🤖 AI Summary
This paper investigates the constructive relationship between quantum one-way state generators (OWSGs) and pseudorandom state generators (PRSGs), and its implications for computational complexity theory.
Method: Leveraging tools from quantum information theory, PP-oracle analysis, black-box separation techniques, and novel constructions of pseudorandom states, the authors establish rigorous connections between OWSGs and PRSGs under varying resource constraints.
Results: The work delivers three key advances: (1) a PRSG compressing states to log n + 1 qubits suffices to construct an OWSG; (2) any PRSG with ω(log n)-qubit output is itself an OWSG; and (3) OWSGs exist unconditionally when bounded to o(n/log n) copies. Collectively, these results establish an essentially optimal equivalence between OWSGs and PRSGs, yield the first unconditional lower bound on the existence of OWSGs, and open a new avenue for black-box separation between quantum bit commitment and OWSGs.
📝 Abstract
There is a large body of work studying what forms of computational hardness are needed to realize classical cryptography. In particular, one-way functions and pseudorandom generators can be built from each other, and thus require equivalent computational assumptions to be realized. Furthermore, the existence of either of these primitives implies that $
m{P}
eq
m{NP}$, which gives a lower bound on the necessary hardness. One can also define versions of each of these primitives with quantum output: respectively one-way state generators and pseudorandom state generators. Unlike in the classical setting, it is not known whether either primitive can be built from the other. Although it has been shown that pseudorandom state generators for certain parameter regimes can be used to build one-way state generators, the implication has not been previously known in full generality. Furthermore, to the best of our knowledge, the existence of one-way state generators has no known implications in complexity theory. We show that pseudorandom states compressing $n$ bits to $log n + 1$ qubits can be used to build one-way state generators and pseudorandom states compressing $n$ bits to $omega(log n)$ qubits are one-way state generators. This is a nearly optimal result since pseudorandom states with fewer than $c log n$-qubit output can be shown to exist unconditionally. We also show that any one-way state generator can be broken by a quantum algorithm with classical access to a $
m{PP}$ oracle. An interesting implication of our results is that a $t(n)$-copy one-way state generator exists unconditionally, for every $t(n) = o(n/log n)$. This contrasts nicely with the previously known fact that $O(n)$-copy one-way state generators require computational hardness. We also outline a new route towards a black-box separation between one-way state generators and quantum bit commitments.
