Optimization by Decoded Quantum Interferometry

TL;DR

Decoded Quantum Interferometry (DQI) reframes optimization as a decoding problem by exploiting quantum Fourier transforms to amplify high-value solutions. The framework reduces a broad class of max-LINSAT problems to syndrome decoding for linear codes, enabling rigorous semicircle-based performance characterizations and, in key cases like Optimal Polynomial Intersection (OPI), potential superpolynomial quantum speedups via efficient decoders such as Berlekamp–Massey. The work provides detailed algorithmic constructions for max-XORSAT and general max-LINSAT, analyzes average-case performance on LDPC-like instances, and compares DQI to classical heuristics and quantum alternatives (QAOA), while also extending to folded codes and multivariate polynomials. It includes concrete resource estimates for implementing the decoding steps and discusses limitations, relation to prior work, and avenues for future exploration such as multivariate OPIs, quantum-decoders, and sampling applications. Overall, the paper lays out a concrete, codified path to quantum speedups in structured optimization problems via decoding primitives.

Abstract

Achieving superpolynomial speedups for optimization has long been a central goal for quantum algorithms. Here we introduce Decoded Quantum Interferometry (DQI), a quantum algorithm that uses the quantum Fourier transform to reduce optimization problems to decoding problems. For approximating optimal polynomial fits over finite fields, DQI achieves a superpolynomial speedup over known classical algorithms. The speedup arises because the problem's algebraic structure is reflected in the decoding problem, which can be solved efficiently. We then investigate whether this approach can achieve speedup for optimization problems that lack algebraic structure but have sparse clauses. These problems reduce to decoding LDPC codes, for which powerful decoders are known. To test this, we construct a max-XORSAT instance where DQI finds an approximate optimum significantly faster than general-purpose classical heuristics, such as simulated annealing. While a tailored classical solver can outperform DQI on this instance, our results establish that combining quantum Fourier transforms with powerful decoding primitives provides a promising new path toward quantum speedups for hard optimization problems.

Figures (15)

• Figure 1: A schematic illustration of the steps of the DQI algorithm. Since the initial Dicke state is of weight $\ell$, the final polynomial $P$ is of degree $\ell$. Here, for simplicity, we take $w_\ell = 1$ and $w_k=0$ for all $k\neq \ell$.
• Figure 2: A stylized example of the Optimal Polynomial Intersection (OPI) problem. For $y_1 \in \mathbb{F}_p$, the orange set above the point $y_1$ represents $F_{y_1}$. Both of the polynomials $Q_1(y)$ and $Q_2(y)$ represent solutions that have a large objective value, as they each intersect all but one set $F_y$.
• Figure 3: Here we plot the expected fraction $\langle s\rangle/p$ of satisfied constraints achieved by DQI with the Berlekamp-Massey decoder and by Prange's algorithm for the OPI problem in the balanced case $r/p = 1/2$, as a function of the ratio of variables to constraints $n/p$. At $n/p=1/10$ Prange's algorithm satisfies a fraction $0.55$ of the clauses whereas DQI satisfies $\langle s \rangle/p = 1/2 + \sqrt{19}/20 \simeq 0.7179$. As a concrete challenge to the classical algorithms community we propose matching or exceeding this value in polynomial time. In our concrete resource estimation in §\ref{['sec:resources']} we consider $n/p = 1/2$, where OPI achieves $\langle s \rangle/p = 1/2 + \sqrt{3}/4 \simeq 0.9330$ and Prange's algorithm achieves $0.75$.
• Figure 4: Decoded Quantum Interferometry over $\mathbb{F}_2$. The algorithm begins with a computation, on a classical computer, of the principal eigenvector $(w_0,\ldots,w_\ell)^T$ of a certain matrix $A^{(m,\ell,0)}$. $P(f)$ is a degree-$\ell$ polynomial that enhances the probability of sampling $\mathbf{x}\in\mathbb{F}_2^n$ with high value of the objective $f$. Index $k$ ranges over $\{0,\ldots,\ell\}$ and $\mathbf{y}$ over the set $\{\mathbf{y} \in \mathbb{F}_2^m:|\mathbf{y}| = k\}$ of $m$-bit strings of Hamming weight $k$. Weight register qubits may be reused for the syndrome register. If $2 \ell + 1 < d^\perp$, the postselection succeeds with probability $\geq 1 - \varepsilon_\ell$, where $\varepsilon_\ell$ is the decoding failure rate on random weight-$\ell$ errors.
• Figure 5: Here we plot the expected fraction $\langle s\rangle/m$ of satisfied constraints, as dictated by Lemma \ref{['thm:genw']}, upon measuring $\ket{P(f)}$ when $P$ is the optimal degree-$\ell$ polynomial. We show the balanced case where $|f_i^{-1}(+1)| \simeq p/2$ for all $i$. Accordingly, the dashed black line corresponds to the asymptotic formula (\ref{['eq:semicircle_general']}) with $r/p = 1/2$.

Theorems & Definitions (43)

• Definition 2.1
• Definition 2.2
• Theorem 4.1
• Remark 5.1: Relationship to the work of Yamakawa and Zhandry
• Remark 5.2: Classical Complexity of OPI
• Lemma 9.1
• Lemma 9.2
• proof
• Lemma 9.3
• proof