arXiv subjects

Maximilian J. Kramer

Publications and source records attributed to Maximilian J. Kramer.

5 recordsLinked to original sources

arXiv2026

Approximability limits for bounded-degree max-LINSAT and implications for decoded quantum interferometry

For general max-k-XORSAT with $k \geq 3$, no polynomial-time algorithm can do substantially better than random guessing on worst-case instances unless $\mathsf{P} = \mathsf{NP}$: approximating beyond the random-assignment value of $1/2$ is $\mathsf{NP}$-hard. The picture changes when each variable appears in at most $D$ constraints. In that bounded-degree setting, polynomial-time algorithms can provably beat the random baseline by an additive amount of order $1/\sqrt{D}$. For Boolean instances, this scaling is known to be optimal: the matching hardness result is due to Trevisan, while the corresponding algorithmic guarantee was established by Barak et al. Whether the same holds over general finite fields, and what it implies for quantum algorithms, has not been established. We make this connection explicit and extend the hardness to max-E$k$-LINSAT$(q,r)$ with bounded degree $D$ and over arbitrary finite fields $\mathbb{F}_q$, proving that it is $\mathsf{NP}$-hard to exceed $r/q + \mathcal{O}_{q,r}(1/\sqrt{D})$. These results provide the complexity-theoretic benchmark for the bounded-degree instances targeted by decoded quantum interferometry (DQI), QAOA, and classical heuristics. Any quantum advantage on bounded-degree instances is therefore confined to the constant prefactor. We further show that in the context of DQI and on $(k,D)$-regular instances, this prefactor is sensitive to the nature of the decoder: DQI with classical decoders faces an information-theoretic $1/\sqrt{D \log D}$ barrier that prevents it from matching the hardness scaling, while DQI with quantum decoders is compatible with the $1/\sqrt{D}$ scaling -- identifying quantum decoding as the key ingredient for matching the complexity-theoretic scaling with DQI.

quant-ph↗

arXiv2026

Optimal algorithmic complexity of inference in quantum kernel methods

Quantum kernel methods are among the leading candidates for achieving quantum advantage in supervised learning. A key bottleneck is the cost of inference: evaluating a trained model on new data requires estimating a weighted sum $\sum_{i=1}^N α_i k(x,x_i)$ of $N$ kernel values to additive precision $\varepsilon$, where $α$ is the vector of trained coefficients. The standard approach estimates each term independently via sampling, yielding a query complexity of $O(N\lVertα\rVert_2^2/\varepsilon^2)$. In this work, we identify two independent axes for improvement: (1) How individual kernel values are estimated (sampling versus quantum amplitude estimation), and (2) how the sum is approximated (term-by-term versus via a single observable), and systematically analyze all combinations thereof. The query-optimal combination, encoding the full inference sum as the expectation value of a single observable and applying quantum amplitude estimation, achieves a query complexity of $O(\lVertα\rVert_1/\varepsilon)$, removing the dependence on $N$ from the query count and yielding a quadratic improvement in both $\lVertα\rVert_1$ and $\varepsilon$. We prove a matching lower bound of $Ω(\lVertα\rVert_1/\varepsilon)$, establishing query-optimality of our approach up to logarithmic factors. Beyond query complexity, we also analyze how these improvements translate into gate costs and show that the query-optimal strategy is not always optimal in practice from the perspective of gate complexity. Our results provide both a query-optimal algorithm and a practically optimal choice of strategy depending on hardware capabilities, along with a complete landscape of intermediate methods to guide practitioners. All algorithms require only amplitude estimation as a subroutine and are thus natural candidates for early-fault-tolerant implementations.

quant-ph↗

arXiv2026

Using deep learning to construct stochastic local search SAT solvers with performance bounds

The Boolean Satisfiability problem (SAT), as the prototypical $\mathsf{NP}$-complete problem, is crucial in both theoretical computer science and practical applications. To address this problem, stochastic local search (SLS) algorithms, which iteratively and randomly update candidate assignments, present an important and theoretically well-studied class of solvers. Recent theoretical advancements have identified conditions under which SLS solvers efficiently solve SAT instances, provided they have access to suitable ``oracles'', i.e., instance-specific distribution samples. We propose leveraging machine learning models, particularly graph neural networks (GNN), as oracles to enhance the performance of SLS solvers. Our approach, evaluated on random and pseudo-industrial SAT instances, demonstrates a significant performance improvement regarding step counts and solved instances. Our work bridges theoretical results and practical applications, highlighting the potential of purpose-trained SAT solvers with performance guarantees.

cs.AI↗

arXiv2026

Tight inapproximability of max-LINSAT and implications for decoded quantum interferometry

We establish tight inapproximability bounds for max-LINSAT, the problem of maximizing the number of satisfied linear constraints over the finite field $\mathbb{F}_q$, where each constraint accepts $r$ values. Specifically, we prove by a direct reduction from Håstad's theorem that no polynomial-time algorithm can exceed the random-assignment ratio $r/q$ by any constant, assuming $\mathsf{P} \neq \mathsf{NP}$. This threshold coincides with the $\ell/m \to 0$ limit of the semicircle law governing decoded quantum interferometry (DQI), where $\ell$ is the decoding radius of the underlying code. Together, these observations delineate the boundary between worst-case hardness and potential quantum advantage, showing that any algorithm surpassing $r/q$ must exploit instance structure beyond what is present in the hard instances produced by PCP reductions.

quant-ph↗

arXiv2025

A measurement-driven quantum algorithm for SAT: Performance guarantees via spectral gaps and measurement parallelization

The Boolean satisfiability problem (SAT) is of central importance in both theory and practice. Yet, most provable guarantees for quantum algorithms rely exclusively on Grover-type methods that cap the possible advantage at only quadratic speed-ups, making the search for approaches that surpass this quadratic barrier a key challenge. In this light, this work presents a rigorous worst-case runtime analysis of a recently introduced measurement-driven quantum SAT solver. Importantly, this quantum algorithm does not exclusively rely on Grover-type methods and shows promising numerical performance. Our analysis establishes that the algorithm's runtime depends on an exponential trade-off between two key properties: the spectral gap of the associated Hamiltonian and the success probability of the driving measurements. We show that this trade-off can be systematically controlled by a tunable rotation angle. Beyond establishing a worst-case runtime expression, this work contributes significant algorithmic improvements. First, we develop a new readout routine that efficiently finds a solution even for instances with multiple satisfying assignments. Second, a measurement parallelization scheme, based on perfect hash families, is introduced. Third, we establish an amplitude-amplified version of the measurement-driven algorithm. Finally, we demonstrate the practical utility of our framework: By suitably scheduling the algorithm's parameters, we show that its runtime collapses from exponential to polynomial on a special class of SAT instances, consistent with their known classical tractability. A problem we leave open is to establish a non-trivial lower bound on the spectral gap as a function of the rotation angle. Resolving this directly translates into an improved worst-case runtime, potentially realizing a super-quadratic quantum advantage.

quant-ph↗