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arXiv:2601.18773

Hamiltonian Decoded Quantum Interferometry for General Pauli Hamiltonians

Kaifeng Bu, Weichen Gu, Xiang Li

quant-phSubmitted Jan 26, 2026• Announced Jan 27, 2026• 0 citations
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Abstract

In this work, we study the Hamiltonian Decoded Quantum Interferometry (HDQI) for the general Hamiltonians $H=\sum_ic_iP_i$ on an $n$-qubit system, where the coefficients $c_i\in \mathbb{R}$ and $P_i$ are Pauli operators. We show that, given access to an appropriate decoding oracle, there exist efficient quantum algorithms for preparing the state $ρ_{\mathcal P}(H) = \frac{\mathcal P^2(H)}{\text{Tr}[\mathcal P^2(H)]}$, where $\mathcal P(H)$ denotes the matrix function induced by a univariate polynomial $\mathcal P(x)$. Such states can be used to approximate the Gibbs states of $H$ 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.

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28 pages

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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.
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.
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.
Decoded Quantum Interferometry (DQI) is a recently introduced quantum algorithm that reduces discrete optimization to decoding with potential advantages over the best-known polynomial-time classical algorithms for certain Max-LINSAT problems. In its original formulation, however, DQI treats all constraints uniformly and cannot exploit the weight structure present in most optimization problems of interest. In this work, we develop multivariate Decoded Quantum Interferometry (multivariate DQI) for weighted optimization problems, focusing on the weighted Max-LINSAT problem over a prime field. Grouping constraints into NNN blocks by distinct weights, we introduce multivariate DQI states built from NNN-variable polynomials of bounded total degree, and derive a closed-form asymptotic expression for both their optimal expectation value and their concentration behavior. We give an explicit preparation circuit using a single decoder call, and extend the analysis to imperfect decoding. We also show that, for certain weighted OPI problems, multivariate DQI outperforms a natural weighted analogue of Prange's algorithm, which serves as the weighted counterpart of the classical benchmark used in the unweighted setting. Finally, we extend the ideas to Hamiltonian DQI, obtaining approximate Gibbs states for commuting Pauli Hamiltonians with block structure.
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.
We consider a class of redudancy-free Hamiltonians, which are those where the trace vanishes for any product of Hamiltonian terms in which at least one term appears an odd number of times. This includes Hamiltonians which are sums of products of Paulis with no relations between them. These Hamiltonians arise naturally in Hamiltonian Decoded Quantum Interferometry, which separates the hardness of preparing quantum states into two steps: a classical decoding step, and a step preparing the thermofield double state of such a redundancy-free Hamiltonian. We focus on the hardness of these Hamiltonians, both for intrinsic interest and to understand the complexity of the state preparation step in Hamiltonian DQI. We present several results showing that the problem becomes easy at quadratically lower temperature than for a general Hamiltonian, with the complexity at given inverse temperature βββ depending on β2dβ^2 dβ2d, where ddd is the degree of the anticommutation graph. We give an efficient classical algorithm for small β2dβ^2 dβ2d to approximate the partition function, while for large β2dβ^2 dβ2d we show that approximating the partition function is NP-hard. To this end, we introduce what we call an anticommutation glass, where frustration arises purely from anticommutation relations. We give a subexponential time quantum algorithm to prepare the thermofield state for small β2dβ^2 dβ2d, and present some results toward a polynomial time algorithm based on the Feiguin-Klich Hamiltonian. As a result of possible independent interest, we show that this Hamiltonian leads to a polynomial time algorithm for preparing thermofield double states of general Hamiltonians on an interaction graph of degree ddd for β≲1/dβ\lesssim 1/dβ≲1/d. Finally, we show that estimating the ground state energy of redundancy-free Hamiltonians is QMA-complete.