Trapdoored Clifford Operators and Applications

2026-10-01

AI Summary

This study addresses the high computational complexity bottleneck in sampling and implementing random Clifford operators caused by the large group size. To overcome this, it introduces a backdoored Clifford distribution based on the Learning Parity with Noise (LPN) assumption, constructing for the first time finite-field backdoored matrices that support efficient multiplication and inversion, thereby resolving the open problem posed by Vaikuntanathan and Zamir. Furthermore, quantum circuit synthesis is optimized through lookup table array operations. The proposed approach achieves near-linear time sampling and polylogarithmic-depth circuit implementations, significantly reducing the costs of classical simulation and quantum verification. Finally, this work establishes that the hardness of batched Clifford circuits is no less than that of worst-case matrix multiplication.

Abstract

Random Clifford operators have numerous applications in quantum computing, including randomized benchmarking, classical shadows, and quantum authentication. However, sampling and implementing uniformly random $n$-qubit Clifford incur near-quadratic complexity due to the size of Clifford group. We introduce a cryptographic way to overcome these barriers: trapdoored Clifford operator distributions whose samples are computationally indistinguishable from uniformly random Cliffords, yet implementing them can be much faster given the trapdoor. We construct a distribution of trapdoored Clifford operators whose elements can be sampled and implemented in near-linear time under a variant of the learning parity with noise assumption. Our constructions allow fast tableau action on Pauli labels for classical simulation, and also can be optimized to admit polylogarithmic-depth implementation. Along the way, we construct trapdoored matrices over finite fields that support efficient multiplication by both a matrix and its inverse, resolving an open question left by Vaikuntanathan and Zamir [SODA'26]. We use these constructions to obtain faster protocols based on random Cliffords. We also explore their applications to the worst-case to average-case reductions for matrix and Clifford problems including the iterated matrix multiplication and Clifford circuit synthesis. In particular, we show the hardness of batching Clifford circuits: synthesizing circuits that apply the same Clifford to multiple registers is at least as hard as worst-case matrix multiplication, even when synthesis succeeds on a small constant fraction of random Cliffords. This extends to approximate implementations by general quantum circuits.

Problem

• Clifford operators
• trapdoor
• computational complexity
• quantum computing
• circuit synthesis

Innovation

• Trapdoored Clifford operators
• Learning parity with noise
• Randomized benchmarking
• Worst-case to average-case reductions
• Clifford circuit synthesis