Semiclassical formulation of the Gottesman-Knill theorem and universal quantum computation

Classical simulation of quantum computation, the gottesman-Knill theorem, and slightly beyond

The Gottesman-Knill theorem states that every “Clifford” quantum circuit, i.e., a circuit composed of Hadamard, CNOT and phase gates, can be simulated efficiently on a classical computer. It is was later found that a highly restricted classical computer (using only NOT and CNOT gates) suffices to simulate all Clifford circuits, implying that these circuits are most likely even significantly wea...

Chow’s theorem and universal holonomic quantum computation

A theorem from control theory relating the Lie algebra generated by vector fields on a manifold to the controllability of the dynamical system is shown to apply to Holonomic Quantum Computation. Conditions for deriving the holonomy algebra are presented by taking covariant derivatives of the curvature associated to a non-Abelian gauge connection. When applied to the Optical Holonomic Computer, ...

Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems

Normalizer circuits [1, 2] are generalized Clifford circuits that act on arbitrary finitedimensional systems Hd1 ⊗· · ·⊗Hdn with a standard basis labeled by the elements of a finite Abelian group G = Zd1 × · · · × Zdn . Normalizer gates implement operations associated with the group G and can be of three types: quantum Fourier transforms, group automorphism gates and quadratic phase gates. In t...

Semiclassical Fourier transform for quantum computation.

Shor’s algorithms for factorization and discrete logarithms on a quantum computer employ Fourier transforms preceding a final measurement. It is shown that such a Fourier transform can be carried out in a semi-classical way in which a “classical” (macroscopic) signal resulting from the measurement of one bit (embodied in a twostate quantum system) is employed to determine the type of measuremen...

Semiclassical Dynamics with Quantum Trajectories: Formulation and Comparison with the Semiclassical Initial Value Representation Propagator