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A lower bound on the quantum query complexity of read-once functions [article]

Howard Barnum, Michael Saks
2002 arXiv   pre-print
We establish a lower bound of Ω(√(n)) on the bounded-error quantum query complexity of read-once Boolean functions, providing evidence for the conjecture that Ω(√(D(f))) is a lower bound for all Boolean  ...  The number of queries is bounded by comparing the required total amount of decoherence of a judiciously selected set of input-output pairs to an upper bound on the amount achievable in a single query step  ...  Our results provide a quantum counterpart to the lower bounds on the randomized decision tree complexity of read-once functions given in [11] and [12] .  ... 
arXiv:quant-ph/0201007v1 fatcat:an663syfufdc7pzzt2xpwldcbe

A lower bound on the quantum query complexity of read-once functions

Howard Barnum, Michael Saks
2004 Journal of computer and system sciences (Print)  
We establish a lower bound of Oð ffiffi ffi n p Þ on the bounded-error quantum query complexity of read-once Boolean functions.  ...  We then prove our result for read-once functions by induction on the number of variables, where the induction step involves a careful choice of weights depending on f to optimize the lower bound attained  ...  Acknowledgments We thank Ronald de Wolf for helpful comments on an earlier version.  ... 
doi:10.1016/j.jcss.2004.02.002 fatcat:gsbj7e4vzrfqdpvdy4qt6wnoby

The Quantum Query Complexity of Read-Many Formulas [chapter]

Andrew M. Childs, Shelby Kimmel, Robin Kothari
2012 Lecture Notes in Computer Science  
The quantum query complexity of evaluating any read-once formula with n black-box input bits is Theta(sqrt(n)).  ...  Applications of these results include a Omega (n^19/18) lower bound for Boolean matrix product verification, a nearly tight characterization of the quantum query complexity of evaluating constant-depth  ...  Then apply the read-once formula evaluation algorithm. Lemma: Using queries, we can produce a formula of size with the same value on the given input.  ... 
doi:10.1007/978-3-642-33090-2_30 fatcat:sj4wkylm7bb5jmdrou7wvhuf4i

Lower Bounds on Quantum Query Complexity for Read-Once Formulas with XOR and MUX Operators

Hideaki FUKUHARA, Eiji TAKIMOTO
2010 IEICE transactions on information and systems  
We introduce a complexity measure r for the class F of read-once formulas over the basis {AND, OR, NOT, XOR, MUX} and show that for any Boolean formula F in the class F , r(F) is a lower bound on the quantum  ...  We also show that for any Boolean function f represented by a formula in F , the deterministic query complexity of f is only quadratically larger than the quantum query complexity of f .  ...  Later we will show that for any read-once formula F in F , r(F) is a lower bound on the quantum query complexity of F and d(F) is an upper bound on the deterministic query complexity of F.  ... 
doi:10.1587/transinf.e93.d.280 fatcat:ruoeckb2nfaw5ghwvosteh6ftm

Reflections for quantum query algorithms [chapter]

Ben W. Reichardt
2011 Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms  
Our proof goes via the general adversary bound, a semidefinite program (SDP) that lower-bounds the quantum query complexity of a function.  ...  It efficiently characterizes the quantum query complexity of a read-once formula over any finite gate set. It further shows that span programs are equivalent to quantum query algorithms.  ...  Previous work has shown that the quantum query complexity of evaluating a read-once AND-OR formula on n inputs is O( √ n log n) [29, 6, 15] , and has characterized the query complexity of read-once formulas  ... 
doi:10.1137/1.9781611973082.44 dblp:conf/soda/Reichardt11a fatcat:qxlsmqsdofeirmtqquvbcp4pe4

Depth-Independent Lower Bounds on the Communication Complexity of Read-Once Boolean Formulas [chapter]

Rahul Jain, Hartmut Klauck, Shengyu Zhang
2010 Lecture Notes in Computer Science  
We show lower bounds of Ω( √ n) and Ω(n 1/4 ) on the randomized and quantum communication complexity, respectively, of all nvariable read-once Boolean formulas.  ...  Our results complement the recent lower bound of Ω(n/8 d ) by Leonardos and Saks [LS09] and Ω(n/2 O(d log d) ) by Jayram, Kopparty and Raghavendra [JKR09] for randomized communication complexity of read-once  ...  Acknowledgments: Research of Rahul Jain and Hartmut Klauck is supported by the internal grants of the Centre for Quantum Technologies (CQT), which is funded by the Singapore Ministry of Education and the  ... 
doi:10.1007/978-3-642-14031-0_8 fatcat:zgw77zfaargctpdjunxxc7tf5q

Depth-Independent Lower bounds on the Communication Complexity of Read-Once Boolean Formulas [article]

Rahul Jain, Hartmut Klauck, Shengyu Zhang
2009 arXiv   pre-print
We show lower bounds of Ω(√(n)) and Ω(n^1/4) on the randomized and quantum communication complexity, respectively, of all n-variable read-once Boolean formulas.  ...  Our results complement the recent lower bound of Ω(n/8^d) by Leonardos and Saks and Ω(n/2^Ω(d d)) by Jayram, Kopparty and Raghavendra for randomized communication complexity of read-once Boolean formulas  ...  Acknowledgments: Research of Rahul Jain and Hartmut Klauck is supported by the internal grants of the Centre for Quantum Technologies (CQT), which is funded by the Singapore Ministry of Education and the  ... 
arXiv:0908.4453v1 fatcat:mhv43mywpjdz7clfnrmnpbdi4e

The Quantum Query Complexity of AC0 [article]

Paul Beame, Widad Machmouchi
2012 arXiv   pre-print
These results yield a nearly linear Ω(n/ n) lower bound on the quantum query complexity of AC^0.  ...  The same lower bound holds for determining whether or not a function f from [2n-2] to [n] is surjective.  ...  Acknowledgements We thank Parikshit Gopalan for suggesting the problem of the approximate degree of AC 0 which motivated this work. We also thank Scott Aaronson and Dave Bacon for helpful pointers.  ... 
arXiv:1008.2422v2 fatcat:ef2zcpeqifa2nid4vitmtjecla

On the computational power of probabilistic and quantum branching program

Farid Ablayev, Aida Gainutdinova, Marek Karpinski, Cristopher Moore, Christopher Pollett
2005 Information and Computation  
More generally, we define syntactic models for quantum and stochastic branching programs of bounded width and prove upper and lower bounds on their power.  ...  For read-oncequantum branching programs (QBPs), we give a symmetric Boolean  ...  Acknowledgments We are grateful to Sasha Razborov for numerous suggestions on the results and presentations.  ... 
doi:10.1016/j.ic.2005.04.003 fatcat:mncsgr5c5rhn3h7yh7izipf6du

Tight adversary bounds for composite functions [article]

Peter Hoyer
2006 arXiv   pre-print
of read-once functions.  ...  The quantum adversary method is a versatile method for proving lower bounds on quantum algorithms.  ...  To demonstrate how Theorem 3 can be applied, we give a simple proof of the Ω( √ n) lower bound due to Barnum and Saks [BS04] on the bounded-error quantum query complexity of read-once functions.  ... 
arXiv:quant-ph/0509067v3 fatcat:a3zgceeasjfghkpqr2grrnpdqa

A Super-Grover Separation Between Randomized and Quantum Query Complexities [article]

Shalev Ben-David
2015 arXiv   pre-print
Assuming a conjecture of Aaronson and Ambainis about optimal quantum speedups for partial functions, we improve this to R(f)=Ω̃(Q(f)^3).  ...  We construct a total Boolean function f satisfying R(f)=Ω̃(Q(f)^5/2), refuting the long-standing conjecture that R(f)=O(Q(f)^2) for all total Boolean functions.  ...  Acknowledgements I would like to thank Scott Aaronson and Robin Kothari for checking an early draft of this result.  ... 
arXiv:1506.08106v1 fatcat:vhyaeypfkfgrzicgnhwenzjw6q

Lower Bounds on Quantum Query Complexity [article]

Peter Hoyer
2005 arXiv   pre-print
We cover the main known techniques for proving lower bounds, and exemplify and compare the methods.  ...  We discuss here what quantum computers_cannot_ do, and specifically how to prove limits on their computational power.  ...  Acknowledgments We thank Michal Koucký and Kolja Vereshchagin for discussions on the proof of the spectral adversary bound.  ... 
arXiv:quant-ph/0509153v1 fatcat:vxihlyiwabe4xdobrtk7mlu2qu

Reflections for quantum query algorithms

Ben W. Reichardt
2010 arXiv   pre-print
Our proof goes via the general adversary bound, a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function.  ...  We give a direct and simplified quantum algorithm based on the dual SDP, with a bounded-error query complexity that matches the general adversary bound.  ...  I also thank Sergio Boixo, Stephen Jordan, Julia Kempe and Rajat Mittal for helpful comments, and the Institute for Quantum Information for hospitality. Research supported by NSERC, ARO and MITACS.  ... 
arXiv:1005.1601v1 fatcat:ou7trlkrebedzket6tpjykj46i

On the Tightness of the Buhrman-Cleve-Wigderson Simulation [chapter]

Shengyu Zhang
2009 Lecture Notes in Computer Science  
This also implies that the classical and quantum communication complexities of certain blockcomposed functions are polynomially related.  ...  In this paper we show that the simulation is actually polynomially tight up to the choice of (g1, . . . , gn).  ...  The work is supported by the Hong Kong grant RGC-419309.  ... 
doi:10.1007/978-3-642-10631-6_45 fatcat:ofvkpjz5hnb4taimrj2262dzx4

On the Exponential Sample Complexity of the Quantum State Sign Estimation Problem [article]

Arthur G. Rattew, Marco Pistoia
2021 arXiv   pre-print
Thus the quantum sample complexity of sign estimation must be exponential: Ω(2^n/2/n).  ...  We demonstrate that the ability to estimate the relative sign of an arbitrary n-qubit quantum state (with real amplitudes), given only k copies of that state, would yield a kn-query algorithm for unstructured  ...  This paper is not a product of the Research Department of JPMorgan Chase & Co. or its affiliates.  ... 
arXiv:2108.03193v2 fatcat:rigveosvh5byvbqkbjog2raymy
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