Pseudorandom Isometries

We introduce a new notion called \({\mathcal {Q}}\) -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 \( \rho \in {\mathcal {Q}}\) , for any polynomial q, should be computationally indistinguishable from the output of a q-fold Haar isometry on \(\rho \) . By fine-tuning \({\mathcal {Q}}\) , 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 \({\mathcal {Q}}\) -secure pseudorandom isometries (PRI) for different interesting settings of \({\mathcal {Q}}\) . 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.