The quantum simulation of quantum chemistry is a promising application of quantum computers. However, for
 Award ID(s):
 1955907
 NSFPAR ID:
 10336135
 Date Published:
 Journal Name:
 Symmetry
 Volume:
 14
 Issue:
 3
 ISSN:
 20738994
 Page Range / eLocation ID:
 457
 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 The variational quantum eigensolver is one of the most promising approaches for performing chemistry simulations using noisy intermediatescale quantum (NISQ) processors. The efficiency of this algorithm depends crucially on the ability to prepare multiqubit trial states on the quantum processor that either include, or at least closely approximate, the actual energy eigenstates of the problem being simulated while avoiding states that have little overlap with them. Symmetries play a central role in determining the best trial states. Here, we present efficient state preparation circuits that respect particle number, total spin, spin projection, and timereversal symmetries. These circuits contain the minimal number of variational parameters needed to fully span the appropriate symmetry subspace dictated by the chemistry problem while avoiding all irrelevant sectors of Hilbert space. We show how to construct these circuits for arbitrary numbers of orbitals, electrons, and spin quantum numbers, and we provide explicit decompositions and gate counts in terms of standard gate sets in each case. We test our circuits in quantum simulations of the
and$${H}_{2}$$ ${H}_{2}$ molecules and find that they outperform standard state preparation methods in terms of both accuracy and circuit depth.$$LiH$$ $LiH$ 
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
${2}^{n}$ dimensional qubit subspace of a circuit model of$n$ qubits. The novelty of this adiabatic theorem is that, unlike previous rigorous results, it does not contain${2}^{n}$ as a factor in the adiabatic timescale, and it allows one to obtain an expression for the adiabatic timescale independent of the cutoff of the infinitedimensional Hilbert space of the circuit Hamiltonian. As an application, we present an explicit dependence of this timescale on circuit parameters for a superconducting flux qubit and demonstrate that leakage out of the qubit subspace is inevitable as the tunnelling barrier is raised towards the end of a quantum anneal. We also discuss a method of obtaining a${2}^{n}\times {2}^{n}$ effective Hamiltonian that best approximates the true dynamics induced by slowly changing circuit control parameters.This article is part of the theme issue ‘Quantum annealing and computation: challenges and perspectives’.

Quantum chemistry is a promising application for noisy intermediatescale quantum (NISQ) devices. However, quantum computers have thus far not succeeded in providing solutions to problems of real scientific significance, with algorithmic advances being necessary to fully utilise even the modest NISQ machines available today. We discuss a method of ground state energy estimation predicated on a partitioning the molecular Hamiltonian into two parts: one that is noncontextual and can be solved classically, supplemented by a contextual component that yields quantum corrections obtained via a Variational Quantum Eigensolver (VQE) routine. This approach has been termed Contextual Subspace VQE (CSVQE), but there are obstacles to overcome before it can be deployed on NISQ devices. The problem we address here is that of the ansatz  a parametrized quantum state over which we optimize during VQE. It is not initially clear how a splitting of the Hamiltonian should be reflected in our CSVQE ansätze. We propose a 'noncontextual projection' approach that is illuminated by a reformulation of CSVQE in the stabilizer formalism. This defines an ansatz restriction from the full electronic structure problem to the contextual subspace and facilitates an implementation of CSVQE that may be deployed on NISQ devices. We validate the noncontextual projection ansatz using a quantum simulator, with results obtained herein for a suite of trial molecules.more » « less

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.$$O(n \ln n \ln \ln n)$$ $O(nlnnlnlnn)$