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Using Quantum Computers for Quantum Simulation

Katherine L Brown, William J Munro, Vivien M Kendon

arXiv:1004.5528v3quant-ph

TL;DR

Quantum simulation seeks to study important quantum systems whose models are too difficult to solve accurately with classical methods. This survey synthesizes theoretical and experimental approaches using quantum computers, including universal Hamiltonian simulation, data extraction, and state preparation. It identifies useful applications within reach of modest quantum computers while documenting efficiency, architecture, error-correction, and precision constraints.

  • Problem

    Important systems in quantum chemistry and superconducting materials cannot be simulated with sufficient accuracy by analytical or classical numerical methods.

  • Method

    The paper surveys theoretical and experimental quantum-computer methods for Hamiltonian evolution, information extraction, state preparation, and implementation across quantum architectures.

  • Results

    The survey concludes that quantum simulation can efficiently represent and study local quantum systems, with useful quantities including energy gaps, eigenvalues, eigenvectors, correlation functions, expectation values, and spectra.

  • Takeaways & Limitations

    Quantum simulation is a potentially early practical application because useful results may be accessible with fewer than a hundred qubits, although implementation overheads remain substantial.

  • Takeaways & Limitations

    Precision requirements, Trotterization, and error-correction overheads can threaten the viability of simulations even at fairly modest sizes.

Abstract

from arXiv · show

Numerical simulation of quantum systems is crucial to further our understanding of natural phenomena. Many systems of key interest and importance, in areas such as superconducting materials and quantum chemistry, are thought to be described by models which we cannot solve with sufficient accuracy, neither analytically nor numerically with classical computers. Using a quantum computer to simulate such quantum systems has been viewed as a key application of quantum computation from the very beginning of the field in the 1980s. Moreover, useful results beyond the reach of classical computation are expected to be accessible with fewer than a hundred qubits, making quantum simulation potentially one of the earliest practical applications of quantum computers. In this paper we survey the theoretical and experimental development of quantum simulation using quantum computers, from the first ideas to the intense research efforts currently underway.

1 Introduction

Quantum simulation addresses quantum systems whose classical simulation becomes infeasible as Hilbert spaces grow exponentially. The survey introduces quantum-computer approaches, emphasizing efficient evolution, information extraction, and practical implementation constraints.

  • 1 Introduction: A general n-qubit state requires 2^n complex amplitudes, making classical simulation feasible only for very small systems.At 27 qubits, storing the state requires 1 Gbyte; each additional qubit doubles the memory requirement.
  • 1 Introduction: Quantum computers can encode general quantum states efficiently in their naturally available superpositions, motivating quantum simulation as an early practical application.The paper identifies useful simulations as potentially requiring upwards of 36 qubits, far fewer than other proposed quantum algorithms.
  • 1 Introduction: Quantum simulation is not a drop-in replacement for classical simulation because measurements reveal limited information and useful results require tailored processing and repeated runs.Classical strong simulations expose the full probability distribution, whereas quantum simulations must concentrate desired information before measurement.
  • 1 Introduction: Efficient quantum simulation requires initialization, quantum processing, and data extraction to use resources scaling polynomially with problem size.The paper notes that polynomial scaling is not always a reliable guide to practical performance.
  • 1 Introduction: The survey reviews theoretical and experimental development, while noting that practical experience is still needed to refine proposed quantum-simulation methods.It presents universal quantum simulation on architectures based on available registers and gate operations.

2. Universal Quantum Simulation

Universal quantum simulation uses a quantum computer to implement Hamiltonian time evolution efficiently when the target Hamiltonian decomposes into a polynomial number of bounded local terms. The approach is broadly applicable in theory, but precision requirements, control assumptions, and error correction create important practical constraints.

  • 2.1. Lloyd’s method: First-order Trotter-Suzuki simulation requires τ ∝ t^2/ε steps to achieve error ε.The total error decreases as the number of discrete time steps increases, with ε ∝ t^2/τ.
  • 2.1. Lloyd’s method: A Hamiltonian decomposed into n polynomially many local terms can be simulated with a number of operations polynomial in system size.Each local term acts on at most ℓ variables, with fixed ℓ, so n is polynomial in N; the operation count is bounded by τng^2.
  • 2.1. Lloyd’s method: The Lloyd method provides a general procedure for evolving a quantum state under a Hamiltonian by decomposing time evolution into implementable local evolutions.The method addresses the core task of calculating |Ψ(t)⟩ from an initial state and Hamiltonian, including time-dependent Hamiltonians.
  • 2.2. Errors and efficiency: Quantum simulation errors scale inversely with ε, and error-correction requirements can become exponentially more costly than for comparable binary computations.The resource dependence is proportional to t^2ng^2/ε, while analyses identify inverse error scaling and substantial error-correction overheads for useful problem sizes.
  • 2.3. Universal Hamiltonians: Entangling Hamiltonians can provide universal simulation with local controls, but optimal control sequences are computationally hard to determine in general.For qubits, even-body interactions can be universal, while any two-body entangling qudit Hamiltonian is efficiently universal with local unitaries.
  • 2.4. Efficient Hamiltonian simulation: The standard assumption of arbitrary efficient local control and switchable interactions is not experimentally feasible for every architecture, restricting the Hamiltonians that can be simulated.NMR systems have always-on interactions, and limited local control may allow only a restricted class of Hamiltonians despite their usefulness as experimental test beds.

3. Data extraction

Quantum simulation provides limited direct access to the evolving wavefunction, so algorithms must be designed around efficiently extracting selected polynomial-sized properties. The surveyed methods obtain quantities including energy gaps, eigenvalues, eigenvectors, correlation functions, expectation values, spectra, and characteristics of quantum chaos.

  • 3. Data extraction: Quantum simulations extract only polynomial-sized results, including energy gaps, eigenvalues, eigenvectors, correlation functions, expectation values, and operator spectra.These outputs use related techniques such as phase estimation and quantum Fourier transforms.
  • 3. Data extraction: Energy gaps are obtained by evolving a ground–excited-state mixture so that the relative phase encodes the gap for phase estimation.The phase difference is directly proportional to the energy gap.
  • 3. Data extraction: Abrams–Lloyd methods find a polynomial fraction of Hamiltonian eigenvalues and eigenvectors when the corresponding time-evolution unitary can be efficiently simulated.The method requires an approximate eigenvector with non-exponentially-small overlap with the target eigenvector.
  • 3. Data extraction: Correlation functions are estimated by controlling operator applications around time evolution and repeatedly measuring a single ancilla.The ancilla measurement initially gives one bit of accuracy; combining repeated outcomes improves the estimate, while spatial translation yields spatial correlations.
  • 3.4. Quantum chaos: Quantum-chaos simulations can yield selected properties efficiently, but most extraction methods provide only polynomial speedups despite exponentially costly classical simulation.Exceptions include one-bit chaos-versus-regularity tests, efficient fidelity-decay measurement, and potentially diffusion constants.

4. Initialization

Initialization is as important as time evolution because useful simulations often require complex states such as ground or thermal states. The surveyed approaches include explicit state preparation, adiabatic evolution, approximate ground-state preparation, thermalization, and quantum Metropolis sampling.

  • 4. Initialization: Preparing complex initial states is a central challenge because ground and Gibbs thermal states are often unknown or difficult to specify.An arbitrary state requires exponentially many parameters, although efficiently described states can admit efficient preparation methods.
  • 4.1. State preparation: Explicitly described states can be prepared with general register-state methods, including Grover-based preparation when the description is sufficiently efficient.Optimized techniques can reduce the CNOT-gate prefactor close to the optimal value.
  • 4.2. Adiabatic evolution: Adiabatic preparation evolves an easy Hamiltonian into the target Hamiltonian slowly enough to remain near its ground state, requiring polynomial time when the gap is not exponentially small.Inhomogeneous transitions can allow finite systems to cross phase transitions in polynomial time without ending in an excited state.
  • 4.2. Adiabatic evolution: Exact ground-state preparation is often unnecessary: phase estimation can use a ground–first-excited-state superposition or a state with substantial ground-state weight.Approximate adiabatic evolution can therefore remain useful for estimating gaps and low-lying eigenvalues.
  • 4.3. Preparing thermal equilibrium states: Thermal-state methods face efficiency limits, while quantum Metropolis sampling provides a quantum analogue of classical Metropolis sampling for Gibbs distributions.The thermalization running-time bound scales as D^a, with a ≤ 1/2 related to the Helmholtz free energy and D the Hilbert-space dimension.

5. Hamiltonian evolution

The section surveys Hamiltonian-evolution techniques for quantum simulation, emphasizing Fourier-based methods, lattice-gas dynamics, and applications to chemistry and noisy environments. These approaches exploit problem structure to reduce operations or represent relevant physical dynamics, while practical simulations remain constrained by available coherence and computation.

  • Quantum pseudo-spectral method: Fourier-based pseudo-spectral evolution uses position- and momentum-diagonal Hamiltonian terms with an efficiently implemented quantum Fourier transform.The method alternates diagonal operations in convenient bases connected by Fourier transformation.
  • Quantum pseudo-spectral method: Tens of thousands of operations are estimated for useful six- to ten-qubit simulations, yet pseudo-spectral evolution remains preferable to Lloyd evolution when Fourier-related diagonal bases apply.The efficiency advantage is therefore algorithmically meaningful despite current coherence limits.
  • Lattice gas automata: Quantum lattice-gas dynamics encode directional amplitudes in two qubits per site and alternate local collision with propagation operations.Taking the continuum limit yields a Schrödinger equation for the total amplitude.
  • Lattice gas automata: The lattice-gas construction generalizes to higher dimensions, more particles, interactions, and external potentials, supporting its prospective practicality for quantum simulation.Its utility is presented as an expectation based on corresponding classical methods.
  • Quantum chemistry: Quantum chemistry methods address molecular dynamics beyond classical exact calculations, including exact electronic simulation for systems with more than four atoms.The approach discretizes position, uses QFT-based time evolution, and supports reaction probabilities and transition quantities.
  • Quantum chemistry: MCSCF initial states improve on Hartree-Fock states for excited states and strong interactions by enabling faster and safer evolution.The stated benefit includes avoiding convergence to unphysical states when energy gaps are small.
  • Noise: Environmental decoherence can be modeled with superoperators, while using hardware decoherence directly is efficient when its statistics and strength sufficiently match the target noise.Small gate imperfections may remain tolerable for perfect-unitary targets, but precision can scale unfavorably with system size.

6. Fermions and bosons

The section develops quantum-simulation approaches for fermionic, bosonic, pairing, and lattice-gauge systems. It emphasizes mappings, state preparation, and Hamiltonian-specific methods for problems that are difficult for classical computation, while identifying operation count and preparation probability as practical constraints.

  • Fermionic systems: The fermionic sign problem limits classical many-body simulation, while quantum computation does not provide a back door to that problem and could target strongly interacting fermion models.High-temperature-superconductor models are identified as important potential applications.
  • Hubbard model: The Hubbard model motivates quantum simulation because classical methods struggle with its analytic and computational treatment, especially beyond one dimension.Its quantum encoding can use second quantization with two qubits per site, while first quantization may be more efficient when particle number is low.
  • The BCS Hamiltonian: Pairing and BCS Hamiltonians require Trotterization because their terms do not commute, making operation savings important for coherence-limited experiments.A qubus architecture reduces the general-case operation count from O(N^5) for NMR to O(N^2).
  • Particle mappings: Qubit simulation requires mappings from physical particles to spin-1/2 systems; fermions map directly, whereas bosons require a bounded-occupation direct state mapping.The bosonic mapping is less efficient but enables simulation when the boson-per-state limit holds.
  • Initial state preparation: Fermionic state preparation uses Slater determinants and ancillas, with the desired all-zero ancilla outcome selected by measurement at probability 1/n.The preparation therefore requires an average of n trials.
  • Initial state preparation: Bosonic product states can be mapped to spin states and prepared by flipping relevant spins before applying a similar linear-combination procedure.Second-quantized Slater-determinant states can also be converted to first-quantized real-space lattice representations for pure and mixed states.
  • Lattice gauge theories: Quantum lattice-gas methods and specialized simulators extend the scope of simulation toward lattice gauge theories, QCD-related models, and tunable Hubbard-plus-dipolar interactions.The review presents these as methods for systems where classical simulations are computationally intensive or physically limited.

7. Overview

The overview presents universal quantum simulation as capable of efficiently representing systems with local or efficiently describable Hamiltonians, while specialized simulators may exploit particular structure. It concludes that implementation remains central and that longer simulations face unresolved error and accuracy concerns.

  • Experimental outlook: The review emphasizes that useful quantum simulation requires implementations large enough to address problems beyond classical computation, not merely theoretically valid algorithms.It identifies multiple methods and systems as options for experimental development.
  • Experimental outlook: Longer simulations remain constrained by the possibility that errors will threaten accuracy, leaving their viability an open question.This limitation is stated as part of the transition from theoretical methods toward useful implementations.
  • Scope and architectures: Universal quantum simulation can efficiently simulate any quantum system with a local or efficiently describable Hamiltonian.The section contrasts this generality with specialized approaches that exploit Hamiltonian properties or symmetries.
  • Scope and architectures: Quantum-simulation architectures overlap broadly with quantum-computing experimental techniques, while the review focuses on implementations corresponding to its theoretical methods.Architecture choice determines which algorithms and physical systems can be addressed.

8. Proof-of-principle experiments

Proof-of-principle experiments have used several quantum-computing platforms to simulate spin, many-body, bosonic, fermionic, and molecular systems, while exposing scalability and control constraints.

  • NMR experiments: NMR experiments simulated spin-chain Hamiltonians by transforming native ZZ interactions into Heisenberg and higher-body interactions.Local unitary rotations enabled X, Y, and Z orientations, while experiments demonstrated three- and four-body interaction simulation.
  • NMR experiments: Three NMR qubits simulated a Fano-Anderson many-body Fermi system after exploiting translational symmetry and mapping fermion modes to qubits.An approximate refocusing scheme reduced decoherence problems during long system evolutions.
  • NMR experiments: NMR simulations of bosonic systems require Hilbert-space truncation, which limits accuracy and makes scaling difficult.Small-system experiments also faced coupling and decoherence constraints that limited their duration and scalability.
  • NMR experiments: Molecular hydrogen simulations used a quantum algorithm to obtain its ground-state energy and illustrate potential quantum-chemistry advantages.The cited approach emphasized simulating dynamics exactly rather than following classical approximations.
  • Photonic experiments: Photonic experiments simulated molecules and four-spin Heisenberg systems using photon polarization, linear optics, measurements, and induced nonlinear control.Photonic platforms offer relatively straightforward experimental requirements but face challenges in obtaining suitable nonlinear interactions and scalability.

9. Atom trap and ion trap architectures

Atom and ion traps offer leading routes to quantum simulation because they provide distinguishable, controllable qubits and exploitable interactions, with trade-offs between control, addressability, parallelism, and scalability.

  • Architectures: Ion traps and optical lattices are promising architectures because their intrinsic couplings can be exploited for quantum simulation.Trapped ions permit individual addressing through spatial separation, while optical lattices provide regularly arranged atoms with tunable interactions.
  • Architecture trade-offs: Optical lattices offer highly parallel manipulation, whereas ion traps provide greater quantum control and more developed simulation research.The architectures therefore trade manipulation parallelism and addressing convenience against control precision.
  • Optical lattices: Optical-lattice proposals use global single-particle and nearest-neighbor interactions to realize fermionic, bosonic, and spin systems without individual atom addressing.Different trapped atoms and lattice-parameter regimes provide the physical realizations of these subsystem types.
  • Ion-trap simulations: Trapped-ion proposals target spin, Bose-Hubbard, Ising, and Heisenberg models, including quantum phase transitions.An experiment with two trapped ions traversed from a quantum paramagnetic regime to quantum ferro- and antiferromagnetic regimes.
  • Optical-lattice simulations: Optical lattices are positioned as special-purpose simulators that can reach larger many-body systems before universal quantum computers can do so.Their natural atom-atom interactions are the central resource for this approach.
  • Coupled cavity arrays: Coupled cavity arrays improve individual-atom control relative to optical lattices and support proposed Heisenberg-model simulations across physical implementations.Their cavities are coupled through photon exchange, combining atom-control features with array-based architectures.

10. Electrons and excitons

Electron-, hole-, and superconducting-qubit architectures extend quantum simulation to spin lattices, artificial atoms, chemical reactions, and molecular collisions, but often impose structural restrictions.

  • Architectures: Electron and hole systems can be controlled in traps or quantum dots, while superconducting circuits use collective electronic states or quantized flux as qubits.These platforms provide alternative physical implementations for quantum simulation beyond atom and ion traps.
  • Spin lattices: Spin-lattice proposals trap electrons above helium surfaces, using induced charges to create double-well potentials for electron confinement.The electron spin serves as the qubit in these proposed arrays.
  • Quantum dots: Quantum dots can act as artificial atoms whose whole-dot degrees of freedom may be suitable for simulating chemical reactions.The main challenge is controlling quantum-dot parameters and locations collectively and predictably.
  • Superconducting architectures: A Josephson-junction proposal simulates molecular collisions by restricting an n-qubit system to the single-excitation subspace.This reduces the Hamiltonian to n × n dimensions while allowing individual Hamiltonian parameters to vary independently.

11. Outlook

Quantum simulation has substantial potential for materials science and quantum chemistry, but precision demands and long Trotterized control sequences remain major obstacles to useful large-scale experiments.

  • Outlook: Quantum simulation is especially promising for fermionic many-body systems and phase transitions, with applications spanning superconductors, magnetic materials, and quantum chemistry.The motivation includes systems whose classical simulation is limited by the sign problem.
  • Outlook: Precision requirements for larger simulations scale inversely with precision and are more costly than in digital classical or qubit computations.The one-to-one mapping between the simulated system and simulator Hilbert spaces creates this resource burden.
  • Outlook: Long Trotterization control sequences combined with precision costs threaten the viability of simulations even at fairly modest sizes.The review identifies this unresolved issue as a significant barrier to larger-scale quantum simulation.
  • Outlook: Special-purpose simulators with Hamiltonians similar to the target system are considered the leading route to calculations beyond conventional computers.Ion traps and optical lattices are described as the most developed and versatile platforms, while solid-state trap arrays are advancing.
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