Exact Non-Identity Check and Gate-Teleportation-Based Indistinguishability Obfuscation are NP-hard for Low-T-Depth Quantum Circuits

TL;DR

This paper proves that deciding Exact Non-Identity Check and achieving gate-teleportation-based indistinguishability obfuscation are NP-hard problems for quantum circuits with low T-depth, highlighting computational limitations in quantum circuit obfuscation.

Contribution

It establishes NP-hardness of ENIC and obfuscation for Clifford+T circuits with logarithmic T-depth, extending previous hardness results to low T-depth regimes.

Findings

ENIC is NP-hard for T-depth O(log n) circuits.

Gate-teleportation-based obfuscation is computationally infeasible for low T-depth circuits.

Results imply no efficient solutions unless P=NP.

Abstract

In 2021, Broadbent and Kazmi developed a gate-teleportation-based protocol for computational indistinguishability obfuscation of quantum circuits. This protocol is efficient for Clifford+T circuits with logarithmically many T-gates, where the limiting factor in the efficiency of the protocol is the difficulty, on input a quantum circuit $C$, of the classical task of producing a description of the unitary obtained by conjugating a Pauli $P$ (corresponding to a Bell-measurement outcome) by $C$, where this description only depends on the input-output functionality of $CPC^{\dagger}$. The task above, in turn, is at least as hard as the problem of determining whether two $n$-qubit quantum circuits are perfectly equivalent up to global phase. In 2009, Tanaka defined the corresponding decision problem Exact Non-Identity Check (ENIC) and showed that ENIC is NQP-complete in general. Motivated by this, we consider in this work what happens when we pass from low T-count to low T-depth. In particular, we show that, for Clifford+T circuits of T-depth $O(\log(n))$, deciding ENIC is NP-hard. This effectively rules out the possibility, for Clifford+T circuits of logarithmic T-depth, of either efficient ENIC or efficient gate-teleportation based computational indistinguishability obfuscation, unless P=NP.