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Quantum Log-Approximate-Rank Conjecture is also False [article]

Anurag Anshu and Naresh Goud Boddu and Dave Touchette
2018 arXiv   pre-print
f, hence refuting the log approximate rank conjecture of Lee and Shraibman [2009].  ...  between the log approximate rank and quantum communication complexity of f.  ...  D.T. is also supported by an NSERC PDF. IQC and PI are supported in part by the Government of Canada and the Province of Ontario.  ... 
arXiv:1811.10525v1 fatcat:dfdqaivhkjaynpial2sxivjhri

Computational Complexity of Discrete Problems (Dagstuhl Seminar 19121)

Anna Gál, Rahul Santhanam, Till Tantau, Michael Wagner
2019 Dagstuhl Reports  
We show how to exactly learn a k-Fourier-sparse n-bit Boolean function from O(k 1.5 (log k) 2 ) uniform quantum samples from that function.  ...  We also show that any oblivious static data structure for decomposable search problems (like ANN) can be obliviously dynamized with O(log n) overhead in update and query time, strengthening a classic result  ...  The Log-Approximate-Rank Conjecture is False Additionally, our function F refutes Grolmusz's Conjecture [2] and a variant of the Log-Approximate-Nonnegative-Rank Conjecture, suggested recently by Kol  ... 
doi:10.4230/dagrep.9.3.64 dblp:journals/dagstuhl-reports/GalST19 fatcat:gypvueu2kreclllv3n6eaz6ch4

A Lifting Theorem with Applications to Symmetric Functions

Arkadev Chattopadhyay, Nikhil S. Mande, Marc Herbstritt
2018 Foundations of Software Technology and Theoretical Computer Science  
We show that the approximate spectral norm of a symmetric function f characterizes the (quantum and classical) bounded error communication complexity of f • XOR. 4.  ...  We also characterize symmetric functions computable by quasi-polynomial sized Threshold of Parity circuits, resolving a conjecture of Zhang [24]. 3.  ...  The Log Approximation Rank Conjecture [14] is the analogous version of the well-known Log Rank Conjecture, for the randomized communication complexity model.  ... 
doi:10.4230/lipics.fsttcs.2017.23 dblp:conf/fsttcs/ChattopadhyayM17 fatcat:g54zbcwukjax5a6jb7q5uswjyu

Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture [article]

Arkadev Chattopadhyay, Ankit Garg, Suhail Sherif
2020 arXiv   pre-print
A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture.  ...  We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on n input bits, each of which has  ...  The ǫ-approximate rank of f • XOR is known to be at most the ǫ-approximate sparsity of f .  ... 
arXiv:2009.02717v1 fatcat:extnblqgiff7xc66os6czqujzi

Von Neumann entropy from unitarity [article]

P. Boes, J. Eisert, R. Gallego, M. P. Mueller, H. Wilming
2018 arXiv   pre-print
The von Neumann entropy is a key quantity in quantum information theory and, roughly speaking, quantifies the amount of quantum information contained in a state when many identical and independent i.i.d  ...  . copies of the state are available, in a regime that is often referred to as being asymptotic.  ...  entropy conjecture would essentially also prove the quantum version.  ... 
arXiv:1807.08773v1 fatcat:wopiomml7jffxp3jcnj2etqguu

Algebraic Methods in Computational Complexity (Dagstuhl Seminar 16411)

Valentine Kabanets, Thomas Thierauf, Jacobo Tóran, Christopher Umans, Marc Herbstritt
2017 Dagstuhl Reports  
It has often proven true that the best way to argue about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting.  ...  Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures.  ...  We also observe that a weaker notion, that we call weak-regularity, already implies hardness of elimination.  ... 
doi:10.4230/dagrep.6.10.13 dblp:journals/dagstuhl-reports/KabanetsTTU16 fatcat:mxexvi3xmngw7iwfc2bodtsd3y

Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture [article]

Sevag Gharibian, François Le Gall
2022 arXiv   pre-print
We also discuss how this dequantization technique may help make progress on the central quantum PCP conjecture.  ...  The Quantum Singular Value Transformation (QSVT) is a recent technique that gives a unified framework to describe most quantum algorithms discovered so far, and may lead to the development of novel quantum  ...  Conversely, if the NLSS conjecture can be shown false, then the quantum PCP conjecture is false.  ... 
arXiv:2111.09079v2 fatcat:satvcyzqnrbwni5gpm7bvtax5q

Asymptotics of Ground State Degeneracies in Quiver Quantum Mechanics [article]

Clay Cordova, Shu-Heng Shao
2015 arXiv   pre-print
For large ranks, the ground state degeneracy is exponential with slope a modular function that we are able to compute at integral values of its argument.  ...  We also observe that the exponential of the slope is an algebraic number and determine its associated algebraic equation explicitly in several examples.  ...  The work of CC is support by a Junior Fellowship at the Harvard Society of Fellows. The work of SHS is supported by the Kao Fellowship at Harvard University.  ... 
arXiv:1503.03178v1 fatcat:xdhmslggpvhu7evezggqy7kspq

Rényi squashed entanglement, discord, and relative entropy differences

Kaushik P Seshadreesan, Mario Berta, Mark M Wilde
2015 Journal of Physics A: Mathematical and Theoretical  
A 91, 022333 (2015)], we also detailed a procedure to obtain Renyi generalizations of any quantum information measure that is equal to a linear combination of von Neumann entropies with coefficients chosen  ...  Taking as a conjecture that the Renyi conditional quantum mutual information is monotone with respect to local operations on both systems A and B, we prove that the Renyi squashed entanglement and the  ...  If Conjecture 1 is false, then some of the consequences in this paper are directly impacted, such as the LOCC monotonicity of the Rényi squashed entanglement.  ... 
doi:10.1088/1751-8113/48/39/395303 fatcat:u2xwtaqpjjby5dhegypnwilnou

Distillation with Sublogarithmic Overhead

Matthew B. Hastings, Jeongwan Haah
2018 Physical Review Letters  
We show that this conjecture is false.  ...  It has been conjectured [1] that for any distillation protocol for magic states for the T gate, the number of noisy input magic states required per output magic state at output error rate ϵ is Ω((1/ϵ))  ...  We show that this conjecture is false.  ... 
doi:10.1103/physrevlett.120.050504 pmid:29481193 fatcat:czwlroh3wrgfvjfxqh5y2g7sxy

Asymptotics of ground state degeneracies in quiver quantum mechanics

Clay Córdova, Shu-Heng Shao
2016 Communications in Number Theory and Physics  
For large ranks, the ground state degeneracy is exponential with slope a modular function that we are able to compute at integral values of its argument.  ...  We also observe that the exponential of the slope is an algebraic number and determine its associated algebraic equation explicitly in several examples.  ...  The work of CC is support by a Junior Fellowship at the Harvard Society of Fellows. The work of SHS is supported by the Kao Fellowship at Harvard University.  ... 
doi:10.4310/cntp.2016.v10.n2.a4 fatcat:quk5id46jza7vdmm7oxwlgjhc4

Black hole microstates vs. the additivity conjectures [article]

Patrick Hayden, Geoff Penington
2020 arXiv   pre-print
We argue that one of the following statements must be true: (a) extensive violations of quantum information theory's additivity conjectures exist or (b) there exists a set of 'disentangled' black hole  ...  Possibility (a) would be a significant result in quantum communication theory, demonstrating that entanglement can enhance the ability to transmit information much more than has currently been established  ...  Since then the largest gap ever demonstrated is an underwhelming ∆ := S min (N 1 ) + S min (N 2 ) − S min (N 1 ⊗ N 2 ) (9) of log 2 [28] . 2 Thus, the additivity conjecture is false, but in a thoroughly  ... 
arXiv:2012.07861v1 fatcat:4nzix3g7u5evrnuuoquajplmaa

From communication complexity to an entanglement spread area law in the ground state of gapped local Hamiltonians [article]

Anurag Anshu, Aram W. Harrow, Mehdi Soleimanifar
2020 arXiv   pre-print
Our construction also provides evidence for a conjecture in physics by Li and Haldane on the entanglement spectrum of lattice Hamiltonians [Li and Haldane, PRL'08].  ...  The second problem is on the quantum communication complexity of testing bipartite states with EPR assistance, a well-known question in quantum information theory.  ...  Perimeter Institute is also supported in part by the Government of Canada and the Province of Ontario.  ... 
arXiv:2004.15009v1 fatcat:qq7sa4a5prdwhld3fcbg7lxvjy

Fast and robust quantum state tomography from few basis measurements [article]

Fernando G.S.L. Brandão, Richard Kueng, Daniel Stilck França
2021 arXiv   pre-print
The protocol is the first to give provably optimal performance in terms of rank and dimension for state copies, measurement settings and memory.  ...  Classical runtime is also reduced substantially and numerical experiments demonstrate a favorable comparison with other state-of-the-art techniques.  ...  The Institute for Quantum Information and Matter is an NSF Physics Frontiers Center. D.S.F. acknowledges financial support from VILLUM FONDEN via the QMATH Centre of Excellence (Grant no. 10059).  ... 
arXiv:2009.08216v2 fatcat:3ahllytudvemzdunrhsl4ezb4q

Lower bounds on the communication complexity of two-party (quantum) processes [article]

Alberto Montina, Stefan Wolf
2014 arXiv   pre-print
Finally, we conjecture the lower bound n 2^n for a noiseless quantum channel with capacity n qubits.  ...  This bound can have an interesting consequence in the context of the recent quantum-foundational debate on the reality of the quantum state.  ...  This work is supported by the Swiss National Science Foundation, the NCCR QSIT, and the COST action on Fundamental Problems in Quantum Physics.  ... 
arXiv:1401.4126v1 fatcat:ib6c6d4igjeyhf5brwsqg2koru
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