Quantum computers offer the potential to simulate nuclear processes that are classically intractable. With the goal of understanding the necessary quantum resources to realize this potential, we employ state-of-the-art Hamiltonian-simulation methods, and conduct a thorough algorithmic analysis, to estimate the qubit and gate costs to simulate low-energy effective field theories (EFTs) of nuclear physics. Within the framework of nuclear lattice EFT, we obtain simulation costs for the leading-order pionless and pionful EFTs. For the latter, we consider both static pions represented by a one-pion-exchange potential between the nucleons, and dynamical pions represented by relativistic bosonic fields coupled to non-relativistic nucleons. Within these models, we examine the resource costs for the tasks of time evolution and energy estimation for physically relevant scales. We account for model errors associated with truncating either long-range interactions in the one-pion-exchange EFT or the pionic Hilbert space in the dynamical-pion EFT, and for algorithmic errors associated with product-formula approximations and quantum phase estimation. We find that the pionless EFT is the least costly to simulate, followed by the one-pion-exchange theory, then the dynamical-pion theory. We demonstrate how symmetries of the low-energy nuclear Hamiltonians can be utilized to obtain tighter error bounds. By retaining the locality of nucleonic interactions when mapped to qubits, we achieve reduced circuit depth and substantial parallelization. In the process, we develop new methods to bound the algorithmic error for classes of fermionic number-preserving Hamiltonians, and obtain tighter Trotter error bounds by explicitly computing nested commutators of Hamiltonian terms. Compared to previous estimates for the pionless EFT, our results represent an improvement by several orders of magnitude.
Quantum computers offer the potential to simulate nuclear processes that are classically intractable. With the goal of understanding the necessary quantum resources to realize this potential, we employ state-of-the-art Hamiltonian-simulation methods, and conduct a thorough algorithmic analysis, to estimate the qubit and gate costs to simulate low-energy effective field theories (EFTs) of nuclear physics. Within the framework of nuclear lattice EFT, we obtain simulation costs for the leading-order pionless and pionful EFTs. For the latter, we consider both static pions represented by a one-pion-exchange potential between the nucleons, and dynamical pions represented by relativistic bosonic fields coupled to non-relativistic nucleons. Within these models, we examine the resource costs for the tasks of time evolution and energy estimation for physically relevant scales. We account for model errors associated with truncating either long-range interactions in the one-pion-exchange EFT or the pionic Hilbert space in the dynamical-pion EFT, and for algorithmic errors associated with product-formula approximations and quantum phase estimation. We find that the pionless EFT is the least costly to simulate, followed by the one-pion-exchange theory, then the dynamical-pion theory. We demonstrate how symmetries of the low-energy nuclear Hamiltonians can be utilized to obtain tighter error bounds. By retaining the locality of nucleonic interactions when mapped to qubits, we achieve reduced circuit depth and substantial parallelization. In the process, we develop new methods to bound the algorithmic error for classes of fermionic number-preserving Hamiltonians, and obtain tighter Trotter error bounds by explicitly computing nested commutators of Hamiltonian terms. Compared to previous estimates for the pionless EFT, our results represent an improvement by several orders of magnitude.
Background: Quantum simulations offer the potential to predict the structure and dynamics of nuclear many-body systems that are beyond the capabilities of classical computing. Generally, preparing the ground state of strongly interacting many-body systems relevant to nuclear physics is, however, inefficient, even using ideal quantum computers. In addition, currently available noisy intermediate-scale quantum (NISQ) era quantum devices possess modest numbers of qubits, limiting the size of quantum many-body systems that can be simulated. In this context, a reformulation of the quantum many-body problems using truncated model spaces and Hamiltonians is desirable to make them more amenable to near-term quantum computers. The importance of symmetries in low-energy theories, including effective field theories (EFTs), lattice quantum chromodynamics (QCD), and effective model spaces for nuclear systems, in particular their interplay with the reduction of active Hilbert spaces, is well known. Lesser known is the fact that the noncommutivity of some symmetries and truncations of the model space can be profitably combined with variational calculations to rearrange the entanglement into localized structures and enable more efficient simulations.Purpose: The goal of the present study is to explore and utilize the noncommutivity of symmetries and model-space truncations of quantum many-body systems important to nuclear physics, particularly in combination with variational algorithms for quantum simulations and effective Hamiltonian learning.Method: We introduce an iterative hybrid classical-quantum algorithm, the Hamiltonian learning variational quantum eigensolver (HL-VQE), that simultaneously optimizes an effective Hamiltonian, thereby rearranging entanglement into the effective model space, and the associated ground-state wave function. Quantum simulations, using classical computers and IBM's superconducting-qubit quantum computers, are performed to demonstrate the HL-VQE algorithm, in the context of the Lipkin-Meshkov-Glick (LMG) model of interacting fermions, where the Hamiltonian transformation corresponds to an orbital rotation. We use a mapping where the number of qubits scales with the logarithm of the size of the effective model space, rather than the particle number.Results: HL-VQE is found to provide an exponential improvement in LMG-model calculations of the ground-state energy and wave function, compared to naive truncations without Hamiltonian learning, throughout a significant fraction of the Hilbert space. In the context of EFT, this corresponds to counterterms scaling exponentially with the cutoff as opposed to power law. Implementations on IBM's QExperience quantum computers and simulators for one- and two-qubit effective model spaces are shown to provide accurate and precise results, reproducing classical predictions.Conclusions: For a range of parameters defining the LMG model, the HL-VQE algorithm is found to have better scaling of quantum resources requirements than previously explored algorithms. In particular, the HL-VQE scales efficiently over a large fraction of the model space, in contrast to VQE alone. This work constitutes a step in the development of entanglement-driven quantum algorithms for descriptions of nuclear many-body systems. This, in part, leverages the potential of noisy intermediate-scale quantum (NISQ) devices. The exponential scaling of counterterms observed in this study suggests the possibility of more general applicability to other nonperturbative EFTs.
Abstract These lectures are a pedagogical—not comprehensive—introduction to the applications of effective field theory (EFT) in the context of nuclear and atomic physics. A common feature of these applications is the interplay between non-perturbative physics (needed at leading order to produce non-relativistic bound states and resonances) and controlled perturbative corrections (crucial for predictive power). The essential ideas are illustrated with the simplest nuclear EFT, pionless EFT, which contains only contact interactions and, with minor changes, can be adapted to certain atomic systems. This EFT exploits the two-body unitarity limit, where renormalization leads to discrete scale invariance in systems of three and more bodies. Remarkably complex structures then arise from very simple leading-order interactions. It briefly describes some of the challenges and rewards of including long-range forces—pion exchange in chiral EFT for nuclear systems or Van der Waals forces between atoms.
Abstract With a view toward addressing the explosive growth in the computational demands of nuclear structure and reactions modeling, we develop a novel quantum algorithm for neutron–nucleus simulations with general potentials, which provides acceptable bound-state energies even in the presence of noise, through the noise-resilient (NR) training method. In particular, the algorithm can now solve for any band-diagonal to full Hamiltonian matrices, as needed to accommodate a general central potential. While we illustrate the approach for exponential Gaussian-like potentials and ab initio inter-cluster potentials (optical potentials), it can also accommodate the complete form of the chiral effective-field-theory nucleon–nucleon potentials used in ab initio nuclear calculations. In this study, we provide a comprehensive analysis for the efficacy of this approach for three different qubit encodings, including the one-hot, binary, and Gray encodings, in terms of the number of Pauli strings and commuting sets involved. We also discuss the advantages of the algorithm for Hamiltonians of various band-diagonal widths, especially critical for potentials of perturbative nature, leading to a drastically reduced runtime of quantum simulations. We prove that the Gray encoding allows for an efficient scaling of the model-space size N (or number of basis states used) and is more resource efficient for band-diagonal Hamiltonians having bandwidth up to N . We introduce a new commutativity scheme called distance-grouped commutativity (DGC) and compare its performance with the well-known qubit-commutativity (QC) scheme. We lay out the explicit grouping of Pauli strings and the diagonalizing unitary under the DGC scheme, and we prove that it outperforms the QC scheme, at the cost of a more complex diagonalizing unitary. Lastly, we provide first solutions of the neutron–alpha dynamics from quantum simulations suitable for noisy intermediate-scale quantum processors, using an optical potential rooted in first principles, as well as a study of the bound-state physics in neutron–Carbon systems, along with a comparison of the efficacy of the one-hot and Gray encodings.
We present a framework for benchmarking quantum algorithms for nuclear many-body systems based on realistic nuclear Hamiltonians such as chiral effective field theory. To this end, we introduce a workflow that maps nuclear interactions in second quantization formalism to qubit Hamiltonians. This enables the systematic construction of benchmark instances spanning no-core and valence-space formulations with two-body (NN) and selected three-body (3N) interactions. Then, we proceed to provide resource estimates for three representative eigenvalue algorithms: Quantum Phase Estimation, Quantum Krylov methods, and Observable Dynamic Mode Decomposition. We compare their resource requirements in terms of T-gate counts and system size, and examine the impact of model-space choices and many-body interactions. The primitives included in our analysis are Trotterization, Qubitization, and Quantum Singular Value Transformation. Our results quantify scaling trends across algorithms and problem classes, and provide a basis for consistent comparisons of quantum approaches to nuclear many-body problems. The implementation is provided by the NuQuLib software stack.
Simulating the Hamiltonian dynamics of quantum systems is one of the most promising applications of digital quantum computers. In this dissertation, we develop an understanding of quantum simulation algorithms concerning their design, analysis, implementation, and application. We implement three leading simulation algorithms, employing diverse techniques to tighten their error analyses and optimize circuit implementations. We produce concrete resource estimates for simulating a Heisenberg spin system, a problem arising in condensed matter physics that is otherwise difficult to solve on a classical computer. The resulting circuits are orders of magnitude smaller than those for the simplest classically-infeasible instances of factoring and quantum chemistry, suggesting the simulation of spin systems as a promising candidate for an early demonstration of practical quantum computation. We design new simulation algorithms by using classical randomness. We show that by simply randomizing how the terms in the Hamiltonian are ordered, one can prove stronger bounds for product formulas and thereby give more efficient quantum simulations. We also develop a classical sampler for time-dependent Hamiltonians, using which we give a simulation algorithm that substantially improves over previous approaches when the Hamiltonian varies significantly with time. We propose a general theory to analyzing product formulas, an approach to quantum simulation widely used in experimental demonstrations but whose error scaling was poorly understood. Our approach directly exploits the commutativity of Hamiltonian, overcoming the limitations of prior error analyses. We prove new speedups of product formulas for simulating many quantum systems, including simulations of nearest-neighbor lattice systems, second-quantized plane-wave electronic structure, k-local Hamiltonians, rapidly decaying power-law interactions, and clustered Hamiltonians, nearly matching or even outperforming the best previous results in quantum simulation. We accompany our analysis with numerical calculation, which suggests that the bounds also have nearly tight constant prefactors. We identify applications of quantum simulation to designing other quantum algorithms and improving quantum Monte Carlo methods. We develop an algorithmic framework ``quantum singular value transformation'' using techniques from quantum simulation and apply it to implement principal component regression. We also apply our new analysis of product formulas and obtain improved quantum Monte Carlo simulations of the transverse field Ising model and quantum ferromagnets.
Quantum simulation of lattice gauge theories offers a path to first-principles calculations of real-time dynamics and finite-density phenomena that are inaccessible to classical Euclidean Monte Carlo methods. Realizing this potential requires a pipeline of controlled approximations---lattice discretization, truncation, digitization, state preparation, time evolution, measurement, and continuum extrapolation---each contributing errors and computational costs. General algorithmic asymptotic complexity arguments capture only part of the cost picture: the practical resources required for end-to-end simulation depend on numerical pre-factors, the cost of the subroutines that various algorithms call, and structural features of the lattice theory, all of which the asymptotic scaling leaves unspecified. This thesis develops contributions that aim to make such dependencies explicit and quantitative. We begin with a rigorous analysis of the asymptotic gate complexity of product formulas and Quantum Signal Processing (QSP) for the time evolution of general classes of lattice Hamiltonians, identifying the parameter regimes in which each method is favorable and estimating the associated pre-factors numerically. Since the cost of QSP-based simulation is dominated (in regimes of practical interest) by the block-encoding subroutine it calls, we also develop several novel block-encoding constructions tailored to bosonic lattice field theories and show that signal-processing techniques can be repurposed as highly efficient block-encoding subroutines. We then turn to the problem of truncating the Kogut--Susskind electric-field Hilbert space. Existing error bounds rely on energy-conservation arguments and require the truncation to grow with simulation time. By exploiting Hilbert space fragmentation we derive the tightest error bounds known to date: they decay factorially in the truncation parameter and are independent of simulation time. We validate them numerically for U(1) pure gauge theory and the Schwinger model. Finally, we address the continuum limit in the presence of approximate time evolution. For product formulas, we show that Trotter errors correspond to irrelevant operators that vanish in the continuum limit, which yields a renormalization trajectory independent of the Trotter step size. More generally, we introduce the Statistically-Bounded Time Evolution (SBTE) protocol, an algorithm-independent framework that treats approximation errors as a systematic uncertainty to be driven below the working statistical precision. This decouples the continuum-limit prescription from the choice of time-evolution algorithm and provides a basis for end-to-end cost comparisons across simulation methods.