Programmable quantum simulations of spin systems with trapped ions

C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao

Rev. Mod. Phys. 93, 025001 – Published 7 April 2021

Abstract

Laser-cooled and trapped atomic ions form an ideal standard for the simulation of interacting quantum spin models. Effective spins are represented by appropriate internal energy levels within each ion, and the spins can be measured with near-perfect efficiency using state-dependent fluorescence techniques. By applying optical fields that exert optical dipole forces on the ions, their Coulomb interaction can be modulated to produce long-range and tunable spin-spin interactions that can be reconfigured by shaping the spectrum and pattern of the laser fields in a prototypical example of a quantum simulator. Here the theoretical mapping of atomic ions to interacting spin systems, the preparation of complex equilibrium states, and the study of dynamical processes in these many-body interacting quantum systems are reviewed, and the use of this platform for optimization and other tasks is discussed. The use of such quantum simulators for studying spin models may inform our understanding of exotic quantum materials and shed light on the behavior of interacting quantum systems that cannot be modeled with conventional computers.

29 More

Received 1 November 2019
Corrected 11 June 2021

DOI:https://doi.org/10.1103/RevModPhys.93.025001

Physics Subject Headings (PhySH)

Quantum simulation

Trapped ions

Spin

Quantum Information, Science & Technology

Corrections

11 June 2021

Correction: The second sentence of the first complete paragraph after Eq. (50) contained an error and has been fixed.

Authors & Affiliations

C. Monroe^*

Joint Quantum Institute and Joint Center on Quantum Information and Computer Science, University of Maryland Department of Physics and National Institute of Standards and Technology, College Park, Maryland 20742, USA

W. C. Campbell

Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA

L.-M. Duan

Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China

Z.-X. Gong

Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA

A. V. Gorshkov

Joint Quantum Institute and Joint Center on Quantum Information and Computer Science, University of Maryland Department of Physics and National Institute of Standards and Technology, College Park, Maryland 20742, USA

P. W. Hess

Department of Physics, Middlebury College, Middlebury, Vermont 05753, USA

R. Islam

Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

K. Kim

Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084 China

N. M. Linke

Joint Quantum Institute, University of Maryland Department of Physics, College Park, Maryland 20742, USA

G. Pagano

Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA

P. Richerme

Department of Physics, Indiana University, Bloomington, Indiana 47405, USA

C. Senko

Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

N. Y. Yao

Department of Physics, University of California, Berkeley, California 94720, USA

^*Corresponding author. monroe@umd.edu

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Vol. 93, Iss. 2 — April - June 2021

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Figure 1
Reduced energy level diagram of a single atomic ion. Effective spin- $1 / 2$ systems are encoded within each atomic ion as stable electronic energy levels $| ↓ ⟩$ and $| ↑ ⟩$ . A typical quantum simulation is composed of three steps. (a) Resonant radiation (blue solid arrows) connects one of the two spin states to a pair of excited state levels (linewidth $γ$ ) and optically pumps each spin to the $| ↓ ⟩$ state through spontaneous emission (red wavy arrows). Here we assume that the excited state $| e ⟩$ couples only to $| ↑ ⟩$ , while the other excited state $| e^{'} ⟩$ couples to both spin states. (Other sets of selection rules are also possible.) (b) In the case of ground-state (Zeeman or hyperfine) defined spins separated by frequency $ω_{0}$ , two tones of off-resonant radiation (purple solid lines) can drive stimulated Raman transitions between the spin states. The two beams have resonant Rabi frequencies $g_{1, 2}$ connecting respective spin states to excited states and are detuned by $Δ ≫ γ$ and have a difference frequency (beat note) detuned from the spin resonance by $μ$ . This coherently couples the spin states to create superpositions ( $μ = 0$ ) and for non-copropagating beams also couples to the motion of the ion crystal ( $μ \neq 0$ ). These processes can also be driven directly by radio-frequency or microwave signals, or, for optically defined spin states, a single laser tone. (c) Resonant radiation (blue solid arrow) drives the $| ↑ ⟩ \leftrightarrow | e ⟩$ cycling transition, causes the $| ↑ ⟩$ state to fluoresce strongly (red wavy arrow), while the $| ↓ ⟩$ state is far from resonance and therefore dark. This allows the near-perfect detection of the spin state by the collection of this state-dependent fluorescence depicted by the eye.
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Figure 2
(a) Radio-frequency (rf) linear trap used to prepare a 1D crystal of atomic ions. The geometry in this trap has three layers of electrodes, with the central (gray) layer carrying rf potentials to generate a 2D quadrupole along the axis of the trap. The outer (gold) electrodes confine the ions along the axis of the trap. For sufficiently strong transverse confinement, the ions form a linear crystal along the trap axis, with an image of 64 ions shown above with characteristic spacing $5 μ m$ for ${}^{171}{Yb}^{+}$ ions. From [160]. (b) Penning trap used to prepare a 2D crystal of atomic ions. The cylindrical (gold) electrodes provide a static quadrupole field that confines the ions along the vertical axis, and the vertical magnetic field stabilizes their orbits in the transverse plane. For sufficiently strong axial confinement, the lowest energy configuration of the ions is a single plane triangular lattice that undergoes rigid body rotation, with an image of $\sim 200$ ${}^{9}{Be}^{+}$ ions shown above with a characteristic spacing of $20 μ m$ . From [33]. For both traps, wide arrows (green) indicate non-copropagating optical dipole force (ODF) laser beams that provide spin-dependent forces along their wave vector difference, giving rise to Ising couplings. The other arrows indicate cooling and spin detection beams (imaging objectives are not shown).
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Figure 3
Raman upper sideband spectrum of the transverse motion of trapped atomic ion crystals. The spectrum is measured by preparing all of the ions in the (dark) state $| ↓ ⟩$ and driving them with global Raman laser beams with beat-note detuning $μ$ from the spin-flip resonance and measuring the total fluorescence of the chain, which responds when the beat note matches a sideband resonance. (a) 32 trapped ${}^{171}{Yb}^{+}$ atomic ions in a linear chain; see Fig. 2. Here the Raman excitation is sensitive to both the $X$ and $Y$ principal axes of transverse motion, and the theoretical position of both sets of 32 modes along $X$ and $Y$ are indicated at the top. The highest frequency sidebands correspond to center-of-mass (c.m.) modes at 4.19 MHz for the $X$ direction and 4.05 MHz for the $Y$ direction. Based on unpublished data from the University of Maryland. (b) Measured (black) and calculated [red (gray) lines between 0.6 and 0.8 MHz] sideband spectrum for a 2D crystal of $345 \pm 25$ ${}^{9}{Be}^{+}$ ions in a Penning trap; see Fig. 2 with rotation frequency 43.2 kHz. As in the linear chain, the highest frequency sideband at 795 kHz corresponds to c.m. motion. Features at the rotation frequency and its harmonics harmonics at low frequency [green (gray)] are due to residual couplings to in-plane degrees of freedom from imperfect beam alignment. Adapted from [328].
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Figure 4
Theoretical comparison of exact Ising couplings to the inverse power-law form of Eq. (23). (a) Calculated Ising couplings to edge ion in a 1D crystal of ten ions confined with a harmonic external axial potential. The Raman detuning from the c.m. mode, scaled to the bandwidth $Δ ω$ of the transverse modes, is $δ_{c . m .} / Δ ω$ , with the best-fit power law (line) of $α = 1$ . Adapted from [158]. (b) Best-fit inverse power-law exponent $α$ for calculated Ising couplings within a 1D crystal of 25 ions with harmonic axial confinement, plotted as a function of the Raman detuning from the c.m. mode $δ_{c . m .} / Δ ω$ . The spread of the points at a given detuning considers various bandwidths $Δ ω$ (axial c.m. confinement frequencies ranging from 100 to 350 kHz with a transverse c.m. confinement frequency of 5 MHz), and the c.m. sideband Rabi frequency $η_{i, c . m .} Ω$ is always less than $1 / 3$ of the detuning $δ_{c . m .}$ in order to limit direct phonon excitation. Adapted from [158]. (c) Calculated Ising couplings from Eq. (22) in a 2D crystal of 217 ions vs a sampling of the distance $d_{i j}$ between ion pairs (circles). The lines are best-fit power-law exponents $α$ (lines) for various detunings from the c.m. mode of 795 kHz. Adapted from [39].
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Figure 5
Direct measurement of Ising nearest-neighbor ( $J_{1}$ ) and next-nearest-neighbor ( $J_{2}$ ) interactions for $N = 3$ trapped-ion spins. (a) Measured time evolution of the probability of state $| ↓ ↓ ↓ ⟩$ subject to global Ising interactions, with the spins initialized in the state $| ↓ ⟩ | ↓ ⟩ | ↓ ⟩$ . The solid line is a fit to theory with an empirical exponential decay. (b) The Fourier spectrum of the oscillations in (a) exposes the frequency splittings of the Ising Hamiltonian. (c) Extracted Ising couplings $J_{1}$ and $J_{2}$ from the Fourier spectra as a function of the applied beat-note detuning $μ$ , scaled so that the c.m., tilt, and zigzag modes of transverse motion occur at $(μ^{2} - ω_{c . m .}^{2}) / ω_{z}^{2} = 0$ , $- 1$ , and $- 2.4$ , respectively. The squares ( $J_{1}$ ) and circles ( $J_{2}$ ) are experimentally measured couplings, and the lines are calculated from Eq. (22) with no free fit parameters. The particular measurements in (a) and (b) correspond to the scaled laser beat-note detuning $(μ^{2} - ω_{c . m .}^{2}) / ω_{z}^{2} = - 1.2$ indicated by the points highlighted by the rectangle in (c). Adapted from [182].
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Figure 6
Mean-field benchmarking of Ising couplings in a 2D crystal of 206(10) ions confined in a Penning trap. The spins are all initialized in a separable product state slightly tipped from the axis of a subsequently applied (weak) long-range Ising interaction. The resulting global spin precession of the ions is measured as a function of the detuning of the optical dipole force laser beams from the axial c.m. mode; see Sec. 1 and Fig. 3. Each point is generated by measuring the mean Ising coupling at a particular laser beam intensity $I$ . The solid decaying line is the prediction of mean-field theory that accounts for couplings to all $N$ transverse modes, with no free parameters. The line’s breadth reflects experimental uncertainty in the initial tipping angle of the spins. The mean-field prediction for the average value of the power-law exponent $α$ from Eq. (23) is indicated by the expanding solid line (right axis, linear scale). For stronger Ising interactions, the mean-field approach breaks down. Adapted from [39].
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Figure 7
(a) All elements of an Ising-coupling matrix measured with a spectroscopic probe in the form of a modulated transverse field. (b) Measurements of two sets of Ising-coupling matrices demonstrating two different effective interaction ranges across the chain, with the solid lines the best fits to inverse power-law form expected in Eq. (23). (c) Rescaled populations in the approximate ground state vs a static transverse-field offset $B_{0}$ and the modulation frequency of a small additional transverse field. Calculated energy levels based on measurements of trap and laser parameters are overlaid as thin white lines, and the lowest coupled excited state is overlaid as the thick lowest line, showing the critical gap near $B_{0} / 2 π = 1.4 kHz$ . The energy of the ground state is always taken to be zero. From [335].
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Figure 8
After preparing the spins in the ground state of a $y$ -polarized field $B_{y}$ , the spins evolve subject to a long-range transverse Ising model described by Eq. (25), with the condition $B_{y} (t = 0) ≫ J_{i j}$ . During the evolution, the strength of the field (upper decaying curve) is reduced to zero or an intermediate value compared to the Ising couplings (lower constant curve). Finally, the spins are measured along any axis of the Bloch sphere. [In much of the work reported later, the measurements are taken to be along the Ising-coupling ( $x$ ) direction of the Bloch sphere for each ion.] If the evolution were adiabatic, the resulting state should have remained in the ground state of the Hamiltonian throughout. Adapted from [160].
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Figure 9
Low-lying energy eigenvalues of Eq. (25) for $N = 6$ , with the ground-state energy $E_{g}$ set to 0, $B_{0} = 5 J_{\max}$ , and the long-range $J_{i j}$ couplings determined from experimental conditions (see the text). The bold line indicates the first coupled excited state, the minimum of which determines the critical field $B_{c}$ and the critical gap $Δ_{c}$ . From [311].
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Figure 10
(a) Local adiabatic ramp profile calculated for the energy levels in Fig. 9, along with a linear ramp and an exponential ramp with decay constant $τ = t_{f} / 6$ . (b) The slope of the local adiabatic (LA) ramp is minimized at the critical field value $B_{c}$ , and is smaller than the slopes of the exponential and linear ramps at the critical point. (c) The inverse of the adiabaticity parameter $γ$ (see the text) is peaked near the critical point for exponential and linear ramps but constant for the local adiabatic profile. Adapted from [311].
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Figure 11
Probability of preparing the AFM ground state for various times during $t_{f} = 2.4 ms$ simulations with three different ramp profiles. The linear ramp takes $\sim 2.3 ms$ to reach the critical point, while the local adiabatic and exponential ramps need only 1.2 ms. Locally adiabatic ramps yield the highest preparation fidelity at all times. Adapted from [311].
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Figure 12
(a) Observed adiabatic evolution of two trapped-ion spins from a paramagnetic state to a ferromagnetic (FM) state. (b) The observed parity oscillation signal of the resulting FM state upon a subsequent rotation of both spins reveals the amount of entanglement in the ground state. From [108].
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Figure 13
Evolution of each of the eight spin states, measured with a CCD camera, plotted as $B_{y} / J_{rms}$ is ramped down in time. The dotted lines correspond to the populations in the exact ground state and the solid lines represent the theoretical evolution expected from the actual ramp. (a) All interactions are AFM. The FM-ordered states vanish and the six AFM states are all populated as $B_{y} \to 0$ . Because $J_{2} \approx 0.8 J_{1}$ , a population imbalance also develops between symmetric and asymmetric AFM. (b) All interactions are FM, with evolution to the two ferromagnetic states as $B_{y} \to 0$ . (c) Entanglement generation for the case of an all AFM interaction, where the symmetric W-state witness $W_{W}$ is used. The entanglement emerges for $B_{y} / J_{rms} < 1.1$ . (d) Entanglement generation for the all FM interactions, where the GHZ witness $W_{GHZ}$ is used. The entanglement occurs when $| B_{y} | / J_{rms} < 1$ . In both (c) and (d) the error bars represent the spread of the measured expectation values for the witness, likely originating from the fluctuations of experimental conditions. The solid black lines are theoretical witness values for the exact expected ground states, and the black dashed lines describe theoretically expected values at the actual ramps of the transverse field $B_{y}$ . The blue (gray) lines reveal the oscillation and suppression of the entanglement due to the remaining spin-motion couplings, showing better agreement with the experimental results. Adapted from [183].
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Figure 14
Paramagnetic-to-ferromagnetic crossover in a small collection of trapped-ion spins. (a) Theoretical values of order parameters vs ratio of transverse field to average Ising coupling $B / | J |$ for $N = 2$ and 9 spins with nonuniform Ising couplings following the experiment given by [160] and assuming perfect adiabatic time evolution. The order parameters, the Binder cumulant, and the magnetization are calculated by directly diagonalizing the relevant Hamiltonian [Eq. (25)]. Order parameters are also calculated for a moderately large system ( $N = 100$ ) with uniform Ising couplings to show the difference between these order parameters. (b) Measured magnetization vs $B / | J |$ (and simulation time) plotted for $N = 2 - 9$ spins, and scaled to the number of spins. As $B / | J |$ is lowered, the spins undergo a crossover from a paramagnetic to a ferromagnetic phase. The crossover curves sharpen as the system size is increased from $N = 2$ to 9, prefacing a phase transition in the limit of infinite system size. The oscillations in the data arise from the imperfect initial state preparation and nonadiabaticity due to finite ramping time. Measured (c) magnetization and (d) Binder cumulant vs $B / | J |$ for $N = 2$ (circles) and 9 spins (diamonds) with representative detection error bars. The data deviate from unity at $B / | J | = 0$ owing to decoherence driven by the Raman transitions creating the Ising couplings. The theoretical curves (solid line for $N = 2$ and dashed line for $N = 9$ spins) are calculated by averaging over 10 000 quantum trajectories. From [160].
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Figure 15
(a) Structure function $S (k)$ for various ranges of AFM interactions, for $B / J_{0} = 0.01$ in a system of $N = 10$ spins. The increased level of frustration for the longer-range interactions reduces the observed antiferromagnetic spin order. The detection errors may be larger than shown here for the longest range of interactions, owing to spatial cross talk from their closer spacing. (b) Distribution of observed states in the spin system, sorted according to their energy $E_{i}$ (with $E_{0}$ denoting the ground-state energy) calculated exactly from Eq. (25) with $B = 0$ . Data are presented for two ranges (red for $α = 1.05$ and blue for $α = 0.76$ ). The dashed lines indicate the cumulative energy distribution functions for these two ranges. Adapted from [159].
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Figure 16
(a) Fluorescence images of both AFM ground states prepared for $N = 14$ trapped-ion spins, with $bright = | ↑ ⟩$ and $dark = | ↓ ⟩$ along the $x$ direction of the Bloch sphere of each spin. (b) State probabilities of all $2^{14} = 16 384$ spin configurations for a 14-ion system following a local adiabatic ramp, ordered by binary index of the states (i.e., $| ↓ ↓ \dots ↓ ⟩ = 0$ and $| ↑ ↑ \dots ↑ ⟩ = 16 383$ ). The Néel-ordered ground states 5461 and 10 922 are unambiguously the most prevalent, despite a total probability of only 3%. From [311].
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Figure 17
(a) Low-lying energy eigenvalues of Eq. (36) for $B_{y} = 0$ for $N = 6$ spins, with the long-range $J_{i j}$ couplings determined from experimental conditions (see the text). Inset: level crossings indicate the presence of first-order phase transitions in the ground state. (b) The critical gap $Δ_{c}$ shrinks to zero at the three phase transitions (vertical dashed lines). Inset: low-lying energy levels of Eq. (36) with $B_{x} = 0$ . From [310].
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Figure 18
(a) Magnetization ( $M_{x} = N_{↑} - N_{↓}$ ) of six ions for increasing axial field strength. Solid staircase line, magnetization of the calculated ground state, with the step locations indicating the first-order phase transitions; diamond points, average magnetization of 4000 experiments for various $B_{x}$ ; solid smooth line, magnetization calculated by numerical simulation using experimental parameters; black dashed line, magnetization of the most probable state (see inset) found at each $B_{x}$ value. Gray bands indicate the experimental uncertainty in $B_{x} / J_{\max}$ at each observed phase transition. (b) Fluorescence images of the six ions with $bright = | ↑ ⟩$ and $dark = | ↓ ⟩$ along the $x$ direction of the Bloch sphere of each spin, showing the ground states found at each step in (a): $| ↓ ↑ ↓ ↑ ↓ ↑ ⟩$ and $| ↑ ↓ ↑ ↓ ↑ ↓ ⟩$ ( $M_{x} = 0$ ), $| ↓ ↑ ↓ ↓ ↑ ↓ ⟩$ ( $M_{x} = - 2$ ), $| ↓ ↓ ↑ ↓ ↓ ↓ ⟩$ and $| ↓ ↓ ↓ ↑ ↓ ↓ ⟩$ ( $M_{x} = - 4$ ), and $| ↓ ↓ ↓ ↓ ↓ ↓ ⟩$ ( $M_{x} = - 6$ ). Adapted from [310].
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Figure 19
(a) Ground-state phase diagram of the system, along with two different trajectories that end at the same value of $B_{x}$ . (b) Probabilities of the four different ground-state spin phases when $B_{y}$ is ramped in a six-ion system. Circles, $| ↓ ↑ ↓ ↑ ↓ ↑ ⟩$ or $| ↑ ↓ ↑ ↓ ↑ ↓ ⟩$ ; squares, $| ↓ ↑ ↓ ↓ ↑ ↓ ⟩$ ; diamonds, $| ↓ ↓ ↑ ↓ ↓ ↓ ⟩$ or $| ↓ ↓ ↓ ↑ ↓ ↓ ⟩$ ; triangles, $| ↓ ↓ ↓ ↓ ↓ ↓ ⟩$ . Gray bands are the experimental uncertainties of the phase transition locations. (c) Probabilities of creating the four different ground states when $B_{x}$ is ramped. Most of the ground states are classically inaccessible in our zero-temperature system. Adapted from [310].
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Figure 20
(a) Level diagram for ${}^{171}{Yb}^{+}$ highlighting relevant states for spin-1 physics. (b) Sketch of the experimental geometry showing the directions of the laser wave vectors and the real magnetic field relative to the ion chain. Both beams are linearly polarized, one along the $\vec{B}$ field (providing $π$ light) and one orthogonal to the $\vec{B}$ field (providing an equal superposition of $σ^{+}$ and $σ^{-}$ light). Multiple beat notes are applied by imprinting multiple frequencies on one beam (in this case, the $π$ -polarized beam). (c) Detailed level diagram of the ${{}^{2}S}_{1 / 2}$ ground state showing Raman beat notes in relation to Zeeman splittings and motional sidebands for the center-of-mass mode. Level splittings are not drawn to scale. From [334].
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Figure 21
Measurements of the prepared (a)–(c) two-spin and (d) four-spin states after ramping an $S_{z}^{2}$ field (narrow blue bars) compared to the values expected for the calculated ground state (gray bars). (a)–(c) Measured populations when the dark state is set at $| 0 ⟩$ , $| - ⟩$ , or $| + ⟩$ , respectively. The dark state is set at $| 0 ⟩$ in (d). Adapted from [334].
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Figure 22
Following an adiabatic ramp, the parity of the final state is measured as a function of the final rotation phase $φ$ as a witness for spin-1 entanglement (see the text for rotation protocol). The dashed and dot-dashed lines represent the theoretically expected values for the ground state [ $| 00 ⟩ / \sqrt{2} - (| - + ⟩ + | + - ⟩) / 2$ ] and the highest excited state [ $| 00 ⟩ / \sqrt{2} + (| - + ⟩ + | + - ⟩) / 2$ ], respectively. The amplitude and phase of the measured oscillation reveal that the prepared state are consistent with the expected ground state. From [334].
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Figure 23
(a)–(c) Measured magnetization $⟨ σ_{i}^{z} (t) ⟩$ (color coded) following a local quench. From (a) to (c), the interaction ranges are $α \approx 1.41, 1.07, 0.75$ . In (a), an effective light cone is evident and the dynamics is approximately described by nearest-neighbor interactions only. Red lines, fits to the observed magnon arrival times [examples shown in (d)]; white lines, light cone for averaged nearest-neighbor interactions; orange dots, fits after renormalization by the algebraic tail. (b),(c) As the interaction range is increased, the light cone disappears and nearest-neighbor models fail to capture the dynamics. (d) Magnetization of spins (ions) 6 and 13 from (a) (top panel) and (c) (bottom panel). Solid lines, Gaussian fits to measured magnon arrival. Top panel: for $α = 1.41$ , a nearest-neighbor Lieb-Robinson bound captures most of the signal (shaded region). Bottom panel: for $α = 0.75$ , the bound does not. (e) Maximum group velocity. With increasing $α$ , the measured magnon arrival velocities (solid circles) approach the group velocity of the nonrenormalized nearest-neighbor model (gray dash-dotted line). If renormalized by the algebraic tail, the nearest-neighbor group velocity increases at small $α$ (open circles), but much less than the increase of the observed magnon velocity. For small $α$ , the measured arrival times are consistent with the divergent behavior predicted for full power-law interactions (black line). Adapted from [169].
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Figure 24
(a) Spatial and time-dependent correlations, (b) extracted light-cone boundary (chosen as the contour $C_{i, j} = 0.04$ ), and (c) correlation propagation velocity following a global quench of a long-range $X Y$ model with $α = 0.63$ . The curvature of the boundary shows (b) an increasing propagation velocity, quickly exceeding (c) the short-range Lieb-Robinson velocity bound $v_{LR}$ . Solid lines give a power-law fit to the data that slightly depends on the choice of contour $C_{i, j}$ . Complementary plots for (d)–(f) $α = 0.83$ , (g)–(i) $α = 1.00$ , and (j)–(l) $α = 1.19$ . As the range of the interactions decreases, correlations do not propagate as quickly through the chain. (j)–(l) For the shortest-range interaction, the experiment demonstrates a faster-than-linear growth of the light-cone boundary despite having $α > 1$ . Error bars, 1 s.d. Adapted from [309].
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Figure 25
Many-body localization. (a),(b) Temporal dynamics for each of the ten ions’ $z$ magnetizations $⟨ σ_{i}^{z} (t) ⟩$ in the case of zero disorder and strong disorder for (a) and (b), respectively. (c) The normalized Hamming distance $D (t)$ has plateaued to a steady-state value for $J_{0} t \geq 5$ at all measured disorder strengths. (d) The time-averaged steady-state value $⟨ D (t) ⟩$ after the plateau shows the onset of a crossover between a thermalizing regime $[⟨ D (t) ⟩ = 0.5]$ and a localizing regime $[⟨ D (t) ⟩ = 0]$ as disorder increases. (e) The steady-state Hamming distance increases with longer-range interactions. (f) The half-chain entropy growth in the absence of disorder (red points) and for a disordered chain with $W = 6 J_{0}$ (blue points), compared with numerical simulations of unitary dynamics (dotted lines) and including known sources of decoherence (solid lines). (a)–(e) Adapted from [346]. (f) Adapted from [44].
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Figure 26
Breakdown of Anderson localization. Main panel: transport efficiency $η_{8}$ to the target (ion 8) under different strengths of static disorder (blue, $B_{\max} = 0.5 J_{0}$ ; red, $B_{\max} = 2.5 J_{0}$ ) and Markovian-like dephasing with rate $γ$ . Experimental points (shown as dark squares and triangles) result from averaging over 20–40 random realizations of disorder and noise, with 25 experimental repetitions each. The regimes of localization, ENAQT ([307]), and the quantum Zeno effect are indicated in gray. The data agree well with theoretical simulations of the coin-tossing random process (light bullets) realized in the experiment, while simulations with ideal Markovian white noise (lines) underestimate ENAQT at large $γ$ . (a) Sketch of the transport network. The ions experience a long-range coupling, with darker and thicker connections indicating higher coupling strengths. The green arrows denote the source (3) and the target (8) for the excitation in the ion network. (b) Sketch of the ion chain representing interacting spin- $1 / 2$ particles as circles, with the spin states denoted by arrows. The ions are subject to random static and dynamic on-site excitation energies, indicated by $B_{i}$ and $W_{i} (t)$ . From [241].
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Figure 27
(a) An initial spin excitation is prepared on one side of a seven-ion chain subject to open boundary conditions and long-range $X Y$ interactions. As the interaction range increases ( $α$ decreases), the effective potential energy for the single excitation changes from a square-well potential to a potential resembling a double well. (b) The double-well potential gives rise to near-degenerate eigenstates when $α$ is decreased, as seen in the calculated energy difference between all pairs of eigenstates vs $α$ . The height of the plot quantifies the off-diagonal density matrix elements of the initial state. (c) In the case of short-range interactions, either one or two spin flips delocalize across the chain during time evolution, so that the quantity $⟨ C ⟩ \approx 0$ in the long-time limit, consistent with the prediction of GGE. (d) For long-range interactions memory of initial conditions is preserved in a long-lived perthermal state. In both (c) and (d), the open squares and circles plot $⟨ C ⟩$ for initial states prepared on the left and right sides of the spin chain, while the filled circles and squares plot the cumulative time average $⟨ \bar{C} ⟩$ for these data. Adapted from [274].
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Figure 28
Measured time evolution (darker points) and cumulative time average (lighter points) of the averaged center of excitation $⟨ C ⟩$ in a 22-ion chain (ion chain image shown at the top). The ion spins are initialized with a single-spin excitation on the left end (middle ion chain image). After evolving in the $X Y$ spin Hamiltonian for time $t = 36 / J_{0}$ , where $J_{0}$ is the nearest-neighbor coupling and the spin excitation is delocalized, but its average position remains stuck on the left half of the chain (bottom ion chain image), the signature of prethermalization. Effective range of the interaction is $α \approx 0.9$ . Adapted from [274].
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Figure 29
Out-of-time-order-correlators (OTOC) in a 2D Penning trap ion simulator; see Fig. 2 for experimental schematic. (a) Demonstration of the ability to tune the sign of the Ising interaction through the symmetric detuning of the optical dipole force around a single mode of motion. Plotted is the residual spin-phonon couplings vs detuning from the center-of-mass mode. A positive (negative) detuning gives rise to an antiferromagnetic (ferromagnetic) interaction. (b) Experimental sequence used to measure an out-of-time-ordered correlation function with alternation in the sign of the Ising coupling. (c) Measured Fourier amplitudes of collective magnetization dynamics under the Ising spin-spin interactions for $N = 111$ spins, showing the sequential buildup of higher-order spin-spin correlations. Adapted from [111].
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Figure 30
Floquet quantum simulation of a discrete time crystal (DTC). (a) Schematic depiction of the Floquet evolution of a trapped-ion spin chain. Three Hamiltonians are applied sequentially in time: (i) a global nominal $π$ pulse $(2 g t = π)$ around the spin $y$ axis with fractional perturbation $ϵ$ , (ii) long-range Ising interactions (see Sec. 2), and (iii) site-dependent disorder along the spin $x$ axes. (b) The $x$ magnetization of each spin is measured after each Floquet sequence, up to 100 periods. The Fourier transform of the oscillations show a clear peak observed at half the driving frequency, even with a programmed perturbation on the global $π$ pulses of $ϵ = 0.03$ , signaling the discrete breaking of time-translation symmetry and the “rigidity” of the time crystal. (c) When the perturbation is too strong ( $ϵ = 0.11$ ), we cross the boundary from the DTC into a symmetry-unbroken phase. (d) For stronger interactions parametrized by the nearest-neighbor Ising coupling $J_{0}$ , the DTC can tolerate larger imperfections to the global rotation pulse, leading to a qualitative phase diagram ([399]). Adapted from [408].
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Figure 31
Trapped-ion quantum simulation of DQPT type I. (a) Measured rate function $λ$ for three different system sizes at $B / J_{0} \approx 2.38$ , with $τ = t B$ the dimensionless time. The kinks in the evolution become sharper for larger $N$ . To take the $Z_{2}$ degeneracy of the ground state of $H_{0}$ into account, here the rate function is defined based on the return probability to the ground-state manifold, namely, $λ (t) = N^{- 1} \log (P_{| ψ_{0} ⟩} + P_{| - ψ_{0} ⟩})$ , where $| - ψ_{0} ⟩ = | ↑ ↑ ↑ \dots ↑ ⟩_{x}$ . (b) Comparison between rate function $λ (t)$ and magnetization evolution $M_{x} (t)$ . The inversion of the magnetization sign corresponds to the nonanalyticity of the rate function $λ (t)$ . Solid lines are exact numerical predictions based on experimental parameters $(B / J_{0} = 2)$ . Adapted from [170].
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Figure 32
(a)–(d) Trapped-ion quantum simulation of a dynamical quantum phase transition of type II following the long-range transverse Ising Hamiltonian with $α \approx 1$ . Long-time averaged values of the two-body correlations $C_{2}$ for different numbers of spins in the chain ranging from 8 to 53. Solid lines are exact numerical solutions to the Schrödinger equation, and the shaded regions take into account uncertainties from experimental Stark-shift calibration errors. Dashed lines in (a) and (b) are calculations using a canonical thermal ensemble with an effective temperature corresponding to the initial energy density. (e)–(g) Domain statistics and corresponding image samples after late-time evolution for various values of $B / J_{0}$ with 53 spins. The reconstructed ion chain images are based on single-shot binary detection of each ion spin state. The top image shows a chain of 53 ions all in bright spin states as a baseline. The other three images show 53 ions in combinations of bright and dark spin states. (e)–(g) Distribution of domain sizes for three different values of $B / J_{0}$ , plotted on a logarithmic scale. Dashed lines are fits to exponential functions, which are expected for an infinite-temperature thermal state. Long tails of deviations are visible, and their prevalence depends on $B / J_{0}$ . (h) Mean of the largest domain size in each single experimental shot vs $B / J_{0}$ , showing a kink at the phase transition. Dashed lines represent a piecewise linear fit, used to extract the transition point $B / J_{0} = 0.89 (7)$ . Adapted from [407].
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Figure 33
Digital simulations of dynamics in four- and six-spin systems. Initially, all spins point up. (a) Four-spin long-range Ising system. Each digital step is given by operators $C = \exp (- i θ_{C} Σ_{i} σ_{z}^{i})$ with $θ_{C} = π / 32$ and $D = \exp (- i θ_{D} Σ_{i < j} σ_{ϕ_{D}}^{i} σ_{ϕ_{D}}^{j})$ , with $θ_{D} = π / 16$ and $ϕ_{D} = 0$ . (b) Six-body interaction with six spins. $F = \exp (- i θ σ_{z})$ with $θ$ variable, and $4 D = \exp (- i θ_{0} Σ_{i < j} σ_{ϕ_{0}}^{i} σ_{ϕ_{0}}^{j})$ , with $θ_{0} = π / 4$ and $ϕ_{0} = 0$ . Bounds for the quantum process fidelity $F_{p}$ are given at $θ = 0.25 π$ . Lines, exact dynamics; unfilled shapes, ideal digitized; filled shapes, data (squares, $P_{0}$ ; diamonds, $P_{1}$ ; circles, $P_{2}$ ; triangles pointing up, $P_{3}$ ; triangles pointing right, $P_{4}$ ; triangles pointing down, $P_{5}$ ; triangles pointing left, $P_{6}$ , where $P_{i}$ is the total probability of finding $i$ spins pointing down). Adapted from [205].
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Figure 34
Variational quantum eigensolver simulation of the lithium hydride (LiH) molecule with three trapped-ion qubits. (a) LiH molecular orbitals (MOs) formed out of atomic orbitals (AOs) contributed by each element. The active space in which the implemented UCC excitation operators act is highlighted in yellow (light gray). (b) The theoretical LiH potential energy surface calculated for the minimal basis set (black solid line) is shown in comparison to the experimentally obtained results, offset to overlap at maximum distance $R$ to better illustrate the well-depth differences in the grid-scan reconstructions. The data points result from sampling the energy landscape using the VQE algorithm or a parameter grid scan and fitting the explored space with a GPR-based machine learning algorithm (dashed line) or a 2D quadratic fit [blue (gray) solid line]. The error bars are obtained from the fits with the underlying data weighted by quantum projection noise. (c) Abstract quantum circuit implementing each of the two target UCCSD unitary operators for the qubits indexed Q0, Q1, and Q2. A fully entangling gate (MS) acts locally between the enclosed qubits and surrounds a local qubit rotation $z$ quantified by parameters $α$ and $β$ , respectively. Adapted from [145].
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Figure 35
VQE simulation of the ground-state energy of the water molecule ( $H_{2} O$ ). (a) Metrics for each circuit are labeled $HF + N$ , as up to $N$ of the most significant interaction terms are added to the ansatz state. The bosonic terms through $HF + 5$ can be represented as pair excitations to reduce the qubit resource requirements, while the MO selection strategy prunes the two least significant molecular states to reduce the qubit count slightly at the expense of accuracy at the millihartree level. Energies should be compared to the full configuration-interaction (FCI) ground-state energy, which is the exact result of diagonalizing the complete Hamiltonian in the minimal chemical basis. (b) Comparison of ground-state energy estimates as additional interactions are included in the UCC ansatz state (labeled $HF + N$ for $N$ significant determinants). The orange diamonds indicate experimental results from implementations with up to three qubits, with $1 σ$ error bars from the bootstrap distribution. The remaining points are from the in silico VQE simulation and show how the ansatz states converge to the full configuration-interaction ground state (indicated by the dot-dashed line). Computational error equivalent to the bound for chemical accuracy (1.6 mHa) is indicated by the shaded region. Adapted from [270].
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Figure 36
QAOA applied to transverse-field long-range Ising model. (a) QAOA protocol. The system is initialized along the $y$ direction on the Bloch sphere in the ground state of the mixing Hamiltonian $H_{B}$ , namely, $| + ⟩^{\otimes N}$ . The unitary evolution under $H_{A (B)}$ is implemented for angles $γ_{i} (β_{i})$ and repeated $p$ times. At the end of the algorithm global measurements in the $x$ and the $y$ basis are performed to compute the average energy $E (\vec{β}, \vec{γ}) = ⟨ \vec{β}, \vec{γ} | H | \vec{β}, \vec{γ} ⟩$ . The measurement results are then processed by a classical optimization algorithm to update the variational parameters. (b) Closed-loop optimization for $p = 1$ and $N = 20$ qubits. Starting from an initial guess, the local gradient is approximated by performing the energy measurements along two orthogonal directions in parameter space. To quantify the performance of the QAOA, the dimensionless quantity $η \equiv [E (\vec{β}, \vec{γ}) - E_{\max}] / (E_{gs} - E_{\max})$ is used, where $E_{\max} (E_{gs})$ are the energies of the highest (lowest) excited state, therefore mapping the entire many-body spectrum to the [0, 1] interval. The algorithm converges after about seven iterations. Adapted from [286].
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