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Childs, Andrew M.; Cleve, Richard; Jordan, Stephen P.; Yonge-Mallo, David L.

doi:10.4086/toc.2009.v005a005

Cited by 34 publications

(20 citation statements)

References 12 publications

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“…It can also be used to implement other quantum algorithms [3,19,20]. We focus on the case of time-independent Hamiltonians for simplicity, but we also outline a straightforward application of our results to time-dependent Hamiltonians.…”

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confidence: 99%

Simulating Hamiltonian Dynamics with a Truncated Taylor Series

et al. 2015

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We describe a simple, efficient method for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator. Our method can simulate the time evolution of a wide variety of physical systems. As in another recent algorithm, the cost of our method depends only logarithmically on the inverse of the desired precision, which is optimal. However, we simplify the algorithm and its analysis by using a method for implementing linear combinations of unitary operations together with a robust form of oblivious amplitude amplification. DOI: 10.1103/PhysRevLett.114.090502 PACS numbers: 03.67.Ac, 89.70.Eg One of the main motivations for quantum computers is their ability to efficiently simulate the dynamics of quantum systems [1], a problem that is apparently hard for classical computers. Since the mid-1990s, many algorithms have been developed to simulate Hamiltonian dynamics on a quantum computer [2][3][4][5][6][7][8][9][10][11][12], with applications to problems such as simulating spin models [13] and quantum chemistry [14][15][16][17]. While it is now well known that quantum computers can efficiently simulate Hamiltonian dynamics, ongoing work has improved the performance and expanded the scope of such simulations.Recently, we introduced a new approach to Hamiltonian simulation with exponentially improved performance as a function of the desired precision [18]. Specifically, we presented a method to simulate a d-sparse, n-qubit Hamiltonian H acting for time t > 0, within precision ϵ > 0, using O(τ logðτ=ϵÞ= log logðτ=ϵÞ) queries to H and O(nτlog 2 ðτ=ϵÞ= log logðτ=ϵÞ) additional two-qubit gates, where τ ≔ d 2 ∥H∥ max t. This dependence on ϵ is exponentially better than all previous approaches to Hamiltonian simulation, and the number of queries to H is optimal [18]. (For simplicity, we refer to combinations of logarithms like those in the above expressions as logarithmic.) Roughly speaking, doubling the number of digits of accuracy only doubles the complexity.The simulation algorithm of [18] is indirect, appealing to an unconventional model of query complexity. In this Letter, we describe and analyze a simplified approach to Hamiltonian simulation with the same cost as the method of [18]. The new approach is easier to understand, and the reason for the logarithmic cost dependence on ϵ is immediate. The new approach decomposes the Hamiltonian into a linear combination of unitary operations. Unlike the algorithm of [18], these terms need not be self-inverse, so the algorithm is efficient for a larger class of Hamiltonians. The new approach is also simpler to analyze: we give a selfcontained presentation in four pages.The main idea of the new approach is to implement the truncated Taylor series of the evolution operator. Similar to previous approaches for implementing linear combinations of unitary operators [12,13], the various terms of the Taylor series can be implemented by introducing an ancillary superposition and performing controlled operations. The time e...

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Simulating Hamiltonian Dynamics with a Truncated Taylor Series

et al. 2015

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“…(The NAND TREE is equivalent to solving NAND d , although the composition we will use for Theorem 2.1 is not the composition of the NAND function, but of the NAND TREE as a whole.) For arbitrary inputs, Farhi et al showed that there exists an optimal quantum algorithm in the Hamiltonian model to solve the NAND TREE in O(2 0.5d ) time [5], and this was extended to a standard discrete algorithm with quantum query complexity O(2 0.5d ) [4,10]. Classically, the best algorithm requires Ω(2 0.753d ) queries [12].…”

Section: A Nonconstructive Upper Bound On Query Complexitymentioning

confidence: 99%

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Kimmel¹

2013

Chicago J. of Theoretical Comp. Sci.

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We describe a method to upper bound the quantum query complexity of Boolean formula evaluation problems, using fundamental theorems about the general adversary bound. This nonconstructive method can give an upper bound on query complexity without producing an algorithm. For example, we describe an oracle problem which we prove (nonconstructively) can be solved in O(1) queries, where the previous best quantum algorithm uses a polylogarithmic number of queries. We then give an explicit O(1)-query algorithm for this problem based on span programs.

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“…For the upper bound, in one extreme case, Grover search [20,21] evaluates an OR gate of degree N using O( √ N) oracle queries. In the other extreme case, Farhi, Goldstone and Gutmann devised a breakthrough algorithm for evaluating the depth-(log 2 N) balanced binary AND-OR formula using N 1/2+o (1) queries [17,15]. Ambainis et al [4] improved this to O( √ N) queries, even for "approximately balanced" formulas, and also extended the algorithm to arbitrary {AND, OR, NOT} formulas with N 1 2 +o(1) queries and time after a preprocessing step.…”

Section: Introductionmentioning

confidence: 99%