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Quantum states prepared by Clifford circuits with fewer than n/2 non-Clifford single-qubit gates cannot be computationally pseudorandom.

Abstract

The key insights are:
The output state |ψ⟩ of a t-doped Clifford circuit (a Clifford circuit with at most t non-Clifford single-qubit gates) has stabilizer dimension at least n - 2t. This means the Weyl distribution qψ is supported on a subspace of dimension at most 2n - 2.
Haar-random states have minimal stabilizer dimension (0) with overwhelming probability. In contrast, the stabilizer dimension of the output of a t-doped Clifford circuit decreases by at most 2 per non-Clifford gate.
This allows distinguishing the output of a t-doped Clifford circuit from a Haar-random state by sampling from the Weyl distribution qψ and checking the dimension of the subspace spanned by the samples.
As a consequence, any family of Clifford circuits that produces an ensemble of computationally pseudorandom quantum states must use at least n/2 non-Clifford single-qubit gates. This bound is tight up to constant factors under plausible computational assumptions.

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Key Insights Distilled From

by Sabee Grewal... at **arxiv.org** 04-02-2024

Deeper Inquiries

The result that a linear number of T-gates are necessary for any Clifford+T circuit to prepare computationally pseudorandom quantum states has significant implications for the construction of pseudorandom quantum states without relying on one-way functions. This result suggests that linear-time quantum-secure pseudorandom functions are necessary for the existence of linear-time constructible pseudorandom states. If such functions do exist, it implies that pseudorandom quantum states can be efficiently generated without the need for one-way functions. This insight opens up new possibilities for the development of quantum cryptographic primitives and quantum algorithms that rely on pseudorandomness.

The techniques used in the context provided can potentially be extended to obtain lower bounds on the stabilizer complexity of other classes of quantum states, such as those with bounded entanglement entropy. By leveraging symplectic Fourier analysis, Bell difference sampling, and the properties of Weyl and characteristic distributions, it may be possible to analyze the stabilizer complexity of quantum states with specific entanglement properties. Understanding the relationship between entanglement entropy and stabilizer complexity could provide insights into the computational properties of quantum states and their learnability.

There are several other structural properties of the Weyl and characteristic distributions that could be leveraged to obtain further insights into the complexity of learning quantum states. For example, the duality property established in Theorem 3.1 and Theorem 3.2 could be used to analyze the distribution of probability mass on subspaces and their complements. Additionally, the symplectic Fourier transform and convolution properties could be explored to study the relationship between different quantum states and their representations in the Weyl basis. By further investigating these properties, researchers may uncover new connections between quantum state properties and computational complexity.

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