@article {3486, title = {Average R{\'e}nyi Entanglement Entropy in Gaussian Boson Sampling}, year = {2024}, month = {3/27/2024}, abstract = {

Recently, many experiments have been conducted with the goal of demonstrating a quantum advantage over classical computation. One popular framework for these experiments is Gaussian Boson Sampling, where quadratic photonic input states are interfered via a linear optical unitary and subsequently measured in the Fock basis. In this work, we study the modal entanglement of the output states in this framework just before the measurement stage. Specifically, we compute Page curves as measured by various R{\'e}nyi-α entropies, where the Page curve describes the entanglement between two partitioned groups of output modes averaged over all linear optical unitaries. We derive these formulas for α=1 (i.e. the von Neumann entropy), and, more generally, for all positive integer α, in the asymptotic limit of infinite number of modes and for input states that are composed of single-mode-squeezed-vacuum state with equal squeezing strength. We then analyze the limiting behaviors when the squeezing is small and large. Having determined the averages, we then explicitly calculate the R{\'e}nyi-α variance for integers α\>1, and we are able to show that these entropies are weakly typical.

}, url = {https://arxiv.org/abs/2403.18890}, author = {Jason Youm and Joseph T. Iosue and Adam Ehrenberg and Yu-Xin Wang and Alexey V. Gorshkov} } @article {3462, title = {Estimation of Hamiltonian parameters from thermal states}, year = {2024}, month = {1/18/2024}, abstract = {

We upper- and lower-bound the optimal precision with which one can estimate an unknown Hamiltonian parameter via measurements of Gibbs thermal states with a known temperature. The bounds depend on the uncertainty in the Hamiltonian term that contains the parameter and on the term\&$\#$39;s degree of noncommutativity with the full Hamiltonian: higher uncertainty and commuting operators lead to better precision. We apply the bounds to show that there exist entangled thermal states such that the parameter can be estimated with an error that decreases faster than 1/n\−\−\√, beating the standard quantum limit. This result governs Hamiltonians where an unknown scalar parameter (e.g. a component of a magnetic field) is coupled locally and identically to n qubit sensors. In the high-temperature regime, our bounds allow for pinpointing the optimal estimation error, up to a constant prefactor. Our bounds generalize to joint estimations of multiple parameters. In this setting, we recover the high-temperature sample scaling derived previously via techniques based on quantum state discrimination and coding theory. In an application, we show that noncommuting conserved quantities hinder the estimation of chemical potentials.

}, url = {https://arxiv.org/abs/2401.10343}, author = {Luis Pedro Garc{\'\i}a-Pintos and Kishor Bharti and Jacob Bringewatt and Hossein Dehghani and Adam Ehrenberg and Nicole Yunger Halpern and Alexey V. Gorshkov} } @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 {3369, title = {Minimum-entanglement protocols for function estimation}, journal = {Physical Review Research}, volume = {5}, year = {2023}, month = {9/29/2023}, abstract = {

We derive a family of optimal protocols, in the sense of saturating the quantum Cram{\'e}r-Rao bound, for measuring a linear combination of d field amplitudes with quantum sensor networks, a key subprotocol of general quantum sensor network applications. We demonstrate how to select different protocols from this family under various constraints. Focusing primarily on entanglement-based constraints, we prove the surprising result that highly entangled states are not necessary to achieve optimality in many cases. Specifically, we prove necessary and sufficient conditions for the existence of optimal protocols using at most k-partite entanglement. We prove that the protocols which satisfy these conditions use the minimum amount of entanglement possible, even when given access to arbitrary controls and ancilla. Our protocols require some amount of time-dependent control, and we show that a related class of time-independent protocols fail to achieve optimal scaling for generic functions.

}, doi = {10.1103/physrevresearch.5.033228}, url = {https://arxiv.org/abs/2110.07613}, author = {Adam Ehrenberg and Jacob Bringewatt and Alexey V. Gorshkov} } @article {3315, title = {Page curves and typical entanglement in linear optics}, journal = {Quantum}, volume = {7}, year = {2023}, month = {5/18/2023}, pages = {1017}, abstract = {

Bosonic Gaussian states are a special class of quantum states in an infinite dimensional Hilbert space that are relevant to universal continuous-variable quantum computation as well as to near-term quantum sampling tasks such as Gaussian Boson Sampling. In this work, we study entanglement within a set of squeezed modes that have been evolved by a random linear optical unitary. We first derive formulas that are asymptotically exact in the number of modes for the R{\'e}nyi-2 Page curve (the average R{\'e}nyi-2 entropy of a subsystem of a pure bosonic Gaussian state) and the corresponding Page correction (the average information of the subsystem) in certain squeezing regimes. We then prove various results on the typicality of entanglement as measured by the R{\'e}nyi-2 entropy by studying its variance. Using the aforementioned results for the R{\'e}nyi-2 entropy, we upper and lower bound the von Neumann entropy Page curve and prove certain regimes of entanglement typicality as measured by the von Neumann entropy. Our main proofs make use of a symmetry property obeyed by the average and the variance of the entropy that dramatically simplifies the averaging over unitaries. In this light, we propose future research directions where this symmetry might also be exploited. We conclude by discussing potential applications of our results and their generalizations to Gaussian Boson Sampling and to illuminating the relationship between entanglement and computational complexity.

}, doi = {10.22331/q-2023-05-23-1017}, url = {https://arxiv.org/abs/2209.06838}, author = {Joseph T. Iosue and Adam Ehrenberg and Dominik Hangleiter and Abhinav Deshpande and Alexey V. Gorshkov} } @article {3405, title = {Projective toric designs, difference sets, and quantum state designs}, year = {2023}, month = {11/22/2023}, abstract = {

Trigonometric cubature rules of degree t are sets of points on the torus over which sums reproduce integrals of degree t monomials over the full torus. They can be thought of as t-designs on the torus. Motivated by the projective structure of quantum mechanics, we develop the notion of t-designs on the projective torus, which, surprisingly, have a much more restricted structure than their counterparts on full tori. We provide various constructions of these projective toric designs and prove some bounds on their size and characterizations of their structure. We draw connections between projective toric designs and a diverse set of mathematical objects, including difference and Sidon sets from the field of additive combinatorics, symmetric, informationally complete positive operator valued measures (SIC-POVMs) and complete sets of mutually unbiased bases (MUBs) (which are conjectured to relate to finite projective geometry) from quantum information theory, and crystal ball sequences of certain root lattices. Using these connections, we prove bounds on the maximal size of dense Btmodm sets. We also use projective toric designs to construct families of quantum state designs. Finally, we discuss many open questions about the properties of these projective toric designs and how they relate to other questions in number theory, geometry, and quantum information.

}, url = {https://arxiv.org/abs/2311.13479}, author = {Joseph T. Iosue and T. C. Mooney and Adam Ehrenberg and Alexey V. Gorshkov} } @article {3355, title = {Quantum Sensing with Erasure Qubits}, year = {2023}, month = {10/2/2023}, abstract = {

The dominant noise in an \"erasure qubit\" is an erasure -- a type of error whose occurrence and location can be detected. Erasure qubits have potential to reduce the overhead associated with fault tolerance. To date, research on erasure qubits has primarily focused on quantum computing and quantum networking applications. Here, we consider the applicability of erasure qubits to quantum sensing and metrology. We show theoretically that, for the same level of noise, an erasure qubit acts as a more precise sensor or clock compared to its non-erasure counterpart. We experimentally demonstrate this by artificially injecting either erasure errors (in the form of atom loss) or dephasing errors into a differential optical lattice clock comparison, and observe enhanced precision in the case of erasure errors for the same injected error rate. Similar benefits of erasure qubits to sensing can be realized in other quantum platforms like Rydberg atoms and superconducting qubits

}, url = {https://arxiv.org/abs/2310.01512}, author = {Pradeep Niroula and Jack Dolde and Xin Zheng and Jacob Bringewatt and Adam Ehrenberg and Kevin C. Cox and Jeff Thompson and Michael Gullans and Shimon Kolkowitz and Alexey V. Gorshkov} } @article {3399, title = {Transition of Anticoncentration in Gaussian Boson Sampling}, year = {2023}, month = {12/13/2023}, abstract = {

Gaussian Boson Sampling is a promising method for experimental demonstrations of quantum advantage because it is easier to implement than other comparable schemes. While most of the properties of Gaussian Boson Sampling are understood to the same degree as for these other schemes, we understand relatively little about the statistical properties of its output distribution. The most relevant statistical property, from the perspective of demonstrating quantum advantage, is the anticoncentration of the output distribution as measured by its second moment. The degree of anticoncentration features in arguments for the complexity-theoretic hardness of Gaussian Boson Sampling, and it is also important to know when using cross-entropy benchmarking to verify experimental performance. In this work, we develop a graph-theoretic framework for analyzing the moments of the Gaussian Boson Sampling distribution. Using this framework, we show that Gaussian Boson Sampling undergoes a transition in anticoncentration as a function of the number of modes that are initially squeezed compared to the number of photons measured at the end of the circuit. When the number of initially squeezed modes scales sufficiently slowly with the number of photons, there is a lack of anticoncentration. However, if the number of initially squeezed modes scales quickly enough, the output probabilities anticoncentrate weakly.

}, url = {https://arxiv.org/abs/2312.08433}, author = {Adam Ehrenberg and Joseph T. Iosue and Abhinav Deshpande and Dominik Hangleiter and Alexey V. Gorshkov} } @article {3071, title = {Simulation Complexity of Many-Body Localized Systems}, year = {2022}, month = {5/25/2022}, abstract = {

We use complexity theory to rigorously investigate the difficulty of classically simulating evolution under many-body localized (MBL) Hamiltonians. Using the defining feature that MBL systems have a complete set of quasilocal integrals of motion (LIOMs), we demonstrate a transition in the classical complexity of simulating such systems as a function of evolution time. On one side, we construct a quasipolynomial-time tensor-network-inspired algorithm for strong simulation of 1D MBL systems (i.e., calculating the expectation value of arbitrary products of local observables) evolved for any time polynomial in the system size. On the other side, we prove that even weak simulation, i.e. sampling, becomes formally hard after an exponentially long evolution time, assuming widely believed conjectures in complexity theory. Finally, using the consequences of our classical simulation results, we also show that the quantum circuit complexity for MBL systems is sublinear in evolution time. This result is a counterpart to a recent proof that the complexity of random quantum circuits grows linearly in time.\ 

}, url = {https://arxiv.org/abs/2205.12967}, author = {Adam Ehrenberg and Abhinav Deshpande and Christopher L. Baldwin and Dmitry A. Abanin and Alexey V. Gorshkov} } @article {2759, title = {The Lieb-Robinson light cone for power-law interactions}, year = {2021}, month = {3/29/2021}, abstract = {

The Lieb-Robinson theorem states that information propagates with a finite velocity in quantum systems on a lattice with nearest-neighbor interactions. What are the speed limits on information propagation in quantum systems with power-law interactions, which decay as 1/rα at distance r? Here, we present a definitive answer to this question for all exponents α\>2d and all spatial dimensions d. Schematically, information takes time at least rmin{1,α\−2d} to propagate a distance~r. As recent state transfer protocols saturate this bound, our work closes a decades-long hunt for optimal Lieb-Robinson bounds on quantum information dynamics with power-law interactions.

}, url = {https://arxiv.org/abs/2103.15828}, author = {Minh C. Tran and Andrew Y. Guo and Christopher L. Baldwin and Adam Ehrenberg and Alexey V. Gorshkov and Andrew Lucas} } @article {2577, title = {Hierarchy of linear light cones with long-range interactions}, journal = {Physical Review X}, volume = {10}, year = {2020}, month = {5/29/2020}, abstract = {

In quantum many-body systems with local interactions, quantum information and entanglement cannot spread outside of a \"linear light cone,\" which expands at an emergent velocity analogous to the speed of light. Yet most non-relativistic physical systems realized in nature have long-range interactions: two degrees of freedom separated by a distance r interact with potential energy V(r)\∝1/rα. In systems with long-range interactions, we rigorously establish a hierarchy of linear light cones: at the same α, some quantum information processing tasks are constrained by a linear light cone while others are not. In one spatial dimension, commutators of local operators \⟨ψ|[Ox(t),Oy]|ψ\⟩ are negligible in every state |ψ\⟩ when |x\−y|≳vt, where v is finite when α\>3 (Lieb-Robinson light cone); in a typical state |ψ\⟩ drawn from the infinite temperature ensemble, v is finite when α\>52 (Frobenius light cone); in non-interacting systems, v is finite in every state when α\>2 (free light cone). These bounds apply to time-dependent systems and are optimal up to subalgebraic improvements. Our theorems regarding the Lieb-Robinson and free light cones, and their tightness, also generalize to arbitrary dimensions. We discuss the implications of our bounds on the growth of connected correlators and of topological order, the clustering of correlations in gapped systems, and the digital simulation of systems with long-range interactions. In addition, we show that quantum state transfer and many-body quantum chaos are bounded by the Frobenius light cone, and therefore are poorly constrained by all Lieb-Robinson bounds.

}, doi = {https://doi.org/10.1103/PhysRevX.10.031009}, url = {https://arxiv.org/abs/2001.11509}, author = {Minh C. Tran and Chi-Fang Chen and Adam Ehrenberg and Andrew Y. Guo and Abhinav Deshpande and Yifan Hong and Zhe-Xuan Gong and Alexey V. Gorshkov and Andrew Lucas} } @article {2406, title = {Complexity phase diagram for interacting and long-range bosonic Hamiltonians}, year = {2019}, month = {06/10/2019}, abstract = {

Recent years have witnessed a growing interest in topics at the intersection of many-body physics and complexity theory. Many-body physics aims to understand and classify emergent behavior of systems with a large number of particles, while complexity theory aims to classify computational problems based on how the time required to solve the problem scales as the problem size becomes large. In this work, we use insights from complexity theory to classify phases in interacting many-body systems. Specifically, we demonstrate a \"complexity phase diagram\" for the Bose-Hubbard model with long-range hopping. This shows how the complexity of simulating time evolution varies according to various parameters appearing in the problem, such as the evolution time, the particle density, and the degree of locality. We find that classification of complexity phases is closely related to upper bounds on the spread of quantum correlations, and protocols to transfer quantum information in a controlled manner. Our work motivates future studies of complexity in many-body systems and its interplay with the associated physical phenomena.\ 

}, url = {https://arxiv.org/abs/1906.04178}, author = {Nishad Maskara and Abhinav Deshpande and Minh C. Tran and Adam Ehrenberg and Bill Fefferman and Alexey V. Gorshkov} } @article {2458, title = {Locality and Heating in Periodically Driven, Power-law Interacting Systems}, journal = {Phys. Rev. A }, volume = {100}, year = {2019}, month = {2019/11/12}, abstract = {

We study the heating time in periodically driven D-dimensional systems with interactions that decay with the distance r as a power-law 1/rα. Using linear response theory, we show that the heating time is exponentially long as a function of the drive frequency for α\>D. For systems that may not obey linear response theory, we use a more general Magnus-like expansion to show the existence of quasi-conserved observables, which imply exponentially long heating time, for α\>2D. We also generalize a number of recent state-of-the-art Lieb-Robinson bounds for power-law systems from two-body interactions to k-body interactions and thereby obtain a longer heating time than previously established in the literature. Additionally, we conjecture that the gap between the results from the linear response theory and the Magnus-like expansion does not have physical implications, but is, rather, due to the lack of tight Lieb-Robinson bounds for power-law interactions. We show that the gap vanishes in the presence of a hypothetical, tight bound.\ 

}, doi = {https://doi.org/10.1103/PhysRevA.100.052103}, url = {https://arxiv.org/abs/1908.02773}, author = {Minh C. Tran and Adam Ehrenberg and Andrew Y. Guo and Paraj Titum and Dmitry A. Abanin and Alexey V. Gorshkov} }