Closed-Form Optimal Quantum Circuits for Single-Query Identification of Boolean Functions

TL;DR

The paper tackles minimum-error identification of the four one-bit Boolean functions with a single quantum query. It leverages symmetry to show the optimum can be realized by a closed-form, gate-level protocol: a separable probe state, a concrete two-qubit measurement unitary, and a compact CNOT-based decomposition achieving $P^*=3/4$. This constructive realization bridges abstract optimal measurements and practical circuit synthesis, enabling explicit resource accounting and hardware deployment. It also outlines a broader program to seek closed-form strategies for larger (n,m,k) settings, while noting potential barriers and the importance of symmetry in revealing implementable structure.

Abstract

We study minimum-error identification of an unknown single-bit Boolean function given black-box (oracle) access with one allowed query. Rather than stopping at an abstract optimal measurement, we give a fully constructive solution: an explicit state preparation and an explicit measurement unitary whose computational-basis readout achieves the Helstrom-optimal success probability 3/4 for distinguishing the four possible functions. The resulting circuit is low depth, uses a fixed gate set, and (in this smallest setting) requires no entanglement in the input state. Beyond the specific example, the main message is operational. It highlights a regime in which optimal oracle discrimination is not only well-defined but implementably explicit: the optimal POVM collapses to a compact gate-level primitive that can be compiled, verified, and composed inside larger routines. Motivated by this, we discuss a "what if" question that is open in spirit: for fixed (n,m,k), could optimal k-query identification (possibly for large hypothesis classes) admit deterministic, closed-form descriptions of the inter-query unitaries and the final measurement unitary acting on the natural n+m-qubit input--output registers (and, if needed, small work registers)? Even when such descriptions are not compact and do not evade known circuit-complexity barriers for generic Boolean functions, making the optimum constructive at the circuit level would be valuable for theory-to-hardware translation and for clarifying which forms of "oracle access" are physically meaningful.