Optimal Polynomial Intersection (OPI)

• The paper demonstrates that OPI reformulates NP-hard optimization as finding a low-degree polynomial that maximizes constraint intersections, dual to Reed–Solomon decoding.
• It introduces the DQI quantum algorithm which achieves nearly linear gate count and space efficiency compared to classical exponential complexity.
• Advanced circuit architectures—explicit register-sharing and dialog representation—reduce reversible RS decoding overhead, broadening applications in quantum cryptanalysis and coding.

Optimal Polynomial Intersection (OPI) is a framework for encoding and solving hard search and optimization problems as polynomial regression–type constraints, with central relevance in quantum algorithms and coding theory. In the OPI problem, one seeks a low-degree polynomial (typically of bounded degree) that “intersects” or matches the maximum number of prescribed set constraints over a finite field; this is dual to the decoding problem for Reed–Solomon codes. Decoded Quantum Interferometry (DQI) provides a quantum algorithmic framework for solving OPI with an asymptotically optimal quantum gate count. Recent work (Khattar et al., 13 Oct 2025) has established that DQI applied to OPI achieves verifiable quantum advantage with the minimal possible space and gate overhead, positioning this as a primary candidate for demonstrating practical, certifiable quantum supremacy in optimization.

1. Quantum Speedup and Computational Complexity

Fundamental to the paper is the demonstration that OPI, for well-chosen parameters and target sets, is classically intractable—requiring O(2^N) time—while DQI provides a quantum circuit with nearly linear complexity $\widetilde{O}(N)$ in the problem size $N$. Theoretical results (specifically Theorem optimalSpeedup in (Khattar et al., 13 Oct 2025)) show that, for instances of OPI constructed over binary extension fields $\mathrm{GF}(2^b)$, the quantum algorithm matches the optimal lower bound of $\Omega(N)$ quantum gates for any NP-hard search/optimization problem with classical hardness $2^N$. This separation saturates what is possible under known quantum algorithmic lower bounds.

A key ingredient is that the OPI problem, expressed as the following: find $Q \in \mathbb{F}_q[x]$ of degree $< n$ such that $Q(\alpha_j) \in F_j$ for maximal cardinality of $j \in [m]$, is dual (in the sense of syndrome decoding) to Reed–Solomon code decoding at large relative distance, for which the only known classical algorithms require exhaustive or near-exhaustive search in high-entropy regimes.

2. Quantum Circuit Design and Reversible Reed–Solomon Decoding

The main technical bottleneck in the DQI implementation for OPI is the construction of a highly space- and gate-efficient reversible Reed–Solomon decoder circuit. The dual code associated with the OPI Vandermonde system is an RS code, and syndrome decoding involves the Extended Euclidean Algorithm (EEA) for computing error locator and evaluator polynomials.

Two principal circuit architectures are highlighted:

• Explicit register-sharing EEA: By synchronizing the update sequence and storing Bézout coefficients and quotients “in place,” the leading-order logical qubit cost is reduced to $2 n b + O(\log n)$, where $n$ is the degree bound and $b$ is the bit-width per field element. The circuit operates by iterating the recurrence

$$\begin{pmatrix} r_{i-1} \ r_{i} \end{pmatrix} = \begin{pmatrix} 0 & 1 \ 1 & -q_i \end{pmatrix} \begin{pmatrix} r_{i-2} \ r_{i-1} \end{pmatrix}$$

in-place, with ancilla savings via register-reuse for the running quotient $q_i$ and state compression.

• Dialog representation (Editor's term): Rather than explicitly storing all intermediate coefficients, the circuit maintains a time-ordered list (“dialog”) of transition matrices (from, e.g., a Bernstein–Yang division-free GCD algorithm), which can be used to reconstruct any needed polynomial quantity or filtered error locator, with reduced ancilla overhead and linear cost in $n$. This approach is especially suited for quantum implementations where arithmetic must be precisely controlled and space is at a premium.

Both implementations are strict improvements over prior quantum arithmetic circuits and enable the entire OPI+DQI pipeline to operate at the theoretical minimum for both qubit space and gate complexity. Notably, these designs have applications that generalize—optimizing arithmetic cost for other quantum cryptanalytic routines, including Shor's algorithm.

3. Practical Resource Estimates and Application Scope

The paper provides explicit resource analyses to substantiate practical quantum speedup. For representative OPI instances over GF(2^b) with problem sizes $(m=4095, n=70, b=12)$, the optimized DQI circuits require approximately $5.72 \times 10^6$ Toffoli gates and $\sim$1885 logical qubits. These numbers are orders of magnitude smaller than previous estimates for quantum factoring of RSA-2048 (which necessitates close to $10^9$ Toffoli gates), implying that a quantum computer capable of factoring RSA-2048 can solve substantially harder OPI instances—by classical cost—in less runtime.

Classically, no algorithmic strategy, including standard or “extended” Prange algorithms, achieves feasible success probability, as the probability per trial remains exponentially small in $n$ and $b$ due to high-entropy construction. Instances can be tuned so that success requires $10^{23}$ or more exhaustive decoding attempts.

The optimized RS decoder is modular and also directly applicable to any algorithmic context requiring GCD-based arithmetic over finite fields, such as in cryptosystems involving finite field arithmetic (e.g., ECC discrete logarithms).

4. Security, Hardness, and Resilience of OPI Instances

Comprehensive hardness analysis is provided for various OPI instance constructions:

• Classical attack resistance: The baseline is the minimal bias probability using “Extended Prange” (XP) decoding, which includes advanced techniques such as constraint targeting, dynamic programming, and linear programming relaxations to estimate decoding bias. For standard randomly chosen target sets, or for target sets constructed via “bent” functions (notably from the Maiorana–McFarland family), all intersections remain near the statistical minimum, ensuring minimal classical bias and near-worst-case Prange performance.
• Twisted Bent Target OPI: By constructing the sets $F_y$ using bent functions, the maximum affine intersection between any linear function and any $F_y$ is minimized, impeding classical information set strategies and rendering algebraic structure-based attacks ineffective.
• Verifiability: Because OPI solutions can be efficiently checked (i.e., evaluating $Q(\alpha_j) \in F_j$), the quantum advantage for DQI+OPI is verifiable—crucial for practical demonstrations of quantum supremacy.

5. Extensions, Broader Applications, and Open Questions

A broad array of applications is envisioned for the optimized circuit architectures and the DQI+OPI setting:

• Extension to other code families: The techniques underpinning OPI extend directly to more general algebraic geometry codes, such as Hermitian codes, where similar duality properties enable DQI-style circuits to encode larger constraint sets per field element, reducing quantum hardware overhead per data point (Gu et al., 8 Oct 2025).
• Applicability to Shor’s and cryptographic algorithms: The in-place GCD and “dialog” approaches generalize naturally to quantum implementations of group arithmetic over finite fields and elliptic curves.
• Parallelization and state synthesis: The dialog-based circuit design potentially allows more efficient parallel state synthesis, though the limits of dialog size minimization and the possibility of further space savings remain open questions.
• Future quantum cryptanalysis: Structured OPI variants and their resilience offer a path for defining new hard optimization benchmarks for quantum algorithms, as well as informing the security analysis of quantum-resistant cryptosystems.

A plausible implication is that by further refining dialog representations or uncovering additional structure in the reversible arithmetic primitives, further asymptotic or at least constant-factor improvements in quantum resource cost may be attainable, not just for OPI but for the broader class of quantum algorithms with intensive reversible arithmetic components.

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By uniting the DQI algorithmic framework, rigorous circuit optimization, resilient instance generation, and practical resource estimation, the fully optimized OPI quantum algorithm presented in (Khattar et al., 13 Oct 2025) defines the current frontier for certifiable, efficient, and scalable quantum speedup in NP-hard optimization and search.