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Quantum computers as universal quantum simulators: state-of-art and perspectives

Francesco Tacchino, Alessandro Chiesa, Stefano Carretta, Dario Gerace

arXiv:1907.03505v2quant-ph

TL;DR

Classical simulation of complex many-body quantum dynamics is limited by exponential resource scaling, motivating programmable quantum simulators. This review explains digital quantum-simulation methods, surveys trapped-ion and superconducting experiments, and assesses their capabilities and remaining challenges. The reviewed experiments demonstrate progress toward universal simulation, but current devices remain constrained in fidelity, scale, and reliability.

  • Problem

    Complex quantum many-body dynamics can require exponentially increasing classical time and memory resources, particularly for strongly correlated systems.

  • Method

    The review presents digital quantum-simulation theory, maps spin Hamiltonians to quantum circuits, and surveys experimental results from trapped-ion and superconducting platforms.

  • Results

    Up to 6-spin models have been simulated with trapped ions and up to 4-spin models with superconducting circuits, while 5 Trotter steps can reduce superconducting-simulator fidelity to slightly above 60%.

  • Takeaways & Limitations

    Near-term noisy quantum processors have demonstrated proof-of-concept universal quantum simulations, but reaching quantum advantage for complex many-body models remains a future goal.

Abstract

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The past few years have witnessed the concrete and fast spreading of quantum technologies for practical computation and simulation. In particular, quantum computing platforms based on either trapped ions or superconducting qubits have become available for simulations and benchmarking, with up to few tens of qubits that can be reliably initialized, controlled, and measured. The present review aims at giving a comprehensive outlook on the state of art capabilities offered from these near-term noisy devices as universal quantum simulators, i.e. programmable quantum computers potentially able to calculate the time evolution of many physical models. First, we give a pedagogic overview on the basic theoretical background pertaining digital quantum simulations, with a focus on hardware-dependent mapping of spin-type Hamiltonians into the corresponding quantum circuit model as a key initial step towards simulating more complex models. Then, we review the main experimental achievements obtained in the last decade regarding the digital quantum simulation of such spin models, mostly employing the two leading quantum architectures. We compare their performances and outline future challenges, also in view of prospective hybrid technologies, towards the ultimate goal of reaching the long sought quantum advantage for the simulation of complex many body models in the physical sciences.

I. INTRODUCTION

Quantum many-body dynamics can exceed classical simulation resources because Hilbert-space size grows exponentially, motivating controllable quantum simulators. This review introduces digital quantum simulation and surveys near-term platforms, especially trapped ions and superconducting circuits, as potential universal simulators.

  • Digital quantum simulation: Digital simulation maps a physical model onto a spin-type model, slices its evolution using the Trotter-Suzuki formula, and executes the resulting unitary sequence as a quantum circuit.The circuit produces an approximated evolved quantum state from the encoded initial state.
  • Motivation: A quantum simulator reproduces the dynamical behavior of a physical model under externally controlled conditions, including systems with internal correlations or entanglement.
  • Scope: The review focuses on near-term programmable quantum computers as universal quantum simulators and surveys experimental spin-Hamiltonian simulations on trapped-ion and superconducting platforms.These architectures are highlighted as the two leading platforms covered in the review.
  • Theory: Quantum time evolution requires implementing U(t) = e^-iHt, while classical matrix exponentiation demands exponentially increasing resources for composite quantum systems.
  • Motivation: Quantum many-body models become classically intractable because the required time and memory resources scale exponentially with system size.The difficulty is especially pronounced when strong correlations dominate, where approximate classical methods may fail to provide correct answers.
  • Theory: Universal quantum computers can efficiently calculate time evolution for Hamiltonians composed of local terms, whereas arbitrary unitaries generally require exponentially many elementary operations.

A. The quantum computer as a universal quantum simulator

Digital quantum simulation encodes a physical Hamiltonian into qubits, decomposes its time evolution into local terms, and implements the resulting circuit with universal gates. Suzuki–Trotter repetition makes the approximation arbitrarily accurate while retaining polynomial scaling for local interactions.

  • Hamiltonian encoding: Time evolution U(t) = exp(−iHt) is implemented by composing elementary unitaries for the Hamiltonian’s local terms.Universal quantum gates realize the circuit products corresponding to successive unitary operations.
  • Trotterization: The Suzuki–Trotter decomposition repeats local-unitary sequences for time slices t/n to approximate the full evolution.If local terms commute, the decomposition is exact at n = 1; otherwise, increasing n reduces the digital error.
  • Trotterization: For any approximation ϵ, the evolution can be computed in at most nϵLmmax^2 operations, which is polynomial in N when L = poly(N).Nearest-neighbor interactions provide an example of the required polynomial locality condition.
  • Hamiltonian encoding: A target Hamiltonian is mapped onto the Pauli algebra of N qubits, with efficiency when the mapped Hamiltonian consists of local terms.The mapping is direct for spin-1/2 systems and extends to broader physical models through suitable encodings.
  • Circuit construction: A practical simulation prepares an initial state, translates local unitaries into elementary gates, composes the circuit, and measures observables.These steps form general instructions for designing quantum simulation algorithms.

III. QUANTUM CIRCUITS

Quantum-circuit simulation requires mapping target Hamiltonians onto qubit Pauli operators and then decomposing their local dynamics into hardware-compatible gates. This framework covers spin models and mappings from higher-spin, fermionic, and fermionic-bosonic systems when locality is retained.

  • Hardware dependence: Hardware-native gate sets and qubit connectivity determine how efficiently the same target unitary can be realized on different platforms.Non-native connectivity can be compensated with SWAP operations, but this may add circuit overhead.
  • Qubit algebra: Qubits obey the Pauli-matrix algebra, including commutation and anticommutation rules that organize the circuit representation.The identity and Levi-Civita and Kronecker tensors specify the relevant algebraic relations.
  • Hamiltonian mapping: Any target Hamiltonian must be mapped to an equivalent Hamiltonian of interacting spin-1/2 operators for qubit-based simulation.The mapping is straightforward for spin-1/2 models and is also known for higher-spin and fermionic systems.
  • Hamiltonian mapping: Local Hamiltonian terms can be reduced to single- and two-spin contributions whose time evolutions are translated into elementary gate sequences.Maintaining locality allows efficient gate decomposition for many physically relevant models.

B. Single-qubit rotations

Single-qubit terms act as effective magnetic fields and generate rotations of the Bloch vector. Arbitrary single-qubit operations can be assembled from coordinate-axis rotations or standard gates such as Hadamard and phase gates.

  • Universal single-qubit operations: The most general single-qubit SU(2) operation can be obtained by combining elementary gates including the Hadamard gate.The construction uses standard single-qubit transformations described in the circuit model.
  • Universal single-qubit operations: Rotations around the x, y, and z axes are represented through specific choices of parameters in U(θ, φ, λ).The mappings include Rz(λ) = U(0, 0, λ), Rx(θ) = U(θ, −π/2, π/2), and Ry(θ) = U(θ, 0, 0).
  • Universal single-qubit operations: Platforms capable of coordinate-axis rotations can realize arbitrary U(θ, φ, λ) operations through a suitable decomposition.Finite fixed-phase gate sets can also provide approximate implementations.
  • Physical interpretation: A single-spin term represents a magnetic field applied to qubit i along a direction h, producing a rotation of its Bloch vector.This connects the Hamiltonian’s one-body terms with the geometric action on a qubit.

C. Two-qubits gates

Two-qubit interactions are implemented by combining single-qubit basis changes with native entangling gates. The appropriate decomposition depends on the hardware platform, and optimization can reduce gate counts for particular Hamiltonians.

  • Interaction decomposition: Two-spin Pauli interactions are generally implemented as combinations of single- and two-qubit gates.These building blocks can be combined to simulate paradigmatic spin models such as Heisenberg, XY, and Ising models.
  • CNOT-based gates: With a CNOT-based universal set, terms generated by σα ⊗ σβ are implemented through a two-qubit entangling circuit and single-qubit reference-frame changes.The construction is typical of superconducting qubit technology with cross-resonance interactions.
  • Superconducting platforms: Superconducting realizations may use XX + YY interactions or controlled-phase gates as native two-qubit operations.Controlled-phase gates are closely related to Ising interactions, while XX terms can be reduced using single-qubit changes of frame.
  • Trapped-ion platforms: Trapped-ion platforms use collective entangling gates that can address subsets of qubits and naturally support long-range and many-body interactions.On two qubits, a collective gate can generate an XX interaction.
  • Optimization and constraints: The elementary decompositions are not always optimal, because combining two-qubit operations can reduce the total gate count for particular target Hamiltonians.Hardware constraints on achievable phases can also require single-qubit corrections or additional constructions.

D. Multiple-qubit interactions

Multi-qubit interaction terms can be decomposed into single- and two-qubit operations, with the circuit pattern generalizable beyond three qubits. Trapped-ion hardware natively supports many-body interactions, enabling efficient decompositions using Mølmer–Sørensen gates.

  • N-qubit interaction terms can in principle be decomposed into single- and two-qubit operations.
  • The decomposition pattern extends to arbitrary N > 3, with individual-qubit reference-frame changes available.
  • Trapped-ion processors natively include many-body interactions, allowing efficient decomposition with Mølmer–Sørensen gates.The hardware scalability itself sets the practical limit on N in principle.

E. Suzuki-Trotter decomposition and digital error

Suzuki–Trotter simulations must balance digital-approximation error against circuit length and hardware noise. Increasing the Trotter-step count can control error, but fixed-length circuits may be preferable on noisy processors.

  • Acceptable digital error must be assessed because increasing Trotter steps also increases gate count and can reduce result quality on noisy processors.
  • Second-order Suzuki–Trotter formulas improve digital-error scaling at the cost of an additional factor per iteration.
  • The ratio r_ϵ = δ^p/n^q controls digital error as a function of target phase and Trotter-step count.
  • Fixed n = 5 fails after a very short phase evolution, whereas quadratic scaling n = δ^2/2ϵ keeps digital error fully under control.For ϵ = 0.1, the quadratic-scaling case reaches n ≃ 10^4.
  • Fixed digital precision requires increasing Trotter steps while keeping r_ϵ fixed, and each step’s phase decreases as 1/n.If each step takes time proportional to 1/δ_n, total hardware computation time remains linear in total phase.
  • When circuit length is limited, keeping n fixed yields phase-dependent digital error scaling, such as δ^2 for first-order formulas.

F. Extracting physical observables

The review describes measurement procedures that reconstruct observables, dynamical correlations, and spectra from quantum-circuit outputs. Ancilla-assisted protocols combine controlled unitaries, time evolution, measurement rotations, and classical Fourier processing.

  • Expectation values are reconstructed by mapping observables to spin operators and combining measurement unitaries with computational-basis readout.
  • Ancilla-assisted circuits compute dynamical correlations by preparing the register and ancilla, applying controlled unitaries around time evolution, and measuring the ancilla.
  • The same correlation protocol supports equal-time correlations, n-point time-correlation functions, and operators expressible as sums of unitary products.
  • Applying a classical FFT to measurements over θ extracts the spectrum of a Hermitian operator.For Q = H, the operator exponential is implemented as a unitary time-evolution-like circuit.
  • Two-qubit Heisenberg evolution admits circuit realizations using either six CNOTs, three CNOTs, or three Uxy operations.

G. Examples

The examples apply digital circuits to Heisenberg and transverse-field Ising models, dynamical correlations, and fermionic systems. They illustrate observable extraction, Trotterized evolution, and mappings whose circuit demands depend on hardware-native interactions.

  • Heisenberg and Ising examples: The two-qubit Heisenberg interaction can be decomposed into elementary gates and used to extract individual-spin magnetizations.The reported numerical example contains no digital error for the measured observable.
  • Heisenberg and Ising examples: A three-spin open Heisenberg chain and a two-qubit transverse-field Ising model provide digital-simulation examples for occupation probability and total magnetization.The Ising simulation uses single-qubit x rotations and ZZ operations, with magnetization obtained from σ_z measurements.
  • Dynamical correlations: The three-spin Heisenberg circuit computes next-to-nearest-neighbor dynamical correlations using an ancilla-based protocol and repeated Trotterized evolution.
  • Dynamical correlations: The three-spin correlation simulation uses n = 5 Trotter steps, initializes |ψ⟩ = |↓↓↓⟩, and compares circuit data with the continuous-phase expectation.
  • Scaling and fermionic mappings: The presented modules can extend to arbitrary spin numbers with pairwise interactions, while correlation quantities can be extracted by repeating modified circuits polynomially many times.
  • Scaling and fermionic mappings: The Jordan–Wigner transformation maps fermionic models such as the two-site Fermi–Hubbard model to Pauli Hamiltonians implementable on qubit registers.
  • Scaling and fermionic mappings: After mapping, one-dimensional fermionic chains may involve one- and two-spin terms, but more general mappings can demand substantial circuit depth.Native many-body interactions can benefit practical fermionic simulation in the NISQ era.

IV. EXPERIMENTAL ACHIEVEMENTS AND PROSPECTIVE TECHNOLOGIES

Trapped-ion and superconducting platforms have enabled digital quantum simulations, but increasing circuit depth and system size remain central challenges. Trapped ions generally support deeper circuits with better performance, while superconducting simulations face fidelity and scalability limits.

  • Trapped ions and superconducting circuits are the two leading platforms for experimental digital quantum simulations.The review focuses on their development as universal quantum simulators.
  • Digital-step count remains correlated with the fidelity of the final simulated state.The comparison is complicated by differing platforms, initial conditions, and reported figures of merit.
  • Trapped-ion simulators generally allow deeper circuits with better performance, including more Trotter steps.
  • 5 Trotter steps limit superconducting circuit simulators, where fidelity falls slightly above 60%.The review describes this level as far from acceptable for scalability.
  • Up to 6-spin models have been simulated with trapped ions, compared with up to 4-spin models on superconducting processors.
  • No current quantum technology increases qubit number arbitrarily without degrading preparation, readout, and gate fidelities.Addressing these scalability and reliability issues remains necessary before fault-tolerant quantum computation.

A. UQS with trapper ions

Trapped-ion hardware combines long coherence and high control with demonstrations of programmable digital simulations, including systems of up to six spins. Scaling remains difficult because larger ion chains reduce gate fidelity and only a few tens of fully controlled qubits are currently practical.

  • Trapped ions provide long coherence times and strong external control for quantum information processing.Programmable trapped-ion processors with up to 11 ions, and commercial systems targeting up to 20 qubits, are described.
  • Trapped-ion qubits can achieve coherence-to-gate-time ratios of 10^5 to 10^6.The review gives coherence times from a few hundred milliseconds to hundreds of seconds and two-qubit gate times of 100–200 µs.
  • Scaling trapped-ion digital simulators beyond a few tens of controlled qubits remains challenging because larger chains limit gate fidelity.Individual addressing and cross-talk avoidance become more difficult as the number of ions increases.
  • 2011 experiments reprogrammed one optical-qubit processor for universal digital quantum simulations.The demonstrations included simulations of up to six spins and multispin interaction terms.
  • The six-spin demonstrations monitored eigenstate populations as a function of the dimensionless phase θ = Et/ℏ.
  • Digital simulation of particle–antiparticle pair creation used up to 4 qubits after mapping fermionic degrees of freedom to Pauli operators with the Jordan–Wigner transformation.The reported toy model also simulated persistent electron–positron entanglement.

B. UQS with superconducting circuits

Superconducting circuits provide a programmable platform for digitally simulating spin Hamiltonians, with experiments demonstrating Heisenberg and Ising dynamics, correlations, and hybrid quantum-classical applications. However, fidelity remains limited by coherence times and systematic circuit errors, especially as circuits deepen.

  • Hardware and digital mapping: Superconducting quantum processors had reached cloud-accessible sizes of up to 20 IBM qubits and 16 Rigetti qubits in the reviewed period.These devices use cryogenic micro-LC resonators with Josephson-junction nonlinearities and have intrinsic connectivity limitations.
  • Experimental demonstrations: 2015 experiments studied two-spin Heisenberg and Ising magnetization on a 4-qubit processor as a function of Suzuki–Trotter steps.These were the first reported digital quantum simulations of spin models on superconducting hardware.
  • Hardware and digital mapping: Superconducting processors with tunable-frequency qubits naturally implement XY interactions that can be digitally programmed into Heisenberg or Ising evolutions.The mapping uses circuit-model sequences built from one- and two-qubit gates.
  • Experimental demonstrations: Experimental fidelity began decreasing after about 2 or 3 digitized steps, although ideal fidelity increased with additional Trotter steps.Five steps still produced limited fidelities because of short coherence times and systematic circuit errors.
  • Experimental demonstrations: Dynamical correlations for a three-spin Heisenberg model showed remarkable agreement with ideal evolution using n = 2 Suzuki–Trotter steps.The comparison included autocorrelations, nearest-neighbor, and next-to-nearest-neighbor cross correlations.
  • Platform comparison: The same hybrid quantum-classical algorithm showed substantial equivalence between trapped-ion and superconducting processors at equal qubit counts, while trapped ions supported larger simulated systems.Superconducting circuits nevertheless offered much faster gates, despite a smaller coherence-time-to-gate-operation ratio.

C. Prospective technologies for UQS

Prospective UQS technologies seek to combine scalability, chip-scale integration, coherence, and fast gates beyond the leading trapped-ion and superconducting platforms. Hybrid spin-photon and electromechanical architectures provide theoretical and early experimental routes toward this goal, while several alternatives remain limited by scalability or interaction strength.

  • Alternative technologies: The leading platforms may struggle to reach more than 100 logical qubits together with substantially larger numbers of error-correcting qubits.This motivates investigation of alternative and hybrid technologies.
  • Alternative technologies: Silicon donor-electron SWAP operations have been reported with 800 ps gating time and approximately 94% fidelity.Further progress remains tied to overcoming scalability challenges.
  • Alternative technologies: Photonic quantum computing remains constrained by weak interactions from intrinsically small material nonlinearities, with single-photon sensitivity not yet measured in the reviewed period.Photonic circuits had been explored extensively as analog quantum simulators.
  • Hybrid platforms: Electromechanical nanoresonators coupled through a superconducting nonlinear element were proposed as scalable UQS building blocks using mechanical degrees of freedom for qubits.The corresponding two-spin Ising simulation included error and dissipation sources.
  • Hybrid platforms: Hybrid technologies aim to combine scalability, multidimensional integration, chip-scale implementation, long coherence, and short gating times.The proposals merge characteristics of two or more existing approaches.
  • Hybrid platforms: A hybrid spin-photon architecture theoretically simulated a three-spin transverse-field Ising model with approximately 95% average fidelity under realistic dissipation parameters.The design combines long-lived spin ensembles, tunable resonators, and transmon nonlinear elements for two-qubit gates.

V. OUTLOOK AND PERSPECTIVES

The review frames universal quantum simulation as programmable time evolution of locally interacting Hamiltonians, summarizes current hardware and algorithms, and identifies quantum advantage as a longer-term objective. Near-term progress depends on error mitigation and technological improvements, while fault tolerance could enable arbitrary digital precision.

  • Outlook and perspectives: Universal quantum simulators encode a physical model as a local Pauli Hamiltonian and implement its digitized evolution with one- and two-qubit gates.Measurements can access observables including spectra and correlation functions.
  • Outlook and perspectives: The review focuses on Heisenberg and Ising spin models because locally interacting spin terms describe many physically important Hamiltonians and provide entry points to more complex systems.The paper connects these methods to classically intractable models such as the Fermi–Hubbard model.
  • Outlook and perspectives: Current NISQ hardware with few tens of non-error-corrected qubits offers a route toward quantum advantage, but noisy gates and limited qubit coherence constrain present experiments.Error mitigation and further technological improvement are identified as important interim directions before fault-tolerant quantum computing.
  • Outlook and perspectives: Fault-tolerant quantum computing is expected to enable universal quantum simulations with arbitrary digital precision.This is contrasted with the limited precision available from current noisy devices.
  • Outlook and perspectives: Universal quantum simulators are also being explored for quantum neural networks, open-system dynamics, and integration with established classical many-body algorithms.The review presents these as ongoing use cases and algorithmic directions rather than established capabilities.
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