Experimental quantum simulation of non-Hermitian dynamical topological states using stochastic Schrödinger equation

Noise is ubiquitous in real quantum systems, leading to non-Hermitian quantum dynamics, and may affect the fundamental states of matter. Here we report in an experiment a quantum simulation of the two-dimensional non-Hermitian quantum anomalous Hall (QAH) model using the nuclear magnetic resonance processor. Unlike the usual experiments using auxiliary qubits, we develop a stochastic average approach based on the stochastic Schrödinger equation to realize the non-Hermitian dissipative quantum dynamics, which has advantages in saving the quantum simulation sources and simplifying the implementation of quantum gates. We demonstrate the stability of dynamical topology against weak noise and observe two types of dynamical topological transitions driven by strong noise. Moreover, a region where the emergent topology is always robust regardless of the noise strength is observed. Our work shows a feasible quantum simulation approach for dissipative quantum dynamics with stochastic Schrödinger equation and opens a route to investigate non-Hermitian dynamical topological physics.


INTRODUCTION
As a fundamental notion beyond the celebrated Landau-Ginzburg-Wilson framework 1 , the topological quantum matter has stimulated extensive studies in recent years, with tremendous progress having been achieved in searching for various types of topological states [2][3][4][5][6][7] . A most important feature of topological matter is the bulk-surface correspondence [2][3][4] , which relates the bulk topology to boundary states and provides the foundation of most experimental characterizations and observations of topological quantum phases, such as via transport measurements [8][9][10] and angle-resolved photoemission spectroscopy [11][12][13] .
The quench-induced dynamical topological phase has been mainly studied in Hermitian systems, while the system is generally non-Hermitian when coupled to the environment 42 . Recently, the interplay between non-Hermiticity and topology has attracted considerable attention 43,44 , with rich phenomena being uncovered, such as the exotic topological phases driven by exceptional points [45][46][47][48] , the anomalous bulk-boundary correspondence 46,[49][50][51] , and the non-Hermitian skin effect 52 . Experimental observations of the non-Hermitian topological physics have been reported in classical systems with gain and loss, like the photonic systems 53,54 , the active mechanical metamaterial 55 , as well as topolectrical circuits 56 , and in quantum simulators, like the nitrogen-vacancy center 57,58 , where the non-Hermitian effects are engineered by coupling to auxiliary qubits.
As an important source of dissipation and non-Hermiticity, the dynamical noise is ubiquitous and inevitable in the real quantum simulations, especially for the quantum quench dynamics, and can be described by the stochastic Schrödinger equation 59,60 . Without the necessity of applying auxiliary qubits, the quantum simulation using the stochastic Schrödinger equation may enable a direct and more efficient way to explore non-Hermitian dynamical phases, hence facilitating the discovery of non-Hermitian topological physics with minimal quantum simulation sources. In particular, the controllable noise can provide a fundamental scheme to explore non-Hermitian dissipative quantum dynamics, and the noise effects on the quench-induced dynamical topological phase give rise to rich nonequilibrium topological physics 61 . However, the experimental study is currently lacking.
In this article, we report the experimental observation of quench-induced non-Hermitian dynamical topological states by simulating a noising two-dimensional (2D) quantum anomalous Hall (QAH) model on an NMR quantum simulator. Unlike previous experiments using auxiliary qubits 57,58 , we achieve with advantages the non-Hermitian quench dynamics via simulating the 1 stochastic Schrödinger equation and by averaged measurements over different noise configurations [59][60][61] . We observe the dynamical topology emerging in the non-Hermitian dissipative quench dynamics on BISs by measuring the time-averaged spin textures in momentum space, and identify two types of dynamical topological transitions classified by distinct dynamical exceptional points by varying the noise strength. Moreover, the existence of a sweet spot region with the emergent topology being robust under arbitrarily strong noise is experimentally verified. Our experiment demonstrates a feasible technique in simulating dynamical topological physics with minimal sources.

Non-Hermitian QAH model
We consider the non-Hermitian 2D QAH model with the magnetic dynamical white noise described by the Hamiltonian where H QAH ðkÞ ¼ hðkÞ Á σ describes the QAH phase 62,63 , with Bloch vector h ¼ ðξ so sin k x ; ξ so sin k y ; m z À ξ 0 cos k x À ξ 0 cos k y Þ.
Here ξ 0 (or ξ so ) simulates the spin-conserved (spin-flipped) hopping coefficient, and m z is the magnetic field. The white noise w i (k, t) of strength ffiffiffiffiffi w i p couples to the Pauli matrix σ i and satisfies hhw i ðk; tÞii noise ¼ 0 and hhw i ðk; tÞw j ðk; t 0 Þii noise ¼ w i δ ij δðt À t 0 Þ, where 〈〈⋅〉〉 noise is the stochastic average over different noise configurations. Without noise, the Hamiltonian H QAH hosts nontrivial QAH phase for 0 < |m z | < |ξ 0 | with Chern number C 1 = sgn(m z ), and the phase is trivial for |m z | > |ξ 0 | or m z = 0 62 . The random noise can change the topology of the QAH model, and plays a vital role on the quantum dynamics induced in the present system. We start with the simple situation with a single noise configuration. In this case the quantum dynamics governed by the stochastic Schrödinger equation i∂ t ψðk; tÞ j i¼ Hðk; tÞ ψðk; tÞ j idescribes a random unitary evolution, which can be further converted into the so-called Itô form 59,60 in simulation (see "Methods" for details) d ψðk; tÞ j i¼ Ài½H eff ðkÞdt þ X i ffiffiffiffiffi w i p σ i dW i ðk; tÞ ψðk; tÞ j i : Here H eff ¼ H QAH À ði=2Þ P i w i is the effective non-Hermitian Hamiltonian, such that the increment of a Wiener process W i ðk; tÞ ð1= ffiffiffiffiffi w i p Þ R t 0 ds w i ðk; sÞ is independent from the wavefunction function ψðtÞ j i, for which we have the Itô rules dtdW i (t) = 0 and dW i (t)dW j (t) = δ ij dt, and the corresponding expectation value is zero. The formal solution of the above equation reads ψðtÞ j i¼ UðtÞ ψð0Þ j i with where T denotes the time ordering. Note that while the equation (3) describes a random unitary evolution in the regime with single noise configuration, after the noise configuration averaging the non-Hermitian dissipative quantum dynamics emerges and is captured by the master equation where ρðk; tÞ hh ψðk; tÞ j iψðk; tÞ h jii noise is the stochastic averaged density matrix; see "Methods" for details. The configuration averaging is a key point for the present quantum simulation of non-Hermitian dynamical topological phases.

Quantum simulation approach
We next develop the quantum simulation approach by introducing the discrete Stochastic Schrödinger equation for the non-Hermitian dissipative quantum dynamics, since the continuous evolution cannot be directly emulated with digital quantum simulators. Specifically, we discretize the continuous time as t n = nτ with small-time step τ, where the integer n ranges from zero to the total number of time steps M. The increment of the Wiener process can be simulated by random numbers ΔW i ðt n Þ ¼ N i ðt n Þ ffiffi ffi τ p for each noise configuration, and we obtain the discretized stochastic Schrödinger equation Here N i (t n ) is sampled from the standard normal distribution to match the expectation and variance of dW i , and the wavefunction is normalized in each time step. The corresponding unitary evolution operator from time t n to t n+1 reads leading to the discrete equation of motion in the linear order of τ after stochastic average, which describes the desired non-Hermitian quantum dynamics. We shall analyse the quality of this discretization versus time step τ in the experiment. The stochastic average of a physical operatorÔ at time t n can now be obtained by Oðk; t n Þ hhhÔðk; t n Þiii noise ¼ Tr½ρðt n ÞÔ: This formalism can be directly simulated in experiment. The above presents the essential idea for simulating the non-Hermitian systems based on the stochastic Schrödinger equation. This method is fundamentally different from that applied in the previous experiments 57,58 using auxiliary qubits, where the non-Hermiticity is obtained from a Hermitian Hamiltonian in the extended Hilbert space by tracing the auxiliary degrees of freedom and careful designs of the quantum circuit with complex unitary operations are required 64,65 . In contrast, our temporal average approach based on the stochastic Schrödinger equation saves the resources of qubits and avoids the implementation of complex gates, which benefits the experimental platforms in various scenarios. Moreover, this quantum simulation approach can be directly extended to exploring higher dimensional non-Hermitian topological phases and phase transitions.

Non-Hermitian dynamical topological phases
Before presenting the experiment, in this section, we briefly introduce the non-Hermitian dynamical topological phases emerging in the quench dynamics described by Eq. (4) and to be studied in this work.
The system is initially prepared at the fully polarized ground state ρ 0 ¼ # j i # h j of a deep trivial Hamiltonian with |m z | ≫ |ξ 0 |. After quenching m z to a nontrivial value at time t n = 0, the system starts to evolve under the post-quench Hamiltonian Hðk; t n Þ; see Fig. 1a. Without noise, the spin polarization hσðk; t n Þi Tr½σ Q nÀ1 i¼0 Uðt nÀi ; t nÀiÀ1 Þρ 0 Q nÀ1 i¼0 U y ðt iþ1 ; t i Þ precesses with respect to the Hamiltonian vector h; see Fig. 1a. The post-quench QAH phase can be determined by the dynamical topology emerging on BISs 29 , identified as the momentum subspaces with h z = 0, where the initial state is perpendicular to the SO field h so ≡ (h x , h y ), leading to vanishing time-averaged spin polarizations.
In the presence of non-Hermiticity, the precession axis for each noise configuration is distorted, leading to the deformation for the BISs and dissipative effect. To characterize the noise effect, the spin polarization needs to be stochastically averaged as sðk; t n Þ hhhσðk; t n Þiii noise ¼ Tr½ρðt n Þσ (9) over different noise configurations (Fig. 1b). Compared to the spin polarization 〈σ(k, t n )〉 without noise, the stochastic averaged s(k, t n ) follows the non-Hermitian dynamics and exhibits dephasing and amplitude decaying effects. We compensate the amplitude decay by rescaling s(k, t n ), leading to the rescaled spin polarizationsðk; t n Þ s 0 ðkÞ þ s þ ðkÞe ÀiωðkÞtn þ s À ðkÞe þiωðkÞtn , where the coefficients s 0,± and oscillation frequency ω are extracted from the experimental data by fitting; see "Methods". Similar to the noiseless case, the time averagẽ vanishes on the deformed BISs (dubbed as dBISs) 61 , with the number of steps M being large enough to minimize the error. The non-Hermitian dynamical topological phase is captured by the dynamical invariant W 1 2π H dBIS gðkÞdgðkÞ, which describes the winding of dynamical field gðkÞ ¼ ð1=N k Þ∂ k? ðs x ðkÞ;s y ðkÞÞ on the dBISs. Here k ⊥ is perpendicular to the dBISs and N k is a normalization factor. Under the dynamical noise, the non-Hermitian dynamical topological phases and phase transitions may be induced, as studied in the experiment presented below.

Experimental setup
The demonstration is performed on the NMR quantum simulator. The sample is the 13 C-labeled chloroform dissolved in acetone-d6, with 13 C and 1 H nuclei denoted as two qubits. The 2D QAH model is simulated by the qubit 13 C, while the other qubit 1 H enhances the signal by Overhauser effect (see Fig. 1c and "Methods"). In the double-rotating frame, the total Hamiltonian of this sample is where J = 215 Hz is the coupling strength, B i is the amplitude of the control pulse, and ϕ i is the phase. We firstly initialize the system into the fully polarized state # j i using the nuclear Overhauser effect 66 . Then we quench m z to the nontrivial region with |m z | < 2ξ 0 and allow the system to evolve under the effective HamiltonianH, in which the non-Hermitian constant term i∑ i w i can be ignored. The evolution is realized by the Trotter approximation combined with control pulse optimizations as follows.
We study the non-Hermitian dissipative quantum dynamics from time t = 0-30 ms. For each noise configuration, numerical results show that the discrete evolution approximates the continuous evolution of the stochastic Schrödinger equation quite well, when the total number of time steps is greater than 100; see Fig. 2. In the experiment, we discretize the time into 300 segments, such that the Hamiltonian in each interval is approximately time-independent. As the interval τ is sufficiently small, the evolution in the n-th step can be realized using the firstorder Trotter decomposition: Uðt nþ1 ; t n Þ % e Àiη x σx τ e Àiη y σy τ e Àiη z σzτ (12) with η Here ξ 0 is set to 1 kHz. Each term on the right-hand side represents a single-qubit rotation with a rotating angle 2η i τ along axis σ i , which can be experimentally realized by tuning the amplitude and phase of the control pulse in Eq. (11) (z-rotation can be indirectly realized via x-and y-rotations), with further pulse optimization techniques to reduce control errors; see Fig. 1c.
We measure the spin polarization 〈σ(k, t)〉 for single noise configuration at every 20τ interval. After averaging over all noise In the presence of noise, the precession axis is distorted, leading to the dissipative dynamics of stochastic averaged spin polarization s(k, t), as shown by the red arrows and distorted blue trajectories. b The non-Hermitian dissipative dynamics can be interpreted as the stochastic average over different noise configurations. Here the solid black arrows represent the Hamiltonian vector h without noise, and the dashed black arrows denote the Hamiltonian vector distorted by time-dependent noise. For each noise configuration (small spheres), the spin polarization is on the surface of Bloch sphere and obeys the unitary dynamics, in spite of the irregular blue trajectory caused by the noise-distorted Hamiltonian vector. However, the average over all noise configurations (big sphere) leads to a globally dissipative effect and a deformation of BISs. c Pulse sequence for simulating the 2D non-Hermitian QAH model. 1 H is initially decoupled, and 13 C is rapidly prepared to the # j i state using the nuclear Overhauser effect. The control pulse is designed according to Eq. (12), where the green and blue circles represent rotations about the x-axis and y-axis with A and φ the amplitude and phase, respectively.
configurations, we obtain the stochastic averaged spin polarization s(k, t), from which the rescaled spin polarizationsðk; tÞ can be constructed by fitting. We repeat the above procedures for the whole momentum space to obtain the time-averaged spin texturessðkÞ.

Experimental results
We start from the weak noise regime, where the noise strength is chosen as w x = 0.05ξ 0 , w y = 0, and w z = 0.01ξ 0 with ξ so = 0.2ξ 0 . The system is quenched to the topological phase with m z = 1.2ξ 0 . In Fig. 3a, we plot the spin polarization 〈σ(t)〉 at the momentum k = (1.286, −0.257) for four different noise configurations. For each noise configuration, no notable decay exists in the spin polarization, manifesting the unitary evolution. However, after averaged over all noise configurations, the system clearly exhibits the non-Hermitian dissipative quantum dynamics; see Fig. 3b. Figure 3c shows the measured time-averaged spin texturess i ðkÞ with fixed k y = −0.257 and k x ∈ [−1.8, 1.8], obtained by rescaling the stochastic averaged spin polarization s(k, t). The momenta with vanishing values represent dBIS points. To obtain the 2D time-averaged spin texture, we discretize the whole momentum space k x , k y ∈ [−1.8, 1.8] into a 15 × 15 lattice and repeat the above measurements. The results are shown in Fig. 3d, from which the dBIS momenta can be identified. Although the corresponding shape is slightly deformed from the ideal BIS with h z = 0 in the absence of noise (see "Methods"), it is obvious that under weak noise, the dynamical field g(k) can be defined everywhere on dBIS and characterizes the nontrivial non-Hermitian dynamical topological phase (Fig. 3e). Indeed, this emergent dynamical topology is robust against the weak noise and is protected by the finite minimal oscillation frequency on the dBISs, serving as a bulk gap for the dynamical topological phase. The experimental minimum oscillation frequency on dBISs is given by ω min ¼ 0:4175 kHz, close to the theoretical value 0.4063 kHz (Fig. 3b). Further, this non-Hermitian dynamical topological phase may break down under strong noise, with two types of dynamical transition being observed below.
We now increase the noise strength to a strong regime with w x = 0.1ξ 0 , w y = 0.05ξ 0 , and w z = 0.45ξ 0 . The averaged spin polarization is measured in the same way as in the weak noise regime. However, the quench dynamics are essentially different, where the spin polarization s(t) at certain momenta, for instance k x = −1.286 and k y = −0.257, displays pure decay without oscillation; see Fig. 4a, b. For these momenta, the dynamical field g vanishes. In Fig. 4c, we show the corresponding spin textures. From the result fors z , we find that singularities emerges on the dBISs and interrupt their continuity. Thus the dBIS breaks down, while the deformation of the shape of dBIS is small, and the non-Hermitian dynamical topological phase transition occurs. In Fig.  4d, we increase the noise strength to w x = 1.6ξ 0 , w y = 0, w z = 0.8ξ 0 and set a strong SO coupling coefficient with ξ so = 2ξ 0 . A qualitatively different dynamical transition is uncovered, where the dBISs are dramatically deformed by the noise and are connected to the topological charge at k = 0. Due to this singularity, the dynamical topology also breaks down. The above two qualitatively different phenomena are referred to as type-I and type-II dynamical transitions, respectively, which we examine below in more detail.
We notice that the equilibrium topological phase transition usually corresponds to the close of energy gap. In the nonequilibrium regime, the analogous quantity is the oscillation frequency. Here we observe the corresponding momentum distribution in Fig. 5a. One can see that the oscillation frequency is in general nonzero but may vanish on certain dBISs momenta when these two types of dynamical transition occur, i.e., ω min ðk c Þ ! 0. Indeed, the momenta (k c ) with just vanishing oscillation frequency are exceptional points of the Liouvillian superoperator, on which the eigenvectors s LðRÞ ± coalesce 61 . Thus the dynamical transitions are driven by exceptional points with vanishing oscillation frequency on dBISs. To further distinguish these two types of dynamical transition and the corresponding exceptional points, we treat the Liouvillian superoperator as a three-level system; see "Methods". The coefficient s + of rescaled dynamical spin polarizationsðk; tÞ contains the information of corresponding eigenvectors s LðRÞ ± . Like the spin-1 system, we measure the Liouvillian polarization hL α i s y þ L α s þ to characterize the Liouvillian superoperator. Here the operator L α is defined as and L z = i[L y , L x ], which satisfies [L α , L β ] = iϵ αβγ L γ . The measured momentum distribution of these quantities in the experiment is shown in Fig. 5b, c, from which an important feature of exceptional points is observed that the component 〈L x 〉 ≈ 0 and 〈L y 〉 ≈ 0 vanish on these points while 〈L z 〉 is in general nonzero (e.g., see Fig. 5c). Therefore, the exceptional points are actually the singularities in the two-component vector field (〈L x 〉, 〈L y 〉). With this observation and to characterize the exceptional points, we consider the Liouvillian polarization on a small loop S enclosing the exceptional points, as shown in Fig. 6a, b. Although the component 〈L z 〉 is nonzero on this loop, the trajectory projected on the 〈L x 〉-〈L y 〉 plane indeed defines a winding number 61 which distinguishes the two types of dynamical transitions. We observe that for type-I transition, the winding number N E = 0 is trivial, while the winding N E = 1 is nontrivial for the type-II dynamical transition. Consequently, these distinct exceptional points on dBISs shows the fundamental difference between the type-I and type- c, respectively. e Dynamical field g = (g x , g y ) (black arrows) obtained from the time-averaged spin texture. Here we set w x = 0.05ξ 0 , w y = 0, w z = 0.01ξ 0 , and ξ so = 0.2ξ 0 . see Fig. 6a, b. Here we note that the topological charges are always singularities of the field (〈L x 〉, 〈L y 〉) and have nontrivial winding number 61 (Fig. 6c, d). The loop S should be introduced without enclosing any non-exceptional charge momentum in characterizing the dynamical transitions and corresponding exceptional points. This also tells that the type-II dynamical transition is similar to the equilibrium topological phase transition, in which the topological charges serve as singular points and the transition occurs when they pass through the BISs 29,30 . On the other hand, the type-I transition is a peculiar feature of the quench-induced non-Hermitian dynamical topological phase transition.
Although the non-Hermitian dynamical topological phase may typically be destroyed in the strong noise regime, a quite interesting feature of the present system is the existence of a sweet spot region satisfying 61 max½ðw y À w x Þξ 2 0 =ξ 2 so À 2jξ so j < w z À w x < min½ðw y À w x Þξ 2 0 =ξ 2 so þ 2jξ so j; (15) in which regime the dynamical topology is always robust at any finite noise strength, as characterized by the taper-type region in Fig. 7. In particular, for the central line with w x = w y = w z , we experimentally increase the noise strength w i in each direction from 0.5ξ 0 to a very large value w i ≃ 10ξ 0 (points O 1,2,3 ) and measure the corresponding time-averaged dynamical spin textures. We observe that although the noise strength is much large compared with all other energy scales, the dBIS ins z remains stable, without suffering singularities. Inside the taper-type region the dynamical topology is well-defined on the dBIS, in sharp contrast to outside points (P 1,2 ). The experimental confirmation of this sweet spot region may offer guidance in designing noisetolerant topological devices.

DISCUSSION
We have experimentally reported the quantum simulation of non-Hermitian quantum dynamics for a 2D QAH model coupled to dynamical noise based on a stochastic average approach of the stochastic Schrödinger equation, and simulated non-Hermitian dynamical topological phases and phase transitions. Our method does not require the ancillary qubits and careful designs of complex unitary gates, hence saving the simulation sources and avoiding the implementation of complex gates in the experiment. The dynamical topological physics driven by dynamical noise has been observed, including the stability of non-Hermitian dynamical topological states protected by the minimal oscillation frequency of quench dynamics under weak noise and two basic types of dynamical topological transitions driven by strong noise and classified by distinct exceptional points. Moreover, a sweet spot region is observed, where the non-Hermitian dynamical topological phase survives at arbitrarily strong noise. Our experiment has shown an advantageous quantum simulation approach to explore the non-Hermitian dynamical topological physics, in which only a minimal number of qubits are used. This approach is directly applicable to high dimensions by taking into account more, but still minimal number of qubits, in which the rich phenomena are expected, and also to other digital quantum simulators.   Fig. 4 Strong noise regime of the non-Hermitian QAH model. a For w x = 0.1ξ 0 , w y = 0.05ξ 0 , w z = 0.45ξ 0 and ξ so = 0.2ξ 0 , the stochastically averaged spin polarization s y (k, t) presents no oscillation. b At the same noise level, s z (k, t) decays to 0 without oscillation. c Time-averaged rescaled spin texturess i ðkÞ, with i = x, y, z. Other than small deformation, singularities (black dot) emerge on the dBIS momenta (type-I dynamical transition). ds i ðkÞ under stronger noise of w x = 1.6ξ 0 , w y = 0, w z = 0.8ξ 0 , and ξ so = 2ξ 0 . The dBIS deforms drastically and connects to the topological charge at k = 0 (type-II dynamical transition).

Stratonovich stochastic Schrödinger equation
We consider the non-Hermitian 2D QAH model (1) with the magnetic dynamical white noise w i (k, t). Since the dynamical white noise is in some sense infinite, the dynamical equation ∂ t ψðk; tÞ j i¼À iHðk; tÞ ψðk; tÞ j i cannot be considered as an ordinary differential equation. Instead, it should be regarded as an integral equation The basic difference is that the integrand f(t) and the increment dW(t) are independent of each other in the Itô form, namely 〈〈f(t)dW(t)〉〉 noise = f(t) 〈〈dW(t)〉〉 noise = 0, while they are not independent in the Stratonovich form. The Schrödinger equation (17) must be interpreted as a Stratonovich stochastic differential equation 59,60 , such that the quantum mechanical probability is preserved, i.e., d〈ψ(t)|ψ(t)〉 = 0.

Converting into the Itô form
Since the wavefunction ψðtÞ j i and the increment dW i (t) are not independent in the Stratonovich form, it is usually convenient to convert the Stratonovich stochastic Schrödinger equation (17) into the Itô form, which takes the form ðIÞd ψðtÞ j i¼Ài½H eff dt þ X i α i dW i ðtÞ ψðtÞ j i: Due to ð1=2Þð ψðt þ dtÞ j iþψðtÞ j iÞ ¼ ½1 À ði=2ÞðH eff dt þ P i α i dW i ðtÞÞ ψðtÞ j i, we have the following relation between the Stratonovich integral and the where we have used the Itô rules dtdW i (t) = 0 and dW i (t)dW j (t) = δ ij dt for the increment of a Wiener process. Substituting this into the Itô stochastic Schrödinger equation (20), we obtain ðSÞd ψðtÞ j i¼ Ài½ðH eff þ i 2 Compared with the original Stratonovich stochastic Schrödinger equation (17), it is easy to find In the main text, we have shown that the formal solution of the Itô stochastic Schrödinger equation (20) is given by a unitary evolution U(t) [see Eq. (3)]. To prove that U(t) is indeed the solution of the Itô equation, we shall note that Uðt þ dt; tÞ ¼ e where the terms other than dt and dW i dW i = dt vanish according to the Itô rules. Thus we recover the Itô stochastic Schrödinger equation, i.e., dUðtÞ ¼ Uðt þ dtÞ À UðtÞ ¼ Ài½H eff dt þ P i ffiffiffiffiffi w i p σ i dW i ðtÞUðtÞ.

Non-Hermitian dissipative quantum dynamics
We now consider the equation of motion for the stochastic density operator ϱðtÞ ¼ ψðtÞ j i ψðtÞ h j, namely dϱðtÞ ¼Uðt þ dt; tÞϱðtÞU y ðt þ dt; tÞ À ϱðtÞ Since the increments dW i (t) are independent of ϱ(t) in the Itô form, after average over different noise configurations the last term vanishes and we arrive at the Lindblad master equation (4) for the stochastic averaged density matrix ρ(t) ≡ 〈〈ϱ(t)〉〉 noise , which describes the non-Hermitian dissipative quantum dynamics.

Stochastically averaged spin dynamics
In this section, we show the stochastically averaged spin dynamics. According to the master equation (4) The solution to this dissipative quantum dynamics can be written as with the coefficients s α ðkÞ ¼ ½s L α ðkÞ Á sðk; 0Þs R α for α = 0, ±. Here s LðRÞ α satisfying s L α ðkÞ Á s R β ðkÞ ¼ δ αβ are the left (right) eigenvectors of the Liouvillian superoperator L T s L α ¼ Àλ α s L α , Ls R α ¼ Àλ α s R α with eigenvalues λ 0 and λ ± = λ 1 ± iω, respectively. The oscillation frequency is denoted as ω.
In experiments, the coefficients s α , decay rates λ 0,1 , and oscillation frequency can be extracted by fitting the experimental data. By ignoring λ 0,1 , we obtain the rescaled spin polarizationsðk; tÞ.

NMR sample
The experiment is performed on the nuclear magnetic resonance processor (NMR). The sample we used is the 13 C-labeled chloroform dissolved in rmacetone-d6. The 13 C spin is used as the working qubit, which is controlled by radio-frequency (RF) fields. The 1 H is decoupled throughout the experiment by Overhauser effect which can enhance the signal strength of 13 C.

Overhauser effect
Applying a weak RF field at the Larmor frequency of one nuclear spin for a sufficient duration may enhance the longitudinal magnetization of the others, this is the steady-state nuclear Overhauser effect (NOE). In modern NMR, the steady-state NOE is mainly exploited in heteronuclear spin systems, where the enhancement of magnetization is useful and dramatic.
For an ensemble of heteronuclear systems made up with a nuclei I with gyromagnetic ratio γ I and a nuclei S with gyromagnetic ratio γ S , with |γ I | > |γ S |, the thermal equilibrium state of the heteronuclear system iŝ where β I /β S = γ I /γ S , 1 4Î z ¼σ z 1, 1 4Ŝ z ¼1 σ z . Assume that a continuous RF field is applied at the I-spin Larmor frequency, inducing transitions across two pairs of energy levels. After sufficient time, the RF field equalizes the populations across the irradiated transitions. At that time, the populations settle into steady-state values, which do not change any more, as long as the RF field is left on. The steady-state spin density operator iŝ By comparing with thermal equilibrium Eq. (29), the S-spin magnetization is enhanced by factor ϵ NOE . For our experiment I = 1 H and S = 13 C.

Noise configurations
For the stochastic average, it is clear that the more noise configurations are considered, the more reasonable result we obtain, as shown in Fig. 8. On the other hand, the large number of noise configurations takes a lot of time. We have performed numerical simulations, and found that the average of 5000 noise configurations can precisely approximate the non-Hermitian dissipative quantum dynamics; see Fig. 8d. However, in NMR experiments, as the relaxation time is in the magnitude of seconds, a complete implementation of all 5000 noise configurations requires an extremely long-running time that we cannot afford. An alternative method to solve the issue is to reduce the number of noise configurations by numerical simulation prior to the implementation of experiments. We test different number of noise configurations, and plot their average dynamics in comparison with the ideal dynamics of the non-Hermitian Hamiltonian; see Fig. 8a-d. The simulated results show that with the increase of the number of noise configurations, the stochastic averaged spin polarization 〈〈〈σ(k, t)〉〉〉 noise would eventually approach to the spin polarization s(k, t) solved by the Lindblad master equation 61 . The opposite is that with the decrease of the number of noise configurations, the performance of the approximation becomes more fluctuating (Fig. 8e). But the 〈〈〈σ(k, t)〉〉〉 noise always fluctuates above and below the theoretical spin polarization s(k, t). After a sufficient number of averaging, the stochastic averaged spin polarization that in the opposite side of theoretical value will be offset by each other. We randomly generated 5000 noise configurations N(t n ) that satisfy the normal distribution and separate these noise configurations into two subgroups in which the noise has an opposite effect on 〈〈〈σ(k, t)〉〉〉 noise . Then we use numerical simulations to select two noise configurations from these two subgroups respectively such that the 〈〈〈σ(k, t)〉〉〉 noise obtained from these four noise configurations precisely approximate the one obtained from the 5000 configurations (Fig. 8f). From experimental results and the corresponding fidelities, it can be concluded that the experiment is in excellent accordance with the simulations. And the theory and experiment results of each group of noise are in good agreement (Fig. 8g) So, it is somehow reasonable to utilize four noise configurations to replace a full description of the non-Hermitian dynamics under 5000 noise configurations. We would like to emphasize that the above numerical simulations to reduce the number of noise configurations does not affect the applicability of the method. In many other quantum systems such as the superconducting circuits or nitrogen-vacancy centers in diamond, the implementation of experiments takes a much shorter time, so they can realize the stochastic average with a larger number of noise configurations.

Experimental results vs. theoretical results
In this section, we show the agreement of our experimental results with the theoretical calculations. In Fig. 9, we compare the experimental spin textures with theoretical ones. Although the resolution of experimental data is lower than that of numerical calculations, the experimental results and the theoretical simulations reach the same conclusion. In Fig. 10 The noise strength is w x = 0.1t 0 , w y = 0.05t 0 , w z = 0.45t 0 . b Exceptional point (white dot) emerges at the charge momentum (green dot) at k = 0, to which the dBIS connects for type-II transition. The noise strength is w x = 1.6t 0 , w y = 0t 0 , w z = 0.8t 0 .
we show the numerical calculations for exceptional points and the corresponding winding numbers, which are consistent with our experimental results (Fig. 7).