Symmetry and Transformations: Rotation Matrices

Edward B. Manoukian

Grant Davis, James K. Freericks

Using a global rotation by theta about the z-axis in the spin sector of the Jordan-Wigner transformation rotates Pauli matrices X and Y in the x-y-plane, while it adds a global complex phase to fermionic quantum states that have a fixed number of particles. With the right choice of angles, this relates expectation values of Pauli strings containing products of X and Y to different products, which can be employed to reduce the number of measurements needed when simulating fermionic systems on a quantum computer. Here, we derive this symmetry and show how it can be applied to systems in Physics and Chemistry that involve Hamiltonians with only single-particle (hopping) and two-particle (interaction) terms. We also discuss the consequences of this for finding efficient measurement circuits in variational ground state preparation.

Bruria Kaufman, C. C. Noack

The Hamiltonian of an isotropic harmonic oscillator is invariant under unitary transformations in three dimensions. This well-known invariance is exploited in a treatment of the Talmi transformation, viz., the transformation of two-particle oscillator functions to center-of-mass and relative coordinates. A simple and transparent form of this transformation in terms of rotation matrices and Wigner coefficients of SU3 is given. The calculation of these Wigner coefficients is described and the problem of degeneracies discussed.

Alberto Galindo, P. Pascual

Grant Davis, James K. Freericks

Using a global rotation by θ about the z-axis in the spin sector of the Jordan–Wigner transformation rotates Pauli matrices X^ and Y^ in the x−y-plane, while it adds a global complex phase to fermionic quantum states that have a fixed number of particles. With the right choice of angles, this relates expectation values of Pauli strings containing products of X^ and Y^ to different products, which can be employed to reduce the number of measurements needed when simulating fermionic systems on a quantum computer. Here, we derive this symmetry and show how it can be applied to systems in Physics and Chemistry that involve Hamiltonians with only single-particle (hopping) and two-particle (interaction) terms. We also discuss the consequences of this for finding efficient measurement circuits in variational ground state preparation.