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Quantum Log-Approximate-Rank Conjecture is also False
[article]
2018
arXiv
pre-print
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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)
2019
Dagstuhl Reports
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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
2018
Foundations of Software Technology and Theoretical Computer Science
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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]
2020
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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)
2017
Dagstuhl Reports
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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]
2022
arXiv
pre-print
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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]
2015
arXiv
pre-print
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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
2015
Journal of Physics A: Mathematical and Theoretical
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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
2018
Physical Review Letters
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2017 accepted arXiv:1709.03543v2 fatcat:t5kq2rkp5zhvdccb4hxanbcziuOther Versions
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
2016
Communications in Number Theory and Physics
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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]
2020
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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]
2021
arXiv
pre-print
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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]
2014
arXiv
pre-print
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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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