Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.03670

Classical Algorithms for Bipartite Quantum Max-Cut on Dense Expanders

Stuart Wayland·Zackary Jorquera·Alexandra Kolla
View original source ↗Read source document ↗Export metadata ↓

Abstract

Quantum Max-Cut is a well-studied local Hamiltonian problem for which the classical complexity of the bipartite case remains open. We give a polynomial-time classical randomized algorithm for the spin-$\tfrac12$ antiferromagnetic Heisenberg model, equivalently Quantum Max-Cut, on dense balanced bipartite expanders. This class includes $G(n,n,p)$ with high probability for every fixed $p>0$. The algorithm estimates the ground energy and any ground state edge correlation to additive error $\varepsilon$ in time polynomial in $n$ and $1/\varepsilon$. We construct a Markov chain on perfect matchings of the complete bipartite graph, each matching representing a product of singlets, that converges to the ground state, and use the expansion of the graph to show that polynomially many steps and samples of this chain suffice.

Explore related subjects

quant-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Stuart Wayland, Zackary Jorquera, Alexandra Kolla. 2026-10-02. Classical Algorithms for Bipartite Quantum Max-Cut on Dense Expanders. https://arxiv.org/abs/2610.03670

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Hamiltonian-Level Certificate for Network-Free Distributed Quantum Simulation:Exact Tensor-Separability Criterion and Approximate Residual Bounds

Circuit cutting allows quantum circuits to be evaluated on smaller devices at the cost of additional sampling and classical reconstruction. For Hamiltonian-simulation circuits, the target unitary $\exp[it\mathbf H]$ connects this decomposition problem to the structure of the generator. We develop a Hamiltonian-level certificate that assesses exact separability and the accuracy of independent module evolution before a gate decomposition is chosen. For a time-independent Hamiltonian and a fixed module partition, the Hilbert--Schmidt projection onto sums of module-local terms isolates a cross-module residual $H_{ns}$. Its vanishing is necessary and sufficient for product evolution at all times; for Pauli-list inputs, this condition is checked in one pass over the combined coefficients. When $H_{ns}\ne0$, the projected evolution has operator-norm error at most $|t| ||H_{ns}||_{op}$. Across a bipartition, an explicit first-order algebraic correction has the same operator-Schmidt rank as $H_{ns}$ and leaves a second-order residual. The same Hamiltonian residual connects this approximation to short-time entanglement, determining the leading state-space and operator-space entangling powers and, through its Schmidt spectrum, the leading Choi tripartite-information terms. Numerical experiments test these relations and demonstrate block-factorised simulation up to $20$ qubits.

quant-ph↗
arXiv2026

Bare-ancilla fault-tolerant syndrome extraction: General extensions of distance-three codes and structural criteria for graph codes

The reliability of quantum computation critically depends on the performance of quantum error-correcting codes (QECCs). Performance of QECCs can be severely degraded by hook errors, which effectively reduce the code distance. We develop a systematic framework for bare-ancilla syndrome extraction in $[[n,k,3]]$ stabilizer codes. It provides a criterion for a specific single ancilla fault model and jointly searches for suitable stabilizer generators and data-ancilla interaction orders to correct hook errors. We call such stabilizer generator sets bare-ancilla codes (BACs). We also present two constructions that enlarge the set of valid BACs while preserving minimum distance, either by keeping the number of logical qubits fixed or by increasing it. Under the correlated-data error model, $3177$ $[[7,1,3]]$ graph codes are found to be BACs out of $3379$ and all $335415$ $[[8,1,3]]$ graph codes are BACs. We tested $500$ $[[7,1,3]]$ and $1000$ $[[8,1,3]]$ graph codes in the presence of anisotropic and depolarizing noise. The number of data-ancilla interactions is an important predictor of the pseudo-threshold of these graph codes. In the analyzed codes, the BAC, which works with the bare ancilla method, usually performs well or better than the flag method, particularly for depolarizing noise. Notably, we report new bare ancilla codes, namely $[[6,1,3]]$ and $[[7,1,3]]$ with improved code rate compared to the bare code used in the work of Muyuan Li \emph{et al.} and Maheshwari \emph{et al.} respectively.

quant-ph↗
arXiv2026

Q-PIPE: A Practical Quantum Phase Encoding Method

Efficient loading of classical data into quantum states remains a central bottleneck of quantum computing. We propose Quantum-Gray Phase Injection for Pixel Encoding (Q-PIPE), which writes continuous data into the eigenphases of a diagonal oracle by phase kickback and converts them coherently into computational-basis states by quantum phase estimation. Sums and differences between data sets are obtained by sequential oracle application, without quantum arithmetic circuits. We apply Q-PIPE to quantum image processing, using edge detection by directional finite differences as a proof of concept. Synthesized with uniformly controlled rotations, the oracle requires O(qN) controlled-NOT gates without ancillas for N pixels and q estimation qubits. This is the same scaling as optimized preparation of the novel enhanced quantum representation, whose output form Q-PIPE reproduces. A half-spectrum normalization removes phase aliasing, and a probability threshold derived from the phase-estimation kernel and scaled with image size controls spectral leakage. In ideal simulation, edge detection is exact for inputs commensurate with the estimation register and has low mean absolute error for continuous data. Under device noise models, multi-controlled-phase synthesis drives the output close to the maximally mixed state. The uniformly controlled synthesis instead reduces the compiled depth by a factor of 53--59 and preserves the edge structure (Pearson correlation r = 0.99 with the classical gradient). The limiting resource on current hardware is therefore the compiled oracle depth, not the encoding principle.

quant-ph↗