@article {3488, title = {The Second Moment of Hafnians in Gaussian Boson Sampling}, year = {2024}, month = {3/20/2024}, abstract = {

Gaussian Boson Sampling is a popular method for experimental demonstrations of quantum advantage, but many subtleties remain in fully understanding its theoretical underpinnings. An important component in the theoretical arguments for approximate average-case hardness of sampling is anticoncentration, which is a second-moment property of the output probabilities. In Gaussian Boson Sampling these are given by hafnians of generalized circular orthogonal ensemble matrices. In a companion work [arXiv:2312.08433], we develop a graph-theoretic method to study these moments and use it to identify a transition in anticoncentration. In this work, we find a recursive expression for the second moment using these graph-theoretic techniques. While we have not been able to solve this recursion by hand, we are able to solve it numerically exactly, which we do up to Fock sector 2n=80. We further derive new analytical results about the second moment. These results allow us to pinpoint the transition in anticoncentration and furthermore yield the expected linear cross-entropy benchmarking score for an ideal (error-free) device.

}, url = {https://arxiv.org/abs/2403.13878}, author = {Adam Ehrenberg and Joseph T. Iosue and Abhinav Deshpande and Dominik Hangleiter and Alexey V. Gorshkov} } @article {3518, title = {Secret extraction attacks against obfuscated IQP circuits}, year = {2023}, month = {12/15/2023}, abstract = {

Quantum computing devices can now perform sampling tasks which, according to complexity-theoretic and numerical evidence, are beyond the reach of classical computers. This raises the question of how one can efficiently verify that a quantum computer operating in this regime works as intended. In 2008, Shepherd and Bremner proposed a protocol in which a verifier constructs a unitary from the comparatively easy-to-implement family of so-called IQP circuits, and challenges a prover to execute it on a quantum computer. The challenge problem is designed to contain an obfuscated secret, which can be turned into a statistical test that accepts samples from a correct quantum implementation. It was conjectured that extracting the secret from the challenge problem is NP-hard, so that the ability to pass the test constitutes strong evidence that the prover possesses a quantum device and that it works as claimed. Unfortunately, about a decade later, Kahanamoku-Meyer found an efficient classical secret extraction attack. Bremner, Cheng, and Ji very recently followed up by constructing a wide-ranging generalization of the original protocol. Their IQP Stabilizer Scheme has been explicitly designed to circumvent the known weakness. They also suggested that the original construction can be made secure by adjusting the problem parameters. In this work, we develop a number of secret extraction attacks which are effective against both new approaches in a wide range of problem parameters. The important problem of finding an efficient and reliable verification protocol for sampling-based proofs of quantum supremacy thus remains open.

}, url = {https://arxiv.org/abs/2312.10156}, author = {David Gross and Dominik Hangleiter} } @article {3312, title = {A sharp phase transition in linear cross-entropy benchmarking}, year = {2023}, month = {5/8/2023}, abstract = {

Demonstrations of quantum computational advantage and benchmarks of quantum processors via quantum random circuit sampling are based on evaluating the linear cross-entropy benchmark (XEB). A key question in the theory of XEB is whether it approximates the fidelity of the quantum state preparation. Previous works have shown that the XEB generically approximates the fidelity in a regime where the noise rate per qudit ε satisfies εN<<1 for a system of N qudits and that this approximation breaks down at large noise rates. Here, we show that the breakdown of XEB as a fidelity proxy occurs as a sharp phase transition at a critical value of εN that depends on the circuit architecture and properties of the two-qubit gates, including in particular their entangling power. We study the phase transition using a mapping of average two-copy quantities to statistical mechanics models in random quantum circuit architectures with full or one-dimensional connectivity. We explain the phase transition behavior in terms of spectral properties of the transfer matrix of the statistical mechanics model and identify two-qubit gate sets that exhibit the largest noise robustness.

}, url = {https://arxiv.org/abs/2305.04954}, author = {Brayden Ware and Abhinav Deshpande and Dominik Hangleiter and Pradeep Niroula and Bill Fefferman and Alexey V. Gorshkov and Michael Gullans} } @article {3136, title = {Scalably learning quantum many-body Hamiltonians from dynamical data}, year = {2022}, month = {9/28/2022}, abstract = {

The physics of a closed quantum mechanical system is governed by its Hamiltonian. However, in most practical situations, this Hamiltonian is not precisely known, and ultimately all there is are data obtained from measurements on the system. In this work, we introduce a highly scalable, data-driven approach to learning families of interacting many-body Hamiltonians from dynamical data, by bringing together techniques from gradient-based optimization from machine learning with efficient quantum state representations in terms of tensor networks. Our approach is highly practical, experimentally friendly, and intrinsically scalable to allow for system sizes of above 100 spins. In particular, we demonstrate on synthetic data that the algorithm works even if one is restricted to one simple initial state, a small number of single-qubit observables, and time evolution up to relatively short times. For the concrete example of the one-dimensional Heisenberg model our algorithm exhibits an error constant in the system size and scaling as the inverse square root of the size of the data set.

}, keywords = {FOS: Computer and information sciences, FOS: Physical sciences, Machine Learning (cs.LG), Quantum Gases (cond-mat.quant-gas), Quantum Physics (quant-ph), Strongly Correlated Electrons (cond-mat.str-el)}, doi = {10.48550/ARXIV.2209.14328}, url = {https://arxiv.org/abs/2209.14328}, author = {Wilde, Frederik and Kshetrimayum, Augustine and Roth, Ingo and Dominik Hangleiter and Sweke, Ryan and Eisert, Jens} } @article {3178, title = {Sharp complexity phase transitions generated by entanglement}, year = {2022}, month = {12/20/2022}, abstract = {

Entanglement is one of the physical properties of quantum systems responsible for the computational hardness of simulating quantum systems. But while the runtime of specific algorithms, notably tensor network algorithms, explicitly depends on the amount of entanglement in the system, it is unknown whether this connection runs deeper and entanglement can also cause inherent, algorithm-independent complexity. In this work, we quantitatively connect the entanglement present in certain quantum systems to the computational complexity of simulating those systems. Moreover, we completely characterize the entanglement and complexity as a function of a system parameter. Specifically, we consider the task of simulating single-qubit measurements of k--regular graph states on n qubits. We show that, as the regularity parameter is increased from 1 to n\−1, there is a sharp transition from an easy regime with low entanglement to a hard regime with high entanglement at k=3, and a transition back to easy and low entanglement at k=n\−3. As a key technical result, we prove a duality for the simulation complexity of regular graph states between low and high regularity.

}, keywords = {Computational Complexity (cs.CC), FOS: Computer and information sciences, FOS: Physical sciences, Quantum Physics (quant-ph)}, doi = {10.48550/ARXIV.2212.10582}, url = {https://arxiv.org/abs/2212.10582}, author = {Ghosh, Soumik and Deshpande, Abhinav and Dominik Hangleiter and Alexey V. Gorshkov and Fefferman, Bill} } @article {3065, title = {A single T-gate makes distribution learning hard}, year = {2022}, month = {7/7/2022}, abstract = {

The task of learning a probability distribution from samples is ubiquitous across the natural sciences. The output distributions of local quantum circuits form a particularly interesting class of distributions, of key importance both to quantum advantage proposals and a variety of quantum machine learning algorithms. In this work, we provide an extensive characterization of the learnability of the output distributions of local quantum circuits. Our first result yields insight into the relationship between the efficient learnability and the efficient simulatability of these distributions. Specifically, we prove that the density modelling problem associated with Clifford circuits can be efficiently solved, while for depth d=nΩ(1) circuits the injection of a single T-gate into the circuit renders this problem hard. This result shows that efficient simulatability does not imply efficient learnability. Our second set of results provides insight into the potential and limitations of quantum generative modelling algorithms. We first show that the generative modelling problem associated with depth d=nΩ(1) local quantum circuits is hard for any learning algorithm, classical or quantum. As a consequence, one cannot use a quantum algorithm to gain a practical advantage for this task. We then show that, for a wide variety of the most practically relevant learning algorithms -- including hybrid-quantum classical algorithms -- even the generative modelling problem associated with depth d=ω(log(n)) Clifford circuits is hard. This result places limitations on the applicability of near-term hybrid quantum-classical generative modelling algorithms.

}, url = {https://arxiv.org/abs/2207.03140}, author = {Marcel Hinsche and Marios Ioannou and Alexander Nietner and Jonas Haferkamp and Yihui Quek and Dominik Hangleiter and Jean-Pierre Seifert and Jens Eisert and Ryan Sweke} }