  Virtual distillation (VD) using measurements of multiple copies of a quantum
circuit have recently been proposed as a method of noise mitigation of
expectation values. Circuit decompositions known as B gates were found only for
single qubit expectation values however practical calculations require
multi-qubit expectation values which cannot be corrected with B gates. We
discover low depth circuit decompositions for multi-qubit expectation values by
combining multiple projections to recover the correct measurement statistics or
expectation values. Our method adds linear entangling gates with number of
qubits, but requires extra measurements. Furthermore, in applications to find
ground states such as the variational quantum eigensolver (VQE) algorithm, the
variational principle is required which states the energy cannot go below the
ground state energy. We discover that the variational principle is violated if
noise is higher on single expectation values than multi-qubit which renders VQE
useless. We show this occurs when using B gates and is preserved if using our
low depth decomposition on all expectation values. We perform demonstration on
real devices and demonstrate our decomposition can mitigate real experimental
noise in VQE for the H$_2$ molecule with a two qubit tapered mapping, H$_3$
with three qubits, and H$_2$ with four qubits. Our decomposition provides a way
to perform duplicate circuit virtual distillation on real devices at
significantly lower depth and for arbitrary observables.
