A Criterion for Post-Selected Quantum Advantage

TL;DR

It is proved that both instantaneous quantum polynomial (IQP) circuits and conjugated Clifford circuits and conjugated Clifford circuits (CCCs) afford a quantum advantage, which settles two questions of Bouland, Fitzsimons, and Koh.

Abstract

Assuming the polynomial hierarchy is infinite, we prove a sufficient condition for determining if uniform and polynomial size quantum circuits over a non-universal gate set are not efficiently classically simulable in the weak multiplicative sense. Our criterion exploits the fact that subgroups of $\mathrm{SL}(2;\mathbb{C})$ are essentially either discrete or dense in $\mathrm{SL}(2;\mathbb{C})$. Using our criterion, we give a new proof that both instantaneous quantum polynomial (IQP) circuits and conjugated Clifford circuits (CCCs) afford a quantum advantage. We also prove that both commuting CCCs and CCCs over various fragments of the Clifford group afford a quantum advantage, which settles two questions of Bouland, Fitzsimons, and Koh. Our results imply that circuits over just $(U^\dagger \otimes U^\dagger) \mathrm{CZ} (U \otimes U)$ afford a quantum advantage for almost all $U \in \mathrm{U}(2)$.

Figures (4)

• Figure 1: (Left) A flowchart illustrating our algorithm for checking if a finite subset $\Gamma \subset \mathrm{SL}(2;\mathbb{C})$ generates a dense subgroup of $\mathrm{SL}(2;\mathbb{C})$. (Right) An identical flowchart, but placed in the context of our criterion, Theorem \ref{['thm:maintwo']}. In this case, $\Gamma$ is a finite subset of $\mathrm{gad}_1(\mathcal{S})$ for some gate set $\mathcal{S}$ and "$\langle \Gamma \rangle$ dense in $\mathrm{SL}(2;\mathbb{C})$" is replaced by "classically intractable" in the sense of Theorem \ref{['thm:maintwo']}.
• Figure 2: A dependency diagram for our hardness results, where an arrow from A to B means "A is used in the proof of B" and "class." means the associated theorem is a classification result.
• Figure 3: A dependency diagram for our classification results, where an arrow from A to B means "A is used in the proof of B".
• Figure 4: The inclusion lattice of the 27 fragments of the multi-qubit Clifford operations. Here, "ALL" denotes every such Clifford operation, $\Gamma$ is the "vertex rotation" $(I - iX - iY - iZ) / 2$, and all other generators are defined in the main text. Red, green, and blue denote $X$-, $Y$-, and $Z$-preserving, respectively. See GS22 for complete details.

Theorems & Definitions (122)

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