Maximizing products of linear forms, and the permanent of positive semidefinite matrices

Yuan, Chenyang; Parrilo, Pablo A.

doi:10.1007/s10107-021-01616-3

Cited by 4 publications

(5 citation statements)

References 13 publications

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“…This result is complementary to [28], where Yuan and Parrilo show that one can approximate within a factor of exp((1 + γ + o(1))n) where γ is the Euler-Mascheroni constant. This leaves the question of what the optimal base c is that can still be efficiently approximated.…”

Section: Inapproximability Of Quantum-bayesian-updatesupporting

confidence: 53%

Inapproximability of Positive Semidefinite Permanents and Quantum State Tomography

Meiburg¹

2021

Preprint

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Quantum State Tomography is the task of estimating a quantum state, given many measurements in different bases. We discuss a few variants of what exactly "estimating a quantum state" means, including maximum likelihood estimation and computing a Bayesian average. We show that, when the measurements are fixed, this problem is NP-Hard to approximate within any constant factor. In the process, we find that it reduces to the problem of approximately computing the permanent of a Hermitian positive semidefinite (HPSD) matrix. This implies that HPSD permanents are also NP-Hard to approximate, resolving a standing question with applications in quantum information and BosonSampling.

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Section: Inapproximability Of Quantum-bayesian-updatesupporting

confidence: 53%

Inapproximability of Positive Semidefinite Permanents and Quantum State Tomography

Meiburg¹

2021

Preprint

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“…Thus we can approximate per(M ) by analyzing the approximation quality of rel(M ) as a relaxation of r(M ). It is easy to see that r(M ) is equivalent to a special case of (1) when the A i are all rank-1, and the result of Theorem 1.3 applied to this problem gives the same approximation factor to the permanent as [YP21].…”

Section: Permanents Of Psd Matricesmentioning

confidence: 95%

“…There has been recent attention on problems similar to (1). The authors of this paper analyzed a special case of (1) where the A i are rank-1 matrices, used in an approximation algorithm for the permanent of PSD matrices [YP21] (see Section 2.4 for more details). Barvinok [Bar93] reduced the problem of certifying feasibility for systems of quadratic equations to finding the optimum of (1), and provided a polynomial time algorithm for solving (1) when d is fixed.…”

Section: Related Workmentioning

confidence: 99%

“…Where the sum is over all permutations of n elements. If M is Hermitian positive semidefinite (PSD), [AGGS17] and [YP21] analyzed a SDP-based approximation algorithm that produces a simply exponential approximation factor to per(M ). Let M = V † V and v i are the columns of V .…”

Section: Permanents Of Psd Matricesmentioning

confidence: 99%

“…Let M = V † V and v i are the columns of V . In [YP21], the problem of approximating per(M ) is related to the problem of maximizing a product of linear forms over the complex sphere r(M ) := max…”

Section: Permanents Of Psd Matricesmentioning

confidence: 99%

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Semidefinite Relaxations of Products of Nonnegative Forms on the Sphere

Yuan,

Parrilo

2021

Preprint

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We study the problem of maximizing the geometric mean of d low-degree non-negative forms on the real or complex sphere in n variables. We show that this highly non-convex problem is NP-hard even when the forms are quadratic and is equivalent to optimizing a homogeneous polynomial of degree O(d) on the sphere. The standard Sum-of-Squares based convex relaxation for this polynomial optimization problem requires solving a semidefinite program (SDP) of size n O(d) , with multiplicative approximation guarantees of Ω( 1n ). We exploit the compact representation of this polynomial to introduce a SDP relaxation of size polynomial in n and d, and prove that it achieves a constant factor multiplicative approximation when maximizing the geometric mean of non-negative quadratic forms. We also show that this analysis is asymptotically tight, with a sequence of instances where the gap between the relaxation and true optimum approaches this constant factor as d → ∞. Next we propose a series of intermediate relaxations of increasing complexity that interpolate to the full Sum-of-Squares relaxation, as well as a rounding algorithm that finds an approximate solution from the solution of any intermediate relaxation. Finally we show that this approach can be generalized for relaxations of products of non-negative forms of any degree.

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Inapproximability of Positive Semidefinite Permanents and Quantum State Tomography

Meiburg

2023

Algorithmica

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Matrix permanents are hard to compute or even estimate in general. It had been previously suggested that the permanents of Positive Semidefinite (PSD) matrices may have efficient approximations. By relating PSD permanents to a task in quantum state tomography, we show that PSD permanents are NP-hard to approximate within a constant factor, and so admit no polynomial-time approximation scheme (unless P = NP). We also establish that several natural tasks in quantum state tomography, even approximately, are NP-hard in the dimension of the Hilbert space. These state tomography tasks therefore remain hard even with only logarithmically few qubits.

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Maximizing products of linear forms, and the permanent of positive semidefinite matrices

Cited by 4 publications

References 13 publications

Inapproximability of Positive Semidefinite Permanents and Quantum State Tomography

Inapproximability of Positive Semidefinite Permanents and Quantum State Tomography

Semidefinite Relaxations of Products of Nonnegative Forms on the Sphere

Inapproximability of Positive Semidefinite Permanents and Quantum State Tomography

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