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Universal digital quantum simulation with trapped ions
B. P. Lanyon, C. Hempel, D. Nigg, M. Müller, R. Gerritsma, F. Zähringer, P. Schindler, J. T. Barreiro, M. Rambach, G. Kirchmair, M. Hennrich, P. Zoller, R. Blatt, C. F. Roos
TL;DR
The paper addresses whether a trapped-ion device can digitally simulate a range of quantum spin dynamics with programmable operations. It implements universal operations and evaluates the simulations using process reconstruction and fidelity bounds. The experiments reproduce interactions beyond the simulator’s native ones, while higher-order accuracy trades against additional operation-induced error.
Problem
Quantum process tomography becomes impractical for n-qubit processes because reconstructing the process matrix requires 12^n expectation values.
Method
The authors use a universal trapped-ion operation set to digitally simulate spin systems and quantify simulations through process tomography and reduced-measurement fidelity bounds.
Results
0.850(8) ≤ process fidelity is obtained for the bounded three-body interaction simulation.
Takeaways & Limitations
The experiments demonstrate universal digital simulation principles in trapped ions, including accurate reproduction of interactions not naturally present in the device.
Takeaways & Limitations
Higher-order digital approximations improve resolution but introduce additional experimental error through the use of more operations.
Abstract
from arXiv · showhide
A digital quantum simulator is an envisioned quantum device that can be pro- grammed to efficiently simulate any other local system. We demonstrate and investigate the digital approach to quantum simulation in a system of trapped ions. Using sequences of up to 100 gates and 6 qubits, the full time dynamics of a range of spin systems are digitally simulated. Interactions beyond those naturally present in our simulator are accurately reproduced and quantitative bounds are provided for the overall simulation quality. Our results demon- strate the key principles of digital quantum simulation and provide evidence that the level of control required for a full-scale device is within reach.
1 Experimental details
The experiments use trapped 40Ca+ ions in linear Paul traps, with qubits encoded in internal states and controlled through global and individually addressed laser beams. The implemented operations form a universal set capable of realizing arbitrary unitary qubit evolution.
- Preparation and measurement: The ions are measured through state-dependent fluorescence, either collectively with a photomultiplier or individually with a CCD camera.Single-qubit rotations before measurement map other observables onto the logical measurement basis.
- Preparation and measurement: Each simulation experiment is repeated at least 200 times per data point to obtain measurement statistics.Collective fluorescence yields probabilities for different numbers of bright ions, while CCD imaging resolves individual ions.
- Ion-trap platform: Two similar linear Paul traps are used, with the second supporting larger ion strings and all experiments involving more than two ions.The second trap’s improved vacuum quality enables larger strings.
- Universal operation set: The operation set includes single-qubit rotations, light-shift operations, and an effective spin-spin interaction generated by a Mølmer-Sørensen pulse.Global beams act on all ions, while addressed beams target individual ions.
- Universal operation set: The unitaries in equations 1–4 form a universal set, allowing arbitrary unitary qubit evolution using only these operations.The operations are implemented with global and individually addressed laser paths.
2 Two-spin simulations
Two-spin simulations compare digital approximations with ideal dynamics for time-independent, time-dependent, and higher-order Trotterized evolutions. Increasing digital resolution improves agreement, while additional operations introduce experimental error and overhead.
- 2.1.1 Time-independent dynamics: Increasing digital resolution improves the reconstructed dynamics, with simulations remaining high quality across the full Hilbert space.The comparison uses experimentally reconstructed quantum process matrices for a two-spin Ising system.
- 2.1.2 Time-dependent dynamics: The time-dependent simulation increases the spin-spin interaction strength so the initial ground state evolves toward an approximation of the joint, highly entangled ground state.The continuous evolution is approximated by an 8-step digital sequence.
- 2.1.3 Higher-order Trotter approximation: The first-order Trotter approximation has errors on the order of t^2/n, whereas the second-order approximation has errors on the order of t^3/n.The second-order sequence splits and rearranges an evolution operator for a closer approximation.
- 2.1.3 Higher-order Trotter approximation: Higher-order approximations improve digital resolution at the expense of more operations and additional experimental error.This trade-off is especially relevant when evolution operators must first be constructed, as in the XYZ model.
3.1 Ising-type models
The basic operations naturally simulate long-range Ising interactions and can be combined to realize asymmetric and more general coupling networks. Refocusing sequences extend this capability beyond the native all-to-all symmetric interaction.
- Long-range Ising interactions: The O4 operation naturally couples every pair of spins with equal strength, matching the long-range Ising model.Each digital time step for this simulation uses the sequence O4O2.
- Experimental demonstrations: The experiments simulate time dynamics for three- and four-spin Ising-type systems, including complementary measurement bases and different transverse-field strengths.The field strength is adjusted by varying the phase evolution of each O2 operation.
- General coupling networks: Refocusing with O4(θ/2, φ)O1(π/2, n)O4(θ/2, φ) excludes ion n from the long-range interaction.Repeating such sequences can produce arbitrary spin-spin coupling networks.
- Asymmetric interactions: An asymmetric interaction is constructed from one all-pair operation followed by three pair-specific operations.The resulting sequence is equivalent to evolution under the desired asymmetric Hamiltonian for phase θ.
- Experimental limitations: Nearest-neighbour simulations require significantly more operations than the long-range Ising model, causing dynamics to damp through decoherence.Laser-intensity fluctuations are identified as a major source of this decoherence.
3.2 3-body interaction with additional transverse field
The digital simulator realizes a three-body interaction and extends it with a transverse field, using operation sequences whose cost reflects the need for Trotter decomposition. The added field makes the simulation particularly operation-intensive and exposes decoherence before all spin states become equally populated.
- Additional transverse field: Adding a transverse field requires a Trotter approximation costing 4 operations per digital step.Three operations implement the three-body interaction and one implements the magnetic-field interaction.
- Three-body interaction: The three-body interaction can be simulated for any phase evolution using only three operations.This is shown as a special case of a broader scheme for n-body spin interactions.
- Observed dynamics: A coarse digital resolution of π/4 is selected to observe dynamics before decoherence mechanisms equally distribute population among the possible spin states.The results include a stroboscopic sequence with fixed settings and a non-zero-field case.
- Additional transverse field: The transverse field is simulated with O3 rather than O2 for the three-body interaction.An alternative would rotate one spin-spin interaction axis using two additional pulses per digital step.
3.3 Process bounding method
The method bounds overall process fidelity without reconstructing the process matrix, reducing measurements while retaining quantitative validation of multiqubit operations.
- Motivation: 12^n expectation values make full process-matrix reconstruction impractical beyond two qubits.The measurement burden grows exponentially and requires maintaining precise experimental control.
- Method: The method bounds process fidelity using two complementary input bases and measured probabilities of correct output states.For n qubits, it prepares two complementary sets of 2^n input states and measures each corresponding output-state fidelity.
- Method: Quantum process tomography reconstructs a complete process matrix, whereas the bounding procedure estimates fidelity from substantially fewer measurements.The technique uses truth-table fidelities and bounds the process fidelity above and below rather than reconstructing the full matrix.
- Measurement cost: 40 instead of 1728 measurements are required for three qubits, and 512 instead of 2,985,984 for six qubits.The bound requires 2^n(n + 1) + 2^n expectation values, compared with 12^n for full process tomography.
- Error analysis: For six-qubit operations, decoherence of GHZ-like states reduced parity amplitudes while leaving total populations comparatively similar.Average absolute parity amplitudes for groups 6, 4, 2 and 0 were 0.58(2), 0.67(1), 0.71(1) and 0.76(1), respectively.
3.4 Fourier transform to extract energy gaps
Fourier transforms of measured observables reveal energy gaps in the simulated Hamiltonian. In the four-ion example, the initial state populates three energy levels, so at most three oscillation frequencies can appear.
- Spectral interpretation: Oscillation frequencies in observable dynamics correspond to energy gaps in the underlying Hamiltonian.A Fourier transform extracts this spectral information from the time-dependent signal.
- Four-ion example: The four-ion long-range Ising example uses the observable probability of finding all combinations of two spins up and two spins down.Its Fourier transform is compared with the ideal Hamiltonian spectrum and the initial state's energy-level distribution.
- Four-ion example: Three of the nine energy levels are populated, limiting the dynamics to at most three observable energy gaps.The number of possible frequencies follows from the populated levels, while their amplitudes also depend on the observable's coupling strengths.
- Result: One of the three fundamental frequencies is clearly resolved in the Fourier transform.The spectrum therefore provides direct evidence of a Hamiltonian energy-gap feature in the measured dynamics.
3.5 Error sources in gate operations
The main identified error sources are fluctuations in laser-ion coupling strength and inaccuracies in gate-operation settings. Coupling fluctuations reproduce damping, while small phase-setting errors can shift simulated frequencies.
- Laser-ion coupling strength fluctuations: Laser-ion coupling-strength fluctuations are a significant source of experimental error in the digital simulations.Possible sources include laser-intensity noise and thermally occupied vibrational modes; operation phases depend on Ω^2.
- Frequency shifts in simulated dynamics: A slight frequency shift in the three-spin transverse Ising simulation could result from gate-operation setup or optimization errors.A 1% error in the O4 phase angle is expected to produce the observed frequency mismatch.
- Frequency shifts in simulated dynamics: The sensitivity to phase-setting errors motivates more accurate methods for optimizing gate operations.The paper notes that other error sources may also contribute to deviations between observed and ideal simulations.