  We introduce a new notion called ${\cal Q}$-secure pseudorandom isometries
(PRI). A pseudorandom isometry is an efficient quantum circuit that maps an
$n$-qubit state to an $(n+m)$-qubit state in an isometric manner. In terms of
security, we require that the output of a $q$-fold PRI on $\rho$, for $ \rho
\in {\cal Q}$, for any polynomial $q$, should be computationally
indistinguishable from the output of a $q$-fold Haar isometry on $\rho$. By
fine-tuning ${\cal Q}$, we recover many existing notions of pseudorandomness.
We present a construction of PRIs and assuming post-quantum one-way functions,
we prove the security of ${\cal Q}$-secure pseudorandom isometries (PRI) for
different interesting settings of ${\cal Q}$. We also demonstrate many
cryptographic applications of PRIs, including, length extension theorems for
quantum pseudorandomness notions, message authentication schemes for quantum
states, multi-copy secure public and private encryption schemes, and succinct
quantum commitments.
