The Wigner Rotation Matrices

The Pennsylvania State Univ.

Anthony G. St. Pierre, William A. Steele

Blagoje Oblak

Nikolai L. Manakov, Alexei V. Meremianin, Anthony F. Starace

Our recent results [N. L. Manakov et al., Phys. Rev. A 57, 3233 (1998); 61, 022103 (2000)] on the invariant representations of finite rotation matrices (FRM's) of integer rank j (in terms of tensor products of vectors connected with a space-fixed reference frame) are generalized here for the general case of arbitrary (i.e., integer or half-integer) rank j. This extension is carried out by using new spinor representations of FRM's in terms of specially introduced spinor-annihilation operators. We demonstrate that all widely used, standard representations of FRM's follow as special cases of our invariant representation for particular parametrizations of the rotation parameters. As the simplest application of invariant spinor representations of FRM's, the factorized form of Wigner ${\mathbf{d}}^{j}(\ensuremath{\beta})$ matrices with an arbitrary rank j is obtained as a product of two triangular matrices composed of various powers of $\mathrm{cos}(\ensuremath{\beta}/2).$

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.