Approximate Span Programs
Span programs are a model of computation that have been used to design quantum algorithms, mainly in the query model. It is known that for any decision problem, there exists a span program that leads to an algorithm with optimal quantum query complexity, however finding such an algorithm is generall...
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| Vydáno v: | Algorithmica Ročník 81; číslo 6; s. 2158 - 2195 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.06.2019
Springer Nature B.V |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | Span programs are a model of computation that have been used to design quantum algorithms, mainly in the query model. It is known that for any decision problem, there exists a span program that leads to an algorithm with optimal quantum query complexity, however finding such an algorithm is generally challenging. We consider new ways of designing quantum algorithms using span programs. We show how any span program that decides a function
f
can also be used to decide “threshold” versions of the function
f
, or more generally, approximate a quantity called the
span program witness size
, which is some property of the input related to
f
. We achieve these results by relaxing the requirement that 1-inputs hit some
target
exactly in the span program, which could potentially make design of span programs significantly easier. In addition, we give an exposition of span program structure, which increases the general understanding of this important model. One implication of this is alternative algorithms for estimating the witness size when the phase gap of a certain unitary can be lower bounded. We show how to lower bound this phase gap in certain cases. As an application, we give the first upper bounds in the adjacency query model on the quantum time complexity of estimating the effective resistance between
s
and
t
,
R
s
,
t
(
G
)
. For this problem we obtain
O
~
(
1
ε
3
/
2
n
R
s
,
t
(
G
)
)
, using
O
(
log
n
)
space. In addition, when
μ
is a lower bound on
λ
2
(
G
)
, by our phase gap lower bound, we can obtain an upper bound of
O
~
1
ε
n
R
s
,
t
(
G
)
/
μ
for estimating effective resistance, also using
O
(
log
n
)
space. |
|---|---|
| AbstractList | Span programs are a model of computation that have been used to design quantum algorithms, mainly in the query model. It is known that for any decision problem, there exists a span program that leads to an algorithm with optimal quantum query complexity, however finding such an algorithm is generally challenging. We consider new ways of designing quantum algorithms using span programs. We show how any span program that decides a function
f
can also be used to decide “threshold” versions of the function
f
, or more generally, approximate a quantity called the
span program witness size
, which is some property of the input related to
f
. We achieve these results by relaxing the requirement that 1-inputs hit some
target
exactly in the span program, which could potentially make design of span programs significantly easier. In addition, we give an exposition of span program structure, which increases the general understanding of this important model. One implication of this is alternative algorithms for estimating the witness size when the phase gap of a certain unitary can be lower bounded. We show how to lower bound this phase gap in certain cases. As an application, we give the first upper bounds in the adjacency query model on the quantum time complexity of estimating the effective resistance between
s
and
t
,
R
s
,
t
(
G
)
. For this problem we obtain
O
~
(
1
ε
3
/
2
n
R
s
,
t
(
G
)
)
, using
O
(
log
n
)
space. In addition, when
μ
is a lower bound on
λ
2
(
G
)
, by our phase gap lower bound, we can obtain an upper bound of
O
~
1
ε
n
R
s
,
t
(
G
)
/
μ
for estimating effective resistance, also using
O
(
log
n
)
space. Span programs are a model of computation that have been used to design quantum algorithms, mainly in the query model. It is known that for any decision problem, there exists a span program that leads to an algorithm with optimal quantum query complexity, however finding such an algorithm is generally challenging. We consider new ways of designing quantum algorithms using span programs. We show how any span program that decides a function f can also be used to decide “threshold” versions of the function f, or more generally, approximate a quantity called the span program witness size, which is some property of the input related to f. We achieve these results by relaxing the requirement that 1-inputs hit some target exactly in the span program, which could potentially make design of span programs significantly easier. In addition, we give an exposition of span program structure, which increases the general understanding of this important model. One implication of this is alternative algorithms for estimating the witness size when the phase gap of a certain unitary can be lower bounded. We show how to lower bound this phase gap in certain cases. As an application, we give the first upper bounds in the adjacency query model on the quantum time complexity of estimating the effective resistance between s and t, Rs,t(G). For this problem we obtain O~(1ε3/2nRs,t(G)), using O(logn) space. In addition, when μ is a lower bound on λ2(G), by our phase gap lower bound, we can obtain an upper bound of O~1εnRs,t(G)/μ for estimating effective resistance, also using O(logn) space. |
| Author | Jeffery, Stacey Ito, Tsuyoshi |
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| References | CR4 CR3 CR6 Harrow, Hassidim, Lloyd (CR12) 2009; 103 CR5 Cleve, Ekert, Macchiavello, Mosca (CR8) 1998; 454 CR19 CR7 CR18 Chandra, Raghavan, Ruzzo, Smolensky, Tiwari (CR9) 1996; 6 CR16 CR15 Beals, Buhrman, Cleve, Mosca, de Wolf (CR2) 2001; 48 CR14 CR13 Reichardt, Špalek (CR21) 2012; 8 CR23 CR11 CR22 CR20 Levin, Peres, Wilmer (CR17) 2009 Doyle, Snell (CR10) 1984 Bennett, Bernstein, Brassard, Vazirani (CR1) 1997; 26 AK Chandra (527_CR9) 1996; 6 527_CR7 527_CR6 527_CR5 527_CR4 527_CR3 527_CR18 527_CR15 527_CR16 DA Levin (527_CR17) 2009 R Cleve (527_CR8) 1998; 454 527_CR19 PG Doyle (527_CR10) 1984 B Reichardt (527_CR21) 2012; 8 527_CR20 AW Harrow (527_CR12) 2009; 103 527_CR13 527_CR14 CH Bennett (527_CR1) 1997; 26 R Beals (527_CR2) 2001; 48 527_CR11 527_CR22 527_CR23 |
| References_xml | – year: 2009 ident: CR17 publication-title: Markov Chains and Mixing Times – ident: CR22 – volume: 6 start-page: 312 issue: 4 year: 1996 end-page: 340 ident: CR9 article-title: The electrical resistance of a graph captures its commute and cover times publication-title: Comput. Complex. doi: 10.1007/BF01270385 – ident: CR19 – ident: CR18 – volume: 48 start-page: 778 year: 2001 end-page: 797 ident: CR2 article-title: Quantum lower bounds by polynomials publication-title: J. ACM doi: 10.1145/502090.502097 – ident: CR3 – ident: CR4 – ident: CR14 – ident: CR15 – volume: 26 start-page: 1510 year: 1997 end-page: 1523 ident: CR1 article-title: Strengths and weaknesses of quantum computing publication-title: SIAM J. Comput. (special issue on quantum computing) – ident: CR16 – ident: CR13 – ident: CR11 – volume: 103 start-page: 150502 year: 2009 ident: CR12 article-title: Quantum algorithm for linear systems of equations publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.103.150502 – volume: 454 start-page: 339 issue: 1969 year: 1998 end-page: 354 ident: CR8 article-title: Quantum algorithms revisited publication-title: Proc. Royal Soc. A Math. Phys. Eng. Sci. doi: 10.1098/rspa.1998.0164 – year: 1984 ident: CR10 publication-title: Random Walks and Electrical Networks, volume 22 of the Carus Mathematical Monographs – ident: CR6 – ident: CR5 – ident: CR7 – volume: 8 start-page: 291 issue: 13 year: 2012 end-page: 319 ident: CR21 article-title: Span-program-based quantum algorithm for evaluating formulas publication-title: Theory Comput. doi: 10.4086/toc.2012.v008a013 – ident: CR23 – ident: CR20 – volume: 8 start-page: 291 issue: 13 year: 2012 ident: 527_CR21 publication-title: Theory Comput. doi: 10.4086/toc.2012.v008a013 – volume: 26 start-page: 1510 year: 1997 ident: 527_CR1 publication-title: SIAM J. Comput. (special issue on quantum computing) – ident: 527_CR4 doi: 10.1109/FOCS.2012.18 – ident: 527_CR16 doi: 10.1109/FOCS.2011.75 – ident: 527_CR11 doi: 10.1145/237814.237866 – ident: 527_CR18 – ident: 527_CR5 doi: 10.1145/2213977.2213985 – ident: 527_CR6 – volume: 6 start-page: 312 issue: 4 year: 1996 ident: 527_CR9 publication-title: Comput. Complex. doi: 10.1007/BF01270385 – ident: 527_CR15 – volume-title: Random Walks and Electrical Networks, volume 22 of the Carus Mathematical Monographs year: 1984 ident: 527_CR10 doi: 10.5948/UPO9781614440222 – ident: 527_CR14 – ident: 527_CR13 – ident: 527_CR19 doi: 10.1109/FOCS.2009.55 – ident: 527_CR7 doi: 10.1007/978-3-642-33090-2_18 – ident: 527_CR20 doi: 10.1137/1.9781611973082.44 – volume: 48 start-page: 778 year: 2001 ident: 527_CR2 publication-title: J. ACM doi: 10.1145/502090.502097 – ident: 527_CR3 doi: 10.1007/978-3-642-39206-1_10 – volume-title: Markov Chains and Mixing Times year: 2009 ident: 527_CR17 – volume: 103 start-page: 150502 year: 2009 ident: 527_CR12 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.103.150502 – ident: 527_CR22 – volume: 454 start-page: 339 issue: 1969 year: 1998 ident: 527_CR8 publication-title: Proc. Royal Soc. A Math. Phys. Eng. Sci. doi: 10.1098/rspa.1998.0164 – ident: 527_CR23 |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Estimation Lower bounds Mathematics of Computing Queries Theory of Computation Upper bounds |
| Title | Approximate Span Programs |
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