Low-Stabilizer-Complexity Quantum States Are Not Pseudorandom
- URL: http://arxiv.org/abs/2209.14530v1
- Date: Thu, 29 Sep 2022 03:34:03 GMT
- Title: Low-Stabilizer-Complexity Quantum States Are Not Pseudorandom
- Authors: Sabee Grewal, Vishnu Iyer, William Kretschmer, Daniel Liang
- Abstract summary: We show that quantum states with "low stabilizer complexity" can be efficiently distinguished from Haar-random.
We prove that $omega(log(n))$ $T$-gates are necessary for any Clifford+$T$ circuit to prepare computationally pseudorandom quantum states.
- Score: 1.0323063834827415
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: We show that quantum states with "low stabilizer complexity" can be
efficiently distinguished from Haar-random. Specifically, given an $n$-qubit
pure state $|\psi\rangle$, we give an efficient algorithm that distinguishes
whether $|\psi\rangle$ is (i) Haar-random or (ii) a state with stabilizer
fidelity at least $\frac{1}{k}$ (i.e., has fidelity at least $\frac{1}{k}$ with
some stabilizer state), promised that one of these is the case. With black-box
access to $|\psi\rangle$, our algorithm uses $O\!\left( k^{12}
\log(1/\delta)\right)$ copies of $|\psi\rangle$ and $O\!\left(n k^{12}
\log(1/\delta)\right)$ time to succeed with probability at least $1-\delta$,
and, with access to a state preparation unitary for $|\psi\rangle$ (and its
inverse), $O\!\left( k^{3} \log(1/\delta)\right)$ queries and $O\!\left(n k^{3}
\log(1/\delta)\right)$ time suffice.
As a corollary, we prove that $\omega(\log(n))$ $T$-gates are necessary for
any Clifford+$T$ circuit to prepare computationally pseudorandom quantum
states, a first-of-its-kind lower bound.
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