This report describes an approach for representation of quantum operators through rotations and rotation through quantum operators. The approach of the proposed method transforms rotation in a kind of a unitary matrix that corresponds to the rotation. Operations with qubits are very similar to the rotation, but with an added phase coefficient. This fact is used to create a process for rotation between unitary matrices. This approach could be used to modifying the controls to apply in a different basis.This report describes an approach for representation of quantum operators through rotations and rotation through quantum operators. The approach of the proposed method transforms rotation in a kind of a unitary matrix that corresponds to the rotation. Operations with qubits are very similar to the rotation, but with an added phase coefficient. This fact is used to create a process for rotation between unitary matrices. This approach could be used to modifying the controls to apply in a different basis.

Journal of Quantum Information ScienceJan 1, 20154 citations

Arbitrary unitary rotation of three-dimensional pixellated images

Alejandro R. Urzúa, Kurt Bernardo Wolf

Using the coefficients introduced by Bargmann and Moshinsky for the reduction of the su( $3$ ) algebra of Cartesian three-dimensional oscillator multiplet states into so( $3$ ) angular momentum submultiplets, we implement unitary rotations of three-dimensional Cartesian arrays that form finite pixellated "volume images." Transforming between the Cartesian and spherical bases, the subgroup of rotations in the latter is converted into rotations of the former, allowing for proper concatenation and inversion of these unitary transformations, which entail no loss of information.Using the coefficients introduced by Bargmann and Moshinsky for the reduction of the su($3$) algebra of Cartesian three-dimensional oscillator multiplet states into so($3$) angular momentum submultiplets, we implement unitary rotations of three-dimensional Cartesian arrays that form finite pixellated "volume images." Transforming between the Cartesian and spherical bases, the subgroup of rotations in the latter is converted into rotations of the former, allowing for proper concatenation and inversion of these unitary transformations, which entail no loss of information.

arXiv:2207.13308Jul 27, 20220 citations

Unitary Matrices

Taylor & Francis eBooksSep 30, 20100 citations

Rotationally Invariant Circuits: Universality with the Exchange Interaction and Two Ancilla Qubits

Iman Marvian, Hanqing Liu, Austin Hulse

Universality of local unitary transformations is one of the cornerstones of quantum computing with many applications and implications that go beyond this field. However, it has recently been shown that this universality does not hold in the presence of continuous symmetries: generic symmetric unitaries on a composite system cannot be implemented, even approximately, using local symmetric unitaries on the subsystems. In this Letter, we show that, despite these constraints, any SU(2) rotationally invariant unitary can be realized with the Heisenberg exchange interaction, which is 2-local and rotationally invariant, provided that the system interacts with a pair of ancilla qubits. We also show that a single ancilla is not enough to achieve universality. Furthermore, we study qubit circuits formed from k-local rotationally invariant unitaries and fully characterize the constraints imposed by locality on the realizable unitaries. We also find an interpretation of these constraints in terms of the average energy of states with a fixed angular momentum.Universality of local unitary transformations is one of the cornerstones of quantum computing with many applications and implications that go beyond this field. However, it has recently been shown that this universality does not hold in the presence of continuous symmetries: generic symmetric unitaries on a composite system cannot be implemented, even approximately, using local symmetric unitaries on the subsystems. In this Letter, we show that, despite these constraints, any SU(2) rotationally invariant unitary can be realized with the Heisenberg exchange interaction, which is 2-local and rotationally invariant, provided that the system interacts with a pair of ancilla qubits. We also show that a single ancilla is not enough to achieve universality. Furthermore, we study qubit circuits formed from k-local rotationally invariant unitaries and fully characterize the constraints imposed by locality on the realizable unitaries. We also find an interpretation of these constraints in terms of the average energy of states with a fixed angular momentum.

arXiv:2202.01963Physical Review LettersMar 29, 202413 citations

Unitary Matrices

WORLD SCIENTIFIC eBooksJul 20, 20160 citations

Using the coefficients introduced by Bargmann and Moshinsky for the reduction of the su(

3

) algebra of Cartesian three-dimensional oscillator multiplet states into so(

3

) angular momentum submultiplets, we implement unitary rotations of three-dimensional Cartesian arrays that form finite pixellated "volume images." Transforming between the Cartesian and spherical bases, the subgroup of rotations in the latter is converted into rotations of the former, allowing for proper concatenation and inversion of these unitary transformations, which entail no loss of information.