Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem

Hallgren, Sean

doi:10.1145/509907.510001

Cited by 160 publications

(164 citation statements)

References 14 publications

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“…Quantum computers promise a substantial speed-up over classical ones for certain number-theoretic problems and the simulation of quantum systems [1][2][3]. Experimental efforts to build a quantum computer remain in their infancy though, limited to proof-of-principle experiments on a handful of qubits.…”

Section: Introductionmentioning

confidence: 99%

Trading Classical and Quantum Computational Resources

2016

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We propose examples of a hybrid quantum-classical simulation where a classical computer assisted by a small quantum processor can efficiently simulate a larger quantum system. First, we consider sparse quantum circuits such that each qubit participates in Oð1Þ two-qubit gates. It is shown that any sparse circuit on n þ k qubits can be simulated by sparse circuits on n qubits and a classical processing that takes time 2 OðkÞ polyðnÞ. Second, we study Pauli-based computation (PBC), where allowed operations are nondestructive eigenvalue measurements of n-qubit Pauli operators. The computation begins by initializing each qubit in the so-called magic state. This model is known to be equivalent to the universal quantum computer. We show that any PBC on n þ k qubits can be simulated by PBCs on n qubits and a classical processing that takes time 2OðkÞ polyðnÞ. Finally, we propose a purely classical algorithm that can simulate a PBC on n qubits in a time 2 αn polyðnÞ, where α ≈ 0.94. This improves upon the brute-force simulation method, which takes time 2 n polyðnÞ. Our algorithm exploits the fact that n-fold tensor products of magic states admit a low-rank decomposition into n-qubit stabilizer states.

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Section: Introductionmentioning

confidence: 99%

Trading Classical and Quantum Computational Resources

2016

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“…Since Shor's breakthrough discovery in 1994 [28], quantum algorithms with an exponential speedup over the best known classical algorithms have been shown for a number of problems (e.g., [10,30,15,22]). All these problems and algorithms share some common features: the problems are group or number theoretic in nature and the key component of each algorithm is the quantum Fourier transform 1 .…”

Section: Introductionmentioning

confidence: 99%

A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

2008

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The Jones polynomial, discovered in 1984 [18], is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten [32]) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang [13,14] provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, e 2πi/5 , and moreover, that this problem is BQP-complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results in [13,14] are heavily based on TQFT, which makes the algorithm essentially inaccessible to computer scientists.We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an n strands braid with m crossings at any primitive root of unity e 2πi/k , where the running time of the algorithm is polynomial in m, n and k. Our algorithm is based, rather than on TQFT, on well known mathematical results (specifically, the path model representation of the braid group and the uniqueness of the Markov trace for the Temperly Lieb algebra). By the results of [14], our algorithm solves a BQP complete problem.The algorithm we provide exhibits a structure which we hope is generalizable to other quantum algorithmic problems. Candidates of particular interest are the approximations of other downwards selfreducible #P-hard problems, most notably, the important open problem of efficient approximation of the partition function of the Potts model, a model which is known to be tightly connected to the Jones polynomial [33].

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“…Between other works finding modal value of data, generating random number, finding the solution of Pell's equation, pattern matching, different type of satisfiability (SAT) problems etc are noteworthy [16]. …”

Section: Sorting By Quantum Algorithms: Time-space Trade Offmentioning

confidence: 99%

Basic Quantum Algorithms and Applications

Rahmi¹,

Shamanta²,

Tasnim³

2012

IJCA

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Quantum computation, the ultimate goal of future computing, is an interesting field for researchers. The concept of quantum computation is based on basics of quantum mechanics. A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena such as superposition and entanglement, to perform operations on data. The basic principle behind quantum computation is that quantum properties can be used to represent data and perform operations on these data. A quantum computer operates by manipulating the qubits with a fixed sequence of quantum logic gates. The sequence of gates to be applied is called a quantum algorithm. The field of quantum computation algorithm is fast moving and the scope is vast. Major quantum algorithms are summarized in this paper along with their applications. General TermsQuantum Algorithms.

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Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem

Cited by 160 publications

References 14 publications

Trading Classical and Quantum Computational Resources

Trading Classical and Quantum Computational Resources

A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

Basic Quantum Algorithms and Applications

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