Quantum Advantage from Conjugated Clifford Circuits

A well-known result of Gottesman and Knill [Got99] states that Clifford circuits – i.e. circuits composed of only CNOT, Hadamard, and π/4 phase gates – are efficiently classically simulable. We show that in contrast, “conjugated Clifford circuits” (CCCs) – where one additionally conjugates every qubit by the same one-qubit gate U – can perform hard sampling tasks. In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford conjugating unitary U can give rise to sampling tasks which cannot be simulated classically to constant multiplicative error, unless the polynomial hierarchy collapses. Furthermore, we show that this hardness result can be extended to allow for the more realistic model of constant additive error, under a plausible complexity-theoretic conjecture.