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Accelerating Lattice Quantum Field Theory Calculations Via Interpolator Optimization Using Noisy Intermediate-Scale Quantum Computing

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Accelerating Lattice Quantum Field Theory Calculations Via Interpolator Optimization Using Noisy Intermediate-Scale Quantum Computing Accelerating lattice quantum field theory calculations via interpolator optimization using noisy intermediate-scale quantum computing The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Avkhadiev, A., P.E. Shanahan, and R.D. Young, "Accelerating lattice quantum field theory calculations via interpolator optimization using noisy intermediate-scale quantum computing." Physical Review Letters 124 (Aug. 2020): no. 080501 doi 10.1103/ PhysRevLett.124.080501 ©2020 Author(s) As Published 10.1103/PhysRevLett.124.080501 Publisher American Physical Society Version Final published version Citable link https://hdl.handle.net/1721.1/125042 Terms of Use Creative Commons Attribution 3.0 unported license Detailed Terms http://creativecommons.org/licenses/by/3.0 PHYSICAL REVIEW LETTERS 124, 080501 (2020) Accelerating Lattice Quantum Field Theory Calculations via Interpolator Optimization Using Noisy Intermediate-Scale Quantum Computing A. Avkhadiev,1,2 P. E. Shanahan ,1,2 and R. D. Young 3 1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 3CSSM, Department of Physics, University of Adelaide, Adelaide, South Australia 5005, Australia (Received 22 August 2019; revised manuscript received 25 November 2019; accepted 31 January 2020; published 26 February 2020) The only known way to study quantum field theories in nonperturbative regimes is using numerical calculations regulated on discrete space-time lattices. Such computations, however, are often faced with exponential signal-to-noise challenges that render key physics studies untenable even with next generation classical computing. Here, a method is presented by which the output of small-scale quantum computations on noisy intermediate-scale quantum era hardware can be used to accelerate larger-scale classical field theory calculations through the construction of optimized interpolating operators. The method is implemented and studied in the context of the 1 þ 1-dimensional Schwinger model, a simple field theory which shares key features with the standard model of nuclear and particle physics. DOI: 10.1103/PhysRevLett.124.080501 Numerical approaches to quantum field theory are the direct searches for dark matter to neutrino physics, will only known way to make predictions for a wide range of remain intractable, even with the advent of exascale physical quantities from the standard model of particle classical computing in the next years; progress on this physics, our best current theory of nature at the smallest front will require a revolutionary approach, and there is scales. Standard model calculations of nuclear physics great interest in the potential applications of quantum processes—such as those needed to interpret experiments computing to overcome this challenge [5,6]. Hybrid meth- using nuclei as targets—are particularly challenging. In ods coupling classical and quantum computing offer a particular, the strong-interaction component of the standard natural pathway to exploit quantum computation despite model, which is encoded in the theory of quantum the small number of qubits, sparse qubit connectivity, lack chromodynamics (QCD), cannot be approached analyti- of error correction, and noisy quantum gates that are cally at the relevant energy scales. The only first-principles hallmarks of current and near-term quantum computing approach to QCD at these scales is numerical: a discretized in the noisy intermediate-scale quantum (NISQ) era [7]. form of the QCD equations can be solved using super- A significant contribution to the computational cost of computers through Monte Carlo integration on a finite LQFT studies could be eliminated by the construction of four-dimensional grid representing space-time [1,2]. This optimized interpolating operators, corresponding in broad technique, named lattice quantum field theory (LQFT), terms to approximations to the quantum wave function of plays an important role in modern particle and nuclear the desired state. Precisely, to determine matrix elements of physics and has been essential in testing the standard model interest in some state in a LQFT computation, such as those against precise measurements of the decays and inter- describing an interaction or decay process, correlation func- actions of particles at frontier machines such as the Large tions are calculated which encode the creation, interaction, Hadron Collider [3,4]. Calculations of nuclei, however, and annihilation, of the state in question. These correlation are limited by exponentially bad scaling of computational functions, however, receive contaminating contributions from cost with the atomic number of the system being studied. the many other states with the quantum numbers of the state of Using current methods, direct studies of nuclei with tens interest. In order to reliably extract the desired piece, the of nucleons, as relevant to diverse physics programs from contributions from all of these unwanted higher-energy states must be suppressed. Typically, this is achieved via an evolution in the Euclidean time of the calculation; the unwanted states are exponentially suppressed by the energy Published by the American Physical Society under the terms of gap to the ground state at large times, but at the cost of an the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to exponential growth in the statistical noise of the Monte Carlo the author(s) and the published article’s title, journal citation, sampling used in the computation (and thus computational and DOI. Funded by SCOAP3. cost). By using optimized interpolating operators for state 0031-9007=20=124(8)=080501(6) 080501-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 124, 080501 (2020) creation and annihilation, constructed to have significant vector meson), while the second excited state is the even- overlap onto the state of interest, this Euclidean time parity scalar eþe− “meson.” evolution, and thus exponential growth in noise, can be Classical computations of ground-state energies.—Using reduced. In this Letter, it is demonstrated for the 1 þ 1- classical computation, energy levels of the Schwinger model dimensional Schwinger model how one can construct such can be obtained using standard Monte Carlo (MC) methods. interpolating operators for classical LQFT calculations Here a local Hamiltonian MC method is studied [10]; details using small-scale quantum computation. Ultimately, the of the application of this approach to the 1 þ 1DSchwinger extension of this approach to the more complex theory of model are given in Ref. [11]. In this formalism, ground-state QCD, along with advancement in quantum hardware, could energy levels can be determined by the analysis of corre- enable a significant acceleration of LQFT computations for lation functions GðτÞ, defined in terms of the expectation ˆ nuclear physics. values of interpolating operators Oðx; τÞ, which are con- The Schwinger model.—The Schwinger model [8],which structed to create and annihilate states with the quantum describes the theory of quantum electrodynamics in one numbers of a target state of interest at some Euclidean space and one time dimension, is a prototypical lattice gauge position (x, τ): theory that shares a number of key features with QCD. This X ˆ ˆ † ˆ ˆ † model thus provides a simplified framework to test new GðτÞ¼ ½hOðx;τÞO ð0;0Þi−hOðx;τÞihO ð0;0Þi: ð4Þ algorithms and approaches to LQFT studies. The theory x describes fermions as a two-component spinor field, with mass m, coupled via charge g to an electromagnetic field, Aμ. Here, a state is created at some initial spatial position and A discretized formulation of the 1 þ 1D Schwinger model time ðx ¼ 0;t¼ 0Þ, and annihilated τ Euclidean time steps can be defined on a staggered space-time lattice via the later. The sum over x projects onto the zero-momentum Kogut-Susskind prescription [9], with the staggered fermion sector. This correlation depends on the energy gaps between Ω field operators denoted by ϕˆ ðx Þ¼ϕˆ . the ground state (vacuum) of the system j i and the tower of n n ˆ In temporal gauge (A0 ¼ 0), the single spatial compo- excitations coupled to the vacuum through O: nent of the gauge field is encoded on links connecting X 2 − − τ τ O Ω ðEn EΩÞ adjacent staggered sites xn and xnþ1. The associated electric Gð Þ¼ jhnj j ij e : ð5Þ ˆ n flux operators can be defined in terms of operators ln, which, together with corresponding raising and lower Numerically, the energy gap to the lowest state of interest is operators, act on the space of links connecting sites: determined from the asymptotic value of the effective mass function: lˆ jl i¼l jl i; l ∈ Z ∀ n; ð1Þ n n n n n 1 GðτÞ ˆ Æ τ → − L jl i¼jl Æ 1i: ð2Þ Meff ð Þ¼ log ðEO EΩÞ: ð6Þ n n n a Gðτ þ aÞ τ→∞ The eigenvalue ln describes the value of the electric flux at 1 Interpolating operators for lattice field theories can be the link connecting sites n and n þ . constructed by inspection, and often the simplest operators Combining the link space with fermionic occupation which have the quantum numbers of the state of interest numbers, a complete Fock space of states in this theory can ⃗ are chosen. For the Schwinger model, the lowest-energy be expressed as fjn;⃗lig. On a lattice with N staggered sites excitation is described by the lightest vector meson, V−,a (N=2 spatial sites), the Hamiltonian of this theory can be massive photon. An interpolating operator for this state can expressed in terms of these operators as be constructed as for odd-parity meson states in staggered lattice formulations of QCD [12]: XN−2 ˆ ϕˆ † ˆ þϕˆ − − ϕˆ † ˆ þ ϕˆ − Hlat ¼ iw ð nLn nþ1 H:c:Þ iwð N−1LN−1 0 H:c:Þ ˆ ˆ † ˆ þ ˆ ˆ † ˆ − ˆ O ðx ; τ Þ¼ϕ L ϕ 1 − ϕ L −1 ϕ −1 : ð7Þ n¼0 V n j n;j n;j nþ ;j n;j n ;j n ;j XN−1 XN−1 ˆ − n ˆ † ˆ ˆ 2 Physically, the operator O creates an eþe pair on sites þ m ð−1Þ ϕnϕn þ J ln: ð3Þ V − þ n¼0 n¼0 xn−1 and xn, and a second e e pair on sites xn and xnþ1.
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