We prove that poly(t) • n 1/D -depth local random quantum circuits with two qudit nearest-neighbor gates on a D-dimensional lattice with n qudits are approximate t-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 poly(t) • n due to Brandão-Harrow-Horodecki [10] for D = 1. We also improve the "scrambling" and "decoupling" bounds for spatially local random circuits due to Brown and Fawzi [14].One consequence of our result is that assuming the polynomial hierarchy (PH) is infinite and that certain counting problems are #P-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 the assumption that PH is infinite. However, to show the hardness of approximate sampling using this strategy requires that the quantum circuits have a property called "anti-concentration", meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Thus 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 proposal by the Google Quantum AI group to perform such a sampling task with 49 qubits on a two-dimensional lattice [7], and confirms their conjecture that O( √ n) depth suffices for anti-concentration.The proof is based on a previous construction of t-designs by [10], an analysis of how approximate designs behave under composition, and an extension of the quasi-orthogonality of permutation operators developed by [10]. 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 O(n ln 2 n) corresponding to depth O(ln 3 n). We also show a lower bound of Ω(n ln n) for the size of such circuit in this case. We also prove that anti-concentration is possible in depth O(ln n ln ln n) (size O(n ln n ln ln n)) using a different model.
We present a quasi-polynomial time classical algorithm that estimates the partition function of quantum many-body systems at temperatures above the thermal phase transition point. It is known that in the worst case, the same problem is NP-hard below this point. Together with our work, this shows that the transition in the phase of a quantum system is also accompanied by a transition in the hardness of approximation. We also show that in a system of n particles above the phase transition point, the correlation between two observables whose distance is at least Ω(log n) decays exponentially. We can improve the factor of log n to a constant when the Hamiltonian has commuting terms or is on a 1D chain. The key to our results is a characterization of the phase transition and the critical behavior of the system in terms of the complex zeros of the partition function. Our work extends a seminal work of Dobrushin and Shlosman on the equivalence between the decay of correlations and the analyticity of the free energy in classical spin models. On the algorithmic side, our result extends the scope of a recent approach due to Barvinok for solving classical counting problems to quantum many-body systems.
The permanent is #P -hard to compute exactly on average for natural random matrices including matrices over finite fields or Gaussian ensembles. Should we expect that it remains #P -hard to compute on average if we only care about approximation instead of exact computation?In this work we take a first step towards resolving this question: We present a quasipolynomial time deterministic algorithm for approximating the permanent of a typical n × n random matrix with unit variance and vanishing mean µ = O(ln ln n) −1/8 to within inverse polynomial multiplicative error. Alternatively, one can achieve permanent approximation for matrices with mean µ = 1/polylog(n) in time 2 O(n ε ) , for any ε > 0.The proposed algorithm significantly extends the regime of matrices for which efficient approximation of the permanent is known. This is because unlike previous algorithms which require a stringent correlation between the signs of the entries of the matrix [1, 2] it can tolerate random ensembles in which this correlation is negligible (albeit non-zero). Among important special cases we note:1. Biased Gaussian: each entry is a complex Gaussian with unit variance 1 and mean µ.2. Biased Bernoulli: each entry is −1 + µ with probability 1/2, and 1 with probability 1/2.
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