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Quantum Simulation of Nuclear Dynamics in First Quantization

Luca Spagnoli, Chiara Lissoni, Alessandro Roggero

TL;DR

This work addresses simulating real-time nuclear dynamics at low energies using a first-quantized pionless EFT on a lattice. It develops and analyzes first-quantized algorithms via product formulas and Quantum Signal Processing (QSP), including an efficient block-encoding of the Hamiltonian and a generalized QSP scheme. The authors provide a complete resource characterization, proving polynomial scaling in the number of particles $\eta$ and polylogarithmic scaling in the single-particle space size $\Omega$, with concrete T-gate and qubit counts and comparisons to second-quantization approaches. The results indicate that early fault-tolerant quantum devices could perform meaningful nuclear-dynamics simulations for simple reactions with tens of millions of T gates and a few hundred logical qubits, pointing to a plausible near-term quantum advantage in nuclear physics.

Abstract

The study of real time dynamics of nuclear systems is of great importance to provide theoretical predictions of cross sections relevant for both terrestrial experiments as well as applications in astrophysics. First principles simulations of these dynamical processes is however hindered by an exponential cost in classical resources and the possibility of performing scalable simulations using quantum computers is currently an active field of research. In this work we provide the first complete characterization of the resource requirements for studying nuclear dynamics with the full Leading Order (LO) pionless EFT Hamiltonian in first quantization employing simulation strategies using both product formulas as well as Quantum Signal Processing. In particular, we show that time evolution of such an Hamiltonian can be performed with polynomial resources in the number of particles, and logarithmic resources in the number of single-particle basis states. This result provides an exponential improvement compared with previous work on the same Hamiltonian model in second quantization. We find that interesting simulations for low energy nuclear scattering could be achievable with tens of millions of T gates and few hundred logical qubits suggesting that the study of simple nuclear reactions could be amenable for early fault tolerant quantum platforms.

Quantum Simulation of Nuclear Dynamics in First Quantization

TL;DR

This work addresses simulating real-time nuclear dynamics at low energies using a first-quantized pionless EFT on a lattice. It develops and analyzes first-quantized algorithms via product formulas and Quantum Signal Processing (QSP), including an efficient block-encoding of the Hamiltonian and a generalized QSP scheme. The authors provide a complete resource characterization, proving polynomial scaling in the number of particles and polylogarithmic scaling in the single-particle space size , with concrete T-gate and qubit counts and comparisons to second-quantization approaches. The results indicate that early fault-tolerant quantum devices could perform meaningful nuclear-dynamics simulations for simple reactions with tens of millions of T gates and a few hundred logical qubits, pointing to a plausible near-term quantum advantage in nuclear physics.

Abstract

The study of real time dynamics of nuclear systems is of great importance to provide theoretical predictions of cross sections relevant for both terrestrial experiments as well as applications in astrophysics. First principles simulations of these dynamical processes is however hindered by an exponential cost in classical resources and the possibility of performing scalable simulations using quantum computers is currently an active field of research. In this work we provide the first complete characterization of the resource requirements for studying nuclear dynamics with the full Leading Order (LO) pionless EFT Hamiltonian in first quantization employing simulation strategies using both product formulas as well as Quantum Signal Processing. In particular, we show that time evolution of such an Hamiltonian can be performed with polynomial resources in the number of particles, and logarithmic resources in the number of single-particle basis states. This result provides an exponential improvement compared with previous work on the same Hamiltonian model in second quantization. We find that interesting simulations for low energy nuclear scattering could be achievable with tens of millions of T gates and few hundred logical qubits suggesting that the study of simple nuclear reactions could be amenable for early fault tolerant quantum platforms.

Paper Structure

This paper contains 15 sections, 18 theorems, 245 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. First quantization algorithms
  3. Exponential of Hamiltonian terms
  4. Product formula implementation
  5. Quantum Signal Processing implementation
  6. Results
  7. Summary and conclusions
  8. Acknowledgments
  9. Subroutines
  10. Phase kick-back method
  11. Norms
  12. Fermionic Semi-norm
  13. Fermionic semi-norm of operators in the Hamiltonian
  14. Commutator bounds
  15. Higher order Trotter-Suzuki formulas

Key Result

Lemma 1

Consider the kinetic energy term of Eq. eq:Tdiscrete for a system with $\eta$ particles on a $d$ dimensional lattice with $M=2^{m}$ sites per direction. Let $\lambda_T \ge {\|T\|}$ an upper bound on its spectral norm. For any $t\in \mathbb{R}$ and $\epsilon > 0$ there exists a quantum circuit that i and a total number of ancilla qubits given by where we have defined The explicit definition of th

Figures (11)

  • Figure 1: Circuit implementation of the exponential of a single two body term for $m=1$ and $d=3$.
  • Figure 2: Alternative implementation of the exponential of a single two body term for $m=1$ and $d=3$.
  • Figure 3: Circuit implementation of the exponential of a single two body term, for the pair $(i,j)$, and a single three-body term, for the triple $(i,j,k)$, for $m=1$ and $d=3$.
  • Figure 4: Circuit implementation of the $SELECT$ unitary for the kinetic energy operator.
  • Figure 5: Circuit implementation of the controlled $SN$ unitary for a case with $\eta=3$ and $dm=3$.
  • ...and 6 more figures

Theorems & Definitions (32)

  • Lemma 1: Exponential of Kinetic Energy
  • proof
  • Lemma 2: Exponential of Potential Energy
  • proof
  • Lemma 3: Compact Exponential of Potential Energy
  • proof
  • Theorem 2.1: First-order Trotter
  • proof
  • Theorem 2.2: Second and fourth-order Trotter-Suzuki
  • proof
  • ...and 22 more