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Quantum Simulation

I. M. Georgescu, S. Ashhab, Franco Nori

arXiv:1308.6253v3quant-phcond-mat.other

TL;DR

Large quantum systems are difficult to simulate because their computational representation and evolution scale exponentially. This review explains how controllable quantum systems can emulate less accessible systems, surveys digital and analog implementations and applications, and identifies remaining challenges in controllability, scalability, and reliability.

  • Problem

    Simulating quantum mechanics remains hard for large systems because memory and computational requirements grow exponentially, while approximation methods can have limitations.

  • Method

    The review synthesizes theoretical and experimental research on quantum simulation, covering controllable quantum systems, digital and analog approaches, implementations, and applications.

  • Results

    Quantum simulation has been demonstrated or proposed across platforms including atoms, ions, spins, superconducting circuits, and other systems for problems in physics and chemistry.

  • Takeaways & Limitations

    Quantum simulation provides a research tool for studying difficult problems across condensed-matter physics, high-energy physics, quantum chemistry, cosmology, and nuclear physics.

  • Takeaways & Limitations

    Proposed quantum simulators still require improved controllability and scalability, with platform-specific trade-offs in array size, individual control, and readout.

Abstract

from arXiv · show

Simulating quantum mechanics is known to be a difficult computational problem, especially when dealing with large systems. However, this difficulty may be overcome by using some controllable quantum system to study another less controllable or accessible quantum system, i.e., quantum simulation. Quantum simulation promises to have applications in the study of many problems in, e.g., condensed-matter physics, high-energy physics, atomic physics, quantum chemistry and cosmology. Quantum simulation could be implemented using quantum computers, but also with simpler, analog devices that would require less control, and therefore, would be easier to construct. A number of quantum systems such as neutral atoms, ions, polar molecules, electrons in semiconductors, superconducting circuits, nuclear spins and photons have been proposed as quantum simulators. This review outlines the main theoretical and experimental aspects of quantum simulation and emphasizes some of the challenges and promises of this fast-growing field.

I. INTRODUCTION

Quantum simulation addresses the exponential difficulty of simulating large quantum systems by using controllable quantum devices to emulate less accessible systems. This review surveys digital, analog, and classical quantum-information-inspired approaches, implementations, applications, and challenges.

  • The problem: Simulating large quantum systems requires exponentially growing memory and computational operations, while approximations can be limited or unavailable.For N spin-1/2 particles, representing the state requires 2^N complex amplitudes and time evolution requires exponentiating a 2^N × 2^N matrix.
  • Approaches: Quantum computers can act as universal quantum simulators, whereas simpler analog devices can mimic specific systems without universal programmability.Analog devices require less control and are problem-specific rather than universal.
  • Applications: Quantum simulation has potential applications across condensed-matter physics, high-energy physics, quantum chemistry, cosmology, nuclear physics, and biology.Examples include quantum phase transitions, quantum magnetism, high-Tc superconductivity, relativistic quantum phenomena, and chemical dynamics.
  • Scope: The review provides a self-contained overview of theoretical and experimental quantum simulation because a comprehensive global review was missing.It covers physical implementations, applications, challenges, and prospects, while directing readers to references for technical details.
  • Definitions: Quantum simulation uses quantum mechanical means to simulate another quantum system, including digital, analog, and quantum-information-inspired classical approaches.A quantum simulator is a controllable quantum system with a mapping between the simulated system’s states and dynamics and those of the simulator.

IV. DIGITAL AND ANALOG QUANTUM SIMULATION

Digital quantum simulation encodes quantum states in qubits and reproduces their evolution or properties through sequences of quantum gates. Its scope is broad, but efficient implementation depends on state preparation, Hamiltonian decomposition, and measurement using polynomial resources.

  • Digital and analog quantum simulation: Successful quantum simulation requires initial-state preparation, time-evolution implementation, and measurement to use only polynomial resources.Measurement is essential because the simulator’s usefulness ultimately depends on extracting information from it.
  • Digital quantum simulation (DQS): Digital quantum simulation encodes a system’s wavefunction in computational-basis qubits and implements its evolution with single- and two-qubit gates.For example, spin-up and spin-down states map to |1⟩ and |0⟩, respectively.
  • Digital quantum simulation (DQS): DQS is universal in principle, although arbitrary unitary operations need not be efficiently simulable; finite-dimensional local Hamiltonians can be simulated efficiently.The efficiency qualification distinguishes formal universality from polynomial-resource simulation.
  • Digital quantum simulation (DQS): DQS applications include preparing quantum states, simulating time evolution, estimating molecular energies, measuring operator quantities, and computing eigenvalues or partition functions.Recursive state preparation reverses the target-to-initial transformation, while recursive phase estimation improves molecular-energy accuracy by adding iterations.
  • Digital quantum simulation (DQS): Hamiltonian evolution is decomposed into local gates by breaking time into small steps and approximating each step, commonly with Trotter formulas.When Hamiltonian terms do not commute, direct efficient classical decomposition is generally unavailable.
  • Digital quantum simulation (DQS): First-order Trotter decomposition can require very small time steps and very many gates for high accuracy, while higher-order decompositions may be more efficient.General many-body interactions can also require ancilla qubits when realized through two-body interactions.

B. Analog quantum simulation (AQS)

Analog quantum simulation maps a target system onto a controllable quantum simulator, often reproducing selected properties rather than the full dynamics. Its value includes useful qualitative results despite bounded control errors, but constructing mappings can require substantial ingenuity.

  • Analog quantum simulation (AQS): AQS maps the simulated Hamiltonian onto a controllable simulator Hamiltonian, with state mappings connecting the corresponding initial and evolved states.The simulator may reproduce only part of the target dynamics, depending on the mapping and simulator capabilities.
  • Analog quantum simulation (AQS): A controllable toy model is typically designed to reproduce a target property such as dynamics or a ground state.
  • Analog quantum simulation (AQS): AQS can reveal qualitative phenomena such as quantum phase transitions even when uncertainties affect the simulator’s control parameters.This usefulness persists up to a certain tolerance level because the qualitative transition may remain observable without full quantitative detail.
  • Analog quantum simulation (AQS): Finding an effective AQS mapping is not always straightforward and may require external fields or ancillary systems to mediate interactions.
  • Analog quantum simulation (AQS): The Bose-Hubbard model provides an example in which interacting bosons in a periodic potential are represented using lattice-site creation, annihilation, number, hopping, and interaction terms.The parameters J and U quantify hopping and on-site interaction strengths, respectively, while ϵ_i denotes a site energy offset.
  • Analog quantum simulation (AQS): A trapped-ion simulator can reproduce the form of the Dirac Hamiltonian through parameter identifications, enabling studies of Zitterbewegung and the Klein paradox.The mapping identifies c with 2ηωΔ and mc^2 with ℏΩ, with Ω controlled by the bichromatic-light intensity.
  • Analog quantum simulation (AQS): AQS preparation and measurement are expected to exploit natural relaxation and direct measurement of simulator observables, but these aspects have not been thoroughly discussed.

C. Quantum-information-inspired algorithms for the classical simulation of quantum systems

Quantum-information-inspired classical algorithms use structured representations of many-particle states to calculate selected quantities more efficiently. Matrix product and projected entangled-pair states extend simulations of lattice systems, while quantum Metropolis sampling addresses the sign problem.

  • Quantum-information-inspired algorithms for the classical simulation of quantum systems: Quantum-information-inspired algorithms seek more efficient classical representations of many-particle states for calculating physical quantities.
  • Quantum-information-inspired algorithms for the classical simulation of quantum systems: Matrix product states and projected entangled-pair states enable more efficient simulations of infinite-size quantum lattice systems in one and two dimensions.
  • Quantum-information-inspired algorithms for the classical simulation of quantum systems: These tensor-network methods allow spin systems to be simulated for longer times and phenomena inaccessible to previous methods to be studied.They can also be combined with Monte Carlo techniques.
  • Quantum-information-inspired algorithms for the classical simulation of quantum systems: The quantum Metropolis algorithm directly samples Hamiltonian eigenstates and thereby overcomes the sign problem.

A. Resource estimation

Quantum-simulation resource needs depend strongly on the problem, system size, and desired precision. Estimates show potential quantum advantages but also substantial qubit, gate, and accuracy-scaling requirements.

  • Resource requirements: The resources needed for quantum simulation depend on the simulation type, and useful demonstrations may require a few tens of qubits.Such systems could reach the limits of present-day supercomputers for frustrated-spin simulations or molecular-energy calculations.
  • Resource requirements: At least 100 qubits and over 200,000 quantum gates per step would be required to outperform current classical computers for pairwise-potential dynamics.The estimate concerns direct-digital simulation of N particles with a relatively small error level.
  • Accuracy and efficiency: Quantum simulation algorithms can have poor scaling with desired accuracy, even when their scaling with system size appears efficient.Reducing the Trotter step can rapidly increase gate counts and introduce computational overhead.
  • Accuracy and efficiency: Computation time for the one-dimensional transverse Ising model grows exponentially with desired precision, motivating new algorithms or implementations that avoid the Trotter formula.The resource analysis considered total physical qubits and computation time as functions of system size and numerical precision.
  • Fault tolerance: Fault-tolerant surface codes were concluded to be superior to circuit models with quantum error correction for quantum simulation under present-day experimental parameters.The comparison used the Ising model as a representative example.
  • Accuracy and efficiency: A hydrogen-molecule simulation achieved ±10^-4 E_h precision with about 522 gates, excluding error correction.The gate count included both one- and two-qubit operations and was evaluated using a DQS algorithm.
  • Analog-simulation constraints: Analog simulations may use fewer controls, but their reliability requires accounting for simulator-specific imperfections and validating results where possible.Cross-validation across physical systems is limited by the availability of comparable implementations, while analytical and numerical checks apply only to small systems.

VI. PHYSICAL REALIZATIONS

Quantum simulation can be physically realized with controllable quantum systems that need not support universal quantum computation. Implementations span atoms, ions, molecules, and other platforms, with tunable interactions and demonstrated digital and analog simulations.

  • Physical platforms: A system need not be capable of universal quantum computation to implement analog quantum simulation, whereas quantum-computer-capable systems can implement digital simulation.This distinction broadens the set of candidate physical platforms.
  • Physical platforms: Candidate analog simulators include atoms, ions, photons, nuclear and electronic spins, and superconducting circuits arranged as one- or two-dimensional qubit arrays.These arrays act as enlarged toy models of solid-state lattice structures.
  • Atoms and optical lattices: Optical lattices offer tunable, essentially defect-free systems whose geometry and dimensionality can be adjusted for condensed-matter simulations.Their controllable parameters include tunneling, local and nonlocal interactions, nonuniform potentials, and couplings between internal states.
  • Atoms and optical lattices: The Mott insulator–superfluid transition can be simulated by changing optical-lattice depth or controlling on-site interactions through Feshbach resonances.Lattice depth mainly changes tunneling strength, while Feshbach resonances tune interactions.
  • Atoms and optical lattices: Atomic gases have been used to investigate the BCS–BEC crossover and the unitarity regime by tuning interatomic interactions.The crossover connects weakly attractive fermions with tightly bound fermion pairs.
  • Molecules and Rydberg atoms: Rydberg atoms and polar molecules provide interaction-based routes to effective spin models, with polar molecules offering strong, externally tunable dipole-dipole interactions.Microwave excitations and spin-rotation couplings expand the effective-spin toolbox for polar molecules.
  • Atoms and ions: Trapped ions encode information in internal levels and vibrational modes, manipulated through resonant internal-state and sideband transitions.Laser-driven Hamiltonians can realize effective analog Hamiltonians or digital quantum gates.
  • Atoms and ions: Trapped ions have produced the most advanced digital quantum-simulation implementations to date, alongside demonstrations of analog simulations of frustrated spins and relativistic motion.The cited digital demonstrations include high-fidelity implementations reported by Barreiro et al. and Lanyon et al.

B. Nuclear and electronic spins

Nuclear spins and semiconductor quantum dots offer distinct routes to quantum simulation, combining controllability with limitations in scalability and engineering.

  • NMR nuclear spins provide long coherence times exceeding 1 s, high-fidelity gates, and coherent control of up to 12 qubits.
  • NMR uses magnetic fields, spin-spin couplings, frequency-selective transitions, and RF pulses to implement one-, two-, and possibly multi-qubit gates.
  • NMR scalability is limited by spectral crowding as the number of energy levels grows exponentially with the number of spins.
  • Quantum dots confine charge carriers, producing quantized levels that make them resemble artificial atoms and permitting electrical or optical manipulation and readout.
  • Mesh-gate quantum-dot arrays can realize tunable lattice geometries, while gate voltages, hole sizes, and doping engineer interactions between qubits.

C. Superconducting circuits

Superconducting circuits encode quantum information in tunable circuit degrees of freedom and support precise control, flexible connectivity, and several proposed simulation architectures.

  • Superconducting circuits encode information in charge, loop-current, or oscillatory states that can be manipulated electrically and measured with integrated instruments.
  • Coherence times exceed 100 µs, decoherence rates are below 10 kHz, and individual control, measurement, and high-fidelity gates have been demonstrated.
  • Circuit Hamiltonians can be tuned through level splittings and capacitive or inductive couplings between charge or flux qubits.
  • Superconducting circuits possess more than two energy levels, enabling analog simulation of spins larger than 1/2.
  • Large fabricated circuits and flexible tunable couplers support proposals for resonator arrays, arbitrary dimensions, fractals, and several condensed-matter models.

D. Photons

Photons support several quantum-simulation demonstrations and proposals, but limited flexibility and scalability constrain the range of problems they can address.

  • Photonic systems encode qubits naturally and implement one-qubit gates easily, while two-qubit gates remain difficult.
  • Photon experiments calculated anyon statistics, obtained the hydrogen-molecule energy spectrum to 20 bits of precision, and simulated frustrated spin systems.
  • Photons in atom-doped materials have been proposed for simulating Luttinger liquids and relativistic field theories.
  • Photon propagation through beam-splitter networks is computationally difficult for classical computers even for a few tens of photons, with experiments reaching four photons.
  • Photon-based quantum simulation remains limited by flexibility and scalability, while several other platforms are also being considered.

A. Condensed-matter physics

Quantum simulators provide routes to studying difficult condensed-matter problems, especially Hubbard and spin models, using both digital and analog approaches.

  • High-Tc superconductivity and disordered or frustrated systems remain major condensed-matter challenges motivating quantum simulation.
  • The Hubbard model is difficult for classical computers when many particles occupy more than one dimension.
  • The Bose-Hubbard model has been implemented with optical-lattice atoms, including demonstrations of a Tonks-Girardeau gas and a graphene-like lattice.
  • Digital simulation studies derive Hubbard-model spectra through operator mappings, initialization, evolution, and measurement, including the Jordan-Wigner transformation.
  • Spin Hamiltonians can be simulated digitally or analogically with trapped ions, while tunable laser interactions realize Ising, XY, and XYZ models.
  • Superconducting circuits offer flexible connectivity for higher-dimensional or fractal spin and Hubbard models, and can naturally represent spins with s > 1/2.

3. Quantum phase transitions

Quantum phase transitions are abrupt ground-state changes driven by quantum fluctuations at absolute zero, and analog quantum simulators provide experimental routes to investigate them and related many-body phenomena.

  • 3. Quantum phase transitions: Quantum phase transitions occur at absolute zero when varying a physical parameter causes an abrupt change in a many-body system’s ground state.
  • 3. Quantum phase transitions: The superfluid-to-Mott-insulator transition was first observed in 2002 using rubidium atoms trapped in an optical lattice.
  • 3. Quantum phase transitions: Adjusting the lattice potential depth controls the ratio between tunneling energy J and on-site interaction energy ˜U, inducing the transition between phases.
  • 3. Quantum phase transitions: A transition from a paramagnet to an antiferromagnet was emulated with two trapped calcium ions using the quantum Ising model.
  • 3. Quantum phase transitions: Disordered systems can exhibit localization and other phenomena absent from perfectly ordered systems, motivating quantum-simulation studies of disorder and frustration.
  • 3. Quantum phase transitions: Optical-lattice atoms simulated ferromagnetic, antiferromagnetic, and frustrated spin configurations, revealing a rich phase diagram with different phase transitions.

5. Spin glasses

Quantum simulation approaches address spin glasses, frustrated systems, superconductivity, and tunable quantum materials across analog and digital platforms.

  • 5. Spin glasses: Spin glasses have mixed ferromagnetic and antiferromagnetic interactions, producing spatially nonuniform and nearly frozen spin orientations at low temperatures.
  • 5. Spin glasses: Digital quantum simulation can efficiently simulate spin glasses, including algorithms for constructing Ising-model Gibbs distributions.
  • 5. Spin glasses: Analog proposals use magnetic impurities, superconducting qubits, or Fermi-Bose mixtures to study spin-glass models such as Sherrington-Kirkpatrick systems.
  • 5. Spin glasses: A two-qubit version of an algorithm for the BCS pairing Hamiltonian was experimentally realized using nuclear magnetic resonance.
  • 5. Spin glasses: Experiments observed signatures of the BCS-to-BEC superfluid crossover as attractive interactions between fermions were varied.
  • 5. Spin glasses: Quantum metamaterials use periodically arranged mesoscopic building blocks, including qubit chains in resonators, to control electromagnetic-field propagation.

8. Topological order

Quantum simulators offer proposed platforms for studying topological order, anyons, lattice gauge theories, and ring-exchange models using engineered interactions and optical lattices.

  • 8. Topological order: Anyons are two-dimensional particles with neither bosonic nor fermionic statistics and have been proposed for topological quantum computation.
  • 8. Topological order: The fractional statistics of anyons in the four-body-interaction Kitaev model could be studied with cold atoms in optical lattices or superconducting circuits.
  • 8. Topological order: Optical lattices could realize abelian and non-abelian anyons in topological lattice models using ancilla particles.
  • 8. Topological order: Neutral atoms have been proposed for simulating both abelian and non-abelian lattice gauge theories.
  • 8. Topological order: Atoms in optical lattices can realize ring-exchange models by placing two internal states in square lattices and plaquette centers.
  • 8. Topological order: Additional proposals include photonic simulation of a nucleon state and superconducting-sphere arrays for the O(3) nonlinear sigma model.

C. Cosmology

Quantum simulation extends to analog cosmology and related quantum models, using controllable laboratory systems to emulate relativistic, chemical, open-system, and nuclear dynamics.

  • C. Cosmology: Acoustic waves in a two-component Bose-Einstein condensate could investigate scalar fields in the curved spacetime of an expanding universe.
  • C. Cosmology: The cosmological analog would vary interparticle coupling or expand the condensate through a temporal ramp, though the proposal may be experimentally challenging.
  • C. Cosmology: Analog models have been proposed to test unobserved phenomena including an Unruh-like effect and the Schwinger effect.
  • C. Cosmology: Superconducting circuits act as artificial atoms with discrete energy levels and coherent oscillations, enabling analogues of atomic effects and quantum-optical models.
  • C. Cosmology: Quantum simulation methods have been proposed for molecular energies, chemical reactions, open quantum systems, quantum maps, and nuclear many-body models.

I. Interferometry

Quantum simulation spans diverse applications and physical platforms, from interferometry and quantum thermodynamics to studies of Majorana fermions, graphene, and neutrino oscillations. Despite progress, practical simulators still require improved controllability, scalability, and theoretical understanding of decoherence.

  • Interferometry: Trapped-ion experiments realized early quantum-simulation studies of nonlinear interferometers and later explored Mach-Zehnder interferometry with ion arrays.
  • Interferometry: Photon experiments demonstrated boson sampling, while trapped ions were proposed as an alternative implementation for this related interferometry problem.
  • Interferometry: Superconducting qubits support Landau-Zener-Stückelberg, Fano, and Fabry-Perot interferometry in quasi-one-dimensional open systems with tunable qubit mirrors.
  • Applications: Quantum simulation has been applied or proposed for topics including Majorana fermions, graphene, neutrino oscillations, quantum heat engines, Brownian motion, and quantum thermodynamics.
  • Applications: Applications extend beyond condensed matter to high-energy physics, quantum chemistry, cosmology, nuclear physics, and potentially biology; summary tables list candidate systems and experimental realizations but are not exhaustive.
  • Challenges: Practical quantum simulators still need better controllability and scalability, with optical lattices offering larger arrays but difficult individual control and readout.
  • Challenges: Further theoretical work should characterize decoherence, control requirements, and when simulator decoherence can itself be useful.
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