Beyond Product State Approximations for a Quantum Analogue of Max Cut

Anshu, Anurag; Gosset, David; Morenz, Karen J.

doi:10.4230/lipics.tqc.2020.7

Cited by 7 publications

(9 citation statements)

References 12 publications

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“…Indeed we believe that quadratic instances allow one to glean insights and develop techniques that might otherwise be obscured in more general instances. Some of the first rigorous approximation algorithms for QLH that go beyond product states were recently developed for quantum Max Cut [2,31]. Moreover, maximally entangled instances are strictly quadratic, and we conjecture these are the hardest cases to approximate.…”

Section: :4mentioning

confidence: 93%

“…Hamiltonian. In stark contrast, although QMA-hard quantum optimization problems arise naturally through well-known physically motivated problems [5,35], they have very few known approximation algorithms with provable approximation factors [2,4,8,11,18,19,21,22,31] 1 . The QMA-hard optimization studied in these works, as well as the problem we sill study here, is the 2-Local Hamiltonian problem [24,25].…”

Section: -Localmentioning

confidence: 99%

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Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems.

Parekh¹,

Thompson²

2021

Proposed for Presentation at the European Symposium No Algorithms (ESA 2021) Held September 6-8, 2021 in Lisbon, Portugal

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The quantum k-Local Hamiltonian problem is a natural generalization of classical constraint satisfaction problems (k-CSP) and is complete for QMA, a quantum analog of NP. Although the complexity of k-Local Hamiltonian problems has been well studied, only a handful of approximation results are known. For Max 2-Local Hamiltonian where each term is a rank 3 projector, a natural quantum generalization of classical Max 2-SAT, the best known approximation algorithm was the trivial random assignment, yielding a 0.75-approximation. We present the first approximation algorithm beating this bound, a classical polynomial-time 0.764-approximation. For strictly quadratic instances, which are maximally entangled instances, we provide a 0.801 approximation algorithm, and numerically demonstrate that our algorithm is likely a 0.821-approximation. We conjecture these are the hardest instances to approximate. We also give improved approximations for quantum generalizations of other related classical 2-CSPs. Finally, we exploit quantum connections to a generalization of the Grothendieck problem to obtain a classical constant-factor approximation for the physically relevant special case of strictly quadratic traceless 2-Local Hamiltonians on bipartite interaction graphs, where a inverse logarithmic approximation was the best previously known (for general interaction graphs). Our work employs recently developed techniques for analyzing classical approximations of CSPs and is intended to be accessible to both quantum information scientists and classical computer scientists.

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Section: :4mentioning

confidence: 93%

Section: -Localmentioning

confidence: 99%

Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems.

Parekh¹,

Thompson²

2021

Proposed for Presentation at the European Symposium No Algorithms (ESA 2021) Held September 6-8, 2021 in Lisbon, Portugal

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“…where the expressions in terms of Pauli matrices follow from Eqs. (32) to (34). Then we may rewrite Eq.…”

Section: B the Quadratic Mapping As Quantum Expectation Valuesmentioning

confidence: 99%

“…Within the broader context of quantum information theory, our work here also provides an alternative perspective to relaxations of quantum Hamiltonian problems. There is a growing interest in classical methods for approximating quantum many-body problems based on SDP relaxations [29][30][31][32][33][34][35][36][37][38]. In that context, rounding procedures are more difficult to formulate because the space of quantum states is exponentially large.…”

mentioning

confidence: 99%

“…In that context, rounding procedures are more difficult to formulate because the space of quantum states is exponentially large. For instance, the algorithm may only round to a subset of quantum states with efficient classical descriptions such as product states [29-31, 33, 36] or low-entanglement states [32,34], effectively restricting the approximation from representing the true ground state. Nonetheless, these algorithms can still obtain meaningful approximation ratios of the optimal energy, indicating that such states can at least capture some qualitative properties of the generically entangled ground state.…”

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confidence: 99%

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Quantum relaxation for quadratic programs over orthogonal matrices

Zhao¹,

Rubin²

2023

Preprint

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Quadratic programming over the (special) orthogonal group encompasses a broad class of optimization problems such as group synchronization, point-set registration, and simultaneous localization and mapping. Such problems are instances of the little noncommutative Grothendieck problem (LNCG), a natural generalization of quadratic combinatorial optimization where, instead of binary decision variables, one optimizes over orthogonal matrices. In this work, we establish an embedding of this class of LNCG problems over the orthogonal group onto a quantum Hamiltonian. This embedding is accomplished by identifying orthogonal matrices with their double cover (Pin and Spin group) elements, which we represent as quantum states. We connect this construction to the theory of free fermions, which provides a physical interpretation of the derived LNCG Hamiltonian as a two-body interacting-fermion model due to the quadratic nature of the problem. Determining extremal states of this Hamiltonian provides an outer approximation to the original problem, analogous to classical relaxations of the problem via semidefinite programming. When optimizing over the special orthogonal group, our quantum relaxation naturally obeys additional, powerful constraints based on the convex hull of rotation matrices. The classical size of this convex-hull representation is exponential in matrix dimension, whereas the quantum representation requires only a linear number of qubits. Finally, to project the relaxed solution into the feasible space, we employ rounding procedures which return orthogonal matrices from appropriate measurements of the quantum state. Through numerical experiments we provide evidence that this quantum relaxation can produce high-quality approximations.

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Linear Programming with Unitary-Equivariant Constraints

Grinko,

Ozols

2024

Commun. Math. Phys.

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Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. Optimization problems with such symmetry can often be formulated as semidefinite programs for a $$d^{p+q}$$ d p + q -dimensional matrix variable that commutes with $$U^{\otimes p} \otimes {\bar{U}}^{\otimes q}$$ U ⊗ p ⊗ U ¯ ⊗ q , for all $$U \in \textrm{U}(d)$$ U ∈ U ( d ) . Solving such problems naively can be prohibitively expensive even if $$p+q$$ p + q is small but the local dimension d is large. We show that, under additional symmetry assumptions, this problem reduces to a linear program that can be solved in time that does not scale in d, and we provide a general framework to execute this reduction under different types of symmetries. The key ingredient of our method is a compact parametrization of the solution space by linear combinations of walled Brauer algebra diagrams. This parametrization requires the idempotents of a Gelfand–Tsetlin basis, which we obtain by adapting a general method inspired by the Okounkov–Vershik approach. To illustrate potential applications of our framework, we use several examples from quantum information: deciding the principal eigenvalue of a quantum state, quantum majority vote, asymmetric cloning and transformation of a black-box unitary. We also outline a possible route for extending our method to general unitary-equivariant semidefinite programs.

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Beyond Product State Approximations for a Quantum Analogue of Max Cut

Cited by 7 publications

References 12 publications

Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems.

Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems.

Quantum relaxation for quadratic programs over orthogonal matrices

Linear Programming with Unitary-Equivariant Constraints

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