  Quantum computers are capable of efficiently contracting unitary tensor
networks, a task that is likely to remain difficult for classical computers.
For instance, networks based on matrix product states or the multi-scale
entanglement renormalization ansatz (MERA) can be contracted on a small quantum
computer to aid the simulation of a large quantum system. However, without the
ability to selectively reset qubits, the associated spatial cost can be
exorbitant. In this paper, we propose a protocol that can unitarily reset
qubits when the circuit has a common convolutional form, thus dramatically
reducing the spatial cost for implementing the contraction algorithm on general
near-term quantum computers. This protocol generates fresh qubits from used
ones by partially applying the time-reversed quantum circuit over qubits that
are no longer in use. In the absence of noise, we prove that the state of a
subset of these qubits becomes $|0\ldots 0\rangle$, up to an error
exponentially small in the number of gates applied. We also provide a numerical
evidence that the protocol works in the presence of noise. We also provide a
numerical evidence that the protocol works in the presence of noise, and
formulate a condition under which the noise-resilience follows rigorously.
