A Generalized Sylvester-Gallai Type Theorem for Quadratic Polynomials

In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of ?^{[3]}???^{[2]} circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials ? satisfy that for every two polynomials Q?,Q? ? ? there is a subset ? ? ?, such that Q?,Q? ? ? and whenever Q? and Q? vanish then ?_{Q_i??} Q_i vanishes, then the linear span of the polynomials in ? has dimension O(1). This extends the earlier result [Amir Shpilka, 2019] that showed a similar conclusion when |?| = 1. An important technical step in our proof is a theorem classifying all the possible cases in which a product of quadratic polynomials can vanish when two other quadratic polynomials vanish. I.e., when the product is in the radical of the ideal generated by the two quadratics. This step extends a result from [Amir Shpilka, 2019] that studied the case when one quadratic polynomial is in the radical of two other quadratics

Lower Bounds on Stabilizer Rank

The stabilizer rank of a quantum state ? is the minimal r such that |?? = ?_{j = 1}^r c_j |?_j? for c_j ? ? and stabilizer states ?_j. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the n-th tensor power of single-qubit magic states. We prove a lower bound of ?(n) on the stabilizer rank of such states, improving a previous lower bound of ?(?n) of Bravyi, Smith and Smolin [Bravyi et al., 2016]. Further, we prove that for a sufficiently small constant ?, the stabilizer rank of any state which is ?-close to those states is ?(?n/log n). This is the first non-trivial lower bound for approximate stabilizer rank. Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of ???, and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function

Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials

In this work, we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if $\mathcal{Q}\subset \mathbb{C}[x_1.\ldots,x_n]$ is a finite set, $|\mathcal{Q}|=m$, of irreducible quadratic polynomials that satisfy the following condition: There is $\delta>0$ such that for every $Q\in\mathcal{Q}$ there are at least $\delta m$ polynomials $P\in \mathcal{Q}$ such that whenever $Q$ and $P$ vanish then so does a third polynomial in $\mathcal{Q}\setminus\{Q,P\}$, then $\dim(\text{span}({\mathcal{Q}}))=\text{poly}(1/\delta)$. The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of $O(1/\delta)$ on the dimension (in the first work an upper bound of $O(1/\delta^2)$ was given, which was improved to $O(1/\delta)$ in the second work).Comment: arXiv admin note: text overlap with arXiv:2006.0826

Sylvester-Gallai Type Theorems for Quadratic Polynomials

We prove Sylvester-Gallai type theorems for quadratic polynomials. Specifically, we prove that if a finite collection Q, of irreducible polynomials of degree at most 2, satisfy that for every two polynomials Q1,Q2 â Q there is a third polynomial Q3â Q so that whenever Q1 and Q2 vanish then also Q3 vanishes, then the linear span of the polynomials in Q has dimension O(1). We also prove a colored version of the theorem: If three finite sets of quadratic polynomials satisfy that for every two polynomials from distinct sets there is a polynomial in the third set satisfying the same vanishing condition then all polynomials are contained in an O(1)-dimensional space. This answers affirmatively two conjectures of Gupta [Electronic Colloquium on Computational Complexity (ECCC), 21:130, 2014] that were raised in the context of solving certain depth-4 polynomial identities. To obtain our main theorems we prove a new result classifying the possible ways that a quadratic polynomial Q can vanish when two other quadratic polynomials vanish. Our proofs also require robust versions of a theorem of Edelstein and Kelly (that extends the Sylvester-Gallai theorem to colored sets).Non UBCUnreviewedAuthor affiliation: Tel Aviv UFacult

Real-time artificial intelligence-based guidance of echocardiographic imaging by novices: Image quality and suitability for diagnostic interpretation and quantitative analysis

Background:We aimed to assess in a prospective multicenter study the quality of echocardiographic exams performed by inexperienced users guided by a new artificial intelligence software and evaluate their suitability for diagnostic interpretation of basic cardiac pathology and quantitative analysis of cardiac chamber and function. Methods:The software (UltraSight, Ltd) was embedded into a handheld imaging device (Lumify; Philips). Six nurses and 3 medical residents, who underwent minimal training, scanned 240 patients (61±16 years; 63% with cardiac pathology) in 10 standard views. All patients were also scanned by expert sonographers using the same device without artificial intelligence guidance. Studies were reviewed by 5 certified echocardiographers blinded to the imager\u27s identity, who evaluated the ability to assess left and right ventricular size and function, pericardial effusion, valve morphology, and left atrial and inferior vena cava sizes. Finally, apical 4-chamber images of adequate quality, acquired by novices and sonographers in 100 patients, were analyzed to measure left ventricular volumes, ejection fraction, and global longitudinal strain by an expert reader using conventional methodology. Measurements were compared between novices\u27 and experts\u27 images. Results:Of the 240 studies acquired by novices, 99.2%, 99.6%, 92.9%, and 100% had sufficient quality to assess left ventricular size and function, right ventricular size, and pericardial effusion, respectively. Valve morphology, right ventricular function, and left atrial and inferior vena cava size were visualized in 67% to 98% exams. Images obtained by novices and sonographers yielded concordant diagnostic interpretation in 83% to 96% studies. Quantitative analysis was feasible in 83% images acquired by novices and resulted in high correlations (r≥0.74) and small biases, compared with those obtained by sonographers. Conclusions:After minimal training with the real-time guidance software, novice users can acquire images of diagnostic quality approaching that of expert sonographers in most patients. This technology may increase adoption and improve accuracy of point-of-care cardiac ultrasound