Classical and Quantum Computation

Kitaev, Alexei; Shen, Ao; Vyalyi, Mikhail N.

doi:10.1090/gsm/047

Cited by 1,041 publications

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“…The simplest of these complete problems is the 2-local Hamiltonian problem, which is informally the quantum version of the circuit satisfiability problem for unitary circuits with gates of constant size. A formal description of this problem, as well as a proof that the 5-local Hamiltonian problem is QMA-complete can be found in [10]. The improvement of this result to the 2-local case is due to Kempe, Kitaev, and Regev [8].…”

Section: Introductionmentioning

confidence: 93%

Testing Non-isometry Is QMA-Complete

Rosgen

2011

Theory of Quantum Computation, Communication, and Cryptography

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Determining the worst-case uncertainty added by a quantum circuit is shown to be computationally intractable. This is the problem of detecting when a quantum channel implemented as a circuit is close to a linear isometry, and it is shown to be complete for the complexity class QMA of verifiable quantum computation. The main idea is to relate the problem of detecting when a channel is close to an isometry to the problem of determining how mixed the output of the channel can be when the input is a pure state.

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Section: Introductionmentioning

confidence: 93%

Testing Non-isometry Is QMA-Complete

Rosgen

2011

Theory of Quantum Computation, Communication, and Cryptography

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“…Another fact is that the unpublished proof by Kitaev and Watrous for the upper bound PP of the class QMA of problems having single-proof quantum Merlin-Arthur proof systems no longer works well for the multi-proof cases with the most straightforward modification. The simplified proof by Marriott and Watrous [39] for the same statement and even the proof of QMA ⊆ PSPACE [30,31] are also the cases. Furthermore, the existing proofs for the property that parallel repetition of a single-proof system reduces the error probability to be arbitrarily small [32,48,31,39] cannot be applied to the multi-proof cases.…”

Section: Introductionmentioning

confidence: 94%

“…The simplified proof by Marriott and Watrous [39] for the same statement and even the proof of QMA ⊆ PSPACE [30,31] are also the cases. Furthermore, the existing proofs for the property that parallel repetition of a single-proof system reduces the error probability to be arbitrarily small [32,48,31,39] cannot be applied to the multi-proof cases. Of course, these arguments do not imply that using multiple quantum proofs is more powerful than using only a single quantum proof from the complexity theoretical viewpoint.…”

Section: Introductionmentioning

confidence: 94%

Quantum Merlin-Arthur Proof Systems: Are Multiple Merlins More Helpful to Arthur?

Kobayashi

Matsumoto

Yamakami

2003

Algorithms and Computation

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This paper introduces quantum "multiple-Merlin"-Arthur proof systems in which Arthur receives multiple quantum proofs that are unentangled with each other. Although classical multi-proof systems are obviously equivalent to classical single-proof systems (i.e., usual Merlin-Arthur proof systems), it is unclear whether or not quantum multi-proof systems collapse to quantum single-proof systems (i.e., usual quantum Merlin-Arthur proof systems). This paper presents a necessary and sufficient condition under which the number of quantum proofs is reducible to two. It is also proved that, in the case of perfect soundness, using multiple quantum proofs does not increase the power of quantum Merlin-Arthur proof systems. * A preliminary version of this paper has appeared in

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“…The first QMA-complete Hamiltonian was introduced by Kitaev [11] building on ideas of Feynman [7]. The Hamiltonian construction represents an updated version of the Cook-Levin construction which shows any NP-complete problem can be embedded into the Boolean satisfiability problem [21].…”

Section: Hamiltonian Complexitymentioning

confidence: 99%

The Computational Complexity of Density Functional Theory

Whitfield

Schuch

Verstraete

2014

Many-Electron Approaches in Physics, Chemistry and Mathematics

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Density functional theory is a successful branch of numerical simulations of quantum systems. While the foundations are rigorously defined, the universal functional must be approximated resulting in a 'semi'-ab initio approach. The search for improved functionals has resulted in hundreds of functionals and remains an active research area. This chapter is concerned with understanding fundamental limitations of any algorithmic approach to approximating the universal functional. The results based on Hamiltonian complexity presented here are largely based on [20]. In this chapter, we explain the computational complexity of DFT and any other approach to solving electronic structure Hamiltonians. The proof relies on perturbative gadgets widely used in Hamiltonian complexity and we provide an introduction to these techniques using the Schrieffer-Wolff method. Since the difficulty of this problem has been well appreciated before this formalization, practitioners have turned to a host approximate Hamiltonians. By extending the results of [20], we show in DFT, although the introduction of an approximate potential leads to a noninteracting Hamiltonian, it remains, in the worst case, an NP-complete problem.

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Classical and Quantum Computation

Cited by 1,041 publications

References 0 publications

Testing Non-isometry Is QMA-Complete

Testing Non-isometry Is QMA-Complete

Quantum Merlin-Arthur Proof Systems: Are Multiple Merlins More Helpful to Arthur?

The Computational Complexity of Density Functional Theory

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