Operators · VQE Hamiltonians and observables
One-particle reduced density matrix
Compact one-body state descriptor and orbital gradient input. Representative form: γₚq = ⟨a†ₚa_q⟩.
Attested & literatureUpdated
- Primary sourceLiu, Christandl & Verstraete 2007
- 1-RDM consistencyDecidable from eigenvalues alone
- 2-RDM consistencyQMA-complete
- 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 StudioOne-particle reduced density matrix is cataloged as an operator rather than a circuit. Compact one-body state descriptor and orbital gradient input.
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 →
The interesting fact about this object is a negative one about its neighbour. Given a matrix , when is it the one-body reduced density matrix of some antisymmetric -fermion state? That is the -representability question, and Liu, Christandl and Verstraete record that Coulson posed the problem and that its fermionic form takes its name from Coleman.
For the one-body case their paper is explicit, and the sentence is worth having exactly: "checking consistency of 2-body reduced density operators of fermionic states is so hard, while checking consistency of 1-body reduced density operators is simple" — and, they add, consistency in that case "can be decided … based solely on the eigenvalues of the reduced density operators."
The reason is structural rather than lucky. The paper gives it: the extreme points of the convex set of one-body density operators are ground states of Hamiltonians containing only bilinear terms in and — free fermions — and those diagonalize easily. So the one-body consistency problem inherits the tractability of the free-fermion problem, which is exactly why this object is a workable state descriptor and a workable orbital-gradient input.
The neighbouring object is intractable, and the precise claim matters. The paper's result is that deciding -representability of the two-body reduced density matrix is QMA-complete, and hence NP-hard. Two qualifications that a shortened version of this sentence would lose: the result classifies the problem's difficulty rather than solving it, and it is about the 2-RDM specifically — the 1-RDM case remains the easy one described above. The paper separately notes that restricting to the diagonal elements of the 2-RDM leaves an NP-hard problem.
One claim this record refuses to make. The explicit one-body condition often quoted — occupation numbers in summing to — is not stated in the paper cited here, which says only that eigenvalues suffice. It may well be correct and it is attributed elsewhere to Coleman's 1963 paper, which is behind a paywall and was not read. So it is left out. Citing this source for a condition it does not state would be the exact failure the catalog's sourcing rule exists to prevent, and "decidable from the eigenvalues" is the true claim and is enough.
Why it does not join the map. It is measured. observable-estimation names the operator being measured in its contract prose, where a parameter lives and a state does not — this record is one of the seventeen the shelf's observable abstention covers.
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 →
- Primary source
- Liu, Christandl & Verstraete 2007
- 1-RDM consistency
- Decidable from eigenvalues alone
- 2-RDM consistency
- QMA-complete
- Why 1-body is easy
- Extreme points are free-fermion ground states
- Problem posed by
- Coulson, per this paper
- Read depth
- Full text
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: One-particle reduced density matrix
REPRESENTATIVE FORM: γₚq = ⟨a†ₚa_q⟩
ROLE: Compact one-body state descriptor and orbital gradient input.
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.
Related entries
- Variational quantum eigensolver for ground-state energy
Algorithms
- Pauli-X operator
Operators
References
- N-representability is QMA-complete
Yi-Kai Liu, Matthias Christandl, F. Verstraete · 2007
Primary source for this record, read at the depth its deepened outcome states.