Quantum gate optimization strategies using alternate Hilbert space decomposition

Quantum computing has become a field in which increasing research efforts have been invested in recent decades. Current machines have an increasing number of qubits and simulations are be- coming more and more efficient. Despite this fact, any non-trivial algorithm requires a coherence time and a fidelity that are not within reach of near-term hardware. The problem of effectively and efficiently decomposing a unitary transformation is of fundamental importance in order to carry out digital quantum simulations, i.e consisting of a finite sequence of quantum gates according to the Solovay-Kitaev theorem. The decoherence and the low fidelity of transformations involving several qubits are currently the most important limitation on quantum computation. The opti- mization of a circuit is an NP-hard problem that requires more and more research efforts.

I will present some possible decomposition strategies at the basis of a naive compilator for a generic number of qubits. This compiler exploits the rules of unitary decomposition of matrices into 2-level matrices, the gray code definition (that allows to associate a full-controlled gate to each 2-level matrix), and the decomposition of full-controlled gate using CNOTs and single-qubit gates. The number of CNOTs still exponentially grows with the number of qubits when using this strategy, making the resulting compilator inefficient.

In order to rigorously study the separability and entanglement characteristics of transforma- tions, we need a computational formalism different from that used in literature. In fact, working in a modified Hilbert space it is possible to highlight more information regarding a given gate and use the concept of tensor rank to define them. This confusion of formalism lies in the use of the Kroneker product instead of the tensor product.