Ruslan Shaydulin

The Sherrington-Kirkpatrick (SK) model serves as a foundational framework for understanding disordered systems. The Quantum Approximate Optimization Algorithm (QAOA) is a quantum optimization algorithm whose performance monotonically improves with its depth $p$. In this work, we introduce a new equivalence between the task of evaluating the energy of QAOA applied to the SK model in the infinite-size limit and the task of simulating a spin-boson system, which we show can be done with modest cost using matrix product states. Using this equivalence, we optimize QAOA parameters and provide numerical evidence that QAOA obtains a $(1-\epsilon)$ approximation to the optimal energy with circuit depth $\mathcal{O}(n/\epsilon^{\infiniteSizeLimitOneOverEta})$ in the average case, with $\varepsilon\lesssim\infiniteSizeLastpError\%$ at $p=\infiniteSizeLastp$. We then use these optimized QAOA parameters to evaluate the QAOA energy for finite-sized instances with up to $30$ qubits and find convergence to the ground state consistent with the infinite-size limit prediction. Our results provide strong numerical evidence that QAOA can efficiently approximate the ground state of the SK model in the average case.

We present a comprehensive end-to-end quantum algorithm for tensor problems, including tensor PCA and planted kXOR, that achieves potential superquadratic quantum speedups over classical methods. We build upon prior works by Hastings~(\textit{Quantum}, 2020) and Schmidhuber~\textit{et al.}~(\textit{Phys.~Rev.~X.}, 2025), and address key limitations by introducing a native qubit-based encoding for the Kikuchi method, enabling explicit quantum circuit constructions and non-asymptotic resource estimation. Our approach substantially reduces constant overheads through a novel guiding state preparation technique as well as circuit optimizations, reducing the threshold for a quantum advantage. We further extend the algorithmic framework to support recovery in sparse tensor PCA and tensor completion, and generalize detection to asymmetric tensors, demonstrating that the quantum advantage persists in these broader settings. Detailed resource estimates show that 900 logical qubits, $\sim 10^{15}$ gates and $\sim 10^{12}$ gate depth suffice for a problem that classically requires $\sim 10^{23}$ FLOPs. The gate count and depth for the same problem without the improvements presented in this paper would be at least $10^{19}$ and $10^{18}$ respectively. These advances position tensor problems as a candidate for quantum advantage whose resource requirements benefit significantly from algorithmic and compilation improvements; the magnitude of the improvements suggest that further enhancements are possible, which would make the algorithm viable for upcoming fault-tolerant quantum hardware.

Quantum mechanics provides cryptographic primitives whose security is grounded in hardness assumptions independent of those underlying classical cryptography. However, existing proposals require low-noise quantum communication and long-lived quantum memory, capabilities which remain challenging to realize in practice. In this work, we introduce a quantum digital signature scheme that operates with only classical communication, using the classical shadows of states produced by random circuits as public keys. We provide theoretical and numerical evidence supporting the conjectured hardness of learning the private key (the circuit) from the public key (the shadow). A key technical ingredient enabling our scheme is an improved state-certification primitive that achieves higher noise tolerance and lower sample complexity than prior methods. We realize this certification by designing a high-rate error-detecting code tailored to our random-circuit ensemble and experimentally generating shadows for 32-qubit states using circuits with ≥ 80 logical (≥ 582 physical) two-qubit gates, attaining 0.90±0.01 fidelity. With increased number of measurement samples, our hardware-demonstrated primitives realize a proof-of-principle quantum digital signature, demonstrating the near-term feasibility of our scheme.

Quantum algorithms for binary optimization problems have been subject of extensive study. However, the application of quantum algorithms to integer optimization problems remains comparatively unexplored. In this paper, we study the Quantum Approximate Optimization Algorithm (QAOA) applied to integer problems on graphs, with each integer variable encoded in a qudit. We derive a general iterative formula for depth-$p$ QAOA expectation on high-girth $d$-regular graphs of arbitrary size. The cost of evaluating the formula is exponential in the QAOA depth $p$ but does not depend on the graph size. Evaluating this formula for Max-$k$-Cut problem for $p\leq 4$, we identify pararegimes ($k=3$ with degree $d \leq 10$ and $k=4$ with $d \leq 40$) in which QAOA outperforms the Frieze-Jerrum semi-definite programming (SDP) algorithm, which provides the best worst-case guarantee on the approximation ratio. To strengthen the classical baseline, we introduce a new heuristic algorithm based on the degree-of-saturation which empirically outperforms both the Frieze-Jerrum algorithm and shallow-depth QAOA. Nevertheless, we provide numerical evidence that QAOA may overtake this heuristic at depth $p\leq 20$. Our results show that moving beyond binary to integer optimization problems can open up new avenues for quantum advantage.

Achieving computational advantage using NISQ devices on practically useful problems is a long standing challenge. We report two papers that experimentally demonstrate a concrete application, namely certified randomness generation, which could be useful for multi-party cryptographic protocols and improving imperfect physical sources of randomness. Both papers involve substantial theoretical contributions to the protocol. We devise a realistic protocol that maximizes practical hardness. The verifier first asks the server to prepare a quantum state using a random circuit and then sends a random measurement basis right before the result must be received. This is repeated for many rounds. We show complexity theoretic evidence for entropy generation and provide improved entropy bounds against adversaries with oracle access to the random circuits. We also construct an end-to-end application of randomness amplification of imperfect sources into nearly perfect randomness, notably achieving everlasting security which uplifts computational security to information theoretic security.