Supersymmetry and Quantum Computation

TL;DR

This paper explores the connection between supersymmetry and quantum computation, demonstrating the computational complexity of certain supersymmetric problems and introducing supersymmetric quantum circuits with unique robustness properties.

Contribution

It introduces the concept of supersymmetry in quantum computation, analyzes the complexity of computing the Witten index, and discusses supersymmetric quantum circuits and their potential advantages.

Findings

Computing the Witten index is P-complete.

Supersymmetric quantum circuits form a nontrivial subclass with robustness.

Examples include the supersymmetric SYK model and fermion hard-core models.

Abstract

The interplay between supersymmetry and classical and quantum computation is discussed. First, it is shown that the problem of computing the Witten index of $\mathcal N \leq 2$ quantum mechanical systems is $\#P$-complete and therefore intractable. Then, the notions of supersymmetry in the space of qubits and supersymmetric quantum circuits are introduced and some of their properties discussed. In particular, it is shown that these define a nontrivial subclass of quantum algorithms with robustness properties typical of supersymmetric systems. Concrete examples, including the supersymmetric SYK model and fermion hard-core models are discussed. Some applications and open questions are suggested.