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

TL;DR

This paper provides a constructive, low-depth quantum circuit for optimally distinguishing four single-bit Boolean functions with one query, demonstrating explicit state preparation and measurement that achieve the Helstrom success probability.

Contribution

It presents the first explicit, implementable quantum circuit for optimal single-query Boolean function identification, bridging theoretical optimality with practical circuit design.

Findings

Achieves Helstrom-optimal success probability of 3/4

Constructs a low-depth, fixed-gate-set circuit without entanglement

Highlights the operational significance of explicit optimal measurement implementations

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.