Improved Transversal Non-Clifford Gates from Cup Products

Abstract

It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford $C^{r-1}Z$ gate, for every constant $r\ge 3$. In particular, we obtain length-$n$ quantum LDPC codes (with constant-weight stabilizers) of polynomial distance $d\ge n^{(1-\epsilon )/r}$ supporting transversal $C^{r-1}Z$ gates on a close-to-linear number $k\ge n^{1-\epsilon }$ of disjoint tuples of logical qubits, for arbitrarily small $\epsilon >0$. Our construction is the first with constant-weight stabilizers that obtains $dk\gg n$, and as a consequence achieves arbitrarily small magic state overhead exponent $\gamma =\log (n/k)/\log (d)>0$. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many $k=\Omega (n)$ logical $C^{r-1}Z$ gates, though with stabilizer weight and physical circuit depth $n^{\epsilon }$. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits. To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal $C^{r-1}Z$. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.