Unitary circuit synthesis for tomography of generalized coherent states

We present a method that outputs a sequence of simple unitary operations to prepare a given quantum state that is a generalized coherent state. Our method takes as inputs the expectation values of some relevant observables. Such expectations can be estimated by performing projective measurements on O(M3⁡log(M/δ)/ε2) copies of the state, where M is the dimension of an associated Lie algebra, ɛ is a precision parameter, and 1 − δ is the required confidence level. The method can be implemented on a classical computer and runs in time O(M4⁡log(M/ε)). It provides O(M⁡log(M/ε)) simple unitaries that form the sequence. The overall complexity is then polynomial in M, being very efficient in cases where M is significantly smaller than the Hilbert space dimension, as for some fermion algebras. When the algebra of relevant observables is given by certain Pauli matrices, each simple unitary may be easily decomposed into two-qubit gates. We discuss applications to efficient quantum state tomography and classical simulations of quantum circuits.