Given copies of a quantum state ρ, a shadow tomography protocol aims to learn all expectation values from a fixed set of observables, to within a given precision ε. We say that a shadow tomography protocol is triply efficient if it is sample- and time-efficient, and only employs measurements that entangle a constant number of copies of ρ at a time. The classical shadows protocol based on random single-copy measurements is triply efficient for the set of local Pauli observables. This and other protocols based on random singlecopy Clifford measurements can be understood as arising from fractional colorings of a graph G that encodes the commutation structure of the set of observables. Here we describe a framework for two-copy shadow tomography that uses an initial round of Bell measurements to reduce to a fractional coloring problem in an induced subgraph of G with bounded clique number. This coloring problem can be addressed using techniques from graph theory known as chi-boundedness. Using this framework we give the first triply efficient shadow tomography scheme for the set of local fermionic observables, which arise in a broad class of interacting fermionic systems in physics and chemistry. We also give a triply efficient scheme for the set of all n-qubit Pauli observables. Our protocols for these tasks use two-copy measurements, which is necessary: sample- efficient schemes are provably impossible using only single-copy measurements. Finally, we give a shadow tomography protocol that compresses an n-qubit quantum state into a poly(n )-sized classical representation, from which one can extract the expected value of any of the 4n Pauli observables in poly(n ) time, up to a small constant error.

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 [1]. 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 [2]. Spectral sequences will play a role analogous to perturbation theory for combinatorial Lapla-cians. In addition, we develop the simplicial surgery technique used in prior work [3]. Our result provides some suggestion that the quantum TDA algorithm [4] cannot be dequantized. More broadly, we hope that our results will open up new possibilities for quantum advantage in topological data analysis.

The success of modern deep learning hinges on the ability to train neural networks at scale. Through clever reuse of intermediate information, backpropagation facilitates training through gradient computation at a total cost roughly proportional to running the function, rather than incurring an additional factor proportional to the number of parameters -- which can now be in the trillions. Naively, one expects that quantum measurement collapse entirely rules out the reuse of quantum information as in backpropagation. But recent developments in shadow tomography, which assumes access to multiple copies of a quantum state, have challenged that notion. Here, we investigate whether parameterized quantum models can train as efficiently as classical neural networks. We show that achieving backpropagation scaling is impossible without access to multiple copies of a state. With this added ability, we introduce an algorithm with foundations in shadow tomography that matches backpropagation scaling in quantum resources while reducing classical auxiliary computational costs to open problems in shadow tomography. These results highlight the nuance of reusing quantum information for practical purposes and clarify the unique difficulties in training large quantum models, which could alter the course of quantum machine learning.

With FPGAs emerging as a promising accelerator for general-purpose computing, there is a strong demand to make them accessible to software developers. Recent advances in OpenCL compilers for FPGAs pave the way for synthesizing FPGA hardware from OpenCL kernel code. To enable broader adoption of this paradigm, significant challenges remain. This paper presents our efforts in developing dynamic profiling and debugging support in OpenCL for FPGAs. We first propose primitive code patterns, including a timestamp and an event-ordering function, and then develop a framework, which can be plugged easily into OpenCL kernels, to dynamically collect and process run-time information.