Kunal Marwaha

We study the complexity of Decoded Quantum Interferometry (DQI), a recently proposed quantum algorithm for approximate optimization. We argue that DQI is hard to classically simulate, and that the hardness comes from locating an exponentially large hidden subset. This type of hardness is shared by Shor's algorithm, but the hidden subset here has no apparent group structure. We first prove that DQI can be simulated in a low level of the polynomial hierarchy, ruling out hardness arguments related to quantum supremacy. Instead, we show that DQI implements an existential coding theory bound based on the MacWilliams identity, and that it prepares a state within an obfuscated quantum harmonic oscillator. Both viewpoints require a coherent application of a discrete Hermite transform, which has no natural classical analog.

While the role of entanglement in quantum proof systems has been extensively studied, the computational power of unentanglement remains poorly understood. Since entanglement admits many inequivalent multipartite structures, it is natural to ask how more fine-grained structural promises affect computational power. In this work we investigate a mild promise: each proof is internally separable, meaning that after tracing out one register, a designated constant-size subsystem is separable from the rest—even though the overall proof may still be entangled across every bipartition. We prove a qualitative jump from one proof to two: with one internally separable proof, the resulting class is contained in $\EXP$ (even allowing inverse-exponential completeness–soundness gap), whereas with two unentangled internally separable proofs, the class equals $\NEXP$ at constant gap. Notably, in the $\NEXP$ construction, the second proof is used solely to implement a SWAP-based purity test.

We introduce a 0.611-approximation algorithm for Quantum MaxCut (QMC) and a 0.8395-approximation algorithm for the EPR Hamiltonian. A novel ingredient in the QMC approximation is to partially entangle pairs of qubits associated to edges in a matching, while preserving the direction of their single-qubit Bloch vectors. This allows us to interpolate between product states and matching-based states with a tunable parameter. For the EPR Hamiltonian, our improvement comes from a new nonlinear monogamy-of-entanglement bound on star graphs and a refined parameterization of a shallow quantum circuit from previous works. We also prove limitations showing that current methods cannot achieve substantially better approximation ratios, indicating that further progress will require fundamentally new techniques.

We construct and analyze a message-passing algorithm for random constraint satisfaction problems (CSPs) at large clause density, generalizing work of El Alaoui, Montanari, and Sellke for Maximum Cut through a connection between random CSPs and mean-field Ising spin glasses. For CSPs with even predicates, the algorithm asymptotically solves a stochastic optimal control problem dual to an extended Parisi variational principle. This gives an optimal fraction of satisfied constraints among algorithms obstructed by the branching overlap gap property of Huang and Sellke, notably including the Quantum Approximate Optimization Algorithm and all quantum circuits on a bounded-degree architecture of up to $\epsilon cdot \log n$ depth.