Digital Quantum Simulation of Spin Models with Circuit Quantum Electrodynamics

Systems of interacting quantum spins show a rich spectrum of quantum phases and display interesting many-body dynamics. Computing characteristics of even small systems on conventional computers poses significant challenges. A quantum simulator has the potential to outperform standard computers in calculating the evolution of complex quantum systems. Here, we perform a digital quantum simulation of the paradigmatic Heisenberg and Ising interacting spin models using a two transmon-qubit circuit quantum electrodynamics setup. We make use of the exchange interaction naturally present in the simulator to construct a digital decomposition of the model-specific evolution and extract its full dynamics. This approach is universal and efficient, employing only resources that are polynomial in the number of spins, and indicates a path towards the controlled simulation of general spin dynamics in superconducting qubit platforms.

Popular Summary

The properties of magnets and strongly correlated systems are ultimately determined by the interactions of elementary particles possessing a spin that can be imagined to be a small subatomic-sized magnet. However, as opposed to classical magnets, the exact dynamics of these spins can only be described by quantum mechanics, which can be simulated only for a few tens of particles on a classical computer. A quantum simulator—a well-controlled quantum system that mimics the behavior of a less-controllable system—may provide the solution to this problem.

Here, we develop a quantum simulator based on superconducting circuits that are fabricated much like conventional computer chips. We extract the dynamics of spin models on a circuit operating at 30 mK in a dilution refrigerator. Our quantum simulation is digital in the sense that it makes use of a mathematical expansion of the time evolution of the simulated system into small steps containing identical sequences of one- and two-qubit gates. This approach is, in principle, not limited to spin systems but can be applied to general quantum systems with local interactions.

By scaling up our method using optimized pulse shapes and cryogenic components for multiplexed readout and control, our approach holds promise for accurately predicting the features and dynamics of larger spin systems. Based on our results, we expect that larger quantum simulators will be built, which will enable the study of a wide range of physics, chemistry, or biology problems such as quantum magnetism, chemical reactions, and high-energy physics in regimes that are inaccessible to classical simulations.

Figure 1

(a) Circuit diagram to characterize the $XY$ exchange interaction on qubits Q1 and Q2 symbolized by the vertical line ($\times$), which is activated for a time $\tau$. To perform standard process tomography of this interaction, separable initial states are prepared using single-qubit rotations $R_{1,2}^{\text{prep}}$ (green) in the beginning and the final state is characterized using single-qubit basis rotations $R_{1,2}^{\mathrm{tom}}$ and joint two-qubit readout (yellow). (b) Digital quantum simulation of the two-spin Heisenberg ($XYZ$) interaction for time $\tau$. The first step after state preparation is to apply the $XY$ gate for a time $\tau$ (dashed box labeled as $XY$). In the second and third steps (dashed boxes with labels $XZ$ and $YZ$), $XZ$ and $YZ$ gates are realized using single-qubit rotations $R_{x,y}^{\pm \pi /2}$ (blue) by an angle $\pm \pi /2$ about the $x$ or $y$ axis transforming the basis in which the $XY$ gate acts. (c) Protocol to decompose and simulate Ising spin dynamics in a homogeneous transverse magnetic field. The circuit between the bold vertical bars with two dots is repeated $n$ times, invoking each $XY$ and phase gates for a time $\tau /n$. See text for details. The actual pulse scheme is provided in Appendix pp3.

Figure 2

(a) Experimentally determined coordinates of the Bloch vectors during exchange ($XY$) interaction represented by small red (Q1) and blue (Q2) points are compared to the ideal paths shown as dashed lines in the $XY$ model. The ideal paths are in the $YZ$ and $XZ$ planes shown as blue and red planes intersecting the Bloch sphere. The time evolution is indicated by the saturation of the colors as the quantum phase angle $2|J|\tau$ advances from 0 (saturated) to $\pi$ (unsaturated). (b) Measured expectation values of the Pauli operators ${\sigma }_{1,2}^{x,y,z}$ for qubits Q1 (red points) and Q2 (blue points), respectively, for the $XY$ interaction as a function of the quantum phase angle $2|J|\tau$ along with the ideal evolution (dashed line). (c) Evolution of the Bloch vector for the quantum simulation of the isotropic Heisenberg interaction versus quantum phase angles from 0 to $3\pi /4$. The path of the Bloch vectors of qubits Q1 and Q2 spans the plane indicated by the rectangular sheets intersecting the Bloch spheres. (d) As in (b) for the Heisenberg interaction.

Figure 3

(a) Digital quantum simulation of the Ising model with transverse homogeneous magnetic field using 1–3 Trotter steps. Shown are the $z$ components $⟨{\sigma }_1^z⟩$ of qubit Q1 (red) and $⟨{\sigma }_2^z⟩$ of qubit Q2 (blue) and the two-point correlation function in the $x$ direction $⟨{\sigma }_1^x{\sigma }_2^x⟩$ (yellow points) of the spins as a function of the quantum phase angle $2|J|\tau$ for the initial state $|\uparrow ⟩(|\uparrow ⟩-i|\downarrow ⟩)/\sqrt{2}$ and a magnetic field strength $B=3J$. Theoretically expected results take systematic phase offsets and finite coherence of the qubits into account (solid curves). The ideal dynamics are obtained from the time-dependent Schrödinger equation for the Ising Hamiltonian (dashed lines). (b) Fidelity with respect to the exactly solved Ising model for displayed quantum phase angles of the final state after ideal unitary evolution in the simulation protocol for $n$ Trotter steps (wire frames) and experimentally obtained final state (colored bars).

Figure 4

Chip design and false-colored optical image of a superconducting qubit (inset). The chip comprises four superconducting qubits Q1–Q4 (orange) made of aluminium and four niobium coplanar waveguide resonators R1–R4 (deep blue) coupled to input and output ports (red). The qubits have individual microwave drive lines (green) and flux bias lines (blue).

Figure 5

(a) Schematic of the experimental setup with complete wiring of electronic components inside and outside of the dilution refrigerator with the same color code as in Fig. 4. (b) Up-conversion circuit for generating controlled microwave pulses. (c) Quantum-limited parametric amplifier circuit to amplify readout pulses at base temperature. (d) Amplifiers used at room temperature just before down-conversion of the signal. (e) Down-conversion circuit (see text for details).

Figure 6

Implementation of the $XY$ gate. The transition frequency of qubit Q1 (red) is tuned into resonance with qubit Q2 (blue) for an interaction time $\tau$ using a fast flux pulse. Before and after the flux pulse, a 16-ns-long buffer is added at an intermediate level to cancel the dynamic phase accumulated by qubit Q1 relative to Q2 (gray area) during the evolution (see text).

Figure 7

Pulse sequences that are applied on qubit Q1 (red) and qubit Q2 (blue) to implement the (a) Heisenberg and (b) Ising spin models. The Gaussian-shaped microwave pulses, which are based on the derivative removal by adiabatic gate (DRAG) method, are applied to the charge lines of the respective qubits to implement single-qubit rotations $R_{x,y}^{\varphi }$ about the $x$ or $y$ axis of the Bloch vector by an angle $\varphi$. Each sequence starts with the preparation of an initial state (green boxes) and ends with microwave pulses for basis rotations to perform state tomography (yellow boxes). The microwave pulses marked with magenta boxes are used for refocusing. The black vertical bars with the two dots in (b) indicate that the enclosed pulse sequence is repeated $n$ times. The $XY$ gates are realized by applying flux pulses to the flux line of qubit Q1 for a time $\tau /n$. The phase gates $R_z^{\varphi /n}$ are implemented by detuning the transition frequency of each qubit from their idle frequencies applying flux pulses for a time $\tau /n$. The numbers stated below the pulses on qubit Q1 represent time scales in nanoseconds.

Figure 8

(a) Measured real and imaginary parts of the $XY$ process $\chi$ matrix ($\mathrm{Re}\chi$, $\mathrm{Im}\chi$), in the basis $\{I=\text{identity},X={\sigma }_x,\tilde{Y}=-i{\sigma }_y,Z={\sigma }_z\}$, describing the mapping from any initial state to the final state for a quantum phase angle of $2|J|\tau =\pi /2$. The dashed wire frames represent the theoretically optimal matrix elements and the colored bars represent measured positive (blue) and negative (red) matrix elements. The fidelity of the experimentally observed process with respect to the ideal process is indicated in the black boxes. (b) As in (a) for a phase angle $\pi$.

Figure 9

(a) Measured real and imaginary parts of the Heisenberg ($XYZ$) process $\chi$ matrix ($\mathrm{Re}\chi$, $\mathrm{Im}\chi$), in the basis $\{I=\text{identity},X={\sigma }_x,\tilde{Y}=-i{\sigma }_y,Z={\sigma }_z\}$, describing the mapping from any initial state to the final state for a quantum phase angle of $2|J|\tau =\pi /2$. The dashed wire frames represent the theoretically optimal matrix elements and the colored bars represent measured positive (blue) and negative (red) matrix elements. The fidelity of the experimentally observed process with respect to the ideal process is indicated in the black boxes. (b) As in (a) for a phase angle $\pi$.