Operators · VQE Hamiltonians and observables
First-order Trotter product
Product-formula approximation to Hamiltonian evolution. Representative form: e^{-itΣH_j} ≈ ∏_j e^{-itH_j}.
Attested & literatureUpdated
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- Representative forme^{-itΣH_j} ≈ ∏_j e^{-itH_j}
- Record typeOperator definition
- Papers1
This record, used inside a worked example
This page says To use this record in VQE
— here is a concrete, runnable instance of it.
VQE on a 2-qubit transverse-field Ising Hamiltonian
2 qubits, H = Z⊗Z + 0.5*(X⊗I + I⊗X), a stated toy Hamiltonian (not a molecule), minimised with a hardware-efficient ansatz.
Where it ends up
- 1043%
- 0143%
- 007%
- 117%
At these angles the ansatz's energy ⟨H⟩ reaches E0 = −√2, the Hamiltonian's exact ground energy; a real VQE run reaches these angles by repeating measure-the-energy then update-the-angles, not by a closed-form solution.
Open that example in StudioFirst-order Trotter product is cataloged as an operator rather than a circuit. Product-formula approximation to Hamiltonian evolution.
TakesNothingWhat joins here
No input port, deliberately. You measure a state with this entry; you do not apply it and pass a register on.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
No output port, deliberately. An observable is measured with, never applied, so there is no register to hand on.
Nothing in the Atlas meets this end.
Not a stage. You measure a state with this; you do not apply it and pass a register on. It has a width and deliberately no ports. See all 60 →
Product-formula approximation to Hamiltonian evolution. A representative mathematical form is e^{-itΣH_j} ≈ ∏_j e^{-itH_j}. To use this record in VQE, an implementation must specify index ordering, coefficient values and units, basis conventions, fermion-to-qubit mapping where applicable, symmetry sector, and measurement grouping. Those choices can change resource counts and even the physical interpretation. The catalog therefore preserves this as a sourced operator definition and refuses to fabricate seven circuit variants for an object that is not itself an ordered gate program.
TakesNothingWhat joins here
No input port, deliberately. You measure a state with this entry; you do not apply it and pass a register on.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
No output port, deliberately. An observable is measured with, never applied, so there is no register to hand on.
Nothing in the Atlas meets this end.
Not a stage. You measure a state with this; you do not apply it and pass a register on. It has a width and deliberately no ports. See all 60 →
- Representative form
- e^{-itΣH_j} ≈ ∏_j e^{-itH_j}
- Record type
- Operator definition
- Circuit
- Not supplied
- Instance data
- Required for execution
This record is not a concrete circuit.
Quantum vs classical
Classical baseline
Use a classical state-vector or matrix simulation at the same width, precision, and measurement objective.
Quantum claim
The quantum record demonstrates a state or operator behavior; it does not make classical simulation or communication costs disappear.
How to compare
Compare fidelity, samples, gate depth, noise, memory, and the cost of preparing and reading the state.
Nobody has reviewed this record for gaps yet.
OPERATOR: First-order Trotter product
REPRESENTATIVE FORM: e^{-itΣH_j} ≈ ∏_j e^{-itH_j}
ROLE: Product-formula approximation to Hamiltonian evolution.
This is a mathematical operator record, not an executable circuit.A reference record, not runnable source. Leona cannot execute it, so it cannot be saved to your Library as a circuit.
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References
- OpenFermion: The Electronic Structure Package for Quantum Computers
Jarrod R. McClean, Kevin J. Sung, Ian D. Kivlichan, Yudong Cao, Chengyu Dai, E. Schuyler Fried, Craig Gidney, Brendan Gimby, Pranav Gokhale, Thomas Häner, Tarini Hardikar, Vojtěch Havlíček, Oscar Higgott, Cupjin Huang, Josh Izaac, Zhang Jiang, Xinle Liu, Sam McArdle, Matthew Neeley, Thomas O'Brien, Bryan O'Gorman, Isil Ozfidan, Maxwell D. Radin, Jhonathan Romero, Nicholas Rubin, Nicolas P. D. Sawaya, Kanav Setia, Sukin Sim, Damian S. Steiger, Mark Steudtner, Qiming Sun, Wei Sun, Daochen Wang, Fang Zhang, Ryan Babbush · 2017
Provides open-source representations and transformations for fermionic and qubit operators used in quantum simulation.