The 2-RDM on a Quantum Computer

The two-electron reduced density matrix is not only a learning target but also the object a quantum computer natively returns. Because the electronic Hamiltonian is two-body, energy is a linear functional of the 2-RDM, so a quantum algorithm need only prepare a state and read off its two-particle marginal.

Energy is linear in the 2-RDM

A molecular Hamiltonian is two-body, , so the ground-state energy is a linear functional of the 2-RDM ,

with the two-body integral tensor, into which the one-body part folds by contraction. Nothing about the -dimensional wavefunction is needed. Fixing determines the energy together with every one- and two-body observable. On a quantum device this is decisive, because the marginal is measurable while the state vector is not.

What each quantum algorithm does with it

• Variational quantum eigensolver (VQE). Prepare a parametrised state on the qubits, measure its 1- and 2-RDM by sampling the Pauli operators that make up and (under a Jordan–Wigner or Bravyi–Kitaev map), form , and minimise over . The 2-RDM is the quantity read from hardware; the circuit is only how the state is prepared.
• Contracted quantum eigensolver (CQE). Solve for the 2-RDM directly. The anti-Hermitian contraction of the Schrödinger equation, , is a closed condition on the 2-RDM whose residual drives a unitary update. Here the variable of the algorithm is the 2-RDM, and the fixed point is the correlated marginal, the quantum-hardware analogue of solving for it.

N-representability is the quantum-hard part, and hardware sidesteps it

Classically one may try to minimise over directly (variational 2-RDM / SDP), but the constraint that come from a real -electron state, its N-representability, has QMA-complete membership, so the exact feasible set is intractable and one settles for the PQGT outer relaxation. A quantum computer prepares a physical state, so its measured 2-RDM is N-representable by construction. The hardware pays the representability cost that the classical cone condition cannot. This is the sharpest sense in which the object is quantum.

Measurement cost is set by locality

The 2-RDM has entries in orbitals, and estimating each to chemical accuracy dominates the run time. Grouping commuting Pauli terms and exploiting nearsightedness, whereby off-diagonal blocks between distant orbitals are negligible, cuts the measurement budget and makes fragment/embedding schemes (DMET-style) run on near-term hardware.

References

• A. Peruzzo et al., “A Variational Eigenvalue Solver on a Photonic Quantum Processor” (Nature Communications, 2014)
• D. A. Mazziotti, on the contracted quantum eigensolver / anti-Hermitian contracted Schrodinger equation
• Y.-K. Liu, M. Christandl & F. Verstraete, “Quantum Computational Complexity of the N-representability Problem: QMA Complete” (Phys. Rev. Lett., 2007)