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1. Low rank representations for quantum simulation of electronic structure

   https://doi.org/10.1038/s41534-021-00416-z  

   Motta, Mario ; Ye, Erika ; McClean, Jarrod R. ; Li, Zhendong ; Minnich, Austin J. ; Babbush, Ryan ; Chan, Garnet Kin-Lic ( May 2021 , npj Quantum Information)

   The quantum simulation of quantum chemistry is a promising application of quantum computers. However, forNmolecular orbitals, the$${\mathcal{O}}({N}^{4})$$$O\left(N^4\right)$gate complexity of performing Hamiltonian and unitary Coupled Cluster Trotter steps makes simulation based on such primitives challenging. We substantially reduce the gate complexity of such primitives through a two-step low-rank factorization of the Hamiltonian and cluster operator, accompanied by truncation of small terms. Using truncations that incur errors below chemical accuracy allow one to perform Trotter steps of the arbitrary basis electronic structure Hamiltonian with$${\mathcal{O}}({N}^{3})$$$O\left(N^3\right)$gate complexity in small simulations, which reduces to$${\mathcal{O}}({N}^{2})$$$O\left(N^2\right)$gate complexity in the asymptotic regime; and unitary Coupled Cluster Trotter steps with$${\mathcal{O}}({N}^{3})$$$O\left(N^3\right)$gate complexity as a function of increasing basis size for a given molecule. In the case of the Hamiltonian Trotter step, these circuits have$${\mathcal{O}}({N}^{2})$$$O\left(N^2\right)$depth on a linearly connected array, an improvement over the$${\mathcal{O}}({N}^{3})$$$O\left(N^3\right)$scaling assuming no truncation. As a practical example, we show that a chemically accurate Hamiltonian Trotter step for a 50 qubit molecular simulation can be carried out in the molecular orbital basis with as few as 4000 layers of parallel nearest-neighbor two-qubit gates, consisting of fewer than 10⁵non-Clifford rotations. We also apply our algorithm to iron–sulfur clusters relevant for elucidating the mode of action of metalloenzymes.

    

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2. Random Quantum Circuits Transform Local Noise into Global White Noise

   https://doi.org/10.1007/s00220-024-04958-z  

   Dalzell, Alexander M. ; Hunter-Jones, Nicholas ; Brandão, Fernando G. S. L. ( March 2024 , Communications in Mathematical Physics)

   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$$p_{\text {noisy}}$$$p_{\text{noisy}}$and the corresponding noiseless output distribution$$p_{\text {ideal}}$$$p_{\text{ideal}}$shrink exponentially with the expected number of gate-level errors. Specifically, the linear cross-entropy benchmarkFthat measures this correlation behaves as$$F=\text {exp}(-2s\epsilon \pm O(s\epsilon ^2))$$$F=\text{exp}(-2sϵ\pm O\left(s{ϵ}^2\right))$, where$$\epsilon $$$ϵ$is the probability of error per circuit location andsis 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$$p_{\text {noisy}}$$$p_{\text{noisy}}$and the uniform distribution$$p_{\text {unif}}$$$p_{\text{unif}}$decays at precisely the same rate. Consequently, the noisy output distribution can be approximated as$$p_{\text {noisy}}\approx Fp_{\text {ideal}}+ (1-F)p_{\text {unif}}$$$p_{\text{noisy}}\approx F{p}_{\text{ideal}}+(1-F)p_{\text{unif}}$. In other words, although at least one local error occurs with probability$$1-F$$$1-F$, 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$$O(F\epsilon \sqrt{s})$$$O\left(Fϵ\sqrt{s}\right)$. Thus, the “white-noise approximation” is meaningful when$$\epsilon \sqrt{s} \ll 1$$$ϵ\sqrt{s}\ll 1$, a quadratically weaker condition than the$$\epsilon s\ll 1$$$ϵs\ll 1$requirement to maintain high fidelity. The bound applies if the circuit size satisfies$$s \ge \Omega (n\log (n))$$$s\ge \Omega (nlog(n\left)\right)$, which corresponds to onlylogarithmic depthcircuits, and if, additionally, the inverse error rate satisfies$$\epsilon ^{-1} \ge {\tilde{\Omega }}(n)$$${ϵ}^{-1}\ge \tilde{\Omega }\left(n\right)$, which is needed to ensure errors are scrambled faster thanFdecays. 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.

    

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3. Approximate Unitary t-Designs by Short Random Quantum Circuits Using Nearest-Neighbor and Long-Range Gates

   https://doi.org/10.1007/s00220-023-04675-z  

   Harrow, Aram W. ; Mehraban, Saeed ( May 2023 , Communications in Mathematical Physics)

   We prove that$${{\,\textrm{poly}\,}}(t) \cdot n^{1/D}$$$\text{poly}\left(t\right)·n^{1/D}$-depth local random quantum circuits with two qudit nearest-neighbor gates on aD-dimensional lattice withnqudits 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$${{\,\textrm{poly}\,}}(t)\cdot n$$$\text{poly}\left(t\right)·n$due to Brandão–Harrow–Horodecki (Commun Math Phys 346(2):397–434, 2016) for$$D=1$$$D=1$. 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 ($${{\,\mathrm{\textsf{PH}}\,}}$$$\mathrm{PH}$) is infinite and that certain counting problems are$$\#{\textsf{P}}$$$\#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 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$$O(\sqrt{n})$$$O\left(\sqrt{n}\right)$depth suffices for anti-concentration. The proof is based on a previous construction oft-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$$O(n\ln ^2 n)$$$O\left(n{ln}^{2}n\right)$corresponding to depth$$O(\ln ^3 n)$$$O\left({ln}^{3}n\right)$. We also show a lower bound of$$\Omega (n \ln n)$$$\Omega (nlnn)$for the size of such circuit in this case. We also prove that anti-concentration is possible in depth$$O(\ln n \ln \ln n)$$$O(lnnlnlnn)$(size$$O(n \ln n \ln \ln n)$$$O(nlnnlnlnn)$) using a different model.

    

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4. Bounds on eigenfunctions of quantum cat maps

   https://doi.org/10.1088/1402-4896/ad0331  

   Kim, Elena ; Koirala, Robert ( October 2023 , Physica Scripta)

   We studyℓ^∞norms ofℓ²-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Biévre in F Bonechi and S De Bièvre (2000,Communications in Mathematical Physics,211, 659–686)) we show that there exists a sequence of eigenfunctionsuwith$\parallel u{\parallel }_{\infty }\gtrsim {\left(\log N\right)}^{-1/2}$. For general eigenfunctions we show the upper bound$\parallel u{\parallel }_{\infty }\lesssim {\left(\log N\right)}^{-1/2}$. Here the semiclassical parameter is$h={\left(2\pi N\right)}^{-1}$. Our upper bound is analogous to the one proved by Bérard in P Bérard (1977,Mathematische Zeitschrift,155, 249-276) for compact Riemannian manifolds without conjugate points.

    

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5. High-harmonic generation in spin and charge current pumping at ferromagnetic or antiferromagnetic resonance in the presence of spin–orbit coupling

   https://doi.org/10.1088/2515-7639/aceaad  

   Varela-Manjarres, Jalil ; Nikolić, Branislav K. ( August 2023 , Journal of Physics: Materials)

   One of the cornerstone effects in spintronics is spin pumping by dynamical magnetization that is steadily precessing (around, for example, thez-axis) with frequencyω₀due to absorption of low-power microwaves of frequencyω₀under the resonance conditions and in the absence of any applied bias voltage. The two-decades-old ‘standard model’ of this effect, based on the scattering theory of adiabatic quantum pumping, predicts that component$I^{S_z}$of spin current vector$(I^{S_x}(t),I^{S_y}(t),I^{S_z})\propto {\omega }_0$is time-independent while$I^{S_x}(t)$and$I^{S_y}(t)$oscillate harmonically in time with a single frequencyω₀whereas pumped charge current is zero$I\equiv 0$in the same adiabatic$\propto {\omega }_0$limit. Here we employ more general approaches than the ‘standard model’, namely the time-dependent nonequilibrium Green’s function (NEGF) and the Floquet NEGF, to predict unforeseen features of spin pumping: namely precessing localized magnetic moments within a ferromagnetic metal (FM) or antiferromagnetic metal (AFM), whose conduction electrons are exposed to spin–orbit coupling (SOC) of either intrinsic or proximity origin, will pump both spin$I^{S_{\alpha }}(t)$and chargeI(t) currents. All four of these functions harmonically oscillate in time at both even and odd integer multiples$N{\omega }_0$of the driving frequencyω₀. The cutoff order of such high harmonics increases with SOC strength, reaching$N_{\mathrm{m}\mathrm{a}\mathrm{x}}\simeq 11$in the one-dimensional FM or AFM models chosen for demonstration. A higher cutoff$N_{\mathrm{m}\mathrm{a}\mathrm{x}}\simeq 25$can be achieved in realistic two-dimensional (2D) FM models defined on a honeycomb lattice, and we provide a prescription of how to realize them using 2D magnets and their heterostructures.

    

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