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Oracle Separation of BQP and PH

Ran Raz, Avishay Tal
2022 Journal of the ACM  
By well-known reductions, this gives a separation of the classes Promise- BQP and Promise- PH in the black-box model and implies an oracle relative to which BQP is not contained in PH .  ...  distinguishes between \({\mathcal {D}} \) and the uniform distribution with advantage \(\Omega(1/\log N)\) . (2) No Boolean circuit of quasi-polynomial size and constant depth distinguishes between \({  ...  ACKNOWLEDGMENTS We would like to thank Scott Aaronson, Shalev Ben-David, Mika Göös, Johan Håstad, Pooya Hatami and Toni Pitassi for very helpful discussions  ... 
doi:10.1145/3530258 fatcat:s5e7xuaplrh5zj4p3fxwsholgu

A stochastic calculus approach to the oracle separation of BQP and PH [article]

Xinyu Wu
2020 arXiv   pre-print
After presentations of Raz and Tal's oracle separation of BQP and PH result, several people (e.g.  ...  Application to the oracle separation of BQP and PH We now use Theorem 2 to construct D as described in Section 1. The distribution D.  ...  Introduction A recent landmark result of Raz and Tal [RT19] shows there exists an oracle A such that BQP A Ę PH A .  ... 
arXiv:2007.02431v1 fatcat:voleiprdpzgufnclp3xpfmm4hi

BQP and the polynomial hierarchy

Scott Aaronson
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
Second, we show that an oracle decision problem separating BQP from PH would follow from the Generalized Linial-Nisan Conjecture, which we formulate here and which is likely of independent interest.  ...  The relationship between BQP and PH has been an open problem since the earliest days of quantum computing.  ...  First, we achieve an oracle separation between BQP and PH for the case of relation problems.  ... 
doi:10.1145/1806689.1806711 dblp:conf/stoc/Aaronson10 fatcat:zkhdz4kjlrf6paqiooe5me4e6u

BQP and the Polynomial Hierarchy [article]

Scott Aaronson
2009 arXiv   pre-print
Second, we show that an oracle decision problem separating BQP from PH would follow from the Generalized Linial-Nisan Conjecture, which we formulate here and which is likely of independent interest.  ...  The relationship between BQP and PH has been an open problem since the earliest days of quantum computing.  ...  First, we achieve an oracle separation between BQP and PH for the case of relation problems.  ... 
arXiv:0910.4698v1 fatcat:vunu2qquqje2jmor6o4f4uoeju

Quantum leap

Don Monroe
2018 Communications of the ACM  
Quantum Resources Conventional "classical" computers store information as bits than can be in one of two states, denoted 0 and 1.  ...  In contrast, a quantum degree of freedom, such as the spin of an electron or the polarization of a photon, can exist  ...  Raz, R. and Tal, A. Oracle Separation of BQP and PH, Electronic Colloquium on Computational Complexity, Report No. 107 (2018) .  ... 
doi:10.1145/3290407 fatcat:2iaczspknndv7efdrh2mhb7srq

Open Problems Related to Quantum Query Complexity [article]

Scott Aaronson
2021 arXiv   pre-print
I offer a case that quantum query complexity still has loads of enticing and fundamental open problems -- from relativized QMA versus QCMA and BQP versus IP, to time/space tradeoffs for collision and element  ...  distinctness, to polynomial degree versus quantum query complexity for partial functions, to the Unitary Synthesis Problem and more.  ...  oracle separation between BQP and the polynomial hierarchy [43] .  ... 
arXiv:2109.06917v1 fatcat:tz4vfvpcfjb2ljcp2kgfhd3tj4

PH = PSPACE [article]

Valerii Sopin
2022 arXiv   pre-print
The idea is to skolemize, and then use additional formulas from the second level of the polynomial hierarchy inside the skolemized prefix to enforce that the skolem variables indeed depend only on the  ...  in the size of a given ϕ, such that the truth of ϕ can be determined from the truth of ϕ'.  ...  However, note that an oracle separation of BQP and PH does not necessarily imply the ordinary separation. There is no contradiction.  ... 
arXiv:1411.0628v17 fatcat:brzujfuwkffeba5znyb6d6hhia

THE DEUTSCH–JOZSA ALGORITHM REVISITED IN THE DOMAIN OF CRYPTOGRAPHICALLY SIGNIFICANT BOOLEAN FUNCTIONS

SUBHAMOY MAITRA, PARTHA MUKHOPADHYAY
2005 International Journal of Quantum Information  
Let f be an n-variable Boolean function and its Walsh spectra is denoted by W f (ω) at the point ω ∈ {0, 1} n . The Boolean function is available in the form of an oracle.  ...  We also analyse certain classes of cryptographically significant Boolean functions and highlight how the basic Deutsch-Jozsa algorithm performs on them.  ...  Bernstein and Vazirani [3] have further shown the relativized separation of NP and even MA from BQP and conjectured that recursive Fourier sampling is not in PH (related discussion is also available  ... 
doi:10.1142/s0219749905000980 fatcat:3auz6xj77rgb3fi32u75q4jywm

Deutsch-Jozsa Algorithm Revisited in the Domain of Cryptographically Significant Boolean Functions [article]

Subhamoy Maitra, Partha Mukhopadhyay
2004 arXiv   pre-print
Let f be an n-variable Boolean function and its Walsh spectra is denoted by W_f(ω) at the point ω∈{0, 1}^n. The Boolean function is available in the form of an oracle.  ...  We also analyse certain classes of cryptographically significant Boolean functions and highlight how the basic Deutsch-Jozsa algorithm performs on them.  ...  Bernstein and Vazirani [3] have further shown the relativized separation of NP and even MA from BQP and conjectured that recursive Fourier sampling is not in PH (related discussion is also available  ... 
arXiv:quant-ph/0410042v1 fatcat:gocmcovblrexxd56gg5oi3vaby

Complexity-Theoretic Limitations on Blind Delegated Quantum Computation

Scott Aaronson, Alexandru Cojocaru, Alexandru Gheorghiu, Elham Kashefi, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
We conjecture that this containment is unlikely by proving that there exists an oracle relative to which BQP ⊂ MA/O(n d ).  ...  We first show that having such a protocol in which the client and the server exchange O(n d ) bits of communication, implies that BQP ⊂ MA/O(n d ).  ...  It would be interesting to see whether the oracle that Raz and Tal [51] recently used to prove a separation between BQP and PH could also be used in order to separate BQP from NP/poly, or even from PH  ... 
doi:10.4230/lipics.icalp.2019.6 dblp:conf/icalp/AaronsonCGK19 fatcat:67fy7qzn7zc35poryn24d4aafm

Quantum versus Classical Proofs and Advice

Scott Aaronson, Greg Kuperberg
2007 Proceedings - IEEE Conference on Computational Complexity  
We end with some conjectures about quantum versus classical oracles, and about the possibility of a classical oracle separation between QMA and QCMA.  ...  First, we give a "quantum oracle separation" between QMA and QCMA.  ...  Acknowledgments We thank the anonymous reviewers for their suggestions, and Dorit Aharonov, Laci Babai, Robert Beals, Robert Guralnick, Bill Kantor, and Cris Moore for helpful correspondence.  ... 
doi:10.1109/ccc.2007.27 dblp:conf/coco/AaronsonK07 fatcat:5spz4nrbxbekdnlhlcjbg5aqoq

A Counterexample to the Generalized Linial-Nisan Conjecture [article]

Scott Aaronson
2011 arXiv   pre-print
In earlier work, we gave an oracle separating the relational versions of BQP and the polynomial hierarchy, and showed that an oracle separating the decision versions would follow from what we called the  ...  It has been conjectured since the 1980s that PH is infinite relative to a random oracle, but the highest levels of PH previously proved separate were NP and coNP.  ...  Acknowledgments I thank Paul Beame for sharing a draft of the manuscript [3] with me, and Lane Hemaspaandra for pointing me to the paper [8] of Cai.  ... 
arXiv:1110.6126v1 fatcat:xnn7yjvzxzd5hm57ktrdrzrntq

Complexity classes of equivalence problems revisited

Lance Fortnow, Joshua A. Grochow
2011 Information and Computation  
To determine if two lists of numbers are the same set, we sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality.  ...  Other canonical forms arise in graph isomorphism algorithms, and the equality of permutation groups given by generators.  ...  In particular, one of the reviewers suggested that we define some sort of hybrid notion of Cohen and transitive genericity, as well as suggested the notion of UP-transitive genericity.  ... 
doi:10.1016/j.ic.2011.01.006 fatcat:6j7dn6pkg5bypdzeuoym6webk4

Interactive proofs with efficient quantum prover for recursive Fourier sampling [article]

Matthew McKague
2011 arXiv   pre-print
We consider the recursive Fourier sampling problem (RFS), and show that there exists an interactive proof for RFS with an efficient classical verifier and efficient quantum prover.  ...  This gives an oracle relative to which BQP is not in MA, and hope that a relativized separation may be found between BQP and PH.  ...  BQP separation.  ... 
arXiv:1012.5699v2 fatcat:6ndgjadtivbihmspv5zrj4dbly

Quantum Versus Classical Proofs and Advice [article]

Scott Aaronson, Greg Kuperberg
2020 arXiv   pre-print
We end with some conjectures about quantum versus classical oracles, and about the possibility of a classical oracle separation between QMA and QCMA.  ...  First, we give a "quantum oracle separation" between QMA and QCMA.  ...  Acknowledgments We thank the anonymous reviewers for their suggestions, and Dorit Aharonov, Laci Babai, Robert Beals, Robert Guralnick, Bill Kantor, and Cris Moore for helpful correspondence.  ... 
arXiv:quant-ph/0604056v4 fatcat:u6wyswlsibbpxcodwy7s52fuki
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