Robbie King

We present sum-of-squares spectral amplification (SOSSA), a framework for improving quantum simulation relevant to low-energy problems. We show how SOSSA can be applied to problems like energy and phase estimation and provide fast quantum algorithms for these problems that significantly improve over prior art. We analyze the performance of SOSSA on the Sachdev-Ye-Kitaev model, a representative strongly correlated system, and demonstrate asymptotic speedups over generic simulation methods by a factor of the square root of the system size. We then apply SOSSA to electronic structure problems in quantum chemistry, yielding a factor of 4 to 195 speedup over the state of the art in ground-state energy estimation for models of Iron-Sulfur complexes and a CO2-fixation catalyst.

We study the complexity of a classic problem in computational topology, the homology problem: given a description of some space X and an integer k, decide if X contains a k-dimensional hole. The setting and statement of the homology problem are completely classical, yet we find that the complexity is characterized by quantum complexity classes. Our result can be seen as an aspect of a connection between homology and supersymmetric quantum mechanics [Wit82]. We consider clique complexes, motivated by the practical application of topological data analysis (TDA). The clique complex of a graph is the simplicial complex formed by declaring every k+1-clique in the graph to be a k-simplex. Our main result is that deciding whether the clique complex of a weighted graph has a hole or not, given a suitable promise on the gap, is QMA1-hard and contained in QMA. Our main innovation is a technique to lower bound the eigenvalues of the combinatorial Laplacian operator. For this, we invoke a tool from algebraic topology known as spectral sequences. In particular, we exploit a connection between spectral sequences and Hodge theory [For94]. Spectral sequences will play a role analogous to perturbation theory for combinatorial Laplacians. In addition, we develop the simplicial surgery technique used in prior work [CK22]. Our result provides some suggestion that the quantum TDA algorithm [LGZ16] cannot be dequantized. More broadly, we hope that our results will open up new possibilities for quantum advantage in topological data analysis.

The ability of quantum computers to directly manipulate and analyze quantum states stored in quantum memory allows them to learn about aspects of our physical world that would otherwise be invisible given a modest number of measurements. Here we investigate a new learning resource which could be available to quantum computers in the future – measurements on the unknown state accompanied by its complex conjugate ρ⊗ρ*. For a certain shadow tomography task, we surprisingly find that measurements on only copies of ρ⊗ρ* can be exponentially more powerful than measurements on ρ⊗K, even for large K. This expands the class of exponential advantages using only a constant overhead quantum memory, or minimal quantum memory, and we provide a number of examples where the state ρ* is naturally available in both computational and physical applications. In addition, we precisely quantify the power of classical shadows on single copies under a generalized Clifford ensemble and give a class of quantities that can be efficiently learned. The learning task we study in both the single copy and quantum memory is physically natural and corresponds to real-space observables with a limit of bosonic modes, where it achieves an exponential improvement in detecting certain signals under a noisy background. In addition to quantifying a fundamentally new and powerful resource in quantum learning, we believe the advantage may find applications in improving quantum simulation, learning from quantum sensors, and uncovering new physical phenomena.

Understanding and approximating extremal energy states of local Hamiltonians is a central problem in quantum physics and complexity theory. Recent work has focused on developing approximation algorithms for local Hamiltonians, and in particular the ``Quantum Max Cut'' (QMaxCut) problem, which is closely related to the antiferromagnetic Heisenberg model. In this work, we introduce a family of semidefinite programming (SDP) relaxations based on the Navascues-Pironio-Acin (NPA) hierarchy which is tailored for QMaxCut by taking into account its SU(2) symmetry. We show that the hierarchy converges to the optimal QMaxCut value at a finite level, which is based on a characterization of the algebra of SWAP operators. We give several analytic proofs and computational results showing exactness/inexactness of our hierarchy at the lowest level on several important families of graphs. We also discuss relationships between SDP approaches for QMaxCut and frustration-freeness in condensed matter physics and numerically demonstrate that the SDP-solvability practically becomes an efficiently-computable generalization of frustration-freeness. Furthermore, by numerical demonstration we show the potential of SDP algorithms to perform as an approximate method to compute physical quantities and capture physical features of some Heisenberg-type statistical mechanics models even away from the frustration-free regions.