A nearly linear-time Decoded Quantum Interferometry algorithm for the Optimal Polynomial Intersection problem

TL;DR

This work advances Decoded Quantum Interferometry (DQI) for the Optimal Polynomial Intersection (OPI) problem by delivering near-linear-time quantum algorithms under QRAM-enabled models. The authors combine Stages 1 and 2 to bypass dense Dicke-state preparation, implement Stage 3 with a Grover-inspired, nearly optimal amplitude encoding, and accelerate Stage 4 using fast Reed–Solomon decoding and Number-Theoretic Transforms. They show that with QRAMQ access to input data, the overall runtime for solving OPI scales as Ō(p polylog p), while achieving an expected constraint-satisfaction ratio around (1/2 + √19/20)/2 ≈ 0.718 in the asymptotic regime, outperforming the best known polynomial-time classical methods. The results depend on the OPI-to-max-LINSAT reduction and Reed–Solomon code structure, and are complemented by concurrent independent work achieving similar near-linear-time performance with alternative Dicke-state constructions. Overall, the paper strengthens the case for quantum-advantaged optimization in coding-theoretic settings and clarifies memory-access model implications for practical quantum algorithms.

Abstract

Recently, Jordan et al. (Nature, 2025) introduced a novel quantum-algorithmic technique called Decoded Quantum Interferometry (DQI) for solving specific combinatorial optimization problems associated with classical codes. They presented a constraint-satisfaction problem called Optimal Polynomial Intersection (OPI) and showed that, for this problem, a DQI algorithm running in polynomial time can satisfy a larger fraction of constraints than any known polynomial-time classical algorithm. In this work, we propose several improvements to the DQI algorithm, including sidestepping the quadratic-time Dicke state preparation. Given random access to the input, we show how these improvements result in a nearly linear-time DQI algorithm for the OPI problem. Concurrently and independently with this work, Khattar et al. (arXiv:2510:10967) also construct a nearly linear-time DQI algorithm for OPI using slightly different techniques.

Figures (5)

• Figure 1: Let $m=500$, $\ell=200$, and $c=0.01$. The binomial distribution $B(m,q)$ is shown in blue and orange, the orange part representing the part where we "overshoot" $\ell$. The uniform distribution used in jordan25:DQI-original---they choose $(w'_k)^2=1/\lceil\sqrt{\ell}\rceil$ for $k$ with $\ell-\lceil\sqrt{\ell}\rceil < k \le \ell$ and $(w'_k)^2=0$ for other $k$---is shown in green.
• Figure 2: The Bloch sphere corresponding to basis $\{|\widehat{0}\rangle,|G_i\rangle\}$ from two angles. The blue axis is between $|\widehat{0}\rangle$ and $|G_i\rangle$, and the red axis is between $|S_i\rangle = \sin\theta|\widehat{0}\rangle+\cos\theta|G_i\rangle$ and $|\overline{S}_i\rangle = \cos\theta|\widehat{0}\rangle-\sin\theta|G_i\rangle$. The transformation of the initial state $|\widehat{0}\rangle$ and into the final state $|G_i\rangle$ follows the dashed lines.
• Figure 3: Implementation of controlled-$\Xi_i^\psi$ (left) and controlled-$\Xi_i^\pi$ (right), requiring, respectively, two and a single call to QRAMC. Here $V_\psi:=|0\rangle\langle0|+\mathsf{e}^{\mathsf{i}\psi}|1\rangle\langle1|$ is a Pauli-$Z$ rotation and $Z=V_\pi$ is the Pauli-$Z$ gate. The top wire represents the control qubit, the center wires represent the set element register, and the bottom wire is the interface register. Both circuits return the interface register to its initial state.
• Figure 4: Simulation of $R_{\mathrm{Q}}$ using the one-hot encoding of $a$. The circuit employs exactly $3M - 2$ Fredkin gates. Note that the two negation gates can be omitted if the ancilla is initialized and restored to the state $|1\rangle \otimes |0\rangle^{\otimes M-1}$.
• Figure 5: Simulation of $R_{\mathrm{Q}}$ that uses $\lceil \log_2 M \rceil$ ancilla qubits to sequentially check whether $a = 0$, $a = 1$, $a = 2$, etc., and, if so, to perform the corresponding swap. When $M$ is a power of $2$, the circuit employs $M$ Fredkin gates, $4M - 8$ Toffoli gates, and $4$ CNOT gates. Note that each pair of gates enclosed in a dashed box can be replaced by a single CNOT gate, and the pair of gates enclosed in the dotted box can be replaced by a single negation gate.

Theorems & Definitions (30)

• Theorem 1.1
• Claim 3.1
• Theorem 4.1
• Theorem C.1
• Theorem C.2
• proof
• Theorem C.3
• Lemma C.4
• proof
• Lemma C.5: Quotient equivalence lemma