Noah Shutty

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 $\calP$, HDQI prepares a purification of the density matrix $$\rho_\calP(H) = \calP^2(H)/\Tr[\calP^2(H)]$$ 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 $\calP(x)$ to approximate $\exp(-\beta x/2)$ yields Gibbs states at inverse temperature $\beta$; other choices of $\calP$ prepare approximate ground states, microcanonical ensembles, and other spectral filters. The decoding problem inherits structural properties of $H$; in particular, local Hamiltonians map to 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 a class of nearly commuting Pauli 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, color code, and Haah's cubic code -- but also develop a matching efficient classical algorithm for this task, thereby delineating the boundary of efficient classical simulation. For a non-commuting semiclassical spin glass and commuting stabilizer code Hamiltonians with quantum defects, HDQI provably prepares Gibbs states up to a constant inverse-temperature threshold using polynomial quantum resources and quasi-polynomial classical preprocessing. These results position HDQI as a versatile new algorithmic primitive, connecting quantum state preparation to classical decoding.

Recently, a quantum algorithm called Decoded Quantum Interferometry (DQI) was introduced that achieves an apparent exponential speedup for Optimal Polynomial Intersection (OPI) problem, which has previously been studied in the contexts of cryptography and error correcting codes. However, this left open the question of how many logical gates and logical qubits would be needed to solve a classically intractable instance of OPI. Here, we develop optimized implementations of DQI which greatly reduce its resource requirements. We establish that DQI for OPI is the first known candidate for verifiable quantum advantage with optimal asymptotic speedup: solving instances with classical hardness $O(2^N)$ requires only $\widetilde{O}(N)$ quantum gates, matching the theoretical lower bound. To realize this, we overcome the primary bottleneck of reversible Reed-Solomon decoding by introducing novel quantum circuits for the Extended Euclidean Algorithm (EEA) that reduce the leading-order space complexity to the theoretical minimum of $2nb$ qubits. These improvements are broadly applicable, including to Shor's algorithm for the discrete logarithm. We analyze OPI over binary extension fields $\GF(2^b)$, assess hardness against new classical attacks, and identify resilient instances. Our resource estimates show that classically intractable OPI instances (requiring $>10^{23}$ classical trials) can be solved with approximately 5.72 million Toffoli gates. This is roughly $1000$ times fewer gates than required for factoring RSA-2048 and, remarkably, is also less than the leading interactive protocol for computational proof of quantumness, positioning DQI as a compelling candidate for practical, verifiable quantum advantage.

Abstract It has long been known that the existence of certain superquantum nonlocal correlations would cause communication complexity to collapse. The absurdity of a world in which any function could be evaluated by two players with a constant amount of communication in turn provides a tantalizing way to distinguish quantum mechanics from incorrect theories of physics; the statement ``communication complexity is nontrivial" has even been conjectured to be a concise information-theoretic axiom for characterizing quantum mechanics. We directly address the viability of that perspective with two results. First, we exhibit a nonlocal game such that communication complexity collapses in any physical theory whose maximal winning probability exceeds the quantum value. Second, we consider the venerable CHSH game that initiated this line of inquiry. In that case, the quantum value is about 0.85 but it is known that a winning probability of approximately 0.91 would collapse communication complexity. We show that the 0.91 result is the best possible using a large class of proof strategies, suggesting that the communication complexity axiom is insufficient for characterizing CHSH correlations. Both results build on new insights about reliable classical computation. The first exploits our formalization of an equivalence between amplification and reliable computation, while the second follows from a rigorous determination of the threshold for reliable computation with formulas of noise-free XOR gates and $\epsilon$-noisy AND gates.