Atsuya Hasegawa

In this work, we present new dequantization and hardness results for estimating spectral sums of matrices, such as the log-determinant. Recent quantum algorithms have demonstrated that the logarithm of the determinant of sparse, well-conditioned, positive matrices can be approximated to $\varepsilon$-relative accuracy in time polylogarithmic in the dimension~$N$, specifically in time~$\mathrm{poly}(\mathrm{log}(N), s, \kappa, 1/\varepsilon)$, where $s$ is the sparsity and $\kappa$ the condition number of the input matrix. We provide a simple dequantization of these techniques that preserves the polylogarithmic dependence on the dimension, at the cost of worse scaling in the sparsity, condition number, and accuracy. Our classical algorithm runs in time~$\mathrm{poly}\mathrm{log}(N)\cdot s^{O(\sqrt{\kappa}\log \kappa/\varepsilon)}$ which constitutes an exponential improvement over previous classical algorithms in certain parameter regimes. In addition, we prove $\mathsf{DQC}1$-completeness for estimating specific spectral sums such as the trace of the inverse and the trace of matrix powers with analogous parameter scalings as the quantum algorithms for the log-determinant in the case of log-local Hamiltonians. The latter solves a main open problem posed by Cade and Montanaro (TQC 2018) and is closely related to the complexity of Schatten $p$-norm estimation. It further shows that the parameter scalings of known quantum algorithms are not achievable classically, assuming~$\mathsf{BPP}\subsetneq\mathsf{DQC}1$. Further, we also consider a different input model where instead of a classical description of a sparse matrix, we are given a block-encoding of it and analyze the complexity of spectral sum estimation in this model. We obtain $\mathsf{DQC}1$-completeness of estimating $\mathrm{tr}[f(A)]$ in this model in a very general way, in particular whenever $f$ and $f^{-1}$ are Lipschitz continuous with bounded Lipschitz constants. We conclude our work with $\mathsf{BQP}$-hardness and $\mathsf{PP}$-completeness results on the estimation of the log-determinant.

Quantum nondeterministic distributed computing was recently introduced as dQMA (distributed quantum Merlin-Arthur) protocols by Fraigniaud, Le Gall, Nishimura and Paz (ITCS 2021). In dQMA protocols, with the help of quantum proofs and local communication, nodes on a network verify a global property of the network. Fraigniaud et al. showed that, when the network size is small, there exists an exponential separation in proof size between distributed classical and quantum verification protocols, for the equality problem, where the verifiers check if all the data owned by a subset of them are identical. In this paper, we further investigate and characterize the power of the dQMA protocols for various decision problems. First, we give a more efficient dQMA protocol for the equality problem with a simpler analysis. This is done by adding a symmetrization step on each node and exploiting properties of the permutation test, which is a generalization of the SWAP test. We also show a quantum advantage for the equality problem on path networks still persists even when the network size is large, by considering ``relay points'' between extreme nodes. Second, we show that even in a general network, there exist efficient dQMA protocols for the ranking verification problem, the Hamming distance problem, and more problems that derive from efficient quantum one-way communication protocols. Third, in a line network, we construct an efficient dQMA protocol for a problem that has an efficient two-party QMA communication protocol. Finally, we obtain the first lower bounds on the proof and communication cost of dQMA protocols. To prove a lower bound on the equality problem, we show any dQMA protocol with an entangled proof between nodes can be simulated with a dQMA protocol with a separable proof between nodes by using a QMA communication-complete problem introduced by Raz and Shpilka (CCC 2004).