VQE UCCSD Ansatz
The unitary coupled-cluster singles-and-doubles trial state, a chemistry-motivated ansatz with built-in physical structure.
From coupled cluster to a unitary
Classical coupled cluster writes a correlated wavefunction as exp(T)|HF>, where |HF> is the Hartree-Fock reference and T is a cluster operator of electron excitations. This exp(T) is not unitary, so it cannot be run directly on a quantum computer. Unitary coupled cluster (UCC) instead uses exp(T - T^dagger), which is unitary and can be implemented as a circuit.
Singles and doubles
UCCSD truncates T to single and double excitations: T_1 moves one electron from an occupied to a virtual orbital, T_2 moves a pair. Each excitation carries a variational amplitude. After a fermion-to-qubit mapping (see Jordan-Wigner), each excitation becomes a sum of Pauli strings, and exp of those strings becomes a sequence of rotations and entangling gates.
Structure
- Start from the Hartree-Fock state, a single computational basis state.
- Apply parameterized excitation operators for all single and double excitations.
- Use Trotterization to approximate the exponential of non-commuting terms.
- Optimize the excitation amplitudes to minimize energy.
Strengths
UCCSD respects particle number and spin symmetries by construction, so the optimizer stays in the physically correct subspace. It systematically captures electron correlation and provides a principled, transferable initialization (classical coupled-cluster amplitudes). It typically reaches chemical accuracy for small molecules where it is feasible.
Cost and limits
The number of double excitations scales roughly as (occupied)^2 times (virtual)^2, so circuit depth grows quickly with basis size, straining near-term hardware. Trotter error and gate count are the main obstacles. Adaptive variants (ADAPT-VQE) grow the ansatz operator by operator, adding only the excitations that most lower the energy, producing shorter circuits than full UCCSD. Contrast with the shallow but less structured hardware-efficient ansatz.