Designs from magic-augmented Clifford circuits

TL;DR

This paper introduces magic-augmented Clifford circuits that efficiently generate approximate unitary and state $k$-designs with reduced depth and magic resource, advancing quantum circuit design for complex quantum states.

Contribution

It presents a novel architecture combining Clifford and magic gates to produce approximate $k$-designs with improved depth and magic efficiency, including new theoretical bounds and no-go theorems.

Findings

Shallow Clifford circuits with magic gates generate approximate $k$-designs with logarithmic depth.

The number of magic gates can be reduced for bounded additive error designs.

Classical statistical mechanics models help understand the depth and magic requirements.

Abstract

We introduce magic-augmented Clifford circuits -- architectures in which Clifford circuits are preceded and/or followed by constant-depth circuits of non-Clifford (``magic") gates -- as a resource-efficient way to realize approximate $k$-designs, with reduced circuit depth and usage of magic. We prove that shallow Clifford circuits, when augmented with constant-depth circuits of magic gates, can generate approximate unitary and state $k$-designs with $\epsilon$ relative error. The total circuit depth for these constructions on $N$ qubits is $O(\log (N/\epsilon)) +2^{O(k\log k)}$ in one dimension and $O(\log\log(N/\epsilon))+2^{O(k\log k)}$ in all-to-all circuits using ancillas, which improves upon previous results for small $k \geq 4$. Furthermore, our construction of relative-error state $k$-designs only involves states with strictly local magic. The required number of magic gates is parametrically reduced when considering $k$-designs with bounded additive error. As an example, we show that shallow Clifford circuits followed by $O(k^2)$ single-qubit magic gates, independent of system size, can generate an additive-error state $k$-design. We develop a classical statistical mechanics description of our random circuit architectures, which provides a quantitative understanding of the required depth and number of magic gates for additive-error state $k$-designs. We also prove no-go theorems for various architectures to generate designs with bounded relative error.