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arXiv:2609.35101

From Simple Sources to Quantum Advantage: Homomorphic Polynomial Transduction via Relative Decoding

Zhong-Xia Shang, Daniel Stilck França

quant-phSubmitted Sep 28, 2026• Announced Sep 29, 2026• 0 citations
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Abstract

Decoded quantum interferometry (DQI) and its Hamiltonian extension (HDQI) prepare states whose amplitudes are low-degree polynomials of an objective, using Fourier transforms and coherent decoding. We recast this approach as quantum state transduction through algebra homomorphisms. We transfer an efficiently preparable operator state of a degree-$D$ polynomial in a source Hamiltonian $H_A$ to the corresponding polynomial state of a target Hamiltonian $H_B=π(H_A)$, where $π$ is a unital $*$-homomorphism between the finite-dimensional source and target $C^*$-algebras. The transduction uses operator Fourier transforms and relative decoding to recover source operators rather than individual term-selection labels. The relative distance $d_{\mathrm{rel}}$ is the first degree at which source and target traces disagree. We prove that $2D<d_{\mathrm{rel}}$ is equivalent to preserving all inner products between degree-$D$ polynomials. Moreover, $d_{\mathrm{rel}}\ge d_{\mathrm{ord}}$, where $d_{\mathrm{ord}}$ is the ordinary distance used in (H)DQI, and we give families with $d_{\mathrm{ord}}=O(1)$ but $d_{\mathrm{rel}}=Θ(n)$ and efficient decoders. Our framework replaces the complicated pilot state preparation by the more modular task of preparing a source polynomial state. DQI and HDQI arise as relation-free special cases. The framework accommodates nonuniform coefficients and noncommuting interactions and extends to fermionic, qudit, and bosonic systems, with applications to approximate optimization and Gibbs sampling. As evidence of advantages, for a nonlinear pairwise variant of optimal polynomial intersection where constant $d_{\mathrm{ord}}$ limits DQI-style preparations, relative decoding yields an ideal quantum score of 0.643 versus 0.606 for the best tested classical heuristic, a gap exceeding 3 percentage points.

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84 pages, 1 figure, comments are welcome!

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Decoded quantum interferometry (DQI) and its Hamiltonian extension (HDQI) prepare states whose amplitudes are low-degree polynomials of an objective, using Fourier transforms and coherent decoding. We recast this approach as quantum state transduction through algebra homomorphisms. We transfer an efficiently preparable operator state of a degree-DDD polynomial in a source Hamiltonian HAH_AHA​ to the corresponding polynomial state of a target Hamiltonian HB=π(HA)H_B=π(H_A)HB​=π(HA​), where πππ is a unital ∗*∗-homomorphism between the finite-dimensional source and target C∗C^*C∗-algebras. The transduction uses operator Fourier transforms and relative decoding to recover source operators rather than individual term-selection labels. The relative distance dreld_{\mathrm{rel}}drel​ is the first degree at which source and target traces disagree. We prove that 2D<drel2D<d_{\mathrm{rel}}2D<drel​ is equivalent to preserving all inner products between degree-DDD polynomials. Moreover, drel≥dordd_{\mathrm{rel}}\ge d_{\mathrm{ord}}drel​≥dord​, where dordd_{\mathrm{ord}}dord​ is the ordinary distance used in (H)DQI, and we give families with dord=O(1)d_{\mathrm{ord}}=O(1)dord​=O(1) but drel=Θ(n)d_{\mathrm{rel}}=Θ(n)drel​=Θ(n) and efficient decoders. Our framework replaces the complicated pilot state preparation by the more modular task of preparing a source polynomial state. DQI and HDQI arise as relation-free special cases. The framework accommodates nonuniform coefficients and noncommuting interactions and extends to fermionic, qudit, and bosonic systems, with applications to approximate optimization and Gibbs sampling. As evidence of advantages, for a nonlinear pairwise variant of optimal polynomial intersection where constant dordd_{\mathrm{ord}}dord​ limits DQI-style preparations, relative decoding yields an ideal quantum score of 0.643 versus 0.606 for the best tested classical heuristic, a gap exceeding 3 percentage points.
Provable guarantees for preparing quantum Gibbs states, such as rapid mixing or gapped coherent preparation paths, are known only for restricted Hamiltonian families and usually must be re-derived for each new family. Building on homomorphic polynomial transduction, a generalization of decoded quantum interferometry (DQI) and Hamiltonian DQI (HDQI), we show how relative decoding transfers such a guarantee from a source Hamiltonian HAH_AHA​ to a target HBH_BHB​. Writing each Hamiltonian as a sum of mmm Pauli terms, the subsets of terms whose product is proportional to the identity, its relations, form a binary linear code, KAK_AKA​ for the source and KBK_BKB​ for the target, with the Pauli-label matrix as parity-check matrix as in DQI. Starting from the canonical thermofield double (TFD) of HAH_AHA​, a Bell transform, a reversible label map, and a coherent decoder for the quotient code KB/KAK_B/K_AKB​/KA​ prepare an approximate TFD of HBH_BHB​, and hence its Gibbs state. Because this decoder only resolves target relations absent from the source, the reachable inverse temperature is set by the relative distance dreld_{\rm rel}drel​, the fewest terms in any such relation. It can far exceed the ordinary distance dordd_{\rm ord}dord​, the fewest terms in any target relation, which bounds the uniform exact decoding radius of DQI and HDQI. We show that linear dreld_{\rm rel}drel​ with efficient decoding at linear radius certifies a constant inverse temperature. As an example, starting from a nearest-neighbor spin chain whose TFD is known to be preparable at every finite temperature, a sparse classical parity-check matrix yields bounded-degree, geometrically nonlocal, noncommuting targets with dord=3d_{\rm ord}=3dord​=3 and drel=Θ(m)d_{\rm rel}=Θ(m)drel​=Θ(m). To our knowledge, this gives a new class of efficiently preparable TFDs.
In this work, we study the Hamiltonian Decoded Quantum Interferometry (HDQI) for the general Hamiltonians H=∑iciPiH=\sum_ic_iP_iH=∑i​ci​Pi​ on an nnn-qubit system, where the coefficients ci∈Rc_i\in \mathbb{R}ci​∈R and PiP_iPi​ are Pauli operators. We show that, given access to an appropriate decoding oracle, there exist efficient quantum algorithms for preparing the state ρP(H)=P2(H)Tr[P2(H)]ρ_{\mathcal P}(H) = \frac{\mathcal P^2(H)}{\text{Tr}[\mathcal P^2(H)]}ρP​(H)=Tr[P2(H)]P2(H)​, where P(H)\mathcal P(H)P(H) denotes the matrix function induced by a univariate polynomial P(x)\mathcal P(x)P(x). Such states can be used to approximate the Gibbs states of HHH for suitable choices of polynomials. We further demonstrate that the proposed algorithms are robust to imperfections in the decoding procedure. Our results substantially extend the scope of HDQI beyond stabilizer-like Hamiltonians, providing a method for Gibbs-state preparation and Hamiltonian optimization in a broad class of physically and computationally relevant quantum systems.
Decoded quantum interferometry (DQI) is a novel paradigm for tackling approximate optimization problems on quantum computers. This framework comes with strong performance guarantees and exploits a well-established duality between optimization and coding theory. A central question, however, is whether DQI can actually provably outperform all polynomial-time classical algorithms. In this work, we establish such an advantage in an oracle setting: we consider an optimization task called folded optimal polynomial intersection (folded OPI), where the acceptance sets are chosen randomly and accessed through membership oracles. We establish a strict gap between the approximation ratio achievable by any polynomial-time classical algorithm and the approximation ratio achieved by the DQI algorithm. Our proof builds on Jordan et al.'s DQI framework for approximate optimization and extends the classical lower-bound method underlying Yamakawa and Zhandry's exact-search oracle separation to approximation. Building on recent developments by Sun and Wootters, Horinaga and Yamakawa, and Jo, we further show that a modified version of the DQI algorithm achieves a strictly higher score guarantee on typical sampled folded OPI instances. As a concrete example, for code rate 0.30.30.3, DQI achieves expected scores of approximately 0.850.850.85. In contrast, exceeding the classical threshold of 0.650.650.65 by any fixed amount with constant probability on sampled instances requires super-polynomially many classical membership queries.
We introduce Hamiltonian Decoded Quantum Interferometry (HDQI), a quantum algorithm that utilizes coherent Bell measurements and the symplectic representation of the Pauli group to reduce Gibbs sampling and Hamiltonian optimization to classical decoding. For a signed Pauli Hamiltonian HHH and any degree-ℓ\ellℓ polynomial P{P}P, HDQI prepares a purification of the density matrix ρP(H)∝P2(H)ρ_{P}(H) \propto {P}^2(H)ρP​(H)∝P2(H) by solving a combination of two tasks: decoding ℓ\ellℓ errors on a classical code defined by HHH, and preparing a pilot state that encodes the anti-commutation structure of HHH. Choosing P(x)P(x)P(x) to approximate exp⁡(−βx/2)\exp(-βx/2)exp(−βx/2) yields Gibbs states at inverse temperature βββ; other choices prepare approximate ground states, microcanonical ensembles, and other spectral filters. For local Hamiltonians, the corresponding decoding problem is that of LDPC codes. Preparing the pilot state is always efficient for commuting Hamiltonians, but highly non-trivial for non-commuting Hamiltonians. Nevertheless, we prove that this state admits an efficient matrix product state representation for Hamiltonians whose anti-commutation graph decomposes into connected components of logarithmic size. We show that HDQI efficiently prepares Gibbs states at arbitrary temperatures for a class of physically motivated commuting Hamiltonians -- including the toric code and Haah's cubic code -- but we also develop a matching efficient classical algorithm for this task. For a non-commuting semiclassical spin glass and commuting stabilizer Hamiltonians with quantum defects, HDQI prepares Gibbs states up to a constant inverse-temperature threshold using polynomial quantum resources and quasi-polynomial classical pre-processing. These results position HDQI as a versatile algorithmic primitive and the first extension of Regev's reduction to non-abelian groups.
We propose a Gibbs state preparation algorithm that exploits algebraic structure in the Hamiltonian. Our starting point is Hamiltonian Decoded Quantum Interferometry (HDQI), a recent algorithm that reduces Gibbs state preparation for Pauli Hamiltonians to decoding error-correcting codes. We introduce quantum decoding HDQI (QD-HDQI), replacing HDQI's coherent classical decoder by a quantum decoder that can exploit interference between different errors. For commuting Pauli Hamiltonians, our reduction is efficient and the associated quantum decoding problem corresponds to transmitting the associated symplectic code over i.i.d. pure-state classical-quantum channels. For uniformly random Hamiltonian signs, the vanishing-error threshold of an efficient decoder directly determines the lowest temperature that our algorithm can reach. Even using only classical decoders, this improves upon HDQI's achievable inverse temperature from linear to square-root in the correctable error fraction; quantum decoders can improve further by tolerating still stronger noise. A similar picture extends to the noncommuting regime previously handled by HDQI, where now the noise is correlated within small, independent blocks. We study two such quantum decoders in our QD-HDQI Gibbs sampling framework. The first, based on unambiguous state discrimination, works efficiently for arbitrary binary linear codes. However, we "dequantize" this strategy for commuting Hamiltonians, contributing a classical rejection sampler that matches its temperature threshold. The second is belief propagation with quantum messages (BPQM), a quantum generalization of classical belief propagation. We show that for certain dilute spin glasses, BPQM's decoding threshold exceeds the Shannon limit for classical decoding, reaching temperatures inaccessible to any classical decoder within our framework.
Decoded Quantum Interferometry (DQI) uses coherent decoding and a quantum Fourier transform to find high-quality solutions of structured optimisation problems. Existing analyses are closely tied to Hamming space, which underlies the optimisation objective, Dicke state preparation and the decoding step of the algorithm. Here we extend the core DQI mechanism beyond Hamming space to finite geometries with translation symmetry, where points are grouped into shells by their distance from a basepoint. Mathematically, these geometries are translation association schemes. In this setting the algorithm can be analysed by tracking one amplitude per shell, and biasing the prepared state towards high-quality solutions becomes a finite tridiagonal eigenvalue problem. As a non-Hamming example, we develop an efficient DQI protocol for finding an m x n finite-field matrix with smallest rank difference to a target matrix. Initial states are uniform superpositions over fixed-rank matrices, and Gabidulin codes provide candidates for efficient low-rank decoding up to a cutoff l. For this objective, this finds solutions with an effective-rank proxy near min(m,n)-l, and the corresponding expected score can be converted into a constant-probability bound on the residual rank of a sample. For Gabidulin nearest-codeword instances, a covering-radius obstruction shows that this bound does not imply an additive guarantee for the true optimum, and we do not claim a quantum advantage for the rank-metric construction. The results instead identify the geometric and coding ingredients for DQI beyond Hamming space.