Quantum computation in algebraic number theory: Hallgren’s efficient quantum algorithm for solving Pell’s equation
Pell’s equation is x 2− dy 2=1, where d is a square-free integer and we seek positive integer solutions x, y>0. Let ( x 0, y 0) be the smallest solution (i.e., having smallest A=x 0+y 0 d ). Lagrange showed that every solution can easily be constructed from A so given d it suffices to compute A....
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| Veröffentlicht in: | Annals of physics Jg. 306; H. 2; S. 241 - 279 |
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| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
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01.08.2003
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| ISSN: | 0003-4916, 1096-035X |
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| Abstract | Pell’s equation is
x
2−
dy
2=1, where
d is a square-free integer and we seek positive integer solutions
x,
y>0. Let (
x
0,
y
0) be the smallest solution (i.e., having smallest
A=x
0+y
0
d
). Lagrange showed that every solution can easily be constructed from
A so given
d it suffices to compute
A. It is known that
A can be exponentially large in
d so just to write down
A we need exponential time in the input size
log
d
. Hence we introduce the regulator
R=ln
A and ask for the value of
R to n decimal places. The best known classical algorithm has sub-exponential running time
O(
exp
log
d
,
poly(n))
. Hallgren’s quantum algorithm gives the result in polynomial time
O(
poly(
log
d),
poly(n))
with probability
1/
poly(
log
d)
. The idea of the algorithm falls into two parts: using the formalism of algebraic number theory we convert the problem of solving Pell’s equation into the problem of determining
R as the period of a function on the real numbers. Then we generalise the quantum Fourier transform period finding algorithm to work in this situation of an irrational period on the (not finitely generated) abelian group of real numbers. This paper is intended to be accessible to a reader having no prior acquaintance with algebraic number theory; we give a self-contained account of all the necessary concepts and we give elementary proofs of all the results needed. Then we go on to describe Hallgren’s generalisation of the quantum period finding algorithm, which provides the efficient computational solution of Pell’s equation in the above sense. |
|---|---|
| AbstractList | Pell’s equation is
x
2−
dy
2=1, where
d is a square-free integer and we seek positive integer solutions
x,
y>0. Let (
x
0,
y
0) be the smallest solution (i.e., having smallest
A=x
0+y
0
d
). Lagrange showed that every solution can easily be constructed from
A so given
d it suffices to compute
A. It is known that
A can be exponentially large in
d so just to write down
A we need exponential time in the input size
log
d
. Hence we introduce the regulator
R=ln
A and ask for the value of
R to n decimal places. The best known classical algorithm has sub-exponential running time
O(
exp
log
d
,
poly(n))
. Hallgren’s quantum algorithm gives the result in polynomial time
O(
poly(
log
d),
poly(n))
with probability
1/
poly(
log
d)
. The idea of the algorithm falls into two parts: using the formalism of algebraic number theory we convert the problem of solving Pell’s equation into the problem of determining
R as the period of a function on the real numbers. Then we generalise the quantum Fourier transform period finding algorithm to work in this situation of an irrational period on the (not finitely generated) abelian group of real numbers. This paper is intended to be accessible to a reader having no prior acquaintance with algebraic number theory; we give a self-contained account of all the necessary concepts and we give elementary proofs of all the results needed. Then we go on to describe Hallgren’s generalisation of the quantum period finding algorithm, which provides the efficient computational solution of Pell’s equation in the above sense. |
| Author | Jozsa, Richard |
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| Keywords | Quantum Fourier transform Quantum algorithms Pell’s equation |
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| References | Ireland, Rosen (BIB9) 1990; vol. 84 Jozsa (BIB8) 2001; March/April P. Shor, Polynomial time algorithms for prime factorisation and discrete logarithms on a quantum computer, in: Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, IEEE Computer Society, Los Alamitos, p. 124 (Extended Abstract). Full version of this paper appears in SIAM J. Comput. 26 (1997) 1484–1510 Buchmann, Thiel, Williams (BIB3) 1995; vol. 325 Lenstra (BIB4) 1982; vol. 56 Lenstra (BIB5) 2002; 49 Buchmann, Williams (BIB13) 1989; vol. 435 Cohen (BIB2) 1993; vol. 138 Hardy, Wright (BIB12) 1965 S. Hallgren, Polynomial time quantum algorithms or Pell’s equation and the principal ideal problem, in: Symposium on the theory of computation STOC, May 2002 Williams (BIB7) 2002 D. Shanks, The infrastructure of real quadratic fields and its applications, in: Proc. 1972 Number Theory Conference, Boulder, Colorado, 1972, pp. 217–224 BIB10 10.1016/S0003-4916(03)00067-8_BIB6 Jozsa (10.1016/S0003-4916(03)00067-8_BIB8) 2001; March/April Lenstra (10.1016/S0003-4916(03)00067-8_BIB5) 2002; 49 10.1016/S0003-4916(03)00067-8_BIB11 Hardy (10.1016/S0003-4916(03)00067-8_BIB12) 1965 Lenstra (10.1016/S0003-4916(03)00067-8_BIB4) 1982; vol. 56 Williams (10.1016/S0003-4916(03)00067-8_BIB7) 2002 Buchmann (10.1016/S0003-4916(03)00067-8_BIB13) 1989; vol. 435 10.1016/S0003-4916(03)00067-8_BIB1 Cohen (10.1016/S0003-4916(03)00067-8_BIB2) 1993; vol. 138 Buchmann (10.1016/S0003-4916(03)00067-8_BIB3) 1995; vol. 325 Ireland (10.1016/S0003-4916(03)00067-8_BIB9) 1990; vol. 84 |
| References_xml | – volume: vol. 325 start-page: 159 year: 1995 end-page: 185 ident: BIB3 article-title: Short representation of quadratic integers publication-title: Computational Algebra and Number Theory – reference: D. Shanks, The infrastructure of real quadratic fields and its applications, in: Proc. 1972 Number Theory Conference, Boulder, Colorado, 1972, pp. 217–224 – year: 1965 ident: BIB12 article-title: An Introduction to the Theory of Numbers – year: 2002 ident: BIB7 article-title: Solving the Pell equation publication-title: Proceedings of the Millennial Conference on Number Theory – reference: P. Shor, Polynomial time algorithms for prime factorisation and discrete logarithms on a quantum computer, in: Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, IEEE Computer Society, Los Alamitos, p. 124 (Extended Abstract). Full version of this paper appears in SIAM J. Comput. 26 (1997) 1484–1510 – volume: vol. 138 year: 1993 ident: BIB2 publication-title: A course in computational algebraic number theory – volume: vol. 84 year: 1990 ident: BIB9 article-title: A classical introduction to modern number theory publication-title: Graduate Texts in Mathematics – ident: BIB10 – volume: vol. 56 start-page: 123 year: 1982 end-page: 150 ident: BIB4 article-title: On the computation of regulators and class numbers in quadratic number fields publication-title: London Mathematical Society Lecture Note Series – reference: S. Hallgren, Polynomial time quantum algorithms or Pell’s equation and the principal ideal problem, in: Symposium on the theory of computation STOC, May 2002 – volume: 49 start-page: 182 year: 2002 end-page: 192 ident: BIB5 article-title: Solving Pell’s equation publication-title: Notices Am. Math. Soc. – volume: March/April start-page: 34 year: 2001 end-page: 43 ident: BIB8 article-title: Quantum factoring, discrete logarithms and the hidden subgroup problem publication-title: Comp. Sci. Eng. – volume: vol. 435 start-page: 335 year: 1989 end-page: 343 ident: BIB13 article-title: A key exchange system based on real quadratic fields publication-title: Advances in Cryptology—CRYPTO’89 – ident: 10.1016/S0003-4916(03)00067-8_BIB11 – volume: vol. 84 year: 1990 ident: 10.1016/S0003-4916(03)00067-8_BIB9 article-title: A classical introduction to modern number theory – volume: vol. 138 year: 1993 ident: 10.1016/S0003-4916(03)00067-8_BIB2 – volume: 49 start-page: 182 issue: 2 year: 2002 ident: 10.1016/S0003-4916(03)00067-8_BIB5 article-title: Solving Pell’s equation publication-title: Notices Am. Math. Soc. – ident: 10.1016/S0003-4916(03)00067-8_BIB1 doi: 10.1145/509907.510001 – volume: vol. 435 start-page: 335 year: 1989 ident: 10.1016/S0003-4916(03)00067-8_BIB13 article-title: A key exchange system based on real quadratic fields – year: 1965 ident: 10.1016/S0003-4916(03)00067-8_BIB12 – volume: vol. 56 start-page: 123 year: 1982 ident: 10.1016/S0003-4916(03)00067-8_BIB4 article-title: On the computation of regulators and class numbers in quadratic number fields – ident: 10.1016/S0003-4916(03)00067-8_BIB6 – volume: vol. 325 start-page: 159 year: 1995 ident: 10.1016/S0003-4916(03)00067-8_BIB3 article-title: Short representation of quadratic integers – year: 2002 ident: 10.1016/S0003-4916(03)00067-8_BIB7 article-title: Solving the Pell equation – volume: March/April start-page: 34 year: 2001 ident: 10.1016/S0003-4916(03)00067-8_BIB8 article-title: Quantum factoring, discrete logarithms and the hidden subgroup problem publication-title: Comp. Sci. Eng. doi: 10.1109/5992.909000 |
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| Snippet | Pell’s equation is
x
2−
dy
2=1, where
d is a square-free integer and we seek positive integer solutions
x,
y>0. Let (
x
0,
y
0) be the smallest solution (i.e.,... |
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| StartPage | 241 |
| SubjectTerms | Pell’s equation Quantum algorithms Quantum Fourier transform |
| Title | Quantum computation in algebraic number theory: Hallgren’s efficient quantum algorithm for solving Pell’s equation |
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