We explore an efficient qubit encoding for calculating the ground state energy of Hamiltonians with the variational quantum eigensolver (VQE). In contrast to encodings such as Jordan-Wigner that use N qubits to encode N basis states, we construct a Hamiltonian with analagous action that makes use of all 2^N available states. The Hamiltonian is structured in terms of creation and annihilation operators that act on the basis states in gray code order. We show how this naturally partitions the Pauli terms into N + 1 commuting sets, which can each be measured using at most a single additional rotation. Our encoding also allows for use of hardware-efficient variational ansatze with shorter depth and fewer two-qubit gates than other ansatze such as unitary coupled cluster. Using the well-studied deuteron as an example, I will share some preliminary results and discuss some of the tradeoffs that arise as a consequence of this encoding.