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Arad, Itai; Sántha, Miklós; Sundaram, Aarthi; Zhang, Shengyu

doi:10.4086/toc.2018.v014a001

Cited by 3 publications

(6 citation statements)

References 20 publications

(29 reference statements)

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“…[12] and Arad et al in Ref. [15] are classical. The classical algorithm relies on that for every Q2SAT problem which has solutions, there is a solution that is the tensor product of one-qubit and two-qubit states,…”

Section: Previous Algorithmsmentioning

confidence: 99%

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Quantum algorithm for a set of quantum 2SAT problems*

Zhang

2021

Chinese Phys. B

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We present a quantum adiabatic algorithm for a set of quantum 2-satisfiability (Q2SAT) problem, which is a generalization of 2-satisfiability (2SAT) problem. For a Q2SAT problem, we construct the Hamiltonian which is similar to that of a Heisenberg chain. All the solutions of the given Q2SAT problem span the subspace of the degenerate ground states. The Hamiltonian is adiabatically evolved so that the system stays in the degenerate subspace. Our numerical results suggest that the time complexity of our algorithm is O(n 3.9) for yielding non-trivial solutions for problems with the number of clauses m = dn(n – 1)/2 (d ≲ 0.1). We discuss the advantages of our algorithm over the known quantum and classical algorithms.

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Section: Previous Algorithmsmentioning

confidence: 99%

“…Although its time complexity is better, it tends to find trivial product solutions. [12,15] The quantum algorithm in Ref. [16] can find entangled solutions but with a slower time complexity of O(mn 2 /δ (n)), where the energy gap δ (n) may be in the form of n −g (g positive).…”

Section: Introductionmentioning

confidence: 99%

Quantum algorithm for a set of quantum 2SAT problems*

Zhang

2021

Chinese Phys. B

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“…It can be seen that after the adiabatic evolution, we do not return to the trivial state but reach a non-trivial state with a high probability. Such an ability to find a non-trivial solution is better than that of the classical algorithm proposed by [12] and [13].…”

Section: Numerical Simulationmentioning

confidence: 93%

“…There are now several algorithms for Q2SAT problems. The algorithm proposed by Beaudrap et al [12] and Arad et al [13] is classical. Although its time complexity is O(n + m), it is likely to find a trivial solution.…”

Section: Quantum 2-satisfiability Problemmentioning

confidence: 99%

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Quantum algorithm of a set of quantum 2-sat problem

Hu,

Zhang,

2020

Preprint

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We present a quantum adiabatic algorithm for a set of quantum 2-satisfiability (Q2SAT) problem, which is a generalization of 2-satisfiability (2SAT) problem. For a Q2SAT problem, we construct the Hamiltonian which is similar to that of a Heisenberg chain. All the solutions of the given Q2SAT problem span the subspace of the degenerate ground states. The Hamiltonian is adiabatically evolved so that the system stays in the degenerate subspace. Our numerical results suggest that the time complexity of our algorithm is O(n 3.9 ) for yielding non-trivial solutions for problems with the number of clauses m = dn(n − 1)/2 (d < ∼ 0.1). We discuss the advantages of our algorithm over the known quantum and classical algorithms.

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Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction

Bausch

2019

Ann. Henri Poincaré

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Fundamentally, it is believed that interactions between physical objects are two-body.Perturbative gadgets are one way to break up an effective many-body coupling into pairwise interactions: a Hamiltonian with high interaction strength introduces a low-energy space in which the effective theory appears k-body and approximates a target Hamiltonian to within precision . One caveat of existing constructions is that the interaction strength generally scales exponentially in the locality of the terms to be approximated, i.e. as Ω(1/ k ); if = 1/poly n in the system size n, as is necessary for e.g. QMA-hardness constructions, the energy differences become highly unphysical.In this work we propose a many-body Hamiltonian construction which introduces only a single separate energy scale of order Θ(1/N 2+δ ), for a small parameter δ > 0, and for N terms in the target Hamiltonian-i.e. all local terms of the simulator have either this norm, or one of O(1). In its low-energy subspace, we can approximate any normalized target Hamiltonian H t = N i=1 h i with norm ratios r = h i 2 / h j 2 = O(exp(exp(poly n))) to within relative precision O(N −δ ). This comes at the expense of increasing the locality by at most one, and adding an at most poly-sized ancilliary system for each coupling; the ancillas being qutrits for exponential scaling, and qudits for doubly exponential r; the interactions on the ancilliary system are geometrically local, and can be translationally-invariant.In order to prove this claim, we borrow a technique from high energy physics-where matter fields obtain effective properties (such as mass) from interactions with an exchange particle-and a tiling Hamiltonian to drop all cross terms at higher expansion orders, which simplifies the analysis of a traditional Feynman-Dyson series expansion.As an application, we discuss implications for QMA-hardness of the L H problem, and argue that "almost" translational invariance-defined as arbitrarily small relative variations of the strength of the local terms-is as good as non-translational-invariance in many of the constructions used throughout Hamiltonian complexity theory. We furthermore show that the choice of geared limit of many-body systems, where e.g. width and height of a lattice are taken to infinity in a specific relation, can have different complexity-theoretic implications: even for translationally-invariant models, changing the geared limit can vary the hardness of finding the ground state energy with respect to a given promise gap from computationally trivial, to QMA EXP -, or even BQEXPSPACE-complete.

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Cited by 3 publications

References 20 publications

Quantum algorithm for a set of quantum 2SAT problems*

Quantum algorithm for a set of quantum 2SAT problems*

Quantum algorithm of a set of quantum 2-sat problem

Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction

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