Complexity Classification of Local Hamiltonian Problems

Cubitt, Toby S.; Montanaro, Ashley

doi:10.1109/focs.2014.21

Cited by 59 publications

(98 citation statements)

References 70 publications

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“…Apart from the possibility of realizing a universal set of matchgates, the extended spin Hamiltonian that we can implement (see equation (8)) also has an interesting significance from a quantum Hamiltonian complexity perspective [84]. In general, it is known [85] that for Hamiltonians of the form in equation (8) also have the same property [86,87]. Therefore, the realization of the full Hamiltonian with all parameters J K L M ( , , , ) finite would represent the simplest controllable quantum system that is hard to simulate even on quantum computers.…”

Section: Resultsmentioning

confidence: 99%

Infrared-dressed entanglement of cold open-shell polar molecules for universal matchgate quantum computing

et al. 2014

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Implementing a scalable quantum information processor using polar molecules in optical lattices requires precise control over the long-range dipole-dipole interaction between molecules in selected lattice sites. We present here a scheme using trapped open-shell Σ 2 polar molecules that allows dipolar exchange processes between nearest and next-nearest neighbors to be controlled in order to construct a generalized transverse Ising spin Hamiltonian with tunable XX, YY and XY couplings in the rotating frame of the driving lasers. The scheme requires a moderately strong bias magnetic field together with near-infrared light to provide local tuning of the qubit energy gap, and mid-infrared pulses to perform rotational state transfer via stimulated Raman adiabatic passage. No interaction between qubits occurs in the absence of the infrared driving. We analyze the fidelity of the resulting two-qubit matchgate, and demonstrate its robustness as a function of the driving parameters. We discuss a realistic application of the system for universal matchgate quantum computing in optical lattices., therefore admixing between the electron spin and the rotational motion of the nuclei is only perturbative. The molecular constants B e , γ sr and Δα depend on the vibrational state of the molecule, although the dependence can be weak for the lowest two vibrational states (v = 0 and v = 1) considered in this work [63]. In [43], Σ 2 polar 4 New J. Phys. 16 (2014) 075001 F Herrera et al

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Section: Resultsmentioning

confidence: 99%

Infrared-dressed entanglement of cold open-shell polar molecules for universal matchgate quantum computing

et al. 2014

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“…This requires us to initiate any direct reduction with a specific class of Hamiltonians, in so-called Y-free form (see Definition 6.8); informally, these are local Hamiltonians such that each local term is a tensor product of generalized τ X and τ Z observables. In the absence of general gap-preserving reductions between different variants of the local Hamiltonian problem (perturbation techniques [CM14] do not generally preserve the promise gap) we obtain a reduction to the hardness of constant-factor approximations to the ground energy of local Hamiltonian of this form only. Nevertheless, even though the entanglement test requires a qudit dimension that scales (poly-logarithmically) with n, we show that any qubit Hamiltonian in Y-free form can be embedded in a Hamiltonian in Y-free form over qudits of dimension 2 t for any t ≥ 1.…”

Section: Introductionmentioning

confidence: 95%

“…Proof of Corollary 6.14. We start by recalling that the Local Hamiltonian problem is QMA-complete for qubit Hamiltonians in the XZ model, up to inverse-polynomial promise gap [CM14]. Let H = E j∈{1,...,ℓ} H j be a given Hamiltonian on n qubits from the XZ model (also allowing terms that are multiples of the identity), with ℓ = poly(n) local terms H j , normalized such that 0 ≤ H ≤ Id.…”

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confidence: 99%

Low-Degree Testing for Quantum States, and a Quantum Entangled Games PCP for QMA

Natarajan

Vidick

2018

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

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We show that given an explicit description of a multiplayer game, with a classical verifier and a constant number of players, it is QMA-hard, under randomized reductions, to distinguish between the cases when the players have a strategy using entanglement that succeeds with probability 1 in the game, or when no such strategy succeeds with probability larger than 1 2 . This proves the "games quantum PCP conjecture" of Fitzsimons and the second author (ITCS'15), albeit under randomized reductions.The core component in our reduction is a construction of a family of two-player games for testing n-qubit maximally entangled states. For any integer n ≥ 2, we give such a game in which questions from the verifier are O(log n) bits long, and answers are poly(log log n) bits long. We show that for any constant ε ≥ 0, any strategy that succeeds with probability at least 1 − ε in the test must use a state that is within distance δ(ε) = O(ε c ) from a state that is locally equivalent to a maximally entangled state on n qubits, for some universal constant c > 0. The construction is based on the classical plane-vs-point test for multivariate low-degree polynomials of Raz and Safra (STOC'97). We extend the classical test to the quantum regime by executing independent copies of the test in the generalized Pauli X and Z bases over F q , where q is a sufficiently large prime power, and combine the two through a test for the Pauli twisted commutation relations.Our main complexity-theoretic result is obtained by combining this family of games with techniques from the classical PCP literature. More specifically, we use constructions of PCPs of proximity introduced by Ben-Sasson et al. (CCC'05), and crucially rely on a linear property of such PCPs. Another consequence of our results is a deterministic reduction from the games quantum PCP conjecture to a suitable formulation of the constraint satisfaction quantum PCP conjecture.give two alternate formulations of Theorem 1.2 that would also establish the same QMA-hardness result, under deterministic reductions, provided that either:(i) it is QMA-hard to approximate the minimum energy of a local Hamiltonian in Y-free form (Definition 6.8) to within constant accuracy (this is a variant of the quantum PCP conjecture for local Hamiltonians), or (ii) it is QMA hard to approximate the ground energy of (not necessarily local) frustration-free Hamiltonian whose every term is a tensor product of generalized Pauli τ X or τ Z observables.Note that point (i) amounts to a deterministic reduction from Conjecture 1.2 to the constraint satisfaction quantum PCP conjecture, and establishes the first proven relation between the two conjectures (see [GKP16] for an incomparable result that relates stronger variants of both conjectures). Point (ii) is arguably a weaker assumption, as the gap is not required to be a constant and the terms of the Hamiltonian are not required to be local. However, due to the restriction that the Hamiltonian is frustration-free, it is currently not known whether the problem is QMA...

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“…Kitaev's seminal paper proving quantum-NP-hardness of the local Hamiltonian problem for the case that each interaction couples at most five spins [1] motivated significant progress towards understanding the computational complexity that arises in different variants of the local Hamiltonian problem [2][3][4][5][6][7][8][9][10]. These results are especially interesting from a computational perspective, answering which families of Hamiltonians are "complicated enough" to perform universal quantum computation [11,12].…”

Section: Introductionmentioning

confidence: 99%

The complexity of translationally invariant low-dimensional spin lattices in 3D

Bausch

Piddock

2017

Journal of Mathematical Physics

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In this paper, we consider spin systems in three spatial dimensions, and prove that the local Hamiltonian problem for 3D lattices with face-centered cubic unit cells, 4-local translationally-invariant interactions between spin-3/2 particles and open boundary conditions is QMA EXP -complete. We go beyond a mere embedding of past hard 1D history state constructions, and utilize a classical Wang tiling problem as binary counter in order to translate one cube side length into a binary description for the verifier input. We further make use of a recently-developed computational model especially well-suited for history state constructions, and combine it with a specific circuit encoding shown to be universal for quantum computation. These novel techniques allow us to significantly lower the local spin dimension, surpassing the best translationally-invariant result to date by two orders of magnitude (in the number of degrees of freedom per coupling). This brings our models en par with the best non-translationally-invariant construction.arXiv:1702.08830v1 [quant-ph]

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Complexity Classification of Local Hamiltonian Problems

Cited by 59 publications

References 70 publications

Infrared-dressed entanglement of cold open-shell polar molecules for universal matchgate quantum computing

Infrared-dressed entanglement of cold open-shell polar molecules for universal matchgate quantum computing

Low-Degree Testing for Quantum States, and a Quantum Entangled Games PCP for QMA

The complexity of translationally invariant low-dimensional spin lattices in 3D

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