Hamiltonian Decoded Quantum Interferometry

TL;DR

HDQI generalizes Decoded Quantum Interferometry to non-abelian Pauli Hamiltonians by encoding the noncommuting structure into a symplectic code and using coherent Bell measurements to reduce Gibbs sampling and optimization to classical decoding tasks. The method yields a purified spectral state $\rho_{\mathcal{P}}(H) = \dfrac{\mathcal{P}^2(H)}{\mathrm{Tr}[\mathcal{P}^2(H)]}$, with Gibbs states recovered when $\mathcal{P}$ approximates $\exp(-\beta x/2)$, and other spectral filters possible. For commuting, nearly independent Hamiltonians (e.g., toric/Haah codes), HDQI achieves efficient Gibbs-state preparation at arbitrary temperatures, supported by a matching classical dequantization route; for non-commuting cases, the pilot state becomes a matrix product state with small bond dimension whenever the anti-commutation graph decomposes into logarithmically sized components, enabling polynomial quantum-time after poly(n) classical pre-processing. The work also provides a classical sampling algorithm for stabilizer descriptions under these conditions and extends the framework to Weyl groups and $\mathbb{F}_p$ settings, suggesting broad applicability and a flexible algorithmic primitive for quantum-spectral tasks. Overall, HDQI positions quantum decoding as a versatile primitive for quantum-spectral tasks and highlights new connections between quantum Hamiltonian complexity, LDPC decoding, and tensor-network representations.

Abstract

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 $H$ and any degree-$\ell$ polynomial ${P}$, HDQI prepares a purification of the density matrix $ρ_{P}(H) \propto {P}^2(H)$ by solving a combination of two tasks: decoding $\ell$ errors on a classical code defined by $H$, and preparing a pilot state that encodes the anti-commutation structure of $H$. Choosing $P(x)$ to approximate $\exp(-βx/2)$ yields Gibbs states at inverse temperature $β$; 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.

Figures (5)

• Figure 1: Schematic illustration of Hamiltonian Decoded Quantum Interferometry (HDQI), a quantum algorithm for modulating the spectral distribution of a given Hamiltonian. For a Pauli Hamiltonian $H = \sum_{i=1}^m v_i P_i$, HDQI transforms a source of Bell pairs $\ket{\Phi}$ into a state which is a superposition over the eigenstates $\ket{\lambda}$ of $H$, with amplitudes modulated by a high-degree polynomial $\mathcal{P}(\lambda)$. HDQI implements this transformation by decoding, in superposition, errors on a certain classical linear code $\mathop{\mathrm{Symp}}\nolimits(H)$ constructed from $H$. The output state of $\operatorname{HDQI}(H)$ has probability $\propto \mathcal{P}^2(\lambda)$ of being $\ket{\lambda}$. In this sense HDQI acts as an algorithmic spectral filter, splitting a uniform incoming source of Bell pairs into a certain modulated spectrum of energies. This polynomial can be chosen to filter for desired eigenvectors. For example, high-degree monomials concentrate amplitudes toward high-energy eigenstates, and polynomial approximations of exponential functions yield approximate Gibbs states.
• Figure 2: A simple non-commuting $3$-qubit Pauli Hamiltonian with anti-commutation graph$G=(V,E)$, where $V=\{X_1 Z_2 X_3, Z_1 X_2 Z_3,X_1\}$ and $E=\{(X_1 Z_2 X_3,Z_1 X_2 Z_3), (Z_1 X_2 Z_3,X_1)\}$. The quantity $\alpha_G$ is called the anti-symmetry character of $G$ and is a coarse measure of the anti-commutativity of $H$ (see Definition \ref{['def:alpha_function']}); in particular, it is equal to $1$ whenever $G$ is the empty graph and all of the Pauli operators in $V$ commute.
• Figure 3: Spectrum of hardness for Hamiltonian models studied in this work. To the left are models which are dequantized in the sense that while HDQI can Gibbs sample them to arbitrary temperature, we develop a classical algorithm which efficiently samples stabilizer descriptions of eigenstates from the Gibbs distribution at any temperature. In the middle are models for which HDQI prepares Gibbs states up to a constant temperature (perhaps with a quasipolynomial classical pre-processing step). The asterisk denotes an assumption of high-distance decodability, which we show numerically but not analytically. To the right are models for which we do not currently provide provable guarantees about the performance of HDQI.
• Figure 4: Maximum fraction of randomly-selected bits that can be flipped such that our modified Belief Propagation decoder remains able to recover the original codeword with probability $\geq 1/2$. For each set of parameters we independently sample 50 commuting Hamiltonians from Algorithm \ref{['alg:greedy-commuting-sampler']}. The points and error bars show the mean and standard deviation, respectively. We consider three different ensembles of Hamiltonian with three different ratios $m/n$ of Pauli terms to qubits, and four different choices of $k$. In each case one observes that the error fraction tolerated increases with $n$ and appears to converge toward a constant value as $n \to \infty$, suggesting that the decoder can correct randomly chosen errors up to a threshold linear weight with substantial probability.
• Figure 5: Schematic of the qualitative behavior of typical "waterfall plots" from the decoding literature. Below some threshold error weight, the decoder succeeds with high probability; above the threshold, the decoder fails with high probability. The structure of the success probability curve implies that the exact choice of success probability to define decoder success does not make a significant impact on numerical analysis.

Theorems & Definitions (131)

• Theorem 1: Nearly independent Hamiltonians
• Theorem 2: Classical sampling for nearly independent Hamiltonians
• Theorem 3: Gibbs state preparation
• Theorem 4: Semicircle law for HDQI
• Theorem 5: Threshold theorem, $p$-semiclassical spin glass
• Definition 6: Commuting Pauli Hamiltonians
• Theorem 7: Univariate polynomial symmetric expansion
• proof
• Definition 8: Symplectic vectors
• Definition 9: Hamiltonian symplectic code