Application of the Level-2 Quantum Lasserre Hierarchy in Quantum Approximation Algorithms.

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: 10.2172/1884683

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Application of the Level- $2$ Quantum Lasserre Hierarchy in Quantum Approximation Algorithms

Ojas Parekh, Kevin Thompson

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.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.

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Quantum speed-ups for solving semidefinite relaxations of polynomial optimization

Daniel Stilck França, Ngoc Hoang Anh Mai

We study quantum algorithms for approximating Lasserre's hierarchy values for polynomial optimization. Let $f,g_1,\ldots,g_m$ be real polynomials in $n$ variables and $f^\star$ the infimum of $f$ over the semialgebraic set $S(g)=\{x: g_i(x)\ge 0\}$ . Let $λ_k$ be the value of the order- $k$ Lasserre relaxation. Assume either (i) $f^\star=λ_k$ and the optimum is attained in the $\ell_1$ -ball of radius $1/2$ , or (ii) $S(g)$ lies in the simplex $\{x\ge 0: \sum_j x_j\le 1/2\}$ , and the constraints define this simplex. After an appropriate coefficient rescaling, we give a quantum algorithm based on matrix multiplicative weights that approximates $λ_k$ to accuracy $\varepsilon>0$ with runtime, for fixed $k$ , $O(n^k\varepsilon^{-4}+n^{k/2}\varepsilon^{-5}),\qquad O\!\left(s_g\!\left[n^k\varepsilon^{-4}+\!\left(n^{k}+\!\sum_{i=1}^m n^{k-d_i}\right)^{1/2}\!\varepsilon^{-5}\right]\right),$ where $s_g$ bounds the sparsity of the coefficient-matching matrices associated with the constraints. Classical matrix multiplicative-weights methods scale as $O(n^{3k}\mathrm{poly}(1/\varepsilon))$ even in the unconstrained case. As an example, we obtain an $O(n\varepsilon^{-4}+\sqrt{n}\varepsilon^{-5})$ quantum algorithm for portfolio optimization, improving over the classical $O(n^{ω+1}\log(1/\varepsilon))$ bound with $ω\approx2.373$ . 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.We study quantum algorithms for approximating Lasserre's hierarchy values for polynomial optimization. Let $f,g_1,\ldots,g_m$ be real polynomials in $n$ variables and $f^\star$ the infimum of $f$ over the semialgebraic set $S(g)=\{x: g_i(x)\ge 0\}$. Let $λ_k$ be the value of the order-$k$ Lasserre relaxation. Assume either (i) $f^\star=λ_k$ and the optimum is attained in the $\ell_1$-ball of radius $1/2$, or (ii) $S(g)$ lies in the simplex $\{x\ge 0: \sum_j x_j\le 1/2\}$, and the constraints define this simplex. After an appropriate coefficient rescaling, we give a quantum algorithm based on matrix multiplicative weights that approximates $λ_k$ to accuracy $\varepsilon>0$ with runtime, for fixed $k$, \[ O(n^k\varepsilon^{-4}+n^{k/2}\varepsilon^{-5}),\qquad O\!\left(s_g\!\left[n^k\varepsilon^{-4}+\!\left(n^{k}+\!\sum_{i=1}^m n^{k-d_i}\right)^{1/2}\!\varepsilon^{-5}\right]\right), \] where $s_g$ bounds the sparsity of the coefficient-matching matrices associated with the constraints. Classical matrix multiplicative-weights methods scale as $O(n^{3k}\mathrm{poly}(1/\varepsilon))$ even in the unconstrained case. As an example, we obtain an $O(n\varepsilon^{-4}+\sqrt{n}\varepsilon^{-5})$ quantum algorithm for portfolio optimization, improving over the classical $O(n^{ω+1}\log(1/\varepsilon))$ bound with $ω\approx2.373$. 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.

quant-pharXiv:2511.14389Nov 19, 20250 citations

Hierarchical quantum decoders

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Second order cone relaxations for quantum Max Cut

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Vir B. Bulchandani, Stephen Piddock

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An SU(2)-symmetric Semidefinite Programming Hierarchy for Quantum Max Cut

Jun Takahashi, Sankara Sai Chaithanya Rayudu, Cunlu Zhou +3 more

Understanding and approximating extremal energy states of local Hamiltonians is a central problem in quantum physics and complexity theory. Recent work has focused on developing approximation algorithms for local Hamiltonians, and in particular the ``Quantum Max Cut'' (QMax-Cut) problem, which is closely related to the antiferromagnetic Heisenberg model. In this work, we introduce a family of semidefinite programming (SDP) relaxations based on the Navascues-Pironio-Acin (NPA) hierarchy which is tailored for QMaxCut by taking into account its SU(2) symmetry. We show that the hierarchy converges to the optimal QMaxCut value at a finite level, which is based on a new characterization of the algebra of SWAP operators. We give several analytic proofs and computational results showing exactness/inexactness of our hierarchy at the lowest level on several important families of graphs. We also discuss relationships between SDP approaches for QMaxCut and frustration-freeness in condensed matter physics and numerically demonstrate that the SDP-solvability practically becomes an efficiently-computable generalization of frustration-freeness. Furthermore, by numerical demonstration we show the potential of SDP algorithms to perform as an approximate method to compute physical quantities and capture physical features of some Heisenberg-type statistical mechanics models even away from the frustration-free regions.Understanding and approximating extremal energy states of local Hamiltonians is a central problem in quantum physics and complexity theory. Recent work has focused on developing approximation algorithms for local Hamiltonians, and in particular the ``Quantum Max Cut'' (QMax-Cut) problem, which is closely related to the antiferromagnetic Heisenberg model. In this work, we introduce a family of semidefinite programming (SDP) relaxations based on the Navascues-Pironio-Acin (NPA) hierarchy which is tailored for QMaxCut by taking into account its SU(2) symmetry. We show that the hierarchy converges to the optimal QMaxCut value at a finite level, which is based on a new characterization of the algebra of SWAP operators. We give several analytic proofs and computational results showing exactness/inexactness of our hierarchy at the lowest level on several important families of graphs. We also discuss relationships between SDP approaches for QMaxCut and frustration-freeness in condensed matter physics and numerically demonstrate that the SDP-solvability practically becomes an efficiently-computable generalization of frustration-freeness. Furthermore, by numerical demonstration we show the potential of SDP algorithms to perform as an approximate method to compute physical quantities and capture physical features of some Heisenberg-type statistical mechanics models even away from the frustration-free regions.

arXiv:2307.15688Jul 28, 20233 citations

2

2

hierarchy satisfies many physically-motivated constraints that the level-

1

2

2

hierarchy satisfies many physically-motivated constraints that the level-

1

We study quantum algorithms for approximating Lasserre's hierarchy values for polynomial optimization. Let

f,g_1,\ldots,g_m

be real polynomials in

n

variables and

f^\star

the infimum of

f

over the semialgebraic set

S(g)=\{x: g_i(x)\ge 0\}

. Let

λ_k

be the value of the order-

k

Lasserre relaxation. Assume either (i)

f^\star=λ_k

and the optimum is attained in the

\ell_1

-ball of radius

1/2

, or (ii)

S(g)

lies in the simplex

\{x\ge 0: \sum_j x_j\le 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

λ_k

to accuracy

\varepsilon>0

with runtime, for fixed

k

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),

where

s_g

bounds the sparsity of the coefficient-matching matrices associated with the constraints. Classical matrix multiplicative-weights methods scale as

O(n^{3k}\mathrm{poly}(1/\varepsilon))

even in the unconstrained case. As an example, we obtain an

O(n\varepsilon^{-4}+\sqrt{n}\varepsilon^{-5})

quantum algorithm for portfolio optimization, improving over the classical

O(n^{ω+1}\log(1/\varepsilon))

bound with

ω\approx2.373

. 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.

1

of the quantum Lasserre hierarchy, the relaxation achieves an approximation ratio of

0.526

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.

It is well-known in physics that the limit of large quantum spin

S

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-

S

generalizations of Quantum Max-Cut (

\mathrm{QMaxCut}_S

), equivalent to the problem of finding the ground state energy of an arbitrary spin-

S

quantum Heisenberg antiferromagnet (

\mathrm{QHA}_S

). We prove that approximating the value of

\mathrm{QHA}_S

to inverse polynomial accuracy is QMA-complete for all

S

, extending previous results for

S=1/2

. We also present two distinct families of classical approximation algorithms for

\mathrm{QMaxCut}_S

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

S

and converge to the Briët-Oliveira-Vallentin approximation ratio

α_{\mathrm{BOV}} \approx 0.956

from below as

S \to \infty