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Exact Ising model simulation on a quantum computer
Alba Cervera-Lierta
TL;DR
The paper addresses how quantum computers can exactly simulate a nontrivial Ising spin chain while also testing device quality. It constructs an efficient diagonalizing circuit, accesses the full spectrum for ground-state, time-evolution, and thermal simulations, and compares implementations on IBM and Rigetti devices. The experiments find the best performance on ibmqx5, with systematic errors still preserving proportional time-evolution relationships.
Problem
Quantum-device tests should include sophisticated quantum algorithms, while classical methods have limitations for strongly correlated quantum systems.
Method
The paper implements an efficient circuit that diagonalizes the one-dimensional transverse Ising Hamiltonian and prepares its eigenstates from computational-basis states.
Results
The experiments simulate ground-state magnetization and time evolution on IBM and Rigetti chips, with ibmqx5 performing best and time-evolution relationships remaining proportionally correct despite errors.
Takeaways & Limitations
Full-spectrum access extends the exact circuit approach beyond ground-state observables to time and thermal evolution, while providing a demanding device test.
Takeaways & Limitations
Experimental accuracy is limited mainly by cumulative gate-implementation errors, with readout errors also contributing.
Abstract
from arXiv · showhide
We present an exact simulation of a one-dimensional transverse Ising spin chain with a quantum computer. We construct an efficient quantum circuit that diagonalizes the Ising Hamiltonian and allows to obtain all eigenstates of the model by just preparing the computational basis states. With an explicit example of that circuit for $n=4$ spins, we compute the expected value of the ground state magnetization, the time evolution simulation and provide a method to also simulate thermal evolution. All circuits are run in IBM and Rigetti quantum devices to test and compare them qualitatively.
1 Introduction
The paper presents an exact quantum simulation of a one-dimensional transverse Ising chain as both a device-quality test and a condensed-matter application. Its efficient diagonalizing circuit provides access to the full spectrum, enabling ground-state, time-evolution, and thermal simulations.
- Motivation: Classical quantum-simulation methods face a sign problem or are efficient only for slightly entangled systems.The paper identifies strongly correlated systems as requiring quantum computers for efficient simulation.
- Contribution: Access to the whole energy spectrum enables exact simulation of time and temperature evolution, not only the ground state.This distinguishes the approach from work that accesses only ground-state magnetization.
- Motivation: The experiment uses a four-qubit transverse Ising chain to test and compare quantum-device quality.The paper also frames the experiment as relevant to condensed-matter physics.
- Contribution: The circuit diagonalizes the Ising Hamiltonian with n^2 gates and depth n log n.Its construction follows the analytical solution of the exactly solvable Ising model.
2 Quantum circuit for the Ising Hamiltonian
The circuit is built by reversing the transformations that diagonalize the Ising Hamiltonian, mapping computational-basis product states to its eigenstates. Jordan–Wigner, Fourier, and Bogoliubov steps yield an efficient implementation whose fSWAP overhead depends on chip connectivity.
- Circuit construction: A disentangling unitary maps the Ising Hamiltonian to a noninteracting diagonal Hamiltonian whose eigenstates are computational-basis states.Applying the unitary to product states therefore provides the model spectrum.
- Circuit construction: The Ising circuit follows Jordan–Wigner, Fourier, and Bogoliubov transformations to obtain the diagonal Hamiltonian.The construction mirrors the analytical solution of the model.
- n = 4 implementation: The inverse circuit prepares Ising eigenstates by undoing the Bogoliubov and Fourier transformations, while Jordan–Wigner requires no gates.The n = 4 circuit explicitly implements this reversed sequence.
- n = 4 implementation: fSWAP gates enforce fermionic anticommutation relations and can be omitted when chip connectivity directly supports the required exchanges.For ibmqx5, removing them can save up to 16 gates of depth; ibmqx4 and Rigetti require them for this implementation.
3 Time evolution
Because the circuit exposes all eigenstates and energies, it can construct exact Ising time evolution by applying phase factors in the diagonal basis before returning to the physical basis. The paper demonstrates this with a four-spin aligned initial state and transverse magnetization.
- Method: All eigenstates and energies allow exact time evolution by decomposing an initial state in the computational basis and applying e^-itϵi phases.The transformed state is then mapped back with Udis.
- Method: For an eigenstate, observables remain constant, whereas an observable not commuting with the Hamiltonian can oscillate in time.The time dependence follows from interference among energy eigenstates.
- Example: The example starts from the all-up state |↑↑↑↑⟩, represented as |0000⟩ in the computational basis.The state is expressed in the diagonal-Hamiltonian basis before time evolution is applied.
4 Thermal simulation
The same diagonalizing circuit supports thermal simulation by preparing computational-basis eigenstates and weighting observables with Boltzmann populations. The paper gives exact-averaging and sampling procedures, with a trade-off between experimental runs and statistical error.
- Thermal preparation: Thermal states are simulated by preparing computational-basis eigenstates with Boltzmann weights and applying Udis.Only the corresponding combinations of X gates are needed for state initialization.
- Exact simulation: Exact thermal simulation averages separately measured observables from all computational-basis states using their energies.The averaging is performed classically after measuring each state.
- Sampling: Sampling thermal simulation draws computational-basis states from the Boltzmann distribution before repeated circuit runs.The resulting observable estimate contains statistical error.
- Trade-off: N × 2^n runs are required for exact averaging, whereas sampling uses N runs but introduces statistical error.The first method has no statistical error from the classical averaging step.
- Thermal behavior: For n = 4, the finite-temperature magnetization shows a critical region around λ = 1 that broadens as temperature increases.At zero temperature, the transition occurs at λ = 1; at sufficiently high temperature, the spins become disordered.
5 Implementation on a quantum computer
The implementation uses IBM and Rigetti superconducting devices whose connectivity and two-qubit gate conventions determine how the four-qubit Ising circuit is mapped and how deeply it runs.
- IBM Quantum Experience: IBM devices use rotational and phase gates together with a two-qubit controlled-X (CNOT) gate.
- IBM Quantum Experience: IBM connectivity differs across devices: the five-qubit chip has a central qubit connected to four others, while the 16-qubit chip uses ladder-like connections.
- Rigetti Computing: Forest: Rigetti’s Acorn is treated as a 19-qubit device because qubit 3 is offline and cannot interact with its neighbors.
- Rigetti Computing: Forest: Rigetti uses a bidirectional controlled-Z gate, so its connectivity does not specify a control direction.
- Rigetti Computing: Forest: Rigetti’s zigzag topology requires fSWAP gates, making the circuit deeper than on ibmqx5 and comparable in this respect to ibmqx4.
6 Results and discussion
The experiments compare exact and device-based ground-state magnetization for a four-spin Ising chain, then examine time evolution and device-dependent errors. ibmqx5 gives the best performance, while systematic and gate-related errors explain deviations from theory.
- Ground-state magnetization: ibmqx5 gives the best ground-state magnetization performance among the tested devices.The paper attributes this to avoiding fSWAP gates because of its qubit connectivity.
- Ground-state magnetization: The experimental simulation approaches the exact prediction more closely at low transverse field strength λ.Errors from gate implementation and readout are identified as important contributors, while decoherence is expected to be smaller on the stated timescales.
- Ground-state magnetization: For λ > 1, errors reduce magnetization because the |1111⟩ state has the highest probability amplitude, whereas errors for λ < 1 can partially compensate.This behavior follows from the exact ground-state wavefunction amplitudes.
- Time evolution: The time-evolution simulation produces similar results, but only ibmqx5 is reported because preparing the initial state requires additional circuit depth.The device is described as able to accommodate the extra gates.
- Error structure: Higher-magnetization points show larger errors, while the proportional relations among field values remain approximately correct.The preserved relations and shifted, lower-amplitude oscillations indicate systematic error sources in the quantum device.
- Measured observable: The measured observable is ⟨σz⟩ rather than staggered magnetization because the devices produce states in the σz basis.Computing the antiferromagnetic order parameter Mx would require only a change in classical post-processing.
7 Conclusions
The work implements an exact one-dimensional transverse-field Ising-chain simulation on IBM and Rigetti devices, covering ground-state magnetization, time evolution, and thermal evolution. The circuit accesses the full spectrum efficiently, while experiments reproduce qualitative time-evolution behavior despite device-dependent errors.
- The experiment implements an exact simulation of a one-dimensional Ising spin chain with transverse field on IBM and Rigetti quantum computers.
- Time-evolution oscillations retain the theoretical proportional relationships despite lower magnetization, smaller amplitudes, and a leftward shift.
- The study evaluates ground-state transverse magnetization and time evolution of the all-aligned |↑↑↑↑⟩ state, and provides exact or sampling methods for thermal evolution.
- The circuit computes all Ising eigenstates from computational-basis initialization, with gate count scaling as n^2 and circuit depth as n log n.
- The ibmqx5 chip gives the best performance overall, although its error remains large in the model’s paramagnetic phase.
- Rigetti’s chip outperforms ibmqx4 despite both implemented circuits having the same gate depth.
- The circuit offers exact time- and thermal-evolution methods that could support simulations of other integrable models or models treated with an ansatz.
Notes
The notes record an award for the IBM-device program and a later change in Rigetti’s hardware.
- The IBM quantum-device program received the IBM “Teach Me QISKit” award.
- The reported Rigetti results use its previous 19-qubit device, whereas the company later changed to an 8-qubit device.
Appendix: Gate decomposition
The appendix decomposes the fermionic, Fourier-transform, and Bogoliubov circuit components into device-compatible gates. It also specifies implementation adjustments for IBM and Rigetti hardware and the basis-state preparation used for the n=4 ground state and time-evolution example.
- Fermionic-SWAP: The fermionic SWAP exchanges occupied fermionic modes with a minus sign and is decomposed into basic gates.
- Device-specific implementation: IBM implementations adapt CNOT orientation to qubit connectivity and decompose controlled-Z using Hadamard gates and CNOT.
- Device-specific implementation: Rigetti includes controlled-Z in its basic gate set, while CNOT and Hadamard are available through pyQuil but can increase circuit depth when non-basic.
- Fourier transform: The Fourier-transform building block uses controlled-Hadamard, whose decomposition is given alongside the relevant phase-gate implementation.
- Bogoliubov transformation: The Bogoliubov transformation is decomposed using B_n^k gates and a controlled-RX construction expressed through rotational gates.
- State preparation: For n=4, the diagonal-basis ground state is |0000⟩ when λ>1 and |0001⟩ when λ<1, requiring an X gate on qubit four in the latter case.
- State preparation: The time-evolution initial state is prepared with an RY gate, a phase gate, and a CNOT between the first and second qubits.