- s = 0 → 1 over 20 steps
- Problem term
- Z₀Z₁ coupling
- Initial Hamiltonian
- H₀ = −X₀ − X₁
- Problem Hamiltonian
- Hₚ = Z₀Z₁
- Success condition
- Track instantaneous ground state
Prepare |++⟩on q[0], q[1]H₀ RX stepon q[0], q[1]Hₚ RZZ stepon q[0], q[1]
Quantum vs classical
Classical baseline
Classical annealing or local-search methods can inspect an objective directly, while adiabatic circuits pay for coherent evolution, gap-dependent runtime, and measurement.
Quantum claim
The quantum path may exploit tunneling and interference for some landscapes, but no generic speedup follows from using a smooth interpolation.
How to compare
Estimate or bound the minimum gap and compare final-state success probability with classical baselines at matched evaluation and runtime budgets.
Nobody has reviewed this record for gaps yet.
from qiskit import QuantumCircuit
qc = QuantumCircuit(2)
qc.h(0)
qc.h(1) # prepare the H0 ground state
for step in range(20):
s = (step + 1) / 20
dt = 0.05
qc.rx(-2 * (1 - s) * dt, 0)
qc.rx(-2 * (1 - s) * dt, 1)
qc.rzz(2 * s * dt, 0, 1)
FINAL_CIRCUIT = qcWhere this sits
This record is named by the layer graph at:
- Interpolate slowly to the problem Hamiltonian Method
Takes An Ising or QUBO operator, diagonal in the computational basis, whose extremal eigenvector encodes the problem's solution — given, for QAOA, as a sum of clause terms over bits and clauses; given, for adiabatic evolution, as the final Hamiltonian of an interpolation whose starting point has an easily-constructed ground state. Neither method is told how the operator was built, or by which encoding. Returns A bit string read off the computational basis — the assignment — together with the objective value it achieves, or, for the adiabatic route, the stated promise that this string is (with fidelity approaching 1, for a long enough evolution time) the true minimiser. Neither route returns an energy with an error bar; both return a string.
Related entries
- QAOA MaxCut on a 5-node ring
Algorithms
- Transverse-field Ising simulation
Algorithms
- Quantum singular value transformation
Algorithms
References
- Quantum Computation by Adiabatic Evolution
Edward Farhi, Jeffrey Goldstone, Sam Gutmann, Michael Sipser · 2000
Defines the adiabatic computation model and relates runtime to the spectral gap of the interpolation.