Universality and Optimal Architectures for Layered Programmable Unitary Decompositions

TL;DR

This framework provides a unified method to verify the universality of various proposed architectures and clarifies the nature of the ``generic''mixers required for such constructions and provides a geometry-aware optimization method for finding the parameters of a decomposition.

Abstract

The decomposition of arbitrary unitary transformations into sequences of simpler, physically realizable operations is a foundational problem in quantum information science, quantum control, and linear optics. We establish a 1D Quantum Field Theory model for justifying the universality of a broad class of such factorizations. We consider parametrizations of the form $U = D_1 V_1 D_2 V_2 \cdots V_{M-1}D_M$, where $\{D_j\}$ are programmable diagonal unitary matrices and $\{V_j\}$ are fixed mixing matrices. By leveraging concepts like the anomalies of our effective model, we establish criteria for universality given the set of mixer matrices. This approach yields a rigorous proof grounded on physics for the conditions required for the parametrization to cover the entire group of special unitary matrices. This framework provides a unified method to verify the universality of various proposed architectures and clarifies the nature of the ``generic'' mixers required for such constructions. We also provide a geometry-aware optimization method for finding the parameters of a decomposition.

Figures (3)

• Figure 1: Graph (left) of states of a $N=2,M=3,$$1D$ QFT transfer matrix model: each node represents a state, a diagonal entry $(D_j)_{\mu\mu}$, and each arrow a path between states of amplitude $(V_j)_{\mu\nu}$. The dual graph (right) represents the Feynman diagram of the two different example histories: each arrow is a propagator acquiring a phase $\exp(i\phi_{j,\mu})$, and each node is an interaction vertex for a coupling $(V_j)_{\mu\nu}$. Our matrix element $U(\phi)_{ab}=\langle b|U(\phi)|a\rangle$ represents the transition amplitude from state $a$ to state $b$ summed over all possible intermediate histories.
• Figure 2: Simulations for different matrix dimensions and mixers. (a) Fidelity precision for the DFT mixer for the different Newton-CG optimization setups for the same number of layers $M=N+1$ ("Analytic" in the legend means it employs the analytical gradient, uses it for finite-difference Hesssian estimation, and improves the solution with Newton-Raphson). (b) Plateau of high fidelity (blue) optimization for $N=4, M=5$ using the frDFT mixer with varying parameter $\alpha$. The frDFT becomes the identity and the DFT for $\alpha$=0 and 1, respectively. The maximum value of $\det(C)$ (red) clearly distinguishes the feasibility region.
• Figure 3: (a) Plateaus of high fidelity (blue) optimization for the mixer $V=\exp(-iHz)$ with $H_{k, k+1}=0.25$, $N=4$, $M=5$, and varying parameter $z$. The maximum value of $\det(C)$ (red) clearly distinguishes the feasibility region. (b) Robustness analysis of the fidelity for a fixed optimized phases solution against perturbations $\epsilon$ around $H_{k, k+1}=0.25$ for $N=10$, $M=11$, and $z=15$. The original phases are not re-optimized but the mixers are perturbed. The $\epsilon$ used in each simulation is sampled from a normal distribution centered in $H_{k, k+1}$ with standard deviation $\Delta\epsilon$.