Bounds on Eventually Universal Quantum Gate Sets
Chaitanya Karamchedu, Matthew Fox, Daniel Gottesman
TL;DR
The paper addresses the problem of determining how many ancillary qudits are needed for an $n$-qu$d$it gate set to become universal on larger quantum systems. It introduces a 4th-moment criterion via $\mathcal{M}_{4}$, leverages invariants of finite linear groups and a complete classification of finite unitary $2$-designs (unitary $2$-groups), and applies Lazard's bound on Hilbert functions to bound the required extension. The main result is a sharp bound $\mathcal{K}(\Gamma) \le d^{4}(n-1) + 1$, improving Ivanyos' previous $d^{8}(n-1) + 1$ and implying, for qubits, universality on $16n$ qubits instead of $256n$. This tightens decidability criteria for eventual universality and reveals a deep link between invariant theory, algebraic geometry, and quantum gate synthesis, with implications for understanding ancilla overhead and gate-set design.
Abstract
Say a collection of $n$-qu$d$it gates $Γ$ is eventually universal if and only if there exists $N_0 \geq n$ such that for all $N \geq N_0$, one can approximate any $N$-qu$d$it unitary to arbitrary precision by a circuit over $Γ$. In this work, we improve the best known upper bound on the smallest $N_0$ with the above property. Our new bound is roughly $d^4n$, where $d$ is the local dimension (the `$d$' in qu$d$it), whereas the previous bound was roughly $d^8n$. For qubits ($d = 2$), our result implies that if an $n$-qubit gate set is eventually universal, then it will exhibit universality when acting on a $16n$ qubit system, as opposed to the previous bound of a $256n$ qubit system. In other words, if adding just $15n$ ancillary qubits to a quantum system (as opposed to the previous bound of $255 n$ ancillary qubits) does not boost a gate set to universality, then no number of ancillary qubits ever will. Our proof relies on the invariants of finite linear groups as well as a classification result for all finite groups that are unitary $2$-designs.