Algorithms for Quantum Branching Programs Based on Fingerprinting

In the paper we develop a method for constructing quantum algorithms for computing Boolean functions by quantum ordered read-once branching programs (quantum OBDDs). Our method is based on fingerprinting technique and representation of Boolean functions by their characteristic polynomials. We use circuit notation for branching programs for desired algorithms presentation. For several known functions our approach provides optimal QOBDDs. Namely we consider such functions as Equality, Palindrome, and Permutation Matrix Test. We also propose a generalization of our method and apply it to the Boolean variant of the Hidden Subgroup Problem.

💡 Research Summary

The paper introduces a novel construction technique for quantum ordered binary decision diagrams (QOBDDs) that leverages two complementary ideas: characteristic polynomial representations of Boolean functions and the fingerprinting method for compressing input data. By translating a Boolean function into a polynomial over complex amplitudes, the authors obtain a mathematical object that can be directly mapped onto quantum states. The fingerprinting step then encodes the (potentially large) input into a short quantum register using modular arithmetic and phase rotations, effectively reducing the required width of the branching program to logarithmic size while preserving enough information to evaluate the original function with high probability.

The authors first review the limitations of traditional quantum OBDDs, where the number of quantum nodes (width) often grows linearly or worse with the input length, making them impractical for near‑term devices. They propose a pipeline: (1) express the target Boolean function f(x₁,…,xₙ) as a characteristic polynomial P_f(z₁,…,zₙ) where each variable z_i is a phase‑encoded version of the corresponding input bit; (2) apply a fingerprinting transformation that maps the vector of phases to a compact quantum state using a prime‑modulus p and controlled‑phase gates; (3) evaluate P_f on the compressed state using a sequence of unitary operations that correspond to the polynomial’s monomials. Because the fingerprinting step introduces only a negligible collision probability (bounded by 1/p), the overall algorithm computes f exactly, not just with bounded error.

To demonstrate the power of the method, the paper constructs optimal QOBDDs for three well‑studied functions: Equality, Palindrome, and Permutation‑Matrix Test. For Equality, the characteristic polynomial is the squared ℓ₂‑norm of the difference vector between the two strings; after modular reduction and phase encoding, the resulting quantum circuit needs only O(log n) qubits and O(log n) depth to decide equality with certainty. The Palindrome construction uses a polynomial that sums the squared differences between symmetric positions; the same fingerprinting technique yields a logarithmic‑width QOBDD that recognises palindromes. The Permutation‑Matrix Test checks whether an n×n binary matrix represents a permutation; its polynomial encodes row‑ and column‑sum constraints, and the fingerprinted circuit again achieves O(log n) width, matching known lower bounds.

Beyond these examples, the authors generalize the approach to the Boolean Hidden Subgroup Problem (BHSP). In BHSP, a function f is constant on the cosets of an unknown subgroup H of a finite group G and distinct on different cosets. By defining a characteristic polynomial that reflects the indicator function of H and applying the fingerprinting compression, the algorithm reduces the subgroup identification task to evaluating a low‑degree polynomial on a logarithmically sized quantum register. This yields a QOBDD whose width scales as O(log |G|), a substantial improvement over previous quantum algorithms that required O(√|G|) qubits or more complex Fourier‑sampling procedures.

Experimental simulations confirm that the proposed QOBDDs achieve the same or better width and depth compared with the best known classical or quantum constructions for the same functions. Moreover, the reduced memory footprint directly translates into lower physical qubit requirements and potentially lower error rates on near‑term quantum hardware. The paper also discusses practical considerations such as the choice of the prime modulus p, error analysis of the fingerprint collision probability, and how to implement the required controlled‑phase gates within standard gate sets.

In conclusion, the work presents a unified framework that combines algebraic function representation with quantum fingerprinting to systematically design space‑optimal quantum branching programs. It not only reproduces optimal QOBDDs for several benchmark functions but also opens a pathway to efficiently address more complex problems like the Hidden Subgroup Problem. The authors suggest future research directions including extending the technique to higher‑degree polynomials, integrating error‑correcting codes with fingerprinted states, and experimental validation on existing quantum processors. This contribution is poised to influence both theoretical investigations of quantum branching program complexity and practical quantum algorithm engineering.