Abstract
Understanding how hydrodynamic behaviour emerges from the unitary evolution of the manyparticle Schrödinger equation is a central goal of nonequilibrium statistical mechanics. In this work we implement a digital simulation of the discrete time quantum dynamics of a spin\(\frac{1}{2}\) XXZ spin chain on a noisy nearterm quantum device, and we extract the high temperature transport exponent at the isotropic point. We simulate the temporal decay of the relevant spin correlation function at high temperature using a pseudorandom state generated by a random circuit that is specifically tailored to the ibmqmontreal 27 qubit device. The resulting output is a spin excitation on a homogeneous background on a 21 qubit chain on the device. From the subsequent discrete time dynamics on the device we are able to extract an anomalous superdiffusive exponent consistent with the conjectured KardarParisiZhang (KPZ) scaling at the isotropic point. Furthermore we simulate the restoration of spin diffusion with the application of an integrability breaking potential.
Introduction
The idea that quantum dynamics of manybody physics is better simulated by controllable quantum systems was put forward by Richard Feynman 40 years ago^{1}. This is known as quantum simulation^{2,3} and is expected to be one of the most promising short term goals of near term quantum computing devices^{4} with inevitable applications in diverse areas ranging from quantum chemistry^{5,6,7} and material science^{8} to high energy physics^{9}. Quantum simulators currently come in two different flavours: analogue and digital^{10,11,12}. In an analogue simulator a purpose built controllable quantum manybody system is prepared in the laboratory with the ability to mimic a specific model Hamiltonian of interest. In a digital simulator the quantum dynamics is mapped to a series of discrete time gates that are used to directly manipulate the information encoded in the quantum state^{2}.
While analogue simulators are built with a specific model in mind, digital simulation offers the possibility to programme different Hamiltonian models so that a wide range of quantum dynamics is, in principle, accessible on the same device. The possibility of universal simulation of manybody quantum dynamics afforded by digital quantum simulation is a tantalising one. In reality, however, the current devices are still some distance from this goal with noisy gate operations and readout. Ultimately, significant progress in error correcting techniques is needed^{4}. In fact it has been on analogue devices where the most significant progress has been made in simulating manybody dynamics^{12}. However, recent progress in error mitigation techniques for digital devices has brought us closer to getting quantitative results from noisy simulations^{13,14,15}.
One dimensional interacting quantum spin systems are perhaps the simplest nontrivial models used in the field of manybody physics. Despite the obvious shortcomings on noisy nearterm quantum devices, there have been several interesting digital simulations^{16,17,18,19,20} which are restricted to either small systems or short times. These simulations can be viewed as important benchmarks of device capability. In this work we show how noisy nearterm quantum devices can be used to shed important light on a research topic which is at the forefront of research in lowdimensional quantum spin dynamics. The issue we address concerns the nature of the emergent high temperature anomalous hydrodynamics of the spin\(\frac{1}{2}\) XXZ spin chain at the isotropic point^{21}.
How macroscopic hydrodynamic behaviour emerges from underlying microscopic physics is a question that has been at the forefront of physics for 200 years^{22,23}. This research continues today in quantum manybody dynamics where the finite temperature transport properties of quantum spin systems is under significant analytical and numerical scrutiny^{24,25,26}. A recent development was the discovery of hightemperature spin superdiffusion at the isotropic point of the spin\(\frac{1}{2}\) XXZ model^{27} using an open systems approach. In this work the nonequilibrium steady state was found to have a current scaling \(\langle \hat{J}\rangle \propto 1/\sqrt{L}\) consistent with a space time scaling x ∝ t^{1/ν} with ν = 3/2. A numerical study of the infinite temperature spin auto correlation functions at the isotropic point^{28} has lead to the conjecture that the dynamics is in the KPZ universality class^{29} and further numerical work^{30} has shown the survival of the associated anomalous scaling of the spinspin autocorrelation functions at finite temperatures. There is still no clear consensus on the exact conditions for the emergence of this universal behaviour. Integrability is conjectured to be central in the emergence of this scaling and progress in incorporating anomalous diffusion in the context of generalised hydrodynamics^{21,31,32} has been made. The predicted superdiffusive exponent has been observed in a recent experimental study of neutron scattering off KCuF_{3} which realises an almost ideal XXZ spin chain^{33}. Furthermore, the scaling was recently confirmed in two analogue simulations of spin chains in both ultracold atoms^{34} and in polariton condensates^{35}.
In this work we perform a digital quantum simulation, of the discrete time dynamics, at the isotropic point of the XXZ model. It was recently discovered that the Trotterised version of the XXZ model is also integrable^{36} and the KPZ scaling at the isotropic point remains^{28,37}. This has the distinct advantage on a near term device of being able to simulate for longer times without having to worry about Trotter errors that plague continuous time simulation. We extract the high temperature correlation function following a recent proposal by Richter and Pal^{38,39} that suggests using specially tailored pseudorandom states which are generated from a relatively shallowdepth circuit^{40,41}. The discrete time dynamics of the spin autocorrelation function is then simulated. We apply a zero noise error mitigation strategy (see Supplementary Method 2 for a discussion on this) and remarkably show that the KPZ anomalous exponent can be extracted for over two decades of time evolution. Furthermore we show that the scaling is independent of the time period of the Trotter step and observe the restoration of diffusion, signalled by the emergence of the exponent ν = 2, when an integrability breaking staggered field perturbation is applied.
Results
Classical discrete time results
We first demonstrate, using classical simulations^{42}, that the transport exponents at the isotropic point for both the (a) clean and (b) staggered field discrete time models are independent of step size. In Fig. 1, we do a first order trotter decomposition of the clean and staggered field models with various timesteps. The power law scaling, in both models, is found to be independent of the time step for the steps chosen. In the insets we show the oscillations of the exponent \(\alpha =\frac{d\ln C(t)}{d\ln n}\) around the expected values (2/3 for the clean model and 1/2 for the model with staggered fields) for each model in the insets, where n is the number of Trotter steps. We avoid timesteps near π as the transport behaviour changes drastically due to manybody resonances^{42}.
Quantum discrete time results
We now come to the key finding of our work: the digital simulation on a real near term quantum computer. In Fig. 2, we show our results for the spin autocorrelation function simulated on ibmqmontreal. We have found that the optimal timestep for our simulations is τ = 4J^{−1}^{42}. In (a), we simulate the clean model, while in (b) we add the staggered field. The green lines show the results on the quantum simulator using a first order Trotter decomposition. Remarkably our results in both the integrable and nonintegrable case track the classical simulation well up to two decades in time evolution. This timescale is sufficient to see the emergence of hydrodynamic scaling. The error bars from sampling noise are negligible here compared to the device error, so we omit them. This is the main result of our work.
In order to increase the number of data points for our power law fit we have employed the concept of weaving^{42}. This allows us to look at more data points in time. The idea is to artificially increase our time resolution in our study of the the floquet unitary \({{{\mathcal{U}}}}(n\tau )\)^{42}. This is done by adding smaller \({{{\mathcal{U}}}}(\tau ^{\prime} \,<\, \tau )\) at the start of the circuit as a modified initial condition, and then weaving the evolution of this new initial state (shifted slightly in time) with the original evolution. We add weaves of 1J^{−1}, 1.5J^{−1}, 2J^{−1}. Furthermore, in obtaining these results, we employ a form of error mitigation known as ‘zero noise extrapolation’, or zne^{13}. However, we do not find that it significantly helps at these time scales for our first order Trotter simulation^{42}.
To extract the power law behaviour of the results from the quantum simulations, we analyse the intersection of two regimes in time: (1) Where the power law scaling is present in the classical results, and 2) where the quantum results have little error compared to the classical results. We then fit a power law to the quantum results via least squares. For panel (a) with the clean model, we get that α ≈ − 0.644, and for panel (b) with the staggered external field we get that α ≈ − 0.505. These have relative errors ~3.40% and ~1.00% compared to the expected scalings of −2/3 and −1/2, respectively.
IBM’s quantum devices are calibrated regularly. When these results were collected, the readout error of qubit 0 was 2.26%, while the average error of all the CNOTs (excluding an outlier at 18.4%) used in the simulation was 1.12%, with a standard deviation of 0.52%.
Discussion
We have provided strong evidence that KPZ scaling and the restoration of diffusion through explicit integrability breaking can be simulated digitally on a near term device. Our work is inspired by the proposal of Richter and Pal^{38} which exploits a pseudorandom state as a starting point for the simulation (see Supplementary Discussion 2 for a comparison to other methods). It is remarkable that our digital quantum simulation is able to follow closely the classical simulation to over two decades in time evolution. There are several features of this simulation which are worth pointing out. First of all the nature of the initial state appears to be extremely useful for the extraction of infinite temperature transport exponents on current quantum hardware. The precise interplay between noise channels and such pseudotypical states merits future detailed investigations. Since these states are locally equivalent to the identity, it is plausible that they offer a special resilience to unital channels such as dephasing. We have confirmed, on hardware, the suggestion^{38} that the hydrodynamic scaling is accessible despite inevitable device noise. Secondly and most importantly, the key feature of our simulations is that we work with the discrete time model and this allows us to simulate long times without Trotter error^{28,36}. To our knowledge this is the first extraction of transport exponents of an interacting quantum system on a digital quantum device. Our findings are consistent with recent experiments in a variety of physical platforms^{33,34,35,43}. As hardware improves further and the number of good device qubits increase, we hope that our work will inspire further work on high temperature transport of nonintegrable and integrable quantum manybody models in regimes not accessible to classical numerics.
Methods
Initial state preparation
All the quantum simulations in this paper were performed on the ibmq montreal 27 qubit device based on coupled transmons. This machine was recently benchmarked to have a quantum volume of 128. The connectivity of the device is shown in Fig. 3a and we will use the 21 qubits which are shown in orange for our dynamical simulations. Our first task, following the suggestion of Richter and Pal^{38} is to generate a pseudorandom state state on the device leaving all but one qubit untouched (q_{0}).
The randomisation procedure is split up into two subroutines; the single qubit gate routine, and the entangling routine. A layer of the procedure is made up of a single qubit step, followed by an entangling step. The single qubit gate routine is as follows:

1.
At layer 1, for each qubit q_{j}, apply \({G}_{1}^{j}\), which is chosen randomly from the set of gates {X^{1/2}, Y^{1/2}, T}

2.
At layer n > 1, for each qubit q_{j} apply \({G}_{n}^{j}\), which is chosen randomly from the set \(\{{X}^{1/2},{Y}^{1/2},T\}\backslash {G}_{n1}^{j}\)
Between each single qubit step, we carry out an entangling step. This consists of applying one of two patterns of CX gates across the device. The choice of pattern in alternated between patterns ‘A’ and ‘B’ (shown in Fig. 3b) at each step. The randomisation procedure is performed over multiple layers until the state is deemed sufficiently random. The number of layers that are needed is estimated from a classical simulation of the time evolution of the bipartite entanglement of the random circuit. The results of the classical simulation are shown in Fig. 3c, where we show the half chain von Neumann entropy as a function of the number of layers in our preparation step. We see that already a modest number of layers is enough to saturate the Page value^{44}. Figure 3d shows the spin density profile of the final state, on the actual hardware following one sampling of the random circuit. The data was extracted by performing 30,000 shots after one sampling of the circuit.
In this work, we will be interested in performing dynamical quantum simulation of spin spin autocorrelation functions, which take the form
where the trace is over the entire Hilbert space. Following the proposal of Richter and Pal^{38,39}, we will use the output of our state preparation circuit in the evaluation of this object. Let us assume for a moment that the output state of the entire register would be \(\left\vert {\psi }_{R,0}\right\rangle =\left\vert 0\right\rangle \left\vert {\psi }_{R}\right\rangle\) with \(\left\vert {\psi }_{R}\right\rangle ={\sum }_{n}{c}_{n}\left\vert n\right\rangle\), where the expansion is over the entire computational basis. If c_{n} are Gaussian random numbers with zero mean (i.e the state is drawn randomly from the unitarity invariant Haar measure) then one can approximate the correlation functions by (for details, see ref. ^{38})
This typicality approach is routinely used to evaluate the time evolution of observables in classical simulations^{24,40,45,46,47,48}. Pseudorandom states can be now generated on noisy nearterm quantum devices with relatively shallow circuits^{41}. The state preparation procedure leads to deviations from a Haar random state. However, as argued in refs. ^{38,40} the exact distribution of the coefficients of the states can deviate from Gaussian and still the same result holds^{40}. A key finding of ref. ^{38} is that the state which is output after the initial randomisation phase is robust to modelled device noise. The main purpose of this paper is to use this protocol in order to extract the decay of the spin autocorrelation function on a current quantum hardware.
Discrete time dynamics
The spin\(\frac{1}{2}\) XXZ Hamiltonian which will be the central focus of our simulation is
where \({S}_{\ell }^{\alpha }={\sigma }_{\ell }^{\alpha }/2\) is the spin operator acting on site ℓ and L is the number of sites. We use open boundary conditions, and focus on the isotropic point (Δ = 1). We define \({h}_{\ell ,\ell +1}=J\left({S}_{\ell }^{x}{S}_{\ell +1}^{x}+{S}_{\ell }^{y}{S}_{\ell +1}^{y}+{{\Delta }}{S}_{\ell }^{z}{S}_{\ell +1}^{z}\right)\) and group all of these twosite operators into two sums: H_{1} = ∑_{ℓ odd} h_{ℓ,ℓ+1}, H_{2} = ∑_{ℓ even} h_{ℓ,ℓ+1}. Note that the twosite operators in a given sum all act on disjoint pairs of sites. Therefore all operators commute with all other operators in their respective sums. We now look at the discrete time dynamics given by a Trotter step τ:
where \({U}_{jk}(\tau )={e}^{i{h}_{jk}\tau }\). The implementation of U_{jk}(τ) in a quantum circuit is given by Fig. 4^{49}.
Note that if we keep nτ fixed and take the limit τ → 0, we get that \({{{\mathcal{U}}}}(n\tau )\to {e}^{i{H}_{{{{\rm{XXZ}}}}}n\tau }\). However, we are less interested in this trotterized unitary as an approximation of the continuous time unitary for the XXZ model, but instead as a floquet system with kicking period τ. This model has Hamiltonian given by:
This Hamiltonian has been shown to also give rise to KPZ like scaling in discrete time^{28,37} and is particularly appealing due to the fact that there is no Trotter error. This was recently exploited in a digital simulation of the spin\(\frac{1}{2}\) XXZ chain in the gapped (Δ > 1) phase on the ibm kawasaki 27qubit machine in order to study the effect of noise on conserved charges^{50}. In our simulations we will also be interested in explicitly breaking the integrability of this model by the application of a staggered field which, at high temperatures, is expected to restore diffusion at the isotropic point. To implement this integrability breaking term, we continue like in the previous case, except as well as H_{1} and H_{2} we add the term \({H}_{3}=\frac{J}{2}\mathop{\sum }\nolimits_{\ell = 0}^{L1}{\left(1\right)}^{\ell }{S}_{\ell }^{z}\). The unitary for the discrete time evolution with the staggered field is given by:
where \({{{{\mathcal{U}}}}}_{3}(\tau )={\prod }_{j}{e}^{i{\sigma }_{j}^{z}{\theta }_{j}}\) is implemented as a collection of single qubit rotations. The effective Hamiltonian is now given by
where ϵ is a dummy variable used to ensure we apply the resulting unitaries in the correct order.
Data availability
Data used in this project are available on request.
Code availability
Code used for this project is available on request.
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Acknowledgements
We thank the QuSys group at TCD, A. Purkayastha and A. Silva for useful discussions. J.G. is funded by a Science Foundation IrelandRoyal Society University Research Fellowship, the European Research Council Starting Grant ODYSSEY (Grant Agreement No. 758403). This project was made possible through the TCDIBM predoctoral programme.
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N.K.: investigation, methodology, software, validation, formal analysis, visualisation, writing (original draft), writing (review and editing); N.F.R.: investigation, writing (review and editing); T.M.: investigation; S.Z.: supervision, writing (review and editing); J.G.: conceptualisation, supervision, resources, writing (original draft), writing (review and editing).
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Keenan, N., Robertson, N.F., Murphy, T. et al. Evidence of KardarParisiZhang scaling on a digital quantum simulator. npj Quantum Inf 9, 72 (2023). https://doi.org/10.1038/s41534023007424
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DOI: https://doi.org/10.1038/s41534023007424