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Drucker, Andrew; Wolf, Ronald de

doi:10.4086/toc.gs.2011.002

Cited by 15 publications

(9 citation statements)

References 124 publications

(92 reference statements)

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“…Aaronson's theorem, post-BQP=PP [30], is quite useful to obtain not only quantum complexity results combined with the postselection argument by Bremner et al [38], but also to provide 'quantum proofs' of classical complexity results [39]. For example, in [30], Aaronson provided alternative and much simpler proof (corresponding to single-qubit rotations).…”

Section: Strong Simulation and Post-bqp=pp Theoremmentioning

confidence: 99%

“…This can be viewed as a 'quantum proof' of #P-hardness of approximating the imaginary Ising partition functions. Aaronson's post-BQP=PP theorem [30], which is employed to show the above result, is also utilized to provide a 'quantum proof' [39] of #P-hardness of approximating the permanent [17] and the Jones polynomial [40] with a multiplicative error.…”

Section: Introductionmentioning

confidence: 99%

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Commuting quantum circuits and complexity of Ising partition functions

Fujii

Morimae

2017

New J. Phys.

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Instantaneous quantum polynomial-time (IQP) computation is a class of quantum computation consisting only of commuting two-qubit gates and is not universal. Nevertheless, it has been shown that if there is a classical algorithm that can simulate IQP efficiently, the polynomial hierarchy collapses to the third level, which is highly implausible. However, the origin of the classical intractability is still less understood. Here we establish a relationship between IQP and computational complexity of calculating the imaginary-valued partition functions of Ising models. We apply the established relationship in two opposite directions. One direction is to find subclasses of IQP that are classically efficiently simulatable by using exact solvability of certain types of Ising models. Another direction is applying quantum computational complexity of IQP to investigate (im)possibility of efficient classical approximations of Ising partition functions with imaginary coupling constants. Specifically, we show that a multiplicative approximation of Ising partition functions is #P-hard for almost all imaginary coupling constants even on planar lattices of a bounded degree.Aaronson and Arkhipov introduced BOSONSAMPLING [15], a sampling problem according to the probability distribution of n bosons scattered by linear optical unitary operations. The probability distribution is given by the permanent of a complex matrix, which is determined by the linear optical unitary operations. Calculation of the permanent of complex matrices is known to be #P-hard [16,17]. Since a polynomial-time machine with an oracle for #P can solve all problems in the PH according to Toda's theorem [18], an exact classical simulation (in the strong sense [19,20] meaning a calculation of the probability distribution of the output) of BOSONSAMPLING is highly intractable in a classical computer. They showed under assumptions of plausible conjectures that if there exists an efficient classical approximation of BOSONSAMPLING (classical simulation in the weak sense [19, 20] meaning a sampling according the probability distribution of the output), the PH collapses to the third level, which is unlikely to occur. (The detailed notions of classical simulation are provided in section 3.) This result brings a novel perspective on linear optical quantum computation and drives many researchers into the recent proof-of-principle experiments [21][22][23][24][25][26][27][28].Another subclass of quantum computation of this kind is instantaneous quantum polynomial-time computation (IQP) proposed by Shepherd and Bremner [29]. IQP consists only of commuting unitary gates, such as q  Î [ ] Z exp i k S k . Here q p Î [ ) 0, 2 is a rotational angle, Z k indicates the Pauli operator on the kth qubit, and S indicates a set of qubits on which the commuting gate acts. (A detailed definition will be provided in the next section.) The input is given by +ñ Ä | n with +ñ º ñ + ñ | (| | ) 0 1 2, and the output qubits are measured in the X-basis. Since all unitary operations are commutabl...

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Section: Strong Simulation and Post-bqp=pp Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Commuting quantum circuits and complexity of Ising partition functions

Fujii

Morimae

2017

New J. Phys.

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“…On a final note, it would be fascinating to show if the analysis of computation in generalized probabilistic theories could say something concrete about quantum computing. In an analogous fashion, tools from quantum theory have been used to prove results in classical computer science; see [59] for a nice review of such results. We speculate that by understanding quantum theory better within the framework of more general theories we can use tools from the latter to prove results in the former.…”

Section: Discussionmentioning

confidence: 99%

Bounds on the power of proofs and advice in general physical theories

Lee¹,

Hoban²

2016

Proc. R. Soc. A.

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Quantum theory presents us with the tools for computational and communication advantages over classical theory. One approach to uncovering the source of these advantages is to determine how computation and communication power vary as quantum theory is replaced by other operationally defined theories from a broad framework of such theories. Such investigations may reveal some of the key physical features required for powerful computation and communication. In this paper, we investigate how simple physical principles bound the power of two different computational paradigms which combine computation and communication in a non-trivial fashion: computation with advice and interactive proof systems. We show that the existence of non-trivial dynamics in a theory implies a bound on the power of computation with advice. Moreover, we provide an explicit example of a theory with no non-trivial dynamics in which the power of computation with advice is unbounded. Finally, we show that the power of simple interactive proof systems in theories where local measurements suffice for tomography is non-trivially bounded. This result provides a proof that QMA is contained in PP, which does not make use of any uniquely quantum structure—such as the fact that observables correspond to self-adjoint operators—and thus may be of independent interest.

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“…The closure of the classical complexity class PP under intersection was first shown using classical techniques by Beigel, Reingold, and Spielman [BRS95], but Aaronson showed it could be reproven using quantum techniques in a simpler way [Aar05]. The survey by Drucker and de Wolf provides more examples of this proof technique [DW11].…”

Section: Introductionmentioning

confidence: 99%

Classical Lower Bounds from Quantum Upper Bounds

Ben-David

Bouland

Garg

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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We prove lower bounds on complexity measures, such as the approximate degree of a Boolean function and the approximate rank of a Boolean matrix, using quantum arguments. We prove these lower bounds using a quantum query algorithm for the combinatorial group testing problem.We show that for any function f , the approximate degree of computing the OR of n copies of f is Ω( √ n) times the approximate degree of f , which is optimal. No such general result was known prior to our work, and even the lower bound for the OR of ANDs function was only resolved in 2013.We then prove an analogous result in communication complexity, showing that the logarithm of the approximate rank (or more precisely, the approximate γ 2 norm) of F : X × Y → {0, 1} grows by a factor of Ω( √ n) when we take the OR of n copies of F , which is also essentially optimal. As a corollary, we give a new proof of Razborov's celebrated Ω( √ n) lower bound on the quantum communication complexity of the disjointness problem.Finally, we generalize both these results from composition with the OR function to composition with arbitrary symmetric functions, yielding nearly optimal lower bounds in this setting as well.

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Cited by 15 publications

References 124 publications

Commuting quantum circuits and complexity of Ising partition functions

Commuting quantum circuits and complexity of Ising partition functions

Bounds on the power of proofs and advice in general physical theories

Classical Lower Bounds from Quantum Upper Bounds

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