Bias-Class Discrimination of Universal QRAM Boolean Memories

Abstract

We study the discrimination of Boolean memory configurations via a fixed Universal QRAM (U-QRAM) interface. Given query access to a quantum memory storing an unknown Boolean function $f:[N]\to\{0,1\}$, we ask: what can be inferred about the bias class of $f$ (its imbalance from $1/2$, up to complement symmetry) using coherent, addressable queries? We show that for exact-weight bias classes, the induced single-query ensemble state on the address register has a two-eigenspace structure that yields closed-form expressions for the single-copy Helstrom-optimal measurement and success probability. Because complementing $f$ changes the state $|\psi\rangle$ only by a global phase, hypotheses $p$ and $1-p$ are information-theoretically identical in this model; thus the natural discriminand is the phase-bias magnitude $|\mu|$ (equivalently $\mu^2$). This goes beyond the perfect-discrimination case of Deutsch-Jozsa and complements exact-identification settings such as Bernstein-Vazirani.