We prove that
We analyse the eigenvectors of the adjacency matrix of the Erdős–Rényi graph
- NSF-PAR ID:
- 10486679
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Communications in Mathematical Physics
- Volume:
- 405
- Issue:
- 1
- ISSN:
- 0010-3616
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract -depth local random quantum circuits with two qudit nearest-neighbor gates on a$${{\,\textrm{poly}\,}}(t) \cdot n^{1/D}$$ D -dimensional lattice withn qudits are approximatet -designs in various measures. These include the “monomial” measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was due to Brandão–Harrow–Horodecki (Commun Math Phys 346(2):397–434, 2016) for$${{\,\textrm{poly}\,}}(t)\cdot n$$ . We also improve the “scrambling” and “decoupling” bounds for spatially local random circuits due to Brown and Fawzi (Scrambling speed of random quantum circuits, 2012). One consequence of our result is that assuming the polynomial hierarchy ($$D=1$$ ) is infinite and that certain counting problems are$${{\,\mathrm{\textsf{PH}}\,}}$$ -hard “on average”, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constant-depth quantum circuits was known to be hard for classical computers under these assumptions. However the standard strategy for extending this hardness result to approximate sampling requires the quantum circuits to have a property called “anti-concentration”, meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Our result improves the required depth for this level of anti-concentration from linear depth to a sub-linear value, depending on the geometry of the interactions. This is relevant to a recent experiment by the Google Quantum AI group to perform such a sampling task with 53 qubits on a two-dimensional lattice (Arute in Nature 574(7779):505–510, 2019; Boixo et al. in Nate Phys 14(6):595–600, 2018) (and related experiments by USTC), and confirms their conjecture that$$\#{\textsf{P}}$$ depth suffices for anti-concentration. The proof is based on a previous construction of$$O(\sqrt{n})$$ t -designs by Brandão et al. (2016), an analysis of how approximate designs behave under composition, and an extension of the quasi-orthogonality of permutation operators developed by Brandão et al. (2016). Different versions of the approximate design condition correspond to different norms, and part of our contribution is to introduce the norm corresponding to anti-concentration and to establish equivalence between these various norms for low-depth circuits. For random circuits with long-range gates, we use different methods to show that anti-concentration happens at circuit size corresponding to depth$$O(n\ln ^2 n)$$ . We also show a lower bound of$$O(\ln ^3 n)$$ for the size of such circuit in this case. We also prove that anti-concentration is possible in depth$$\Omega (n \ln n)$$ (size$$O(\ln n \ln \ln n)$$ ) using a different model.$$O(n \ln n \ln \ln n)$$ -
Abstract We study the distribution over measurement outcomes of noisy random quantum circuits in the regime of low fidelity, which corresponds to the setting where the computation experiences at least one gate-level error with probability close to one. We model noise by adding a pair of weak, unital, single-qubit noise channels after each two-qubit gate, and we show that for typical random circuit instances, correlations between the noisy output distribution
and the corresponding noiseless output distribution$$p_{\text {noisy}}$$ shrink exponentially with the expected number of gate-level errors. Specifically, the linear cross-entropy benchmark$$p_{\text {ideal}}$$ F that measures this correlation behaves as , where$$F=\text {exp}(-2s\epsilon \pm O(s\epsilon ^2))$$ is the probability of error per circuit location and$$\epsilon $$ s is the number of two-qubit gates. Furthermore, if the noise is incoherent—for example, depolarizing or dephasing noise—the total variation distance between the noisy output distribution and the uniform distribution$$p_{\text {noisy}}$$ decays at precisely the same rate. Consequently, the noisy output distribution can be approximated as$$p_{\text {unif}}$$ . In other words, although at least one local error occurs with probability$$p_{\text {noisy}}\approx Fp_{\text {ideal}}+ (1-F)p_{\text {unif}}$$ , the errors are scrambled by the random quantum circuit and can be treated as global white noise, contributing completely uniform output. Importantly, we upper bound the average total variation error in this approximation by$$1-F$$ . Thus, the “white-noise approximation” is meaningful when$$O(F\epsilon \sqrt{s})$$ , a quadratically weaker condition than the$$\epsilon \sqrt{s} \ll 1$$ requirement to maintain high fidelity. The bound applies if the circuit size satisfies$$\epsilon s\ll 1$$ , which corresponds to only$$s \ge \Omega (n\log (n))$$ logarithmic depth circuits, and if, additionally, the inverse error rate satisfies , which is needed to ensure errors are scrambled faster than$$\epsilon ^{-1} \ge {\tilde{\Omega }}(n)$$ F decays. The white-noise approximation is useful for salvaging the signal from a noisy quantum computation; for example, it was an underlying assumption in complexity-theoretic arguments that noisy random quantum circuits cannot be efficiently sampled classically, even when the fidelity is low. Our method is based on a map from second-moment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds. -
By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid
has -distortion bounded below by a constant multiple of . We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number , then the 1-Wasserstein metric over has -distortion bounded below by a constant multiple of . We proceed to compute these dimensions for -powers of certain graphs. In particular, we get that the sequence of diamond graphs has isoperimetric dimension and Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the 1-Wasserstein metric over has -distortion bounded below by a constant multiple of . This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of -embeddable graphs whose sequence of 1-Wasserstein metrics is not -embeddable. -
Abstract Leptoquarks (
s) are hypothetical particles that appear in various extensions of the Standard Model (SM), that can explain observed differences between SM theory predictions and experimental results. The production of these particles has been widely studied at various experiments, most recently at the Large Hadron Collider (LHC), and stringent bounds have been placed on their masses and couplings, assuming the simplest beyond-SM (BSM) hypotheses. However, the limits are significantly weaker for$$\textrm{LQ}$$ models with family non-universal couplings containing enhanced couplings to third-generation fermions. We present a new study on the production of a$$\textrm{LQ}$$ at the LHC, with preferential couplings to third-generation fermions, considering proton-proton collisions at$$\textrm{LQ}$$ and$$\sqrt{s} = 13 \, \textrm{TeV}$$ . Such a hypothesis is well motivated theoretically and it can explain the recent anomalies in the precision measurements of$$\sqrt{s} = 13.6 \, \textrm{TeV}$$ -meson decay rates, specifically the$$\textrm{B}$$ ratios. Under a simplified model where the$$R_{D^{(*)}}$$ masses and couplings are free parameters, we focus on cases where the$$\textrm{LQ}$$ decays to a$$\textrm{LQ}$$ lepton and a$$\tau $$ quark, and study how the results are affected by different assumptions about chiral currents and interference effects with other BSM processes with the same final states, such as diagrams with a heavy vector boson,$$\textrm{b}$$ . The analysis is performed using machine learning techniques, resulting in an increased discovery reach at the LHC, allowing us to probe new physics phase space which addresses the$$\textrm{Z}^{\prime }$$ -meson anomalies, for$$\textrm{B}$$ masses up to$$\textrm{LQ}$$ , for the high luminosity LHC scenario.$$5.00\, \textrm{TeV}$$ -
Abstract Approximate integer programming is the following: For a given convex body
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