Error-mitigated simulation of quantum many-body scars on quantum computers with pulse-level control

Quantum many-body scars are an intriguing dynamical regime in which quantum systems exhibit coherent dynamics and long-range correlations when prepared in certain initial states. We use this combination of coherence and many-body correlations to benchmark the performance of present-day quantum computing devices by using them to simulate the dynamics of an antiferromagnetic initial state in mixed-field Ising chains of up to 19 sites. In addition to calculating the dynamics of local observables, we also calculate the Loschmidt echo and a nontrivial unequal-time connected correlation function that witnesses long-range many-body correlations in the scarred dynamics. We find coherent dynamics to persist over up to 40 Trotter steps even in the presence of various sources of error. To obtain these results, we leverage a variety of error-mitigation techniques including noise tailoring, zero-noise extrapolation, dynamical decoupling, and physically motivated postselection of measurement results. Crucially, we also find that using pulse-level control to implement the Ising interaction yields a substantial improvement over the standard controlled-not-based compilation of this interaction. Our results demonstrate the power of error-mitigation techniques and pulse-level control to probe many-body coherence and correlation effects on present-day quantum hardware.

Figure 1

Schematic of the state-preparation circuit and the Trotter circuit for one time step $\mathrm{\Delta }t$ in an $L=5$ site chain. The first part of the circuit prepares the Néel state $|01010\rangle$ from the polarized state $|00000\rangle$ using $X$ gates. The Trotter circuit uses single-qubit rotations $R_X\left({\theta }_i^X\right)=e^{-i{\theta }_i^X{X}_i/2}$ and $R_Z\left({\theta }_i^Z\right)=e^{-i{\theta }_i^Z{Z}_i/2}$ and two-qubit gates $R_{ZZ}\left({\theta }_i^{ZZ}\right)=e^{-i{\theta }_i^{ZZ}{Z}_i{Z}_{i+1}/2}$. Here, ${\theta }_1^Z={\theta }_L^Z=-2V\mathrm{\Delta }t$. The remaining angles are given by ${\theta }_i^X=2\mathrm{\Omega }\mathrm{\Delta }t$, ${\theta }_i^Z=-4V\mathrm{\Delta }t$, and ${\theta }_i^{ZZ}=2V\mathrm{\Delta }t$ for $i=1,2,...,L$.

Figure 2

Benchmarking the two-cnot and scaled $R_{ZX}$ implementations of the $R_{ZZ}\left(\theta \right)$ gate. All data were taken on the IBM QPU Casablanca (ibmq_casablanca). (a) Standard implementation of the $R_{ZZ}\left(\theta \right)$ gate using two cnots. (b) Implementation of the $R_{ZZ}\left(\theta \right)$ gate using an $R_{ZX}\left(\theta \right)$ gate and appropriate single-qubit rotations. (c) Duration in nanoseconds of the pulse schedule on ibmq_casablanca as a function of $\theta =2V\mathrm{\Delta }t\le 2.5$ for the two-cnot and scaled-$R_{ZX}$ implementations of $R_{ZZ}\left(\theta \right)$. The scaled-$R_{ZX}$ implementation has a substantially shorter pulse duration for all $\theta$ considered. (d) Fidelity of the two-cnot and scaled-$R_{ZX}$ implementations of $R_{ZZ}\left(\theta \right)$ obtained using quantum process tomography on ibmq_casablanca as described in the text. The fidelity of the scaled-$R_{ZX}$ implementation decreases with increasing $\theta$ while that of the two-cnot implementation remains roughly constant. Data were accumulated over two days, so fluctuations in the fidelity due to calibration drifts are visible.

Figure 3

Unmitigated Trotter simulation of the staggered magnetization density $\langle Z_{\pi }\left(t\right)\rangle /L$ [(a), (b); see Eq. (1.4)] and accumulated error $D\left(t\right)$ [(c), (d); see Eq. (3.1)] from the initial state $|Z_2\rangle$. Data for chains of [(a), (c)] 12 and [(b), (d)] 19 qubits were obtained using ibmq_guadalupe and ibmq_toronto, respectively. Data for both the two-cnot (red) and scaled-$R_{ZX}$ (blue) implementations of the Trotter circuit are shown, with noiseless Trotter simulation results (black) for reference. No error mitigation was used here to directly compare the two $R_{ZZ}$ implementations. Error bars for each point in (a) and (b) represent the standard deviation of the data over 20 trials. Error bars in (c) and (d) are calculated from those in (a) and (b) by propagation of errors. Oscillations of $\langle Z_{\pi }\left(t\right)\rangle /L$ over roughly one period are observed for the scaled-$R_{ZX}$ implementation and are barely discernible for the two-cnot implementation. The two-cnot implementation also accumulates more error than the scaled-$R_{ZX}$ implementation.

Figure 4

Error-mitigated Trotter simulation of the staggered magnetization density $\langle Z_{\pi }\left(t\right)\rangle /L$ [(a), (b); see Eq. (1.4)] and accumulated error $D\left(t\right)$ [(c), (d); see Eq. (3.1)] from the initial state $|Z_2\rangle$. Data for chains of [(a), (c)] 12 and [(b), (d)] 19 qubits obtained using ibmq_guadalupe and ibmq_toronto, respectively. Data for both the two-cnot (red) and scaled-$R_{ZX}$ (blue) implementations of the $R_{ZZ}$ gate are shown, with noiseless Fibonacci-projected Trotter simulation results (black) for reference. Each data point in (a) and (b) is the result of linear ZNE with scale factors $\lambda \in \{1.0,1.5,2.0\}$ for ten random Pauli-twirling circuits. Error bars for all data points represent uncertainty in the ZNE and are calculated as described in the main text. The error-mitigation strategies outlined in Sec. 3 result in a substantial improvement for both implementations of the $R_{ZZ}$ gate relative to the results shown in Fig. 3.

Figure 5

Loschmidt echo $\mathcal{L}\left(t\right)$ of the initial state $|Z_2\rangle$ for a chain of 12 qubits calculated on ibmq_guadalupe using the same error-mitigation techniques described in Fig. 4 and the text. Data obtained from simulations using the two-cnot implementation of the $R_{ZZ}$ gate (red) barely show any tendency toward a revival around $Vt=20$, whereas data obtained using the scaled-$R_{ZX}$ implementation (blue) show a more pronounced revival. Both revivals are nowhere near as pronounced as the one obtained from the Fibonacci-projected ideal Trotter simulation (black), further indicating the effect of errors on the simulation despite the error-mitigation measures used.

Figure 6

Circuits used to calculate the correlation function ${\mathcal{C}}_Y\left(t\right)$ on the QPU. (a) Circuit to calculate ${\langle {\left(PYP\right)}_i\rangle }_{M_{Y_i}=\pm 1}$ [Eq. (4.3b)]. (b) Circuit to calculate ${\langle {\left(PYP\right)}_i\rangle }_{\pm Y_i}$ [Eq. (4.3c)]. Here, $U_{\text{Trotter}}$ denotes the Trotter circuit. In the beginning of each circuit, a different initial state is prepared. At the end of each circuit, the expectation value of ${\left(PYP\right)}_j$ is measured.

Figure 7

Dynamics of the correlator ${\mathcal{C}}_Y\left(t\right)$ [see Eq. (1.6)] from the initial state $|Z_2\rangle$ in [(a), (c)] the QMBS regime ($V=\mathrm{\Delta }t=1$, $\mathrm{\Omega }=0.24$) and [(b), (d)] the chaotic regime ($V=1$, $\mathrm{\Omega }=2$, $\mathrm{\Delta }t=0.16$). Panels (a) and (b) are calculated for a chain of 5 qubits using ibmq_casablanca, and panels (c) and (d) are calculated for a chain of 12 qubits using ibmq_guadalupe. The calculation uses ZNE for scale factors $\lambda \in \{1.0,1.5,2.0\}$ with [(a), (b)] eight, (c) ten, and (d) nine random circuit instances for Pauli twirling. Error bars representing the uncertainty in the ZNE were calculated as in Fig. 4. For $L=5$, oscillations with relatively slowly decaying amplitude are clearly visible throughout the simulation time window in the QMBS regime (a). For $L=12$, these oscillations remain coherent but exhibit a more rapid decay due to the accumulation of gate and readout errors. In the chaotic regime [(b), (d)], the correlator rapidly decays after a single approximate revival and exhibits good agreement with the ideal Trotter simulation results over the full simulation time window.

Figure 8

Pulse schedules for the two implementations of the $R_{ZZ}$ gate. (a) Pulse schedule for the two-cnot implementation of $R_{ZZ}(\theta =2.0)$ on ibmq_casablanca. Pulse durations are measured in units of $dt=0.2222\text{ns}$. Pulses labeled “$\text{CR}(\pi /4)$” and “$\overline{\text{CR}}(-\pi /4)$” are the cross-resonance pulses described in the text. Other Gaussian pulses correspond to single-qubit gates. On the D1 channel, the overlapping symbols above two circular arrows read $VZ\left(\pi \right)$. This denotes that a “virtual” rotation by $\pi$ around $Z$ [92] has been implemented using a pulse delay. The symbols below the two central Gaussian pulses read $X(\pi /2)$ (dark pulse) and $Y(\pi /2)$ (light pulse), respectively, indicating rotations by $\pi /2$ around the $X$ and $Y$ axes, respectively. On the D3 channel, the overlapping symbols below the two central Gaussian pulses read $Y(-\pi /2)$ (light pulse) and $X(\pi /2)$ (dark pulse), respectively. (b) Pulse schedule for the scaled-$R_{ZX}$ implementation of $R_{ZZ}(\theta =2.0)$ on ibmq_casablanca. Pulses labeled “GaussianSquare” are the scaled cross-resonance pulses discussed in the text. The scaled-$R_{ZX}$ implementation uses half as many cross-resonance pulses as the two-cnot implementation.

Figure 9

Comparison of the dynamics of the staggered magnetization density $\langle Z_{\pi }\left(t\right)\rangle /L$ in the QMBS regime ($V=1$, $\mathrm{\Omega }=0.24$) obtained from exact diagonalization (ED) (light green) and ideal Trotter simulations (grey) with $\mathrm{\Delta }t=1$. Although the large time step incurs substantial Trotter error, clear long-lived oscillations of the staggered magnetization are visible for both evolutions. The green and black curves indicate Fibonacci-projected ED and ideal Trotter dynamics, respectively.

Figure 10

(a) The dynamics of the number of nearest-neighbor pairs of Rydberg excitations, ${\sum }_{i=1}^{L-1}{n}_i{n}_{i+1}$, obtained from ED and ideal Trotter with $\mathrm{\Delta }t=1$ at $L=12$ and $L=19$. The number of such nearest-neighbor pairs is subextensive even for the Trotter dynamics. (b) The weight of the time-evolved state $|\psi \left(t\right)\rangle$ in the Fibonacci Hilbert space, $\langle \psi \left(t\right)|P_{\mathrm{fib}}|\psi \left(t\right)\rangle$, plotted using ED and ideal Trotter at $L=12$ and $L=19$. Even though the Trotter dynamics loses more weight in the subspace over the simulation window than the ED dynamics, the Trotter-evolved state retains a finite weight in the subspace at these system sizes. Taken together, these results justify our use of the postselection technique described in this Appendix.

Figure 11

Postselected QPU results without further error mitigation. See the caption of Fig. 3 for a description of panels (a)–(d). The red and blue curves are calculated using the same QPU data set as Fig. 3 with postselection applied, while the black curve denotes the Fibonacci-projected ideal Trotter dynamics for comparison. The data points on the black curve are used to calculate $D\left(t\right)$.

Figure 12

Implementation of Pauli twirling for (a) a cnot gate and (b) the non-Clifford $R_{ZZ}\left(\theta \right)$ gate. In both cases, the Pauli gates ${\sigma }_c^{\alpha }$ and ${\sigma }_t^{\beta }$ applied to the control and target qubits before the gate being twirled are chosen randomly. Pauli gates applied after the two-qubit gate being twirled are chosen such that the logical action of the two-qubit gate is unaffected. For the cnot gate, $\gamma$ and $\delta$ are chosen as $\gamma =\alpha +\beta (\beta -1)(\frac{7}{2}-\beta )(1-\frac{2}{3}\alpha )$ and $\delta =\beta +\alpha (\alpha -3)(\beta \text{mod}2-\frac{1}{2})$. For the $R_{ZZ}\left(\theta \right)$ gate, the same Pauli gates are applied before and after $R_{ZZ}\left(\theta \right)$, but the rotation angle $\theta \to -\theta$ if the randomly selected Pauli gates anticommute with ${\sigma }_c^z{\sigma }_t^z$.

Figure 13

Complete list of error-mitigated local magnetization results $\langle Z_i\rangle$ versus time $Vt$ for a 12-site chain measured on ibmq_guadalupe using the scaled-$R_{ZX}$ and two-cnot implementations (blue and red, respectively). The ideal Trotter simulation data (black) are also shown for reference.

Figure 14

Complete list of error-mitigated local magnetization results $\langle Z_i\rangle$ versus time $Vt$ for a 19-site chain measured on ibmq_toronto using the scaled-$R_{ZX}$ and two-cnot implementations (blue and red, respectively). The ideal Trotter simulation data (black) are also shown for reference.