  In order to achieve fault-tolerant quantum computation, we need to repeat the
following sequence of four steps: First, perform 1 or 2 qubit quantum gates (in
parallel if possible). Second, do a syndrome measurement on a subset of the
qubits. Third, perform a fast classical computation to establish which errors
have occurred (if any). Fourth, depending on the errors, we apply a correction
step. Then the procedure repeats with the next sequence of gates. In order for
these four steps to succeed, we need the error rate of the gates to be below a
certain threshold. Unfortunately, the error rates of current quantum hardware
are still too high. On the other hand, current quantum hardware platforms are
designed with these four steps in mind. In this work we make use of this
four-step scheme not to carry out fault-tolerant computations, but to enhance
short, constant-depth, quantum circuits that perform 1 qubit gates and
nearest-neighbor 2 qubit gates. To explore how this can be useful, we study a
computational model which we call Local Alternating Quantum Classical
Computations (LAQCC). In this model, qubits are placed in a grid allowing
nearest neighbor interactions; the quantum circuits are of constant depth with
intermediate measurements; a classical controller can perform log-depth
computations on these intermediate measurement outcomes to control future
quantum operations. This model fits naturally between quantum algorithms in the
NISQ era and full fledged fault-tolerant quantum computation. We show that
LAQCC circuits can create long-ranged interactions, which constant-depth
quantum circuits cannot achieve, and use it to construct a range of useful
multi-qubit gates. With these gates, we create three new state preparation
protocols for a uniform superposition over an arbitrary number of states,
W-states, Dicke states and may-body scar states.
