Lex Kemper (NCSU)
Lie algebraic generation of quantum circuits

Unitary synthesis, e.g. in constructing unitary coupled cluster factors or for evolution under a time dependent Hamiltonian is a key component of quantum simulation on quantum computers. Synthesizing the corresponding quantum circuit is typically done by breaking the operator into small circuit elements, named Trotter decomposition, which leads to circuits whose depth often scales unfavorably. We present two algorithms to help overcome these difficulties.

First, it is possible to synthesize exact quantum circuit representations of the desired time evolution unitaries. This is accomplished by considering the Lie algebra generated by the Hamiltonian, and partitioning it via Cartan decomposition. Coupled with an appropriate ansatz, this method yields an exact time evolution unitary, with polynomial or exponential circuit depth depending on the model under consideration.

Second, when the circuit elements of the Trotter decomposition are limited to a subset of SU(4) - or equivalently, when the Hamiltonian may be mapped onto free fermionic models - several identities exist that combine and simplify the circuit. Based on this, we present an algorithm that compresses the circuit elements into a single block of quantum gates. This results, for example, in a fixed depth time evolution for certain classes of Hamiltonians. We explicitly show how this algorithm works for several spin models, and demonstrate its use for adiabatic state preparation of the transverse field Ising model. Based on these ideas, we will also present an extension to controlled gates and imaginary time evolution.