
==== Front
Nat Commun
Nat Commun
Nature Communications
2041-1723
Nature Publishing Group UK London

52082
10.1038/s41467-024-52082-2
Article
Probing spin hydrodynamics on a superconducting quantum simulator
http://orcid.org/0000-0003-1130-4280
Shi Yun-Hao 123
Sun Zheng-Hang 12
http://orcid.org/0000-0003-1440-6354
Wang Yong-Yi 12
Wang Zheng-An 34
Zhang Yu-Ran 5
http://orcid.org/0009-0000-3639-2294
Ma Wei-Guo 12
Liu Hao-Tian 12
Zhao Kui 3
Song Jia-Cheng 12
Liang Gui-Han 12
Mei Zheng-Yang 12
Zhang Jia-Chi 12
http://orcid.org/0000-0002-4334-9283
Li Hao 3
Chen Chi-Tong 12
Song Xiaohui 1
http://orcid.org/0000-0001-5072-3096
Wang Jieci 6
Xue Guangming 3
http://orcid.org/0000-0002-5963-8266
Yu Haifeng 3
http://orcid.org/0000-0001-8845-2770
Huang Kaixuan huangkx@baqis.ac.cn

3
http://orcid.org/0000-0002-0734-1217
Xiang Zhongcheng zcxiang@iphy.ac.cn

124
http://orcid.org/0000-0002-7776-2455
Xu Kai kaixu@iphy.ac.cn

123478
http://orcid.org/0000-0001-5895-0111
Zheng Dongning 12478
http://orcid.org/0000-0002-5328-2449
Fan Heng hfan@iphy.ac.cn

123478
1 grid.9227.e 0000000119573309 Institute of Physics, Chinese Academy of Sciences, Beijing, China
2 https://ror.org/05qbk4x57 grid.410726.6 0000 0004 1797 8419 School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, China
3 https://ror.org/04nqf9k60 grid.510904.9 0000 0004 9362 2406 Beijing Academy of Quantum Information Sciences, Beijing, China
4 grid.59053.3a 0000000121679639 Hefei National Laboratory, Hefei, China
5 https://ror.org/0530pts50 grid.79703.3a 0000 0004 1764 3838 School of Physics and Optoelectronics, South China University of Technology, Guangzhou, China
6 https://ror.org/053w1zy07 grid.411427.5 0000 0001 0089 3695 Department of Physics and Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, Hunan Normal University, Changsha, China
7 https://ror.org/020vtf184 grid.511002.7 Songshan Lake Materials Laboratory, Dongguan, China
8 grid.410726.6 0000 0004 1797 8419 CAS Center for Excellence in Topological Quantum Computation, UCAS, Beijing, China
31 8 2024
31 8 2024
2024
15 757324 3 2024
23 8 2024
© The Author(s) 2024
2024
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Characterizing the nature of hydrodynamical transport properties in quantum dynamics provides valuable insights into the fundamental understanding of exotic non-equilibrium phases of matter. Experimentally simulating infinite-temperature transport on large-scale complex quantum systems is of considerable interest. Here, using a controllable and coherent superconducting quantum simulator, we experimentally realize the analog quantum circuit, which can efficiently prepare the Haar-random states, and probe spin transport at infinite temperature. We observe diffusive spin transport during the unitary evolution of the ladder-type quantum simulator with ergodic dynamics. Moreover, we explore the transport properties of the systems subjected to strong disorder or a tilted potential, revealing signatures of anomalous subdiffusion in accompany with the breakdown of thermalization. Our work demonstrates a scalable method of probing infinite-temperature spin transport on analog quantum simulators, which paves the way to study other intriguing out-of-equilibrium phenomena from the perspective of transport.

Quantum devices offer the potential to simulate quantum phenomena, which are otherwise computationally intractable. Here, Shi, Sun, Wang and coauthors use a superconducting quantum simulator to study spin-transport at infinite temperature.

Subject terms

Quantum simulation
Qubits
501100001809 National Natural Science Foundation of China (National Science Foundation of China) 92265207, T2121001, 11934018 Beijing Natural Science Foundation (Grant No. Z200009), Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0301800)Beijing Nova Program (Nos. 20220484121, 2022000216)issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Transport properties of quantum many-body systems driven out of equilibrium are of significant interest in several active areas of modern physics, including the ergodicity of quantum systems1–4 and quantum magnetism5–7. Understanding these properties is crucial to unveil the non-equilibrium dynamics of isolated quantum systems8,9. One essential property of transport is the emergence of classical hydrodynamics in microscopic quantum dynamics, which shows the power-law tail of autocorrelation functions8. The rate of the power-law decay, referred as to the transport exponent, characterizes the universal classes of hydrodynamics. In d-dimensional quantum systems, in addition to generally expected diffusive transport with the exponent d/2 in non-integrable systems10–12, more attention has been attracted by the anomalous superdiffusive5,13–16 or subdiffusive transport2,3,17–19, with the exponent larger or smaller than d/2, respectively.

Over the last few decades, considerable strides have been made in enhancing the scalability, controllability, and coherence of noisy intermediate-scale quantum (NISQ) devices based on superconducting qubits20–23. With these advancements, several novel phenomena in non-equilibrium dynamics of quantum many-body systems have been observed, such as quantum thermalization24,25, ergodicity breaking26–29, time crystal30–32, and information scrambling33,34. More importantly, in this platform, the beyond-classical computation has been demonstrated by sampling the final Haar-random states of randomized sequences of gate operations35–39. Recently, a method of measuring autocorrelation functions at infinite temperature based on the Haar-random states has been proposed, which opens up a practical application of pseudo-random quantum circuits for simulating hydrodynamics on NISQ devices40,41.

In this work, using a ladder-type superconducting quantum simulator with up to 24 qubits, we first demonstrate that in addition to the digital pseudo-random circuits35–41, a unitary evolution governed by a time-independent Hamiltonian, i.e., an analog quantum circuit, can also generate quantum states randomly chosen from the Haar measure, i.e., the Haar-random states, for measuring the infinite-temperature autocorrelation functions42–44. Subsequently, we study the properties of spin transport on the superconducting quantum simulator via the measurement of autocorrelation functions by using the Haar-random states. Notably, we observe a clear signature of the diffusive transport on the qubit ladder, which is a non-integrable system11,12,25.

Upon subjecting the qubit ladder to disorder, a transition from delocalized phases to many-body localization (MBL) occurs as the strength of the disorder increases45. By measuring the autocorrelation functions, we experimentally probe an anomalous subdiffusive transport with intermediate values of the disorder strength. The observed signs of subdiffusion are consistent with recent numerical results and can be explained as a consequence of a Griffth-like region on the delocalized side of the MBL transition2,3,46–49.

Finally, we explore spin transport on the qubit ladder with a linear potential, and it is expected that Stark MBL occurs when the potential gradients are sufficiently large28,50–54. With a large gradient, the conservation of the dipole moment emerges28,54, associated with the phenomena known as the Hilbert space fragmentation55–57. Recent theoretical works reveal a subdiffusion in the dipole-moment conserving systems17,19. In this experiment, we present evidence of a subdiffusive regime of spin transport in the tilted qubit ladder.

Results

Experimental setup and protocol

Our experiments are performed on a programmable superconducting quantum simulator consisting of 30 transmon qubits with the geometry of a two-legged ladder, see Fig. 1a, b. The nearest-neighbor qubits are coupled by a fixed capacitor, and the effective Hamiltonian of capacitive interactions can be written as22,23 (also see Supplementary Note 1)1 H^I/ℏ=∑m∈{↑,↓}∑j=1L−1Jj,m∥(σ^j,m+σ^j+1,m−+H.c.)+∑j=1LJj⊥σ^j,↑+σ^j,↓−+H.c.,

where ℏ = h/2π, with h being the Planck constant (in the following, we set ℏ = 1), L is the length of the ladder, σ^j,m+ (σ^j,m−) is the raising (lowering) operator for the qubit Qj,m, and Jj,m∥ (Jj⊥) refers to the rung (intrachain) hopping strength. For this device, the averaged rung and intrachain hopping strength are J∥¯/2π≃7.3MHz and J⊥¯/2π≃6.6MHz, respectively. The XY and Z control lines on the device enable us to realize the drive Hamiltonian H^d=∑m∈{↑,↓}∑j=1LΩj,m(e−iϕj,mσ^j,m++eiϕj,mσ^j,m−)/2, and the on-site potential Hamiltonian H^Z=∑m∈{↑,↓}∑j=1Lwj,mσ^j,m+σ^j,m−, respectively. Here, Ωj,m and ϕj,m denote the driving amplitude and the phase of the microwave pulse applied on the qubit Qj,m, and wj,m is the effective on-site potential.Fig. 1 Superconducting quantum simulator and experimental pulse sequences.

a The schematic showing the ladder-type superconducting quantum simulator, consisting of 30 qubits (the blue region), labeled Q1,↑ to Q15,↑, and Q1,↓ to Q15,↓. Each qubit is coupled to a separate readout resonator (the green region), and has an individual control line (the red region) for both the XY and Z controls. b Schematic diagram of the simulated 24 spins coupled in a ladder. The blue and yellow double arrows represent the infinite-temperature spin hydrodynamics without preference for spin orientations. c Schematic diagram of the quantum circuit for measuring the autocorrelation functions at infinite temperature. All qubits are initialized at the state 0. Subsequently, an analog quantum circuit U^R(tR) acts on the set of qubits QR to generate Haar-random states. This is followed by a time evolution of all qubits, i.e., U^H(t)=exp(−iH^t) with H^ being the Hamiltonian of the system, in which the properties of spin transport are of our interest. d Experimental pulse sequences corresponding to the quantum circuit in (c) displayed in the frequency (ω) versus time (T) domain. To realize U^R(tR), qubits in the set QR are tuned to the working point (dashed horizontal line) via Z pulses, and simultaneously, the resonant microwave pulses represented as the sinusoidal line are applied to QR through the XY control lines. Meanwhile, the qubit QA is detuned from the working point with a large value of the frequency gap Δ. To realize the subsequent evolution U^H(t) with the Hamiltonian (1), all qubits are tuned to the working point.

To study spin transport and hydrodynamics, we focus on the equal-site autocorrelation function at infinite temperature, which is defined as2 Cr,r=1DTr[ρ^r(t)ρ^r],

where ρ^r is a local observable at site r,ρ^r(t)=eiH^tρ^re−iH^t, and D is the Hilbert dimension of the Hamiltonian H^. Here, for the ladder-type superconducting simulator, we choose ρ^r=(σ^1,↑z+σ^1,↓z)/2 (r = 1)12, and the autocorrelation function can be rewritten as3 C1,1=14(c1,↑;1,↑+c1,↑;1,↓+c1,↓;1,↑+c1,↓;1,↓),

with cμ;ν=Tr[σ^μz(t)σ^νz]/D (subscripts μ and ν denote the qubit index 1 ↑ or 1, ↓).

The autocorrelation functions (2) at infinite temperature can be expanded as the average of Cr,r(ψ0)=⟨ψ0∣ρ^r(t)ρ^r∣ψ0⟩ over different ψ0 in z-basis. In fact, the dynamical behavior of an individual Cr,r(ψ0) is sensitive to the choice of ψ0 under some circumstances (see Supplementary Note 7 for the dependence of Cr,r(ψ0) on ψ0 in the qubit ladder with a linear potential as an example). To experimentally probe the generic properties of spin transport at infinite temperature, one can obtain (2) by measuring and averaging Cr,r(ψ0) with different ψ015. Alternatively, we employ a more efficient method to measure (2) without the need of sampling different ψ0. Based on the results in ref. 40 (also see “Methods”), the autocorrelation function cμ;ν can be indirectly measured by using the quantum circuit as shown in Fig. 1c, i.e.,4 cμ;ν≃ψνR(t)∣σ^μz∣ψνR(t),

where ψνR(t)=U^H(t)[0ν⊗ψR] with ψR=U^R⨂i∈QR0i, and U^R being a unitary evolution generating Haar-random states. For example, to experimentally obtain c1,↓;1,↑, we choose Q1,↑ as QA, and the remainder qubits as the QR. After performing the pulse sequences as shown in Fig. 1d, we measure the qubit Q1,↓ at z-basis to obtain the expectation value of the observable σ^1,↓z.

Observation of diffusive transport

In this experiment, we first study spin transport on the 24-qubit ladder consisting of Q1,↑, …, Q12,↑ and Q1,↓, …, Q12,↓, described by the Hamiltonian (1). For a non-integrable model, one expects that diffusive transport C1,1 ∝ t−1/2 occurs12. To measure the autocorrelation function C1,1 defined in Eq. (3), we should first perform a quantum circuit generating the required Haar-random states ψR. Instead of using the digital pseudo-random circuits in refs. 35–41, here we experimentally realize the time evolution under the Hamiltonian H^R=H^I+H^d, where the parameters Ωj,m and ϕj,m in H^d have site-dependent values with the average Ω¯/2π≃10.4MHz (Ω¯/J∥¯≃1.4) and ϕ¯=0 (see “Methods” and Supplementary Note 3 for more details), i.e., U^R(tR)=exp(−iH^RtR), which is more suitable for our analog quantum simulator. To benchmark that the final state ψR=U^R(tR)0 can approximate the Haar-random states, we measure the participation entropy SPE=−∑k=1Dpklnpk, with D being the dimension of Hilbert space, pk = ∣〈k∣ψR〉∣2, and {k} being a computational basis. Figure 2a shows the results of SPE with different evolution times tR. For the 23-qubit system, the probabilities pk are estimated from the single-shot readout with a number of samples Ns = 3 × 107. It is seen that the SPE tends to the value for Haar-random states, i.e., SPET=Nln2−1+γ with N = 23 being the number of qubits and γ ≃ 0.577 as the Euler’s constant36. Moreover, for the final state ψR with tR = 200 ns, the distribution of probabilities pk satisfies the Porter-Thomas distribution (see Supplementary Note 4).Fig. 2 Observation of diffusive transport.

a Experimental verification of preparing the states via the time evolution of participation entropy. Here, we chose QR = {Q1,↑, Q2,↑, …, Q12,↑, Q2,↓, Q3,↓, …, Q12,↓} with total 23 qubits. The inset of (a) shows the corresponding quantum circuit. The dotted horizontal line represents the participation entropy for Haar-random states, i.e., SPET≃15.519. b Experimental results of the autocorrelation function C1,1(t) for the qubit ladder with L = 12, which are measured by performing the quantum circuit shown in Fig. 1c, d. Here, we consider the state generated from U^R(tR) with tR = 200 ns, which is approximate to a Haar-random state. Markers are experimental data. The solid line is the numerical simulation of the correlation function C1,1 at infinite temperature. The dashed line represents a power-law decay t−1/2. Error bars represent the standard deviation.

In Fig. 2b, we show the dynamics of the autocorrelation function C1,1 measured via the quantum circuit in Fig. 1c with tR = 200 ns. The experimental data satisfies C1,1 ∝ t−α, with a transport exponent α ≃ 0.5067, estimated by fitting the data in the time window t ∈ [50 ns, 200 ns]. Our experiments clearly show that spin diffusively transports on the qubit ladder H^I(1), and demonstrate that the analog quantum circuit U^R(tR) with tR = 200 ns can provide sufficient randomness to measure the autocorrelation function defined in Eq. (2) and probe infinite-temperature spin transport. We also discuss the influence of tR in Supplementary Note 4, numerically showing that the results of C1,1 do not substantially change for longer tR > 200 ns. Moreover, in Supplementary Note 4, we show that for a shortly evolved time tR ≃ 15 ns, the values of the observable defined in Eq. (4) are incompatible with the infinite-temperature autocorrelation functions. Given that the chosen initial state for generating the Haar-random state exhibits a high effective temperature associated with the Hamiltonian H^R, the state would asymptotically converge to the Haar-random state with a sufficiently extended tR. However, with tR ≃ 15 ns, the time scale is too small to get rid of the coherence, and the value of SPE for the state ψR is much smaller than the SPET (see Fig. 2a), suggesting that ψR with tR ≃ 15 ns is far away from the Haar-random state, and cannot be employed to measure the infinite-temperature autocorrelation function (2). In the following, we fix tR = 200 ns, and study spin transport in other systems with ergodicity breaking.

Subdiffusive transport with ergodicity breaking

After demonstrating that the quantum circuit shown in Fig. 1c can be employed to measure the infinite-temperature autocorrelation function C1,1, we study spin transport on the superconducting qubit ladder with the disorder, whose effective Hamiltonian can be written as H^D=H^I+∑m∈{↑,↓}∑j=1Lwj,mσ^j,m+σ^j,m−, with wj,m drawn from a uniform distribution [ − W, W], and W is the strength of disorder. For each disorder strength, we consider 10 disorder realizations and plot the dynamics of averaged C1,1 with different W are plotted in Fig. 3a. With the increasing of W, and as the system approaches the MBL transition, C1,1 decays more slowly. Moreover, the oscillation in the dynamics of C1,1 becomes more obvious with larger W, which is related to the presence of local integrals of motion in the deep many-body localized phase58.Fig. 3 Subdiffusive transport on the superconducting qubit ladder with disorder.

a The time evolution of autocorrelation function C1,1(t) for the qubit ladder with L = 12 and different values of disorder strength W, ranging from W/2π = 35 MHz (W/J∥¯≃0.5) to W/2π = 70 MHz (W/J∥¯≃9.6). Markers (lines) are experimental (numerical) data. b Transport exponent α as a function of W obtained from fitting the data of C1,1(t). Error bars (experimental data) and shaded regions (numerical data) represent the standard deviation.

We then fit both the experimental and numerical data with the time window t ∈ [50 ns, 200 ns] by adopting the power-law decay C1,1 ∝ t−α. As shown in Fig. 3b, we observe an anomalous subdiffusive region with the transport exponent α < 1/2. For the strength of disorder W/2π ≳ 50 MHz, the transport exponent α ~ 10−2 indicates the freezing of spin transport and the onset of MBL on the 24-qubit system2. Here, we emphasize that the estimated transition point between the subdiffusive regime and MBL is a lower bound since, with longer evolved time, the exponent α obtained from the power-law fitting becomes slightly larger (see Supplementary Note 6).

Next, we explore the transport properties on a tilted superconducting qubit ladder, which is subjected to the linear potential H^L=∑j=1LΔj∑m∈{↑,↓}σ^j,m+σ^j,m−, with Δ = 2WS/(L − 1) being the slope of the linear potential (see the tilted ladder in the inset of Fig. 4a). Thus, the effective Hamiltonian of the tilted superconducting qubit ladder can be written as H^T=H^I+H^L. Different from the aforementioned breakdown of ergodicity induced by the disorder, the non-ergodic behaviors induced by the linear potential arise from strong Hilbert-space fragmentation55–57. The ergodicity breaking in the disorder-free system H^T is known as the Stark MBL28,50–54.Fig. 4 Subdiffusive transport on the superconducting qubit ladder with linear potential.

a Time evolution of autocorrelation function C1,1(t) for the tilted qubit ladder with L = 12 and WS/2π≤20 MHz. b is similar to (a) but for the data with WS/2π≥24 MHz. Markers (lines) are experimental (numerical) data. c Transport exponent α as a function of WS. For WS/2π≤20 MHz and WS/2π≥24 MHz, the exponent α is extracted from fitting the data of C1,1(t) with the time window t ∈ [50 ns, 200 ns] and t ∈ [100 ns, 400 ns], respectively. Error bars (experimental data) and shaded regions (numerical data) represent the standard deviation.

We employ the method based on the quantum circuit shown in Fig. 1c to measure the time evolution of the autocorrelation function C1,1 with different slopes of the linear potential. The results are presented in Fig. 4a, b. Similar to the system with the disorder, the dynamics of C1,1 still satisfies C1,1 ∝ t−α with α < 0.5, i.e., subdiffusive transport. Figure 4c displays the transport exponent α with different strengths of the linear potential, showing that α asymptotically drops as WS increases.

Two remarks are in order. First, by employing the same standard for the onset of MBL induced by disorder, i.e., α ~ 10−2, the results in Fig. 4c indicate that the Stark MBL on the tilted 24-qubit ladder occurs when WS/2π ≳ 80 MHz (Δ/2π ≳ 14.6 MHz). Second, on the ergodic side (WS/2π < 80 MHz and W/2π < 50 MHz for the tilted and disordered systems, respectively), the transport exponent α exhibits rapid decay with increasing WS up to WS/2π ≃ 20 MHz in the tilted system. Subsequently, as WS continues to increase, the decay of z becomes slower. In contrast, for the disordered system, α consistently decreases with increasing disordered strength W. We note that the impact of the emergence of dipole-moment conservation with increasing the slope of linear potential on the spin transport, and its distinction from the transport in disordered systems remains unclear and deserve further theoretical studies.

Discussion

Based on the novel protocol for simulating the infinite-temperature spin transport using the Haar-random state40, we have experimentally probed diffusive transport on a 24-qubit ladder-type programmable superconducting processor. Moreover, when the qubit ladder is subject to sufficiently strong disorder, we observe the signatures of subdiffusive transport, accompanied by the breakdown of ergodicity due to MBL.

It is worthwhile to emphasize that previous experimental studies of the Stark MBL mainly focus on the dynamics of imbalance50,59,60. Different from the disorder-induced MBL with a power-law decay of imbalance observed in the subdiffusive Griffith-like region61, for the Stark MBL, there is no experimental evidence for the power-law decay of imbalance50,59,60. Here, by measuring the infinite-temperature autocorrelation function, we provide solid experimental evidence for the subdiffusion in tilted systems, which is induced by the emergence of strong Hilbert-space fragmentation55–57. Theoretically, it has been suggested that for a thermodynamically large system, non-zero tilted potentials, i.e., Δ > 0, will lead to a subdiffusive transport with α ≃ 1/417,62. In finite-size systems, both results, as shown in Fig. 4 and the cold atom experiments on the tilted Fermi-Hubbard model63 demonstrate a crossover from the diffusive regime to the subdiffusive one. Investigating how this crossover scales with increasing system size is a further experimental task, which requires quantum simulators with a larger number of qubits.

Ensembles of Haar-random pure quantum states have several promising applications, including benchmarking quantum devices42,64 and demonstrating beyond-classical computation35–39. Our work displays a practical application of the randomly distributed quantum state, i.e., probing the infinite-temperature spin transport. In contrast to employing digital random circuits, where the number of imperfect two-qubit gates is proportional to the qubit number36–41, the scalable analog circuit adopted in our experiments can also generate multi-qubit Haar-random states useful for simulating hydrodynamics. The protocol employed in our work can be naturally extended to explore the non-trivial transport properties on other analog quantum simulators, including the Rydberg atoms42,65–67, quantum gas microscopes68,69, and the superconducting circuits with a central resonance bus, which enables long-range interactions21,70,71.

Methods

Derivation of Eq. 4

Here, we present the details of the deviation of Eq. (4), which is based on the typicality12,40,72. According to Eq. (2), cμ;ν=Tr[σ^μz(t)σ^νz]/D, with D = 2N. We define N^ν=(σ^νz+1)/2, and then cμ;ν=1DTr[σ^μz(t)N^ν]. By using N^ν=(N^ν)2, we have cμ;ν=1DTr[N^νσ^μz(t)N^ν]. We note that N^ν is an operator which projects the state of the ν-th qubit to the state 0.

According to the typicality12,40,72, the trace of an operator O^ can be approximated as the expectation value averaged by the pure Haar-random state r, i.e.,5 1DTr[O^]=⟨r∣O^∣r⟩+O(2−N/2),

with N being the number of qubits. It indicates that the infinite-temperature expectation value Tr[O^]/D can be better estimated by the expectation value for the Haar-random state ⟨r∣O^∣r⟩. Thus, cμ;ν≃⟨r∣N^νσ^μz(t)N^ν∣r⟩=⟨ψνR(t)∣σ^μz∣ψνR(t)⟩ for multi-qubit systems. Based on the definition of the projector N^ν,N^νr is a Haar-random state for the whole system except for the ν-th qubit, and in the experiment, only a (N − 1)-qubit Haar-random state is required.

Numerical simulations

Here, we present the details of the numerical simulations. We calculate the unitary time evolution ψ(t+Δt)=e−iH^Δtψ(t) by employing the Krylov method49. The Krylov subspace is panned by the vectors defined as {ψ(t),H^ψ(t),H^2ψ(t),…,H^(m−1)ψ(t)}. Then, the Hamiltonian H^ in the Krylov subspace becomes a m-dimensional matrix Hm=Km†HKm, where H denotes the Hamiltonian H^ in the matrix form, and Km is the matrix whose columns contain the orthonormal basis vectors of the Krylov space. Finally, the unitary time evolution can be approximately simulated in the Krylov subspace as ψ(t+Δt)≃Km†e−iHmΔtKmψ(t). In our numerical simulations, the dimension of the Krylov subspace m is adaptively adjusted from m = 6 to 30, making sure the numerical errors are smaller than 10−14.

For the numerical simulation of the U^R(tR)=e−iH^dtR in Fig. 1c, based on the experimental data of the XY drive, the parameters in H^d are Ωj,m/2π = 10.4 ± 1.6 MHz, and ϕj,m ∈ [ − π/10, π/10].

Details of generating Haar-random states

In this section, we present more details about the generation of faithful Haar-random states. The analog quantum circuit employed to generate Haar-random states is U^R=exp[−i(H^I+H^d)t], where H^I is given by Eq. (1) and H^d=∑m∈{↑,↓}∑j=1LΩj,m(e−iϕj,mσ^j,m++eiϕj,mσ^j,m−)/2 is the drive Hamiltonian.

Here, we first numerically study the influence of the driving amplitude Ωj,m. For convenience, we consider ϕj,m = 0, and isotropic driving amplitude, i.e., Ω = Ωj,m for all (j, m). We chose QR = {Q1,↑, Q2,↑, …, Q12,↑, Q2,↓, Q3,↓, …, Q12,↓} with total 23 qubits. The dynamics of participation entropy SPE for different values of Ω are plotted in Fig. 5a, and the values of SPE with the evolved time t = 200 ns and 1000 ns are displayed in Fig. 5b. It is seen that for small Ω, the growth of SPE is slow and with increasing Ω, it becomes more rapid. In this experiment, we chose Ω¯/J∥¯≃1.4 because the participation entropy can achieve SPET with a relatively short evolved time t ≃ 200 ns. As Ω further increases, the time when SPET is reached does not significantly become shorter. Based on the above discussions, Ω¯/J∥¯≃1.4 is an appropriate choice of the driving amplitude.Fig. 5 Impact of driving amplitude and phases of microwave pulse on the generation of Haar-random states.

a The time evolution of the participation entropy SPE for different driving amplitude Ω. b The value of SPE at two evolved times t = 200 ns and 1000 ns, as a function of Ω. c The dynamics of SPE with the phases of the microwave pulse drawn from [ − π, π]. Here, we present the numerical data of 5 different samples of the phases. The inset shows the dynamics in a shorter time interval. The horizontal dashed line represents the participation entropy for Haar-random states SPET≃15.519.

Next, we numerically study the influence of the randomness for the phases of driving microwave pulse ϕj,m. In this experiment, by using the correction of crosstalk, the randomness of the phases is small, i.e., ϕj,m ∈ [ − π/10, π/10]. Here, we consider the phases with large randomness, i.e., ϕj,m ∈ [ − π, π]. The numerical results for the time evolution of SPE with 5 samples of ϕj,m are plotted in Fig. 5c. With ϕj,m ∈ [ − π, π], the participation entropy can still tend to SPET around 200 ns. Only the short-time behaviors are slightly different from each other for the 5 samples (see the inset of Fig. 5c).

Supplementary information

Supplementary Information

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Supplementary information

The online version contains supplementary material available at 10.1038/s41467-024-52082-2.

Acknowledgements

We thank Hai-Long Shi and H. S. Yan for their helpful discussions. Z.X., D.Z., K.X., and H.F. are supported by the Beijing Natural Science Foundation (Grant No. Z200009), National Natural Science Foundation of China (Grants Nos. 92265207, T2121001, 12122504, 12247168, 11934018, T2322030), Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0301800), Beijing Nova Program (Nos. 20220484121, 2022000216). Y.-H.S. acknowledges the support of the Postdoctoral Fellowship Program of CPSF (Grant No. GZB20240815). Z.-A.W. acknowledges the support of the China Postdoctoral Science Foundation (Grant No. 2022TQ0036).

Author contributions

H.F. supervised the project. Z.-H.S. proposed the idea. Y.-H.S. conducted the experiment with the help of K.H. and K.X. Z.-H.S., Y.-Y.W., and Y.-H.S. performed the numerical simulations. Z.X. and D.Z. fabricated the ladder-type sample. X.S., G.X., and H.Y. provided the Josephson parametric amplifiers. W.-G.M., H.-T.L., K.Z., J.-C.S., G.-H.L., Z.-Y.M., J.-C.Z., H.L., and C.-T.C. helped the experimental setup. Z.-A.W., Y.-R.Z., J.W., K.X., and H.F. discussed and commented on the manuscript. Z.-H.S., Y.-H.S., Y.-Y.W., Y.-R.Z., and H.F. co-wrote the manuscript. All authors contributed to the discussions of the results and development of the manuscript.

Peer review

Peer review information

Nature Communications thanks Amir Karamlou, and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The authors declare that the data supporting the findings of this study are available within the paper and its Supplementary Information files. Should any raw data files be needed in another format, they are available from the corresponding author upon reasonable request. Source data are provided in this paper.

Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

These authors contributed equally: Yun-Hao Shi, Zheng-Hang Sun, Yong-Yi Wang.
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