  One strategy to fit larger problems on NISQ devices is to exploit a tradeoff
between circuit width and circuit depth. Unfortunately, this tradeoff still
limits the size of tractable problems since the increased depth is often not
realizable before noise dominates. Here, we develop qubit-efficient quantum
algorithms for entanglement spectroscopy which avoid this tradeoff. In
particular, we develop algorithms for computing the trace of the n-th power of
the density operator of a quantum system, $Tr(\rho^n)$, (related to the R\'enyi
entropy of order n) that use fewer qubits than any previous efficient algorithm
while achieving similar performance in the presence of noise, thus enabling
spectroscopy of larger quantum systems on NISQ devices. Our algorithms, which
require a number of qubits independent of n, are variants of previous
algorithms with width proportional to n, an asymptotic difference. The crucial
ingredient in these new algorithms is the ability to measure and reinitialize
subsets of qubits in the course of the computation, allowing us to reuse qubits
and increase the circuit depth without suffering the usual noisy consequences.
We also introduce the notion of effective circuit depth as a generalization of
standard circuit depth suitable for circuits with qubit resets. This tool helps
explain the noise-resilience of our qubit-efficient algorithms and should aid
in designing future algorithms. We perform numerical simulations to compare our
algorithms to the original variants and show they perform similarly when
subjected to noise. Additionally, we experimentally implement one of our
qubit-efficient algorithms on the Honeywell System Model H0, estimating
$Tr(\rho^n)$ for larger n than possible with previous algorithms.
