Quantum computing enhanced computational catalysis

Burg, Vera von; Low, Guang Hao; Häner, Thomas; Steiger, Damian S.; Reiher, Markus; Roetteler, Martin; Troyer, Matthias

doi:10.1103/physrevresearch.3.033055

Cited by 144 publications

(131 citation statements)

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“…However, since there needs to be at least one index in common between each Hamiltonian term, the upper bound is actually O(n 7 s ). By multiplying the upper bound for the number of second order commutator terms to evaluate, n 7 s , with the fitted function f (M ), where M = n 3 s , we see that the relationship of this complexity is observed to be ∼ O(n 4 s ), performing better than the upper bound. The derivation of the estimated T-gate complexity is detailed in the following section.…”

Section: Numerical Evaluation Of Momentum Space Commutatorsmentioning

confidence: 93%

“…In recent years, qubitization has emerged as an alternative to Trotter-Suzuki simulations on fault tolerant hardware [7,8,[128][129][130][131]. Unlike Trotter-Suzuki methods, qubitization is known to saturate lower bounds on the query complexity for quantum simulation.…”

Section: Cost Estimates For Qubitizationmentioning

confidence: 99%

“…In contrast, the best known results for simulating jellium using Trotter methods are on the order of 10 9 non-Clifford operations for systems of 27 spin orbitals (54 qubits). The gulfs between these two estimates suggest that further optimization may be needed to allow eQED to reach the same levels of performance that we can reach for non-relativistic electronic structure calculations, however, the gulfs between the two are not so large as to suspect that such simulations will be infeasible once subjected to the same optimizations that lowered the costs of simulation for challenge problems in chemistry from 10 14 non-Clifford gates [13] to on the order of 10 9 non-Clifford gates [7,8]…”

Section: Cost Estimate For Qpementioning

confidence: 99%

“…A great body of literature exists that shows that quantum computers can provide exponential advantages for certain ab initio electronic structure calculations [3][4][5][6][7][8]. The utility of quantum based devices stems from their ability to directly simulate the dynamics of the Coulomb Hamiltonian without making the approximations that are usually required to make their classical counterparts efficiently tractable.…”

Section: Introductionmentioning

confidence: 99%

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Simulating Effective QED on Quantum Computers

Stetina

Ciavarella

et al. 2022

Quantum

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In recent years simulations of chemistry and condensed materials has emerged as one of the preeminent applications of quantum computing, offering an exponential speedup for the solution of the electronic structure for certain strongly correlated electronic systems. To date, most treatments have ignored the question of whether relativistic effects, which are described most generally by quantum electrodynamics (QED), can also be simulated on a quantum computer in polynomial time. Here we show that effective QED, which is equivalent to QED to second order in perturbation theory, can be simulated in polynomial time under reasonable assumptions while properly treating all four components of the wavefunction of the fermionic field. In particular, we provide a detailed analysis of such simulations in position and momentum basis using Trotter-Suzuki formulas. We find that the number of T-gates needed to perform such simulations on a 3D lattice of ns sites scales at worst as O(ns3/ϵ)1+o(1) in the thermodynamic limit for position basis simulations and O(ns4+2/3/ϵ)1+o(1) in momentum basis. We also find that qubitization scales slightly better with a worst case scaling of O~(ns2+2/3/ϵ) for lattice eQED and complications in the prepare circuit leads to a slightly worse scaling in momentum basis of O~(ns5+2/3/ϵ). We further provide concrete gate counts for simulating a relativistic version of the uniform electron gas that show challenging problems can be simulated using fewer than 1013 non-Clifford operations and also provide a detailed discussion of how to prepare multi-reference configuration interaction states in effective QED which can provide a reasonable initial guess for the ground state. Finally, we estimate the planewave cutoffs needed to accurately simulate heavy elements such as gold.

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Section: Numerical Evaluation Of Momentum Space Commutatorsmentioning

confidence: 93%

Section: Cost Estimates For Qubitizationmentioning

confidence: 99%

Section: Cost Estimate For Qpementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Simulating Effective QED on Quantum Computers

Stetina

Ciavarella

et al. 2022

Quantum

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“…composed of Γ terms, the goal of quantum simulation is to implement an approximation of its time evolution operator e iHt at time t, using a quantum computer. As a ubiquitous subroutine, it already finds promising applications in material science and quantum chemistry [1][2][3][4][5]. However, despite the simplicity of the problem statement, developing quantum simulation algorithms that minimize the required resources (e.g., the gate complexity) has drawn tremendous effort [6][7][8].…”

mentioning

confidence: 99%

Concentration for Trotter error

Chen

Brandão²

2021

Preprint

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Quantum simulation is expected to be one of the key applications of future quantum computers. Product formulas, or Trotterization, are the oldest and, still today, an appealing method for quantum simulation. For an accurate product formula approximation in the spectral norm, the state-of-the-art gate complexity depends on the number of terms in the Hamiltonian and a certain 1-norm of its local terms.This work considers the concentration aspects of Trotter error: we prove that, typically, the Trotter error exhibits 2-norm (i.e., incoherent) scaling; the current estimate with 1-norm (i.e., coherent) scaling is for the worst cases. For general k-local Hamiltonians and higher-order product formulas, we obtain gate count estimates for input states drawn from a 1-design ensemble (which includes, e.g., computational basis states). Our gate count depends on the number of terms in the Hamiltonian but replaces the 1-norm quantity by its analog in 2-norm, giving significant speedup for systems with large connectivity. Our typical-case results generalize to Hamiltonians with Fermionic terms and when the input state is drawn from a low-particle number subspace. Further, when the Hamiltonian itself has Gaussian coefficients, such as the SYK models, we show the stronger result that the 2-norm behavior persists even for the worst input state.Our primary technical tool is a family of simple but versatile inequalities from non-commutative martingales called uniform smoothness. We use them to derive Hypercontractivity, namely p-norm estimates for low-degree polynomials (including k-local Pauli operators), which implies concentration via Markov's inequality. In terms of optimality, we give examples that simultaneously match our p-norm estimates and the spectral norm estimates. Therefore, our improvement is due to asking a qualitatively different question from the spectral norm bounds. Our results give evidence that product formulas in practice may generically work much better than expected. CONTENTS III. Non-Random k-Local Hamiltonians A. Heuristic Argument for First-Order Trotter B. Proof Outline C. Bounds on the g-th Order D. Bounds for g -th Order and Beyond. E. Proof of Theorem III.1 1. Constant overhead improvement from another Hypercontractivity F. Spin Models at a Low Particle Number G. k-locality for Fermions

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