The quantum simulation of quantum chemistry is a promising application of quantum computers. However, for
We present a new quantum adiabatic theorem that allows one to rigorously bound the adiabatic timescale for a variety of systems, including those described by originally unbounded Hamiltonians that are made finitedimensional by a cutoff. Our bound is geared towards the qubit approximation of superconducting circuits and presents a sufficient condition for remaining within the
This article is part of the theme issue ‘Quantum annealing and computation: challenges and perspectives’.
more » « less Award ID(s):
 1936388
 NSFPAR ID:
 10473403
 Publisher / Repository:
 Royal Society Publishing
 Date Published:
 Journal Name:
 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
 Volume:
 381
 Issue:
 2241
 ISSN:
 1364503X
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract N molecular orbitals, the 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 twostep lowrank 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$${\mathcal{O}}({N}^{4})$$ $O\left({N}^{4}\right)$ gate complexity in small simulations, which reduces to$${\mathcal{O}}({N}^{3})$$ $O\left({N}^{3}\right)$ gate complexity in the asymptotic regime; and unitary Coupled Cluster Trotter steps with$${\mathcal{O}}({N}^{2})$$ $O\left({N}^{2}\right)$ gate complexity as a function of increasing basis size for a given molecule. In the case of the Hamiltonian Trotter step, these circuits have$${\mathcal{O}}({N}^{3})$$ $O\left({N}^{3}\right)$ depth on a linearly connected array, an improvement over the$${\mathcal{O}}({N}^{2})$$ $O\left({N}^{2}\right)$ 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 nearestneighbor twoqubit gates, consisting of fewer than 10^{5}nonClifford rotations. We also apply our algorithm to iron–sulfur clusters relevant for elucidating the mode of action of metalloenzymes.$${\mathcal{O}}({N}^{3})$$ $O\left({N}^{3}\right)$ 
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 gatelevel error with probability close to one. We model noise by adding a pair of weak, unital, singlequbit noise channels after each twoqubit gate, and we show that for typical random circuit instances, correlations between the noisy output distribution
and the corresponding noiseless output distribution$$p_{\text {noisy}}$$ ${p}_{\text{noisy}}$ shrink exponentially with the expected number of gatelevel errors. Specifically, the linear crossentropy benchmark$$p_{\text {ideal}}$$ ${p}_{\text{ideal}}$F that measures this correlation behaves as , where$$F=\text {exp}(2s\epsilon \pm O(s\epsilon ^2))$$ $F=\text{exp}(2s\u03f5\pm O\left(s{\u03f5}^{2}\right))$ is the probability of error per circuit location and$$\epsilon $$ $\u03f5$s is the number of twoqubit 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$$p_{\text {noisy}}$$ ${p}_{\text{noisy}}$ decays at precisely the same rate. Consequently, the noisy output distribution can be approximated as$$p_{\text {unif}}$$ ${p}_{\text{unif}}$ . In other words, although at least one local error occurs with probability$$p_{\text {noisy}}\approx Fp_{\text {ideal}}+ (1F)p_{\text {unif}}$$ ${p}_{\text{noisy}}\approx F{p}_{\text{ideal}}+(1F){p}_{\text{unif}}$ , 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$$1F$$ $1F$ . Thus, the “whitenoise approximation” is meaningful when$$O(F\epsilon \sqrt{s})$$ $O\left(F\u03f5\sqrt{s}\right)$ , a quadratically weaker condition than the$$\epsilon \sqrt{s} \ll 1$$ $\u03f5\sqrt{s}\ll 1$ requirement to maintain high fidelity. The bound applies if the circuit size satisfies$$\epsilon s\ll 1$$ $\u03f5s\ll 1$ , which corresponds to only$$s \ge \Omega (n\log (n))$$ $s\ge \Omega (nlog(n\left)\right)$logarithmic depth circuits, and if, additionally, the inverse error rate satisfies , which is needed to ensure errors are scrambled faster than$$\epsilon ^{1} \ge {\tilde{\Omega }}(n)$$ ${\u03f5}^{1}\ge \stackrel{~}{\Omega}\left(n\right)$F decays. The whitenoise approximation is useful for salvaging the signal from a noisy quantum computation; for example, it was an underlying assumption in complexitytheoretic 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 secondmoment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds. 
Abstract We prove that
depth local random quantum circuits with two qudit nearestneighbor gates on a$${{\,\textrm{poly}\,}}(t) \cdot n^{1/D}$$ $\phantom{\rule{0ex}{0ex}}\text{poly}\phantom{\rule{0ex}{0ex}}\left(t\right)\xb7{n}^{1/D}$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$${{\,\textrm{poly}\,}}(t)\cdot n$$ $\phantom{\rule{0ex}{0ex}}\text{poly}\phantom{\rule{0ex}{0ex}}\left(t\right)\xb7n$ . 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 ($$D=1$$ $D=1$ ) is infinite and that certain counting problems are$${{\,\mathrm{\textsf{PH}}\,}}$$ $\phantom{\rule{0ex}{0ex}}\mathrm{PH}\phantom{\rule{0ex}{0ex}}$ hard “on average”, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constantdepth 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 “anticoncentration”, meaning roughly that the output has nearmaximal entropy. Unitary 2designs have the desired anticoncentration property. Our result improves the required depth for this level of anticoncentration from linear depth to a sublinear 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 twodimensional 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$$\#{\textsf{P}}$$ $\#P$ depth suffices for anticoncentration. The proof is based on a previous construction of$$O(\sqrt{n})$$ $O\left(\sqrt{n}\right)$t designs by Brandão et al. (2016), an analysis of how approximate designs behave under composition, and an extension of the quasiorthogonality 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 anticoncentration and to establish equivalence between these various norms for lowdepth circuits. For random circuits with longrange gates, we use different methods to show that anticoncentration happens at circuit size corresponding to depth$$O(n\ln ^2 n)$$ $O\left(n{ln}^{2}n\right)$ . We also show a lower bound of$$O(\ln ^3 n)$$ $O\left({ln}^{3}n\right)$ for the size of such circuit in this case. We also prove that anticoncentration is possible in depth$$\Omega (n \ln n)$$ $\Omega (nlnn)$ (size$$O(\ln n \ln \ln n)$$ $O(lnnlnlnn)$ ) using a different model.$$O(n \ln n \ln \ln n)$$ $O(nlnnlnlnn)$ 
Abstract We study
ℓ ^{∞}norms ofℓ ^{2}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 eigenfunctionsu with . For general eigenfunctions we show the upper bound $\parallel u{\parallel}_{\infty}\gtrsim {\left(\mathrm{log}N\right)}^{1/2}$ . Here the semiclassical parameter is $\parallel u{\parallel}_{\infty}\lesssim {\left(\mathrm{log}N\right)}^{1/2}$ . Our upper bound is analogous to the one proved by Bérard in P Bérard (1977, $h={\left(2\pi N\right)}^{1}$Mathematische Zeitschrift ,155 , 249276) for compact Riemannian manifolds without conjugate points. 
Abstract One of the cornerstone effects in spintronics is spin pumping by dynamical magnetization that is steadily precessing (around, for example, the
z axis) with frequencyω _{0}due to absorption of lowpower microwaves of frequencyω _{0}under the resonance conditions and in the absence of any applied bias voltage. The twodecadesold ‘standard model’ of this effect, based on the scattering theory of adiabatic quantum pumping, predicts that component of spin current vector ${I}^{{S}_{z}}$ is timeindependent while $({I}^{{S}_{x}}(t),{I}^{{S}_{y}}(t),{I}^{{S}_{z}})\propto {\omega}_{0}$ and ${I}^{{S}_{x}}(t)$ oscillate harmonically in time with a single frequency ${I}^{{S}_{y}}(t)$ω _{0}whereas pumped charge current is zero in the same adiabatic $I\equiv 0$ limit. Here we employ more general approaches than the ‘standard model’, namely the timedependent 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 $\propto {\omega}_{0}$ and charge ${I}^{{S}_{\alpha}}(t)$I (t ) currents. All four of these functions harmonically oscillate in time at both even and odd integer multiples of the driving frequency $N{\omega}_{0}$ω _{0}. The cutoff order of such high harmonics increases with SOC strength, reaching in the onedimensional FM or AFM models chosen for demonstration. A higher cutoff ${N}_{\mathrm{m}\mathrm{a}\mathrm{x}}\simeq 11$ can be achieved in realistic twodimensional (2D) FM models defined on a honeycomb lattice, and we provide a prescription of how to realize them using 2D magnets and their heterostructures. ${N}_{\mathrm{m}\mathrm{a}\mathrm{x}}\simeq 25$