Carla M. Quispe Flores

arXiv2026

All You Need Is Non-Commutative Words

We represent lexical tokens as unitary matrices and encode each sentence as their ordered product. The noncommutativity of matrix product captures word order without positional encodings (PEs). The same algebra yields several capabilities, including antisymmetric self-attention with no query, key, or value projections, and parallel composition of variable-length text chunks at a reduced attention cost. Furthermore, it provides a canonical-coset readout layer that encodes all true unitary degrees of freedom compactly, while supporting continual learning through nested group extensions that enlarge the operator space with each new task preserving prior representations exactly. Across standard text-classification benchmarks, the method matches or exceeds bag-of-words baselines. Achieving higher accuracy on IMDB and comparable performance on AG News. Notably, this is accomplished by replacing the conventional $\sim$30,000-dimensional vocabulary space with a dense, 64-parameter real-valued encoding, highlighting the expressive efficiency of our parameterization.

cs.CL↗

arXiv2025

Time complexity in preparing metrologically useful quantum states

We investigate the fundamental time complexity, as constrained by Lieb-Robinson bounds, for preparing entangled states useful in quantum metrology. We relate the minimum time to the Quantum Fisher Information ($F_Q$) for a system of $N$ quantum spins on a $d$-dimensional lattice with $1/r^α$ interactions with $r$ being the distance between two interacting spins. We focus on states with $F_Q \sim N^{1+γ}$ where $γ\in (0,1]$, i.e., scaling from the standard quantum limit to the Heisenberg limit. For short-range interactions ($α> 2d+1$), we prove the minimum time $t$ scales as $t \gtrsim L^γ$, where $L \sim N^{1/d}$. For long-range interactions, we find a hierarchy of possible speedups: $t \gtrsim L^{γ(α-2d)}$ for $2d < α< 2d+1$, $t \gtrsim \log L$ for $(2-γ)d < α< 2d$, and $t$ may even vanish algebraically in $1/L$ for $α< (2-γ)d$. These bounds extend to the minimum circuit depth required for state preparation, assuming two-qubit gate speeds scale as $1/r^α$. We further show that these bounds are saturable, up to sub-polynomial corrections, for all $α$ at the Heisenberg limit ($γ=1$) and for $α> (2-γ)d$ when $γ<1$. Our results establish a benchmark for the time-optimality of protocols that prepare metrologically useful quantum states.

quant-ph↗