Unitarization of Pseudo-Unitary Quantum Circuits in the S-matrix Framework

TL;DR

The paper develops a novel partial inversion operation that maps S-matrices to T-matrices, enriching pseudo-unitary circuit theory with a 3D diagrammatic representation and a deformed metric framework that preserves dot-product conservation. It formalizes a matrix-operator approach to transition between unitary and pseudo-unitary groups, introduces a special zero-handling set to manage singularities, and presents a renormalized-growth protocol to extend circuit dimensions without increasing matrix size. Key contributions include the operator definition of partial inversion, a diagrammatic 3D visualization, a deformed metric algebra, and the renormalized-growth algorithm, all enabling inter-unitary circuit constructions and potential relaxation of No-Go Theorems in large scattering lattices. The work connects S-matrix theory to inter-unitary circuit design, with implications for scalable quantum devices and tensor-network-inspired renormalization techniques.

Abstract

Pseudo-unitary circuits are recurring in both S-matrix theory and analysis of No-Go theorems. We propose a matrix and diagrammatic representation for the operation that maps S-matrices to T-matrices and, consequently, a unitary group to a pseudo-unitary one. We call this operation ``partial inversion'' and show its diagrammatic representation in terms of permutations. We find the expressions for the deformed metrics and deformed dot products that preserve physical constraints after partial inversion. Subsequently, we define a special set that allows for the simplification of expressions containing infinities in matrix inversion. Finally, we propose a renormalized-growth algorithm for the T-matrix as a possible application. The outcomes of our study expand the methodological toolbox needed to build a family of pseudo-unitary and inter-pseudo-unitary circuits with full diagrammatic representation in three dimensions, so that they can be used to exploit pseudo-unitary flexibilization of unitary No-Go Theorems and renormalized circuits of large scattering lattices.

Figures (4)

• Figure 1: S-matrix and T-matrix scattering diagrams for two-component spinors.
• Figure 2: Graph that defines the relation between $U(2^n,I^{\otimes n})$ and $U(2^n,\sigma_3^{\otimes n})$ (magenta link between nodes of respective metrics), and defines $SU(2,\eta)$ metric-deformation operators in $\hat{\Omega}(i,k,\nu,\rho)$ (full black and red edges) as links between metric vectors (nodes $\vec{ \hat{ \alpha}}$, $\vec{ \hat{\phi} }$). $\vec{ \hat{\phi} }$ and $\vec{ \hat{\epsilon} }$ are linked by powers, not $\hat{\Omega}$ (dashed edges). The magenta edge marks a sequence of equal indices as the first $2^{n-1}$ odious numbers in the odious series. Black edges mark $i=k$, and red edges mark $i\neq k$, with $i,k$ the partial inversion indices. The nodes of pseudo-unitary metrics are linked to the nodes of pseudo-special-unitary metrics by dotted edges to represent their relation with simpler and more symmetric sets of metrics.
• Figure 3: Equality of two renormalized-growth paths shows agreement with graphical interpretation of scattering lattice.
• Figure 4: Landauer conductance of real T-matrix for diagrammatic dimensions from $(2,2)$ to $(10,10)$ after application of the renormalized-growth algorithm.