A Field-Theoretical Approach to the P vs. NP Problem via the Phase Sign of Quantum Monte Carlo

Abstract

I present here a new method that allows the introduction of a discrete auxiliary symmetry in a theory in such a way that the eigenvalue spectrum of the fermion functional determinant is made up of complex conjugated pairs. The method implies a particular way of introducing and integrating over auxiliary fields related to a set of artificial shift symmetries. Gauge fixing the artificial continuous shift symmetries in the direct and dual sectors leads to the appearance of direct and dual Becchi-Rouet-Stora-Tyutin (BRST)-type global symmetries and of a symplectic structure over the field space. Such a method may allow the extension of the applicability of quantum Monte Carlo methods to some problems plagued by the fermionic sign problem.

The hidden quantum origin of gauge connections

“…However, to probe such a curvature directly we would require either a path in the form of a closed loop, or two distinct paths intersecting in some points. The impact of this observation I tried to analyse in an old article of mine [19] and then [20], but my new understanding on the gauge-quantum relation may shed new lights on that and also show how curvature in inner space may play some interesting roles in solving computational problems considered hard.…”

“…For example, we cannot control an infinite number of elements integrated over in each equivalence class but what about a finite number of them, but not only one? I was thinking about this problem since [19] and [20] where indeed, I tried to control the integration over several paths in which two (or more) choices of gauge representatives were made simultaneously. While it looked strange, it appeared that at least one NP-type problem was solved quite quickly (at least at the level of a simulation, there was no way of accessing a quantum computer of any sort at that time and at that time, I actually didn't know it was a quantum problem to begin with).…”

“…The mathematics of this has been described in [19] and [20] in great detail so I will just comment here on the interpretation thereof and of several things I am now understanding slightly better compared to [19] and [20]. First, in [19] and [20] I wasn't aware of the connection I see now between quantum mechanics and gauge theory, therefore I considered the simultaneous double choice of representatives a mathematical tool to create additional structure in the problem. With only one path going through each equivalence class once, we are somewhat limited in terms of symmetries we could add to the problem.…”

Cosmological constant as quantum error correction from generalised gauge invariance in double field theory

“…The introduction of gauge symmetries in quantum information theory has been recently explored in [1] and [2]. Quantum error correction codes were required because the classical error correction based on analysing copies of the same information for discrepancies cannot be applied in quantum computing due to the no-cloning theorem.…”

“…Extending the space of states in order to obtain auxiliary symmetries that could simplify certain computations has been used in ref. [2] and [19]. There, the extension was based on the Batalin-Vilkovisky quantisation of gauge theories with non-closing gauge algebras and the extensions in the form of field-anti-field formalisms [20], [21].…”