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arXiv:2105.05698 · Leibniz international proceedings in informatics

Application of the Level-222 Quantum Lasserre Hierarchy in Quantum Approximation Algorithms

Ojas Parekh, Kevin Thompson

Published Jan 1, 2021• 6 citations
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Abstract

The Lasserre Hierarchy is a set of semidefinite programs which yield increasingly tight bounds on optimal solutions to many NP-hard optimization problems. The hierarchy is parameterized by levels, with a higher level corresponding to a more accurate relaxation. High level programs have proven to be invaluable components of approximation algorithms for many NP-hard optimization problems. There is a natural analogous quantum hierarchy, which is also parameterized by level and provides a relaxation of many (QMA-hard) quantum problems of interest. In contrast to the classical case, however, there is only one approximation algorithm which makes use of higher levels of the hierarchy. Here we provide the first ever use of the level-$2$ hierarchy in an approximation algorithm for a particular QMA-complete problem, so-called Quantum Max Cut. We obtain modest improvements on state-of-the-art approximation factors for this problem, as well as demonstrate that the level-$2$ hierarchy satisfies many physically-motivated constraints that the level-$1$ does not satisfy. Indeed, this observation is at the heart of our analysis and indicates that higher levels of the quantum Lasserre Hierarchy may be very useful tools in the design of approximation algorithms for QMA-complete problems.

DOI
10.4230/lipics.icalp.2021.102

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We resolve the approximability of the maximum energy of the Quantum Max Cut (QMC) problem using product states. A classical 0.498-approximation, using a basic semidefinite programming relaxation, is known for QMC, paralleling the celebrated 0.878-approximation for classical Max Cut. For Max Cut, improving the 0.878-approximation is Unique-Games-hard (UG-hard), and one might expect that improving the 0.498-approximation is UG-hard for QMC. In contrast, we give a classical 1/2-approximation for QMC that is unconditionally optimal, since simple examples exhibit a gap of 1/2 between the energies of an optimal product state and general quantum state. Our result relies on a new nonlinear monogamy of entanglement inequality on a triangle that is derived from the second level of the quantum Lasserre hierarchy. This inequality also applies to the quantum Heisenberg model, and our results generalize to instances of Max 2-Local Hamiltonian where each term is positive and has no 1-local parts. Finally, we give further evidence that product states are essential for approximations of 2-Local Hamiltonian.

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The Lasserre Hierarchy is a set of semidefinite programs which yield increasingly tight bounds on optimal solutions to many NP-hard optimization problems. The hierarchy is parameterized by levels, with a higher level corresponding to a more accurate relaxation. High level programs have proven to be invaluable components of approximation algorithms for many NP-hard optimization problems. There is a natural analogous quantum hierarchy, which is also parameterized by level and provides a relaxation of many (QMA-hard) quantum problems of interest. In contrast to the classical case, however, there is only one approximation algorithm which makes use of higher levels of the hierarchy. Here we provide the first ever use of the level-222 hierarchy in an approximation algorithm for a particular QMA-complete problem, so-called Quantum Max Cut. We obtain modest improvements on state-of-the-art approximation factors for this problem, as well as demonstrate that the level-222 hierarchy satisfies many physically-motivated constraints that the level-111 does not satisfy. Indeed, this observation is at the heart of our analysis and indicates that higher levels of the quantum Lasserre Hierarchy may be very useful tools in the design of approximation algorithms for QMA-complete problems.
The Lasserre Hierarchy is a set of semidefinite programs which yield increasingly tight bounds on optimal solutions to many NP-hard optimization problems. The hierarchy is parameterized by levels, with a higher level corresponding to a more accurate relaxation. High level programs have proven to be invaluable components of approximation algorithms for many NP-hard optimization problems. There is a natural analogous quantum hierarchy, which is also parameterized by level and provides a relaxation of many (QMA-hard) quantum problems of interest. In contrast to the classical case, however, there is only one approximation algorithm which makes use of higher levels of the hierarchy. Here we provide the first ever use of the level-222 hierarchy in an approximation algorithm for a particular QMA-complete problem, so-called Quantum Max Cut. We obtain modest improvements on state-of-the-art approximation factors for this problem, as well as demonstrate that the level-222 hierarchy satisfies many physically-motivated constraints that the level-111 does not satisfy. Indeed, this observation is at the heart of our analysis and indicates that higher levels of the quantum Lasserre Hierarchy may be very useful tools in the design of approximation algorithms for QMA-complete problems.
Decoders are a critical component of fault-tolerant quantum computing. They must identify errors based on syndrome measurements to correct quantum states. While finding the optimal correction is NP-hard and thus extremely difficult, approximate decoders with faster runtime often rely on uncontrolled heuristics. In this work, we propose a family of hierarchical quantum decoders with a tunable trade-off between speed and accuracy while retaining guarantees of optimality. We use the Lasserre Sum-of-Squares (SOS) hierarchy from optimization theory to relax the decoding problem. This approach creates a sequence of Semidefinite Programs (SDPs). Lower levels of the hierarchy are faster but approximate, while higher levels are slower but more accurate. We demonstrate that even low levels of this hierarchy significantly outperform standard Linear Programming relaxations. Our results on rotated surface codes and honeycomb color codes show that the SOS decoder approaches the performance of exact decoding. We find that Levels 2 and 3 of our hierarchy perform nearly as well as the exact solver. We analyze the convergence using rank-loop criteria and compare the method against other relaxation schemes. This work bridges the gap between fast heuristics and rigorous optimal decoding.
Quantum Max Cut (QMC), also known as the quantum anti-ferromagnetic Heisenberg model, is a QMA-complete problem relevant to quantum many-body physics and computer science. Semidefinite programming relaxations have been fruitful in designing theoretical approximation algorithms for QMC, but are computationally expensive for systems beyond tens of qubits. We give a second order cone relaxation for QMC, which optimizes over the set of mutually consistent three-qubit reduced density matrices. In combination with Pauli level-111 of the quantum Lasserre hierarchy, the relaxation achieves an approximation ratio of 0.5260.5260.526 to the ground state energy. Our relaxation is solvable on systems with hundreds of qubits and paves the way to computationally efficient lower and upper bounds on the ground state energy of large-scale quantum spin systems.
We study quantum algorithms for approximating Lasserre's hierarchy values for polynomial optimization. Let f,g1,…,gmf,g_1,\ldots,g_mf,g1​,…,gm​ be real polynomials in nnn variables and f⋆f^\starf⋆ the infimum of fff over the semialgebraic set S(g)={x:gi(x)≥0}S(g)=\{x: g_i(x)\ge 0\}S(g)={x:gi​(x)≥0}. Let λkλ_kλk​ be the value of the order-kkk Lasserre relaxation. Assume either (i) f⋆=λkf^\star=λ_kf⋆=λk​ and the optimum is attained in the ℓ1\ell_1ℓ1​-ball of radius 1/21/21/2, or (ii) S(g)S(g)S(g) lies in the simplex {x≥0:∑jxj≤1/2}\{x\ge 0: \sum_j x_j\le 1/2\}{x≥0:∑j​xj​≤1/2}, and the constraints define this simplex. After an appropriate coefficient rescaling, we give a quantum algorithm based on matrix multiplicative weights that approximates λkλ_kλk​ to accuracy ε>0\varepsilon>0ε>0 with runtime, for fixed kkk, O(nkε−4+nk/2ε−5),O ⁣(sg ⁣[nkε−4+ ⁣(nk+ ⁣∑i=1mnk−di)1/2 ⁣ε−5]), O(n^k\varepsilon^{-4}+n^{k/2}\varepsilon^{-5}),\qquad O\!\left(s_g\!\left[n^k\varepsilon^{-4}+\!\left(n^{k}+\!\sum_{i=1}^m n^{k-d_i}\right)^{1/2}\!\varepsilon^{-5}\right]\right), O(nkε−4+nk/2ε−5),O(sg​[nkε−4+(nk+∑i=1m​nk−di​)1/2ε−5]), where sgs_gsg​ bounds the sparsity of the coefficient-matching matrices associated with the constraints. Classical matrix multiplicative-weights methods scale as O(n3kpoly(1/ε))O(n^{3k}\mathrm{poly}(1/\varepsilon))O(n3kpoly(1/ε)) even in the unconstrained case. As an example, we obtain an O(nε−4+nε−5)O(n\varepsilon^{-4}+\sqrt{n}\varepsilon^{-5})O(nε−4+n​ε−5) quantum algorithm for portfolio optimization, improving over the classical O(nω+1log⁡(1/ε))O(n^{ω+1}\log(1/\varepsilon))O(nω+1log(1/ε)) bound with ω≈2.373ω\approx2.373ω≈2.373. Our approach builds on and sharpens the analysis of Apeldoorn and Gilyén for the SDPs arising in polynomial optimization. We also show how to implement the required block encodings without QRAM. Under the stated assumptions, our method achieves a super-quadratic speedup in the problem dimension for computing Lasserre relaxations.
It is well-known in physics that the limit of large quantum spin SSS should be understood as a semiclassical limit. This raises the question of whether such emergent classicality facilitates the approximation of computationally hard quantum optimization problems, such as the local Hamiltonian problem. We demonstrate this explicitly for spin-SSS generalizations of Quantum Max-Cut (QMaxCutS\mathrm{QMaxCut}_SQMaxCutS​), equivalent to the problem of finding the ground state energy of an arbitrary spin-SSS quantum Heisenberg antiferromagnet (QHAS\mathrm{QHA}_SQHAS​). We prove that approximating the value of QHAS\mathrm{QHA}_SQHAS​ to inverse polynomial accuracy is QMA-complete for all SSS, extending previous results for S=1/2S=1/2S=1/2. We also present two distinct families of classical approximation algorithms for QMaxCutS\mathrm{QMaxCut}_SQMaxCutS​ based on rounding the output of a semidefinite program to a product of Bloch coherent states. The approximation ratios for both our proposed algorithms strictly increase with SSS and converge to the Briët-Oliveira-Vallentin approximation ratio αBOV≈0.956α_{\mathrm{BOV}} \approx 0.956αBOV​≈0.956 from below as S→∞S \to \inftyS→∞.
We resolve the approximability of the maximum energy of the Quantum Max Cut (QMC) problem using product states. A classical 0.498-approximation, using a basic semidefinite programming relaxation, is known for QMC, paralleling the celebrated 0.878-approximation for classical Max Cut. For Max Cut, improving the 0.878-approximation is Unique-Games-hard (UG-hard), and one might expect that improving the 0.498-approximation is UG-hard for QMC. In contrast, we give a classical 1/2-approximation for QMC that is unconditionally optimal, since simple examples exhibit a gap of 1/2 between the energies of an optimal product state and general quantum state. Our result relies on a new nonlinear monogamy of entanglement inequality on a triangle that is derived from the second level of the quantum Lasserre hierarchy. This inequality also applies to the quantum Heisenberg model, and our results generalize to instances of Max 2-Local Hamiltonian where each term is positive and has no 1-local parts. Finally, we give further evidence that product states are essential for approximations of 2-Local Hamiltonian.