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arXiv:1701.06321

Quantum entanglement, sum of squares, and the log rank conjecture

Boaz Barak, Pravesh K. Kothari, David Steurer

Published Jan 23, 2017• 0 citations
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Abstract

For every $ε>0$, we give an $\exp(\tilde{O}(\sqrt{n}/ε^2))$-time algorithm for the $1$ vs $1-ε$ \emph{Best Separable State (BSS)} problem of distinguishing, given an $n^2\times n^2$ matrix $\mathcal{M}$ corresponding to a quantum measurement, between the case that there is a separable (i.e., non-entangled) state $ρ$ that $\mathcal{M}$ accepts with probability $1$, and the case that every separable state is accepted with probability at most $1-ε$. Equivalently, our algorithm takes the description of a subspace $\mathcal{W} \subseteq \mathbb{F}^{n^2}$ (where $\mathbb{F}$ can be either the real or complex field) and distinguishes between the case that $\mathcal{W}$ contains a rank one matrix, and the case that every rank one matrix is at least $ε$ far (in $\ell_2$ distance) from $\mathcal{W}$. To the best of our knowledge, this is the first improvement over the brute-force $\exp(n)$-time algorithm for this problem. Our algorithm is based on the \emph{sum-of-squares} hierarchy and its analysis is inspired by Lovett's proof (STOC '14, JACM '16) that the communication complexity of every rank-$n$ Boolean matrix is bounded by $\tilde{O}(\sqrt{n})$.

DOI
10.48550/arxiv.1701.06321

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For every ε>0ε>0ε>0, we give an exp⁡(O~(n/ε2))\exp(\tilde{O}(\sqrt{n}/ε^2))exp(O~(n​/ε2))-time algorithm for the 111 vs 1−ε1-ε1−ε \emph{Best Separable State (BSS)} problem of distinguishing, given an n2×n2n^2\times n^2n2×n2 matrix M\mathcal{M}M corresponding to a quantum measurement, between the case that there is a separable (i.e., non-entangled) state ρρρ that M\mathcal{M}M accepts with probability 111, and the case that every separable state is accepted with probability at most 1−ε1-ε1−ε. Equivalently, our algorithm takes the description of a subspace W⊆Fn2\mathcal{W} \subseteq \mathbb{F}^{n^2}W⊆Fn2 (where F\mathbb{F}F can be either the real or complex field) and distinguishes between the case that W\mathcal{W}W contains a rank one matrix, and the case that every rank one matrix is at least εεε far (in ℓ2\ell_2ℓ2​ distance) from W\mathcal{W}W. To the best of our knowledge, this is the first improvement over the brute-force exp⁡(n)\exp(n)exp(n)-time algorithm for this problem. Our algorithm is based on the \emph{sum-of-squares} hierarchy and its analysis is inspired by Lovett's proof (STOC '14, JACM '16) that the communication complexity of every rank-nnn Boolean matrix is bounded by O~(n)\tilde{O}(\sqrt{n})O~(n​).
We study the best separable state problem (BSS), which asks for the maximum acceptance probability of a quantum measurement over unentangled states. In classical terms, the goal is to maximize ⟨(x⊗y),M(x⊗y)⟩\langle(x \otimes y), M (x \otimes y)\rangle⟨(x⊗y),M(x⊗y)⟩ over unit vectors x,yx,yx,y where 0⪯M⪯I0 \preceq M \preceq I0⪯M⪯I; we call this value BSS(M)\mathrm{BSS}(M)BSS(M). We study BSS\mathrm{BSS}BSS in the "perfect completeness" regime, where given MMM such that BSS(M)=1\mathrm{BSS}(M) = 1BSS(M)=1 the goal is to find the best possible solution x,yx,yx,y -- this generalizes the problem of finding a rank-one matrix as close as possible to a given subspace of Rn×n\mathbb{R}^{n \times n}Rn×n guaranteed to contain a rank-one matrix. The strongest known algorithmic guarantees for this problem are: (1) an algorithm which finds a solution with value 1−ε1-\varepsilon1−ε in time exp⁡(n(log⁡n)O(1)/ε2)\exp(\sqrt{n} (\log n)^{O(1)} / \varepsilon^2)exp(n​(logn)O(1)/ε2), due to Barak, Kothari, and Steurer, and (2) an algorithm which finds a solution with value q/nq/nq/n in time roughly nO(q)n^{O(q)}nO(q), due to Bhattiprolu, Ghosh, Guruswami, Lee, and Tulsiani. We give a much simpler approach to rounding the SoS relaxation, generalizing the canonical "global correlation rounding" technique, and obtain a better running time. Given MMM with BSS(M)=1\mathrm{BSS}(M) = 1BSS(M)=1, our algorithm finds a solution with value 1−ε1-ε1−ε in time nO(n/ε)n^{O(\sqrt{n/\varepsilon})}nO(n/ε​), and a solution of value q/nq/nq/n in time nO(q)n^{O(\sqrt q)}nO(q​). Using the same techniques, we prove a new variant of the "pinning lemma", a measure-decomposition theorem widely used in LP/SDP rounding, high-dimensional probability, and statistical physics, which we believe is of independent interest.
For every constant ε>0, we give an exp(Õ(∞n))-time algorithm for the 1 vs 1 - ε Best Separable State (BSS) problem of distinguishing, given an n2 x n2 matrix ℳ corresponding to a quantum measurement, between the case that there is a separable (i.e., non-entangled) state ρ that ℳ accepts with probability 1, and the case that every separable state is accepted with probability at most 1 - ε. Equivalently, our algorithm takes the description of a subspace 𝒲 ⊆ 𝔽n2 (where 𝔽 can be either the real or complex field) and distinguishes between the case that contains a rank one matrix, and the case that every rank one matrix is at least ε far (in 𝓁2 distance) from 𝒲.
We fully determine the communication complexity of approximating matrix rank, over any finite field F. We study the most general version of this problem, where0⩽r<R⩽n0\leqslant r < R\leqslant n0⩽r<R⩽nare given integers and Alice and Bob need to determine whether their respective matricesA,B∈Fn×nA, B\in \mathbb{F}^{n\times n}A,B∈Fn×nsatisfy rk(A+B)=r(A+B)=r(A+B)=rversus rk(A+B)=R(A+B)=R(A+B)=R. We show that this problem has commu-nication costΩ(r2log⁡∣F∣)\Omega(r^{2}\log\vert \mathbb{F}\vert)Ω(r2log∣F∣), which is optimal. Our lower bound holds even for quantum protocols and even for error probability12(1−∣F∣−r/3)\frac{1}{2}(1-\vert \mathbb{F}\vert ^{-r/3})21​(1−∣F∣−r/3), which too is optimal because this problem has a two-bit classical protocol with error12(1−∣F∣−Θ(r))\frac{1}{2}(1-\vert \mathbb{F}\vert ^{-\Theta(r)})21​(1−∣F∣−Θ(r)). Prior to our work, lower bounds were known only for constant-error protocols and only for consecutive integersrrrandRRR, with no implication for the approximation of matrix rank. We also settle an analogous question for subspaces, where Alice has a subspaceSSS, Bob has a subspaceTTT) and they need to approximate the dimension of the subspaceS+TS+TS+Tgenerated bySSSandTTT(equivalently, approximate the dimension ofS∩TS\cap TS∩T). As an application, we obtain anΩ(n2log⁡∣F∣)/k\Omega(n^{2}\log\vert \mathbb{F}\vert)/kΩ(n2log∣F∣)/kmemory lower bound for any streaming algorithm withkkkpasses that approximates the rank of an input matrixM∈Fn×nM\in \mathbb{F}^{n\times n}M∈Fn×nwithin a factor of2−δ\sqrt{2}-\delta2​−δ, for anyδ>0\delta > 0δ>0. Our result is an exponential improvement inkkkover previous work.
One of the best lower bound methods for the quantum communication complexity of a function H (with or without shared entanglement) is the logarithm of the approximate rank of the communication matrix of H. This measure is essentially equivalent to the approximate gamma-2 norm and generalized discrepancy, and subsumes several other lower bounds. All known lower bounds on quantum communication complexity in the general unbounded-round model can be shown via the logarithm of approximate rank, and it was an open problem to give any separation at all between quantum communication complexity and the logarithm of the approximate rank. In this work we provide the first such separation: We exhibit a total function H with quantum communication complexity almost quadratically larger than the logarithm of its approximate rank. We construct H using the communication lookup function framework of Anshu et al. (FOCS 2016) based on the cheat sheet framework of Aaronson et al. (STOC 2016). From a starting function F, this framework defines a new function H=F_G. Our main technical result is a lower bound on the quantum communication complexity of F_G in terms of the discrepancy of F, which we do via quantum information theoretic arguments. We show the upper bound on the approximate rank of F_G by relating it to the Boolean circuit size of the starting function F.
One of the strongest techniques available for showing lower bounds on quantum communication complexity is the logarithm of the approximation rank of the communication matrix--the minimum rank of a matrix which is entrywise close to the communication matrix. This technique has two main drawbacks: it is difficult to compute, and it is not known to lower bound quantum communication complexity with entanglement. Linial and Shraibman recently introduced a norm, called gamma_2^{alpha}, to quantum communication complexity, showing that it can be used to lower bound communication with entanglement. Here the parameter alpha is a measure of approximation which is related to the allowable error probability of the protocol. This bound can be written as a semidefinite program and gives bounds at least as large as many techniques in the literature, although it is smaller than the corresponding alpha-approximation rank, rk_alpha. We show that in fact log gamma_2^{alpha}(A)$ and log rk_{alpha}(A)$ agree up to small factors. As corollaries we obtain a constant factor polynomial time approximation algorithm to the logarithm of approximate rank, and that the logarithm of approximation rank is a lower bound for quantum communication complexity with entanglement.
Abstract We consider the problem of maximizing a homogeneous polynomial on the unit sphere and its hierarchy of sum-of-squares relaxations. Exploiting the polynomial kernel technique, we obtain a quadratic improvement of the known convergence rate by Reznick and Doherty and Wehner. Specifically, we show that the rate of convergence is no worse than O(d2/ℓ2)O(d^2/\ell ^2)O(d2/ℓ2) O ( d 2 / ℓ 2 ) in the regime ℓ=Ω(d)\ell = \Omega (d)ℓ=Ω(d) ℓ = Ω ( d ) where ℓ\ell ℓ ℓ is the level of the hierarchy and d the dimension, solving a problem left open in the recent paper by de Klerk and Laurent ( arXiv:1904.08828 ). Importantly, our analysis also works for matrix-valued polynomials on the sphere which has applications in quantum information for the Best Separable State problem. By exploiting the duality relation between sums of squares and the Doherty–Parrilo–Spedalieri hierarchy in quantum information theory, we show that our result generalizes to nonquadratic polynomials the convergence rates of Navascués, Owari and Plenio.