Prateek P. Kulkarni

Prateek P. Kulkarni

The graph isomorphism problem asks whether two graphs are identical up to vertex relabeling. While the exact problem admits quasi-polynomial-time classical algorithms, many applications in molecular comparison, noisy network analysis, and pattern recognition require a flexible notion of structural similarity. We study the quantum query complexity of approximate graph isomorphism testing, where two graphs on $n$ vertices drawn from the Erdős--Rényi distribution $\mathcal{G} (n,1/2)$ are considered approximately isomorphic if they can be made isomorphic by at most $k$ edge edits. We present a quantum algorithm based on MNRS quantum walk search over the product graph $Γ(G,H)$ of the two input graphs. When the graphs are approximately isomorphic, the quantum walk search detects vertex pairs belonging to a dense near isomorphic matching set; candidate pairings are then reconstructed via local consistency propagation and verified via a Grover-accelerated consistency check. We prove that this approach achieves query complexity $\mathcal{O}(n^{3/2} \log n/\varepsilon)$, where $\varepsilon$ parameterizes the approximation threshold. We complement this with an $Ω(n^2)$ classical lower bound for constant approximation, establishing a genuine polynomial quantum speedup in the query model. We extend the framework to spectral similarity measures based on graph Laplacian eigenvalues, as well as weighted and attributed graphs. Small-scale simulation results on quantum simulators for graphs with up to twenty vertices demonstrate compatibility with near-term quantum devices.

Prateek P. Kulkarni

We introduce list privacy amplification (LPA), a relaxation of the final step of quantum key distribution (QKD) in which Alice and Bob extract a list of $L$ candidate keys from a raw string correlated with an eavesdropper Eve, with the guarantee that at least one key is perfectly secret while Eve cannot identify which. This parallels list decoding in error-correcting codes: relaxing unique decoding to list decoding increases the decoding radius; analogously, list extraction increases achievable key length beyond the standard quantum leftover hash lemma (QLHL). Within the abstract cryptography framework, we formalise LPA and prove the \emph{Quantum List Leftover Hash Lemma} (QLLHL): an $L$-list of $\ell$-bit keys can be extracted from an $n$-bit source with smooth min-entropy $k$ iff \[ \ell \le k + \log L - 2\log(1/ε) - 3, \] yielding a tight additive $\log L$ gain over QLHL. This gain arises because the index of the secure key is chosen after hashing and hidden from Eve, effectively contributing $\log L$ bits of entropy. Applying QLLHL to BB84-type QKD, a list size $L = 2^{αn'}$ increases the tolerable phase-error threshold from $h^{-1}(1 - h(e_b))$ to $h^{-1}(1 - h(e_b) + α)$, exceeding the standard $\approx 11\%$ bound for any $α> 0$. We prove tightness via a matching intercept-resend attack, establish composability with Wegman--Carter authentication, and present two constructions: a polynomial inner-product hash over $\mathbb{F}_{2^m}$ and a Toeplitz-based variant, running in $O(nL)$ and $O(nL \log n)$ time.