01684nas a2200157 4500008004100000245006700041210006700108260001400175520120000189100001601389700002201405700002001427700001701447700002501464856003701489 2024 eng d00aAverage Rényi Entanglement Entropy in Gaussian Boson Sampling0 aAverage Rényi Entanglement Entropy in Gaussian Boson Sampling c3/27/20243 a

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 Rényi-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 Rényi-α variance for integers α>1, and we are able to show that these entropies are weakly typical.

1 aYoum, Jason1 aIosue, Joseph, T.1 aEhrenberg, Adam1 aWang, Yu-Xin1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2403.1889001817nas a2200181 4500008004100000245006100041210006100102260001400163520125300177100003201430700001901462700002201481700002201503700002001525700002801545700002501573856003701598 2024 eng d00aEstimation of Hamiltonian parameters from thermal states0 aEstimation of Hamiltonian parameters from thermal states c1/18/20243 a

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'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.

1 aGarcía-Pintos, Luis, Pedro1 aBharti, Kishor1 aBringewatt, Jacob1 aDehghani, Hossein1 aEhrenberg, Adam1 aHalpern, Nicole, Yunger1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2401.1034301621nas a2200157 4500008004100000245006100041210005700102260001400159520113900173100002001312700002201332700002301354700002401377700002501401856003701426 2024 eng d00aThe Second Moment of Hafnians in Gaussian Boson Sampling0 aSecond Moment of Hafnians in Gaussian Boson Sampling c3/20/20243 a

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.

1 aEhrenberg, Adam1 aIosue, Joseph, T.1 aDeshpande, Abhinav1 aHangleiter, Dominik1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2403.1387801434nas a2200145 4500008004100000245005900041210005800100260001400158490000600172520100600178100002001184700002201204700002501226856003701251 2023 eng d00aMinimum-entanglement protocols for function estimation0 aMinimumentanglement protocols for function estimation c9/29/20230 v53 a

We derive a family of optimal protocols, in the sense of saturating the quantum Cramér-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.

1 aEhrenberg, Adam1 aBringewatt, Jacob1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2110.0761302008nas a2200181 4500008004100000245005800041210005800099260001400157300000900171490000600180520148900186100002201675700002001697700002401717700002301741700002501764856003701789 2023 eng d00aPage curves and typical entanglement in linear optics0 aPage curves and typical entanglement in linear optics c5/18/2023 a10170 v73 a

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 Rényi-2 Page curve (the average Rényi-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 Rényi-2 entropy by studying its variance. Using the aforementioned results for the Rényi-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.

1 aIosue, Joseph, T.1 aEhrenberg, Adam1 aHangleiter, Dominik1 aDeshpande, Abhinav1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2209.0683801855nas a2200145 4500008004100000245007300041210006900114260001500183520138800198100002201586700001901608700002001627700002501647856003701672 2023 eng d00aProjective toric designs, difference sets, and quantum state designs0 aProjective toric designs difference sets and quantum state desig c11/22/20233 a

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.

1 aIosue, Joseph, T.1 aMooney, T., C.1 aEhrenberg, Adam1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2311.1347901590nas a2200217 4500008004100000245004000041210004000081260001400121520100000135100002101135700001601156700001501172700002201187700002001209700001901229700001901248700002101267700002201288700002501310856003701335 2023 eng d00aQuantum Sensing with Erasure Qubits0 aQuantum Sensing with Erasure Qubits c10/2/20233 a

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

1 aNiroula, Pradeep1 aDolde, Jack1 aZheng, Xin1 aBringewatt, Jacob1 aEhrenberg, Adam1 aCox, Kevin, C.1 aThompson, Jeff1 aGullans, Michael1 aKolkowitz, Shimon1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2310.0151201909nas a2200157 4500008004100000245006300041210006300104260001500167520141800182100002001600700002201620700002301642700002401665700002501689856003701714 2023 eng d00aTransition of Anticoncentration in Gaussian Boson Sampling0 aTransition of Anticoncentration in Gaussian Boson Sampling c12/13/20233 a

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.

1 aEhrenberg, Adam1 aIosue, Joseph, T.1 aDeshpande, Abhinav1 aHangleiter, Dominik1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2312.0843301610nas a2200157 4500008004100000245005700041210005600098260001400154520112700168100002001295700002301315700002901338700002301367700002501390856003701415 2022 eng d00aSimulation Complexity of Many-Body Localized Systems0 aSimulation Complexity of ManyBody Localized Systems c5/25/20223 a

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. 

1 aEhrenberg, Adam1 aDeshpande, Abhinav1 aBaldwin, Christopher, L.1 aAbanin, Dmitry, A.1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2205.1296701198nas a2200169 4500008004100000245006000041210005400101260001400155520069100169100001900860700002000879700002900899700002000928700002500948700001800973856003700991 2021 eng d00aThe Lieb-Robinson light cone for power-law interactions0 aLiebRobinson light cone for powerlaw interactions c3/29/20213 a

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.

1 aTran, Minh, C.1 aGuo, Andrew, Y.1 aBaldwin, Christopher, L.1 aEhrenberg, Adam1 aGorshkov, Alexey, V.1 aLucas, Andrew uhttps://arxiv.org/abs/2103.1582802312nas a2200217 4500008004100000245006500041210006400106260001400170490000700184520168700191100001901878700001901897700002001916700002001936700002301956700001601979700001901995700002502014700001802039856003702057 2020 eng d00aHierarchy of linear light cones with long-range interactions0 aHierarchy of linear light cones with longrange interactions c5/29/20200 v103 a

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.

1 aTran, Minh, C.1 aChen, Chi-Fang1 aEhrenberg, Adam1 aGuo, Andrew, Y.1 aDeshpande, Abhinav1 aHong, Yifan1 aGong, Zhe-Xuan1 aGorshkov, Alexey, V.1 aLucas, Andrew uhttps://arxiv.org/abs/2001.1150901697nas a2200169 4500008004100000245008100041210006900122260001500191520115700206100002001363700002301383700001901406700002001425700002001445700002501465856003701490 2019 eng d00aComplexity phase diagram for interacting and long-range bosonic Hamiltonians0 aComplexity phase diagram for interacting and longrange bosonic H c06/10/20193 a

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. 

1 aMaskara, Nishad1 aDeshpande, Abhinav1 aTran, Minh, C.1 aEhrenberg, Adam1 aFefferman, Bill1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/1906.0417801621nas a2200181 4500008004100000245007900041210006900120260001500189490000800204520106600212100001901278700002001297700002001317700001701337700002301354700002501377856003701402 2019 eng d00aLocality and Heating in Periodically Driven, Power-law Interacting Systems0 aLocality and Heating in Periodically Driven Powerlaw Interacting c2019/11/120 v1003 a

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. 

1 aTran, Minh, C.1 aEhrenberg, Adam1 aGuo, Andrew, Y.1 aTitum, Paraj1 aAbanin, Dmitry, A.1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/1908.02773