Determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy

Fenner, Stephen A.; Green, Frederic; Homer, Steven; Pruim, Randall

doi:10.1098/rspa.1999.0485

Cited by 55 publications

(84 citation statements)

References 29 publications

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“…In the standard quantum computational model, the hypothetical ability to determine expectation values with b bits of precision using resources polynomial in b implies the ability to efficiently solve problems in #P , the class of problems associated with the ability to count the number of solutions to NP-complete problems such as satisfiability. This is a consequence of more general results in [10].…”

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confidence: 70%

Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

et al. 2006

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We consider quantum computational models defined via a Lie-algebraic theory. In these models, specified initial states are acted on by Lie-algebraic quantum gates and the expectation values of Lie algebra elements are measured at the end. We show that these models can be efficiently simulated on a classical computer in time polynomial in the dimension of the algebra, regardless of the dimension of the Hilbert space where the algebra acts. Similar results hold for the computation of the expectation value of operators implemented by a gatesequence. We introduce a Lie-algebraic notion of generalized mean-field Hamiltonians and show that they are efficiently (exactly) solvable by means of a Jacobi-like diagonalization method. Our results generalize earlier ones on fermionic linear optics computation and provide insight into the source of the power of the conventional model of quantum computation. Quantum models of computation are widely believed to be more powerful than classical ones. Although this has been shown to be true in a few cases, it is still important to determine when a quantum algorithm for a given problem is more resource efficient than any classical one, or, conversely, when a classical algorithm is just as efficient as any quantum counterpart. In general, one needs to know whether it is worth investing in building a quantum computer (QC) and what is required for success. In this paper, we show close connections between these issues and the efficient (or exact) solvability of Hamiltonians. In particular, we show that a class of quantum models we call generalized mean-field Hamiltonians (GMFHs) [1] is efficiently solvable and furthermore does not provide a stronger-than-classical model of computation: A quantum device engineered to have dynamical gates generated by Hamiltonians from such a set cannot directly simulate universal efficient quantum computation and can be efficiently simulated by a classical computer (CC).An algorithm is a sequence of elementary instructions that solves instances of a problem. It is said to be efficient if the resources required to solve problem instances of size N are polynomial in N (poly(N )) resources. Typically, the size of a problem instance is the number of bits required to represent it, and the relevant resources are time and space. In the last few years it has been shown that many pure-state quantum algorithms can be efficiently simulated on a CC when the extent of entanglement is limited (e.g., [2,3]) or when the quantum gates available are far from allowing us to build a set of universal gates [4,5,6]. Here, we focus on a Lie algebraic analysis to obtain other situations where quantum algorithms can be efficiently simulated by CCs. The so-called generalized coherent states (GCSs) [7] play a decisive role in our analysis.The algorithms considered here make use of the Liealgebraic model of quantum computing (LQC). An LQC algorithm begins with the specification of a semisimple, compact M -dimensional real Lie algebraĥ of skew-Hermitian operators acting on a finite-...

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Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

et al. 2006

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“…In addition, we can get by with only with simple qubit state teleportation [BBC + 93]. Our results immediately show that the class NQNC 0 (the constant-depth analog of NQP, see below), is actually the same as NQP, which is known to be as hard as the polynomial hierarchy [FGHP99]. We give this result in Section 3.1.…”

Section: Introductionmentioning

confidence: 70%

“…As mentioned before, NQP [ADH97] is defined as the class of languages recognized by quantum Turing machines (equivalently, uniform quantum circuit families over a finite set of gates) where the acceptance criterion is that the accepting state appear with nonzero probability. It is known [FGHP99,YY99] that NQP = C = P, which contains NP and is hard for the polynomial hierarchy. Since QNC 0 circuit families must also draw their gates from some finite set, we clearly have NQNC 0 ⊆ NQP.…”

Section: Other Classes Of Constant-depth Quantum Circuitsmentioning

confidence: 99%

Bounds on the Power of Constant-Depth Quantum Circuits

Fenner

Green

Homer

et al. 2005

Lecture Notes in Computer Science

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We show that if a language is recognized within certain error bounds by constantdepth quantum circuits over a finite family of gates, then it is computable in (classical) polynomial time. In particular, our results implywhere EQNC 0 is the constant-depth analog of the class EQP. On the other hand, we adapt and extend ideas of Terhal & DiVincenzo [TD02] to show that, for any family F of quantum gates including Hadamard and CNOT gates, computing the acceptance probabilities of depth-five circuits over F is just as hard as computing these probabilities for arbitrary quantum circuits over F. In particular, this implies thatwhere NQNC 0 is the constant-depth analog of the class NQP. This essentially refutes a conjecture of Green et al. that NQACC ⊆ TC 0 [GHMP02].

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“…Adleman et al [1] introduced the complexity class NQP as a quantum extension of this probabilistic characterization. Subsequently, NQP (even with arbitrary complex amplitudes) was shown to coincide with the classical counting class co-C=P [7,6,20]. This shows the power of quantum computation.…”

Section: What Is Quantum Np?mentioning

confidence: 96%

Quantum NP and a Quantum Hierarchy

Yamakami

2002

Foundations of Information Technology in the Era of Network and Mobile Computing

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The complexity class NP is quintessential and ubiquitous in theoretical computer science. Two different approaches have been made to define "Quantum NP," the quantum analogue of NP: NQP by Adleman, DeMarrais, and Huang, and QMA by Knill, Kitaev, and Watrous.From an operator point of view, NP can be viewed as the result of the 3-operator applied toP. Recently, Green, Homer, Moore, and Pollett proposed its quantum version, called the N-operator, which is an abstraction of NQP. This paper introduces the 3Q-operator, which is an abstraction of QMA, and its complement, the VQ-operator. These operators not only define Quantum NP but also build a quantum hierarchy, similar to the Meyer-Stockmeyer polynomial hierarchy, based on two-sided bounded-error quantum computation.Keywords: quantum quantifier, quantum operator, quantum polynomial hierarchy 1. What is Quantum NP?Computational complexity theory based on a 'lUring machine (TM, for short) was formulated in the 1960s. The complexity class NP was later introduced as the collection of sets that are recognized by nondeterministic TMs in polynomial time. By the earlier work of Cook, Levin, and Karp, NP was quickly identified as a central notion in complexity theory by means of NP-completeness. NP has since then exhibited its rich structure and is proven to be vital to many fields of theoretical computer science.

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Determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy

Cited by 55 publications

References 29 publications

Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

Bounds on the Power of Constant-Depth Quantum Circuits

Quantum NP and a Quantum Hierarchy

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