Large-scale simulations of Floquet physics on near-term quantum computers

Abstract

Periodically driven quantum systems exhibit a diverse set of phenomena but are more challenging to simulate than their equilibrium counterparts. Here, we introduce the Quantum High-Frequency Floquet Simulation (QHiFFS) algorithm as a method to simulate fast-driven quantum systems on quantum hardware. Central to QHiFFS is the concept of a kick operator which transforms the system into a basis where the dynamics is governed by a time-independent effective Hamiltonian. This allows prior methods for time-independent simulation to be lifted to simulate Floquet systems. We use the periodically driven biaxial next-nearest neighbor Ising (BNNNI) model, a natural test bed for quantum frustrated magnetism and criticality, as a case study to illustrate our algorithm. We implemented a 20-qubit simulation of the driven two-dimensional BNNNI model on Quantinuum’s trapped ion quantum computer. Our error analysis shows that QHiFFS exhibits not only a cubic advantage in driving frequency ω but also a linear advantage in simulation time t compared to Trotterization.

Introduction

There has been a recent spike of interest in the dynamical control of quantum phases by driving strongly correlated quantum systems out of equilibrium. In particular, Floquet systems, that is quantum systems subjected to periodic driving, not only exhibit a rich set of physical phenomena, including exotic non-equilibrium topological phases of matter^1,2,3, but can also generate quantum phenomena on demand for technological applications^4,5,6,7,8. For example, time crystalline phases⁹, which break continuous time-translational symmetry have been demonstrated with trapped ions¹⁰, superconducting qubits^11,12,13 and diamond defects¹⁴. Similarly, periodic driving can lead to symmetry-protected edge modes^1,2,3, a phenomenon that has been realized in both trapped ions¹⁵ and superconducting¹⁶ platforms, and has potential applications for error correction. However, the strong correlations present in such systems make their modeling classically challenging.

While analog quantum simulators have been employed to study non-equilibrium dynamics^17,18,19, they suffer from an intrinsic model-dependency. Digital quantum simulation promises universal algorithms for non-equilibrium dynamics with exponentially less resources than known classical methods. In fact, this provided one of the original motivations for developing quantum hardware in the first place²⁰. While a distant dream for decades, with rapid technological progress, the possibility of realizing a quantum advantage for quantum simulation is now in sight^21,22. Nonetheless, error rates are yet to reach the error correction threshold for moderate-sized devices and hence quantum simulation algorithms designed for fault-tolerant devices remain challenging for the near future^23,24,25,26. This has prompted the search for quantum simulation algorithms suitable for implementation on near-term quantum devices^{25,27,28,29,30,31}. However, most of these studies focused on simulating time-independent systems and less attention has so far been paid to how to simulate periodically driven systems.

Standard Trotterization^{1,2,3,9,10,11,12,13,14,15,16,32,33,34,35,36}, where the time-ordered integral governing the dynamics is discretized into short time-steps (implemented via a Trotter approximation), fails to take advantage of the periodic structure of Floquet problems and thus acquires a substantial computational overhead.

Alternative proposals include qubitization of the truncated Floquet-Hilbert space³⁷, which require additional ancillae, truncated Taylor²⁶ and Dyson series³⁸ which are not guaranteed to be unitary or using variational optimization³⁴. The former require deep circuits, typically not accessible in the NISQ era and the latter is limited by (noise-induced) barren plateaus^{39,40,41,42,43,44} as well as other barriers to variational optimization on near-term quantum hardware^45,46. Correspondingly, digital quantum simulations of systems subject to a continuous driving field have thus far been limited to only a few qubits^35,36.

Here we draw motivation from prior work on classical methods for studying Floquet physics^{47,48,49,50,51,52,53} and methods for analog quantum simulation³², to develop a quantum algorithm for simulating periodically driven systems on digital quantum computers. Crucially this approach takes explicit advantage of the periodicity of the driving terms to reduce the resource requirements of the simulation. Key to our approach, is the use of a high-frequency approximation that performs a time-dependent basis transformation into a rotating frame^51,52. In effect we reverse the usual perspective taken by Floquet engineering, where analog simulation of time-dependent Hamiltonians is used to implement the effective time-independent dynamics. Here we instead take advantage of this correspondence to implement the effective time-independent dynamics on quantum hardware using a kick operator. Previously developed time-independent techniques can then be lifted and reused for the Floquet simulation^{23,24,25,26,27,28,29,30}. The Quantum High Frequency Floquet Simulation (QHiFFS - pronounced quiffs) algorithm thus opens up the potential of simulating fast-driven systems using substantially shorter-depth circuits than standard Trotterization.

We use QHiFFS to implement a 20-qubit, 230-gate two-dimensional simulation of the transversely driven biaxial next-nearest neighbor Ising (BNNNI) model on a trapped ion quantum computer. This implementation is, to the best of our knowledge, the largest digital 2D quantum simulation of a periodically driven system to date (see section “Case study: Two-dimensional biaxial next-nearest-neighborIsing (BNNNI) model in a transverse field”). We choose to focus on the BNNNI model because it is one of the most representative spin-exchange models containing both tunable frustration and tunable quantum fluctuations, creating competing quantum phases and criticality. Indeed, this behavior is successfully captured by our hardware implementation. We stress that this simulation would not have been feasible using standard Trotterization, which would have required a 50-fold deeper circuit (see Fig. 3).

Finally, we set QHiFFS on solid analytic foundations by providing an analysis on the scaling of the final simulation errors. In particular, we find that first order QHiFFS not only exhibits a cubic scaling advantage in driving frequency ω, but also a linear one in simulation time t compared to a second-order Trotter sequence. The error scaling in system size n remains linear for local models (see Error Analysis). Thus we see QHiFFS finding use both on moderate-sized error-prone near-term devices and in the fault-tolerant era.

Results

The QHiFFS algorithm

The QHiFFS algorithm simulates quantum systems governed by a Hamiltonian of the form

$$\hat{H}(t)={\hat{H}}_{0}+\hat{V}(t)\quad{\rm{with}}\quad\hat{V}(T+t)=\hat{V}(t)\,,$$

(1)

where ${\hat{H}}_{0}$ is the time independent non-driven Hamiltonian and $\hat{V}(T+t)$ is a periodic driving term. Our aim is to simulate real-time evolution under this Hamiltonian from some initial time t₀ to some final time t. That is, to implement the unitary

$${\hat{U}}_{{\rm{exact}}}({t}_{0},t)={\mathcal{T}}\exp \left(-i{\int\nolimits_{{t}_{0}}^{t}}\hat{H}(s)ds\right)\,,$$

(2)

where ${\mathcal{T}}$ is the time ordering operator.

As is often found in physics, choosing the right reference frame is key to a simple description of the dynamics of a system. To take a paradigmatic example—the motion of planets in our solar system appears rather complex in the earth’s reference frame but becomes much simpler if we transform into the frame of reference of the sun. Our algorithm, as sketched in Fig. 1, takes a similar approach⁵⁴:

• 1. Apply ${e}^{+i\hat{K}({t}_{0})}$ to kick an initial state $\left\vert \psi \right\rangle$ at time t₀ into a reference frame where the dynamics of the driven system is governed by an effective Hamiltonian ${\hat{H}}_{{\rm{eff}}}$.

• 2. Apply ${e}^{-i(t-{t}_{0}){\hat{H}}_{{\rm{eff}}}}$ to evolve the state from times t₀ to t in this new reference frame.

• 3. Apply ${e}^{-i\hat{K}(t)}$ to kick the system back into the original (lab) frame of reference.

Or, more compactly, the total simulation is of the form

$${\hat{U}}_{{\rm{exact}}}({t}_{0},t)={e}^{-i\hat{K}(t)}{e}^{-i(t-{t}_{0}){\hat{H}}_{{\rm{eff}}}}{e}^{+i\hat{K}({t}_{0})}\,.$$

(3)

Crucially, ${\hat{H}}_{{\rm{eff}}}$ is time-independent. Thus, one can use prior methods^{23,24,25,26,27,28,29,30} for time-independent quantum simulation for its implementation. Furthermore, in many cases, as discussed below, the kick operator, $\hat{K}(t)$, will take a simple form enabling it to be easily implemented on quantum hardware.

Fig. 1: Content overview.

(a) QHiFFS Algorithm Concept. To realize a time-dependent periodic time evolution, we transform from the standard basis to a (periodically) rotating frame using ${e}^{+i\hat{K}({t}_{0})}$. Evolution in this frame from t₀ to t is governed by the time-independent effective Hamiltonian ${\hat{H}}_{{\rm{eff}}}$. The adjoint of the kick operator ${e}^{-i\hat{K}(t)}$ is applied at the final simulation time t to kick back into the original frame. (b) Benchmark system. To demonstrate the advantage of QHiFFS, as compared to standard Trotterization, we implement the 20-qubit transversely driven 2D biaxial next-nearest neighbor Ising (BNNNI) model with periodic boundary conditions on Quantinuum’s H1 trapped ion quantum hardware (see Fig. 3). The BNNNI model, Eq. (6), consists of nearest-neighbor coupling with strength J and next-nearest-neighbor coupling with strength κJ. We suppose a 4 by 5 qubit lattice is periodically driven with strength h and at frequency ω.

The kick transformation and corresponding effective Hamiltonian for a periodically-driven system can be obtained by expanding perturbatively in the driving frequency ω = 2π/T ^50,51,52,55. As detailed in the Methods section and our Supplementary Note I, to ${\mathcal{O}}(\frac{1}{{\omega }^{2}})$ the truncated effective Hamiltonian ${\tilde{H}}_{{\rm{eff}}}$ and kick operator $\tilde{K}(t)$ are given by:

$${\tilde{H}}_{{\rm{eff}}}\,\approx\, {\hat{H}}_{0}+\frac{1}{\omega }\mathop{\sum }\limits_{j=1}^{\infty }\frac{1}{j}\left [ {\hat{V}}^{(j)},{\hat{V}}^{(-j)} \right]$$

(4)

$$\tilde{K}(t)\,\approx -\frac{i}{\omega }\mathop{\sum }\limits_{j=1}^{\infty }\frac{1}{j}\left({\hat{V}}^{(j)}{e}^{ij\omega t}-{\hat{V}}^{(-j)}{e}^{-ij\omega t}\right)\,.$$

(5)

with Fourier components ${\hat{V}}^{(j)}$ of the time-dependent potential $\hat{V}(t)=\mathop{\sum }\nolimits_{j = 1}^{\infty }{\hat{V}}^{(j)}{e}^{ij\omega t}+{\hat{V}}^{(-j)}{e}^{-ij\omega t}$. We note that the rotating wave approximation takes an analogous approach in the sense that under the rotating wave approximation evolution is governed by the $0^{\rm th}$ order approximation to the propagator in the interaction picture.

For analog quantum simulators, where Floquet engineering is commonly applied, introducing particular periodic drivings on native analog physical systems^17,18,19 is relatively easy and so Eq. (2) is used to approximate Eq. (3). For digital quantum simulators it is hard to implement continuous time-dependence due to their discretized gate-based ansatz. Hence, Eq. (3) is implemented on digital quantum hardware to effectively perform the continuous time-dependence given in Eq. (2).

The resources required to implement a simulation via the QHiFFS algorithm depend on the forms of $\hat{V}(t)$ and ${\hat{H}}_{0}$. Here we describe a physically motivated setting in which the final simulation is particularly simple. A more detailed account of the resources required for other cases is included in Supplementary Note I.B.

In what follows, we will focus on local driving (i.e., assume $\hat{V}(t)$ is one-local). This is a natural limit to consider since many driving phenomena, for example the optical driving of lattice systems, can be modeled in terms of local driving. In this case, $\hat{K}(t)$ is one-local to second order. Thus the initial and final kicks in Eq. (3) can be implemented using only single qubit rotations. If we further assume that ${\hat{V}}^{(j)}$ and ${\hat{V}}^{(-j)}$ commute for all j, which is the case here, then the first order correction to ${\hat{H}}_{{\rm{eff}}}$ vanishes and we have that ${\tilde{H}}_{{\rm{eff}}}={\hat{H}}_{0}+{\mathcal O}(\frac{1}{{\omega }^{2}})$.

This yields one of the main benefits of the QHiFFS algorithm, as it allows to reuse of previously developed time-independent techniques and lifts them to a Floquet Hamiltonian simulation^{25,26,27,28,29,30}. For example, if ${\hat{H}}_{0}$ is diagonal, or it can be analytically diagonalized, then one can simulate evolution under ${\hat{H}}_{{\rm{eff}}}$, and correspondingly evolution under $\hat{H}(t)$, using a fixed depth circuit. The transversely driven Ising model (independent of topology and dimension) falls into this category. More generally, one could use known methods for simulating the time-independent 1D XY-model²⁵, Bethe diagonalizable models⁵⁶ and translationally symmetric models³⁰, to simulate the effect of local driving on such systems.

We note that in classical simulations of Floquet systems, high frequency expansions are known and commonly used^{48,50,51,52,53} both for simulation and for Floquet engineering. In the latter case, Eq. (2) and Eq. (3) are viewed from the opposite perspective to the one we take here. Namely, the time-dependent driving term, $\hat{V}(t)$, can be designed to implement otherwise unfeasible complex many-body time-independent Hamiltonians, ${\hat{H}}_{{\rm{eff.}}}$. This perspective is also taken in the quantum computing community for gate calibration and optimization^57,58,59. However, on digital quantum computers the harder task is typically to simulate the time-ordered integral. Hence, this is our focus here. Namely, we propose implementing Eq. (3) on quantum hardware to simulate Floquet dynamics.

Case study: two-dimensional biaxial next-nearest-neighbor Ising (BNNNI) model in a transverse field

The transverse field BNNNI model is an example of an Ising model with next-nearest neighbor axial interactions^60,61,62. Interest in such models is largely motivated by the following key facts: Firstly, the zero-temperature critical behavior of the quantum spin Ising system in D-dimensions is connected to the classical critical behavior of the corresponding (D + 1)-dimensional classical system. Secondly, the results of such systems can provide insight into the general role of quantum fluctuations in quantum magnetism and also have direct relevance to experiments on numerous frustrated quantum magnets⁶³. While most work has focused on studying non-driven models, the non-equilibrium quantum dynamics of one-dimensional models has also been explored in a quench protocol⁶⁴.

Here we consider the two-dimensional Floquet driven transverse field BNNNI model⁶⁵ defined on a rectangular lattice with the Hamiltonian

$$\hat{H}(t)=J\left(-\sum\limits_{\langle i,j\rangle }{\hat{Z}}_{i}{\hat{Z}}_{j}+\kappa \sum\limits_{\langle \langle i,j\rangle \rangle }{\hat{Z}}_{i}{\hat{Z}}_{j}\right) -h\cos (\omega t)\sum\limits_{i}{\hat{X}}_{i}\,.$$

(6)

Here $\hat{Z},\hat{X}$ are Pauli operators, ∑_{〈i, j〉} denotes a sum over the nearest neighbors, and ∑_{〈〈i, j〉〉} is a sum over axial next-nearest neighbors as shown in Fig. 1(b). For simplicity, below we consider only the case of J, κ > 0, for which the model is a paradigmatic example of frustrated magnetism.

Classical simulations of two-dimensional frustrated magnets are very challenging. Even their equilibrium properties are subject to long-standing controversy⁶⁶. In general, classically simulating their non-equilibrium properties is even more challenging, which has stimulated widespread interest in their quantum simulation⁶⁷. These challenges also limit our understanding of physics of the transverse field BNNNI model. Its static phase diagram is well-established for special cases. The zero-field (h = 0) system exhibits a phase transition from a ferromagnetic phase to an antiphase at ${\kappa }_{c}=\frac{1}{2}$⁶⁵. In the ferromagnetic phase, all spins point in the same direction, while in the antiphase one finds periodic sequences of 2 spins pointing in the same direction, followed by 2 spins pointing in the opposite direction⁶⁵. The presence of a transverse magnetic field, $h\hat{X}$, introduces additional quantum fluctuations. For κ = 0, we recover the extensively studied transverse field quantum Ising model^68,69 with ferromagnetic order for h < h_c ≈ 3.04⁷⁰. At h_c the model undergoes a phase transition to a paramagnetic phase with all spins aligned in the $\hat{X}$ direction in the limit of h ≫ 1.

To probe dynamical properties and nonequilibrium phases of this model we study the next-nearest-neighbor correlation function averaged over all qubits on the lattice. That is, for an n_x × n_y lattice, we compute

$${C}_{\psi }(t):= \frac{1}{{n}_{x}{n}_{y}}\mathop{\sum }\limits_{{l}_{x}=0}^{{n}_{x}-1}\mathop{\sum }\limits_{{l}_{y}=0}^{{n}_{y}-1}\left\langle \psi \left | {\hat{U}(t)}^{\dagger }{\hat{Z}}_{{l}_{x},{l}_{y}}{\hat{Z}}_{{l}_{x},{l}_{y}+2}\hat{U}(t) \right | \psi \right\rangle$$

(7)

where $\left\vert \psi \right\rangle$ denotes an initial state of interest. To fit the lattice onto Quantinuum’s H1-1 quantum computer, in our hardware implementation, we consider a 4 × 5 lattice with periodic boundary conditions. The correlator $C_{\psi}(t)\, > \,0$ indicates that directions of the next-nearest axial neighbor spins are aligned as in the ferromagnetic phase, while $C_{\psi}(t)\, < \, 0$ indicates that they are anti-aligned (as in the antiphase). This quantity can serve to probe the field-induced phase change. In our case, we generally take $\left\vert \psi \right\rangle$ to be the ground state, denoted by $\left\vert {\psi }_{0}\right\rangle$ of $\hat{H}(t=0)$, setting t₀ = 0 in the following. The choice sets our simulations realistically connected with experimental situations. We note that since $C_{\psi}(t)$ is computed in Eq. (7) as a sum of expectation values of observables diagonal in the computational basis, we use a computational basis measurement to estimate all of them from each shot maximizing shot-efficiency of $C_{\psi}(t)$ estimation.

Numerical simulations

We performed numerical simulations to study how well the QHiFFS algorithm captures short range correlation functions of the periodically driven BNNNI model. Figure 2(a) shows the next-nearest-neighbor correlation function, ${C}_{{\psi }_{0}}(t)$, as a function of h and κ for different simulation times. We chose ω = 30, which ensures the applicability of the kick approximation for all considered parameter values of the model. Regions of ferromagnetic and antiferromagnetic correlation are indicated in red and blue respectively. We see that periodic driving further enriches the physics of the BNNNI model. In particular, when the BNNNI system is driven out of equilibrium by a high-frequency high-strength transverse field, a noticeable unstable ferromagnetic tendency can be induced out of an otherwise antiphase near the $\kappa =\frac{1}{2}$ quantum critical fan. While this instability persisted for all times we studied (i.e., until the limits of our first order QHiFFS simulation), we expect that this is a transient feature of a pre-thermalization phase. Physically, it can be understood as follows: In the non-driving case, the system stays in one of the three major equilibrium states (ferromagnetic, antiphase, or paramagnetic in the h-κ nonthermal parameter space. The low-h Floquet BNNNI system maintains the dominant feature of its original equilibrium magnetic state (ferromagnetic or antiphase). When the magnetic field amplitude becomes sufficiently strong such that the equilibrium state is in the paramagnetic phase, the driven Floquet system effectively experiences a sequence of large and low field. As such, the equilibrium paramagnetic regime is now replaced one carrying the feature of low-field ordered phases. A similar behavior has also recently been discovered in a heavy-fermion Kondo lattice in a classical simulation⁷¹. Although a determination of Floquet dynamical state requires a full evaluation of spatially dependent spin-spin correlator, our result suggests that driven quantum systems are an exciting research area for generating and manipulating quantum phases.

Fig. 2: Numerical simulations of non-equilibrium dynamics of the transverse field BNNNI model.

Two-dimensional slices of the next-nearest-neighbor correlation function, ${C}_{{\psi }_{0}}(t)$, as a function of h and κ for different simulation times t computed exactly (a) and computed via $1^{\rm{st}}$ order QHiFFS (b). Here J = 1 and ω = 30. The red line gives ${C}_{{\psi }_{0}}(t=0)\ne 0$ and the green/blue curves plot the exact/QHiFFS correlation function ${C}_{{\psi }_{0}}(t)$ as a function of time for the parameters h = 2 and κ = 0.25 also used in our hardware implementation (Fig. 3). In (c), we plot in logscale the error in the correlation function computed via QHiFFS - that is, the difference between the slices shown in (a) and (b). Videos corresponding to this figure can be found in the Supplementary Information.

Importantly, these complex changes are well-captured by the QHiFFS algorithm. This is shown in Fig. 2(b) where we plot the correlation function, ${C}_{{\psi }_{0}}(t)$, as computed using the QHiFFS algorithm. Indeed, as shown in Fig. 2(c), the errors in the correlation function (that is, the difference between the exact correlation function and the one computed via QHiFFS) are small. Specifically, the correlation function error on average (over the different h and κ values) is 0.0121 at t = 8.25 T, 0.0175 at t = 15.25 T and 0.0216 at t = 22.25 T. In the Supplementary Note III. A, we provide equivalent numerical results on the 4 × 6 and 5 × 5 transversely driven BNNNI model; indicating that these larger versions could have also worked on quantum hardware.

Hardware implementation

To demonstrate the suitability of the QHiFFS algorithm for real quantum hardware we used all 20 qubits of Quantinuum’s H1-1 trapped ion platform to simulate the 4 × 5 BNNNI model with periodic boundary conditions (see Fig. 1 and for the executed hardware circuit see Fig. 5). This was enabled by the all-to-all connectivity of the platform—on superconducting circuit devices (as compared to the trapped ion device that we used here), periodic boundary conditions on two-dimensional lattice models are typically hard to implement due to the restriction to nearest-neighbor gates.

Due to a finite sampling budget of approximately 30,000 shots, we needed to restrict ourselves to specific model parameters. We decided to implement two studies—the first a simulation of the correlation function as a function of time, the second a slice of the correlation function at a fixed time plotted versus the model parameters. Without loss of generality, we took J = 1. For our plot of the correlation function as a function of time, we then chose h = 2, κ = 0.25, and ω = 30 for which our numerical results indicate strong effects of periodic driving. For our study of model parameter dependence, we took t = 22.25 T and focused on the region 0.25 ≤ κ ≤ 0.475 and 2.0 ≤ h ≤ 2.6. The chosen parameters ensured that all model parameters are of a similar order of magnitude resulting in strong frustration and quantum fluctuations.

In Fig. 3(a), we plot as a function of time: the correlation function obtained from our quantum hardware implementation (black data points), the exact value of the correlation function (green line) and the value predicted from classical simulations of the QHiFFS algorithm (blue line). Furthermore, for the hardware results, we provide error bars quantifying their uncertainty due to a finite shot number as two standard deviations of the sample mean. Figure 3(b) shows the ratio of the number of 2-qubit gates that would have been required for a Trotter simulation as compared to a QHiFFS simulation for the same average simulation fidelity. This varied from 45-fold (at short times) to 53-fold (at long times) and thus the Trotter simulation was not feasible to be implemented on quantum hardware.

Fig. 3: Hardware results.

(a) Here we plot ${C}_{\tilde{{\psi }}_{0}}(t)$, Eq. (7), the next-nearest-neighbor correlation function, for our short depth approximation of the ground (initial) state $\left\vert \tilde{\psi }_{0}\right\rangle$ averaged over all qubit sites k, l. The green line gives the exact correlation function value, the blue line is the correlation function as computed using the first order QHiFFS ansatz, which is implementable on quantum hardware by a shallow circuit (compare Fig. 5) and the orange line a second order standard Trotterization using the same depth circuit as QHiFFS. The black data points are hardware results obtained from a 20-qubit Quantinuum H1-1 quantum computer. Their 2σ-error bars quantify uncertainty of the results due to finite shot number. (b) The red line compares the gate overhead required by Trotterization for the same (noise-free) algorithmic fidelity. That is, we plot R(t) = N_Trotter(t)/N_QHiFFS(t) where N_Trotter(t) (N_QHiFFS(t)) is the number of 2-qubit gates to simulate to time t with a second order Trotterization (the first order QHiFFS). To make R(t) a fair comparison metric, the time step for the Trotterization is set to ensure the algorithmic error is the same for both methods. (c) Here we plot the $C_{\psi}(t)$ dependence on the parameters h and κ for a zoomed in region of the total parameter space (i.) at time t = 22.25 T. (ii.) and (iii.) plot the exact correlation function values and those computed on quantum hardware respectively, with the error between them shown in (iv.). The raw data as well as the corresponding quantum circuits of the shown quantum hardware results in (a) and (c) are provided in the Supplementary Information.

In Fig. 3(c), we plot the zoomed in $C_{\psi}(t)$ dependence on the parameters at time t = 22.25 T computed exactly and as computed on quantum hardware. The hardware error is less than 0.021 for all data points. Our implementation successfully captures the noticeable unstable ferromagnetic tendency induced out of the antiphase at the $\kappa =\frac{1}{2}$ quantum critical fan. This suggests that QHiFFS could open up avenues to study novel driven quantum phases and criticality in strongly correlated electron systems. This could become particularly relevant experimentally when material systems with interesting high energy scales are subject to an electromagnetic field.

The expected values of the correlation function are largely within our shot-noise based error bars, implying that our simulation (despite the considerable circuit depth and qubit number) is predominantly shot-noise limited. Indeed, the simulation is also limited by the size of the available hardware—we chose to study a 20 qubit simulation not because this was the largest QHiFFS could handle, but rather because the Quantinuum H1-1 system was the largest all-to-all device to which we had access. Due to this shot noise limit, we expect the further exemplary models we studied in “Additional Models” and the Supplementary Note III would work similarly well.

Error analysis

To place the algorithm on solid conceptual foundations, as well as better understand the results of our numerical simulations and hardware implementations, we have conducted an analysis of QHiFFS algorithmic errors. To quantitatively judge the quality of a given Floquet simulation, we consider the average simulation infidelity over the uniform distribution of input states

$$\begin{array}{l}\epsilon ({\hat{U}}_{1},{\hat{U}}_{2}):= 1-{\mathbb{E}}{\left[{\left| \left\langle \psi | {\hat{U}}_{1}^{\dagger }{\hat{U}}_{2}| \psi \right\rangle \right| }^{2}\right]\Bigg| }_{\left\vert \psi \right\rangle \sim {{\rm{Haar}}}_{n}}\,.\end{array}$$

(8)

Here we provide a summary of our error analysis for the transversely driven BNNNI model. In Supplementary Notes IV and V we frame our error analysis more generally.

We start by analyzing the scaling of errors for the standard Trotterization approach. As shown in detail in Supplementary Notes III and V, there are three sources of error in this case, one coming from the discretization of the time-ordered integral, one coming from the non-commutativity of the Hamiltonian at different times and another (the standard time-independent Trotter error) coming from the non-commutativity of terms in the Hamiltonian at a fixed time. The dominant contribution in the high-frequency regime comes from discretization, while the errors from non-commutativity are comparably small. We find that the average infidelity for the transverse BNNNI model scales as

$$\epsilon ({\hat{U}}_{{\rm{exact}}}(t),{\hat{U}}_{{\rm{Trotter}}}(t))={\mathcal{O}}\left(\frac{n{h}^{2}{t}^{4}{\omega }^{2}}{8{m}^{2}}\right).$$

(9)

where m is the number of Trotter steps and n is the system size. Thus, in contrast to standard time-independent Trotterization, the error here is independent of ${\hat{H}}_{0}$ and scales as $t^4$ rather than $t^2$.

In comparison, we find that the long time error for the transversely driven BNNNI model when simulated by QHiFFS, can be approximated as

$$\epsilon ({\hat{U}}_{{\rm{exact}}}(t),{\hat{U}}_{{\rm{QHiFFS}}}(t))\,\approx\, \frac{281}{32}\frac{n\cdot {h}^{4}\cdot (1+{\kappa }^{2})\cdot {J}^{2}\cdot {t}^{2}}{{\omega }^{4}}\,.$$

(10)

The dominant contribution here is from the high-frequency truncation of the effective Hamiltonian. At short times, the truncation of the Kick approximation also contributes a significant oscillatory error to the averaged infidelity. However, the relative significance of this effect is washed out at longer times due to the contribution from the effective Hamiltonian which grows quadratically in time (compare Supplementary Note I). We also provide a rigorous upper bound for $\epsilon ({\hat{U}}_{{\rm{exact}}}(t),{\hat{U}}_{{\rm{QHiFFS}}}(t))$ in Supplementary Note IV and show that it admits the same scaling in t and ω as Eq. (10), namely is in ${\mathcal{O}}\left(\frac{{t}^{2}}{{\omega }^{4}}\right)$.

To support our analysis, we numerically computed the exact errors for the transversely driven BNNNI model. Specifically, in Fig. 4 we plot these errors, and our corresponding error estimates for the parameter regime we simulated on quantum hardware. Namely, a 4 × 5 two-dimensional transversely-driven BNNNI model with J = 1, κ = 0.25, h = 2, ω = 30 (in (a)) and t = 22.75T (in (b)). Overall, we find an excellent, nearly indistinguishable, agreement with our analysis.

Fig. 4: Error analysis.

Here we plot the average fidelity as a function of (a) simulation time t, (b) frequency ω and (c) system size n. Unless varied, the parameters equal those of our hardware implementation in Fig. 3; namely J = 1, κ = 0.25, h = 2, ω = 30 and t = 22.75 T. Overall, we find an excellent agreement between our analytical error estimate (black) and the exact numerical calculations (cyan). (Note, we plot here the full 1st-order expression for the error given in Eq. (10) as derived in Supplementary Note IV.B). The yellow line indicates the exact error when ${\hat{H}}_{{\rm{eff}}}$ is expanded to second order in ω and the kick operator is expanded to first order in ω. (d)–(f) show the numerically calculated overhead of standard Trotterization R in terms of simulation time t, frequency ω and system size n. For first order QHiFFS (cyan) we observe linear, quadratic (since t is chosen in units of $T=\frac{2\pi }{\omega }$) and constant scalings with t, ω and n respectively (shown in red) as predicted by Eq. (11).

Our error analysis further allows us to estimate how the circuit depth of a standard Trotterization needs to scale compared to QHiFFS to achieve the same fidelity. Let R be the ratio of the number of two qubit gates used by Trotterization compared to the number of two qubit gates used by QHiFFS. For the driven BNNNI model presented here R = m, the number of Trotter steps used by the Trotterization since the $1^{\rm{st}}$ order QHiFFS requires the same number of two qubit gates as one ($1^{\rm{st}}$ or $2^{\rm{nd}}$ order) Trotter step. Specifically, comparing our upper bound Eq. (9) and estimate Eq. (10) we would expect R to scale as

$$R \sim \frac{t{\omega }^{3}}{Jh\sqrt{1+{\kappa }^{2}}}\,,$$

(11)

assuming our bound and estimate are tight. It turns out that this predicted scaling does indeed well-reproduce the exact scaling for R computed numerically. Namely, the Trotter overhead grows linearly in time, as shown in Figs. 3(b) and 4(d), and cubically in ω, as shown in Fig. 4(e). We stress that this observation is independent of system size, n, as shown in Fig. 4(f).

The advantage of the QHiFFS ansatz can be further boosted by truncation at higher orders in k, (i.e., (1/ω)^k). If we truncate at order k, the ratio of circuit depths scales instead as

$$R \sim \frac{t{\omega }^{2}}{J\sqrt{1+{\kappa }^{2}}}{\left(\frac{\omega }{h}\right)}^{k}\,.$$

(12)

Such higher order expansions open up the simulation of longer times and larger system sizes without increasing the algorithmic approximation infidelity, giving a significant advantage to the QHiFFS approach.

Additional models

In Supplementary Note III, we provide additional numerics to support the broad applicability of QHiFFS. In order to discuss an entangling driving term that leads to faster growth of locality in ${\hat{H}}_{{\rm{eff.}}}$ as well as an effective time-independent Hamiltonian that needs to be trotterized, we analyze two more models each exhibiting at least one of these features. To be precise, we study a model that is derived from the BNNNI Hamiltonian, Eq. (6), by substituting the non-entangling driving term with the following 2-local term

$$\hat{V}(t)=-{J}_{X}\cos (\omega t)\sum\limits_{\langle i,j\rangle }{\hat{X}}_{i}{\hat{X}}_{j}.$$

(13)

Following extensive numerical studies in Supplementary Note III.B, we find no noteworthy difference in the QHiFFS performance of 2-local compared to 1-local driving. In Supplementary Fig. 2 we provide examples of the time-evolved correlators similar to Fig. 3 (a) and (b), and in Supplementary Fig. 3 and Supplementary Fig. 4 we show the algorithmic error scaling (equivalently to Fig. 4).

Further, we consider a driven XY model of the following form

$$\hat{H}(t)=-{J}_{X}\sum\limits_{\langle i,j\rangle }{\hat{X}}_{i}{\hat{X}}_{j}-{J}_{Y}\sum\limits_{\langle i,j\rangle }{\hat{Y}}_{i}{\hat{Y}}_{j} -{J}_{Z}\cos (\omega t)\sum\limits_{\langle i,j\rangle }{\hat{Z}}_{i}{\hat{Z}}_{j}.$$

(14)

In this case the truncated effective Hamiltonian ${\tilde{H}}_{{\rm{eff.}}}$ also needs to be trotterized. We show, that also in these cases QHiFFS yields an advantage over the Trotterization of the time-ordered exponential (see Supplementary Notes III and V). The reason for this improvement lies in the error sources of the Trotterization of time-ordered exponentials. Besides the error from non-zero commutation of terms in ${\tilde{H}}_{{\rm{eff.}}}$ form which both QHiFFS and naive Trotterization suffer, the latter also acquires errors additionally from non-zero commutation of $\hat{H}(t)$ at different times, errors from discretizing the integrals, as well as non-suppressed commutator errors between ${\hat{H}}_{0}$ and $\hat{V}(t)$; compare Eq. (19) (QHiFFS) and Eq. (21) (Trotter) as well as Supplementary Information Eqs. (79) and (80). Equivalently to the other two numerical examples, we give examples of (QHiFFS) time evolved observable in Supplementary Figs. 5 and 6 respectively, including an illustrative explanation why standard Trotterization breaks for single frequency driving within the first half period. Analogously, the algorithm scaling behavior is shown in Supplementary Fig. 7 matching our predictions in Supplementary Information Eq. (81) to Eq. (84). Qualitatively, we find that the QHiFFS error is then typically dominated by the BCH-error of the effective Hamiltonian; while QHiFFS maintains an error scaling advantage given by the avoided time-independent BCH-error and shorter Trotter sequence of ${\hat{H}}_{{\rm{eff.}}}$ compared to $\hat{H}(t)$ (e.g., see Supplementary Information Eq. (30) and compare Eq. (81) with Eq. (82)). We discuss this in more detail in the Supplementary Information.

Discussion

Here we have proposed the QHiFFS algorithm for simulating periodically driven systems on digital quantum computers. Central to the algorithm is the use of a kick operator to transform the problem into a frame of reference where the dynamics of the system is governed by a time-independent effective Hamiltonian. Thus, this method avoids the costly discretization of the time-ordered integral required by standard Trotterization and allows one to re-use previous time-independent Hamiltonian simulation methods^{25,26,27,28,29,30}. In parallel, QHiFFS does not require any form of variational optimization of the sort required by many other near-term simulation methods^27,28,29,30.

Our error scaling analysis demonstrates that QHiFFS outperforms standard Trotterization for high driving frequencies, ω. Specifically, at long simulation times, t, we see a linear advantage in t and at least cubic improvements in ω compared to standard Trotterization, for diagonalizable effective Hamiltonians. In parallel, the algorithmic errors of QHiFFS scale linearly in system size, n, for local models, and at worst quadratically. These favorable scalings again point towards QHiFFS’s suitability for larger-scale implementations.

To benchmark QHiFFS we performed a 20-qubit simulation of the periodically driven two-dimensional BNNNI model on Quantinuum’s 20-qubit H1-1 trapped ion platform. As shown in Fig. 3, the correlation function values computed on the quantum computer capture the true predicted values up to the precision allowed by shot noise (2σ = 1.49 ⋅ 10⁻²). In contrast, standard Trotterization (specifically a $2^{\rm{nd}}$ order Trotterization with the same algorithmic error as a $1^{\rm{st}}$ order QHiFFS) would have required approximately 50 times as many two-qubit gates, making the method completely unfeasible on current hardware due to gate errors and decoherence.

Future interesting applications of QHiFFS on quantum hardware could include the simulation of strongly correlated systems in the presence of linearly or circularly polarized electromagnetic field for the study of the emergence of unconventional superconductivity⁷², quantum spin liquids⁷³, and Kondo coherence collapse⁷⁴. Similarly, one could study even more complicated, approximately fast-forwardable Ising models, either by adding longer range interactions or by studying three spatial dimensions. We believe the latter would be viable with current gate fidelities, e.g., on Quantinuum’s trapped ion device, but (for a non-trivial simulation) would require more qubits.

In practice, QHiFFS shows to be suitable to simulate strongly correlated insulating systems, in which the charge degree of freedom is fully gapped out to a much larger energy scale while the spin becomes the only low-energy degree of freedom. In these systems, the typical exchange interaction is at an energy scale of 10 meV. As such, the frequency of an electromagnetic field can be easily chosen to be much bigger than the exchange interaction strength.

In special cases, namely those where the time-independent Hamiltonian ${\hat{H}}_{0}$ and the driving term $\hat{V}(t)$ are diagonal or efficiently diagonalizable, a fixed depth circuit can be used to simulate arbitrary times with an algorithmic error that grows only quadratically in time; compared to the power of 4 for standard Trotterization. Thus, in this limit, the QHiFFS algorithm provides a method for approximate fast-forwarding quantum simulations^27,28,29 that is highly suitable to near-term hardware.

We stress that while the simulation in such cases is particularly simple, and thus also simpler to simulate classically, QHiFFS could still be used to perform classically intractible simulations. This is because even the fixed depth circuits enabled by the kick approximation could be challenging to simulate classically. In general, apart from special cases like Clifford circuits or one-dimensional low entangled states⁷⁵, the classical cost of simulating quantum circuits grows exponentially with circuit depth. Therefore, one may expect that sufficiently deep and entangling fixed-depth QHiFFS circuits will be challenging to simulate classically in higher spatial dimensions. Given that these gate sequences are good approximations of the dynamics of quantum many-body systems that are forced out of equilibrium by a high-frequency drive, such strong entanglement generation is a natural expectation. Furthermore, even in the case where QHiFFS is classically simulable for some input states, it could still be used to implement classically intractable simulations for non-classically simulable initial states. Such states could plausibly be prepared via analog simulation strategies. This could provide one of the earliest avenues for physically interesting non-classically simulable implementations of QHiFFS on near-term hardware.

Longer term, we expect QHiFFS to find use in the fault-tolerant era as well as the near-term era. In the former case, one could use higher order expansions of the effective Hamiltonian and kick operators to achieve a higher accuracy, and implement the more complex resulting effective Hamiltonian using well-established fault-tolerant simulation methods²⁶.

Methods

Algorithmic methods

Here we provide an overview (for details see Supplementary Note I) on how to obtain an effective time-independent Hamiltonian H_eff for Floquet simulation. Using both Schrödinger equations (for $\hat{H}(t)$ and $\hat H_{\text{eff}}$) and a time dependent transformation ${e}^{-i\hat{K}(t)}$ one gets a defining relation for $\hat H_{\text{eff}}$,

$$\begin{array}{r}{\hat{H}}_{{\rm{eff}}}={e}^{i\hat{K}(t)}\hat{H}(t){e}^{-i\hat{K}(t)}+i\left(\frac{\partial {e}^{i\hat{K}(t)}}{\partial t}\right){e}^{-i\hat{K}(t)}\,.\end{array}$$

(15)

Using the Hadamard Lemma and $\frac{\partial }{\partial x}{e}^{\hat{A}(x)}=\mathop{\int}\nolimits_{0}^{1}{e}^{y\hat{A}(x)}\left(\frac{\partial }{\partial x}\hat{A}(x)\right){e}^{(1-y)\hat{A}(x)}dy$ this can be rewritten in nested commutator form:

$${\hat{H}}_{{\rm{eff}}}=\mathop{\sum }\limits_{q=0}^{\infty }\frac{{i}^{q}}{q!}\left[\hat{K}{(t)}_{(q)},\hat{H}(t)\right] -\mathop{\sum }\limits_{q=0}^{\infty }\frac{{i}^{q}}{(q+1)!}\left[\hat{K}{(t)}_{(q)},\frac{\partial \hat{K}(t)}{\partial t}\right]$$

(16)

with $\left [\hat{K}{(t)}_{(q)},\hat{H}(t) \right]=\left[\hat{K}(t),\left [\hat{K}{(t)}_{(q-1)},\hat{H}(t) \right] \right ]$. So far, no structure of the problem or approximation is used. In the case of periodic time-dependence, Floquet’s theorem implies one may use a high-frequency expansion for the kick operator and effective Hamiltonian,

$$\begin{array}{r}\hat{K}(t)=\mathop{\sum }\limits_{k=1}^{\infty }\frac{1}{{\omega }^{k}}{\hat{K}}^{(k)}(t)\quad{\rm{and}}\quad {\hat{H}}_{{\rm{eff}}}=\mathop{\sum }\limits_{k=0}^{\infty }\frac{1}{{\omega }^{k}}{\hat{H}}_{{\rm{eff}}}^{(k)}\,,\end{array}$$

(17)

and truncate at (small) finite order^51,52,55. We note that for a different time-dependence structure another ansatz for $\hat{K}(t)$ and ${\hat{H}}_{{\rm{eff}}}$ might lead to preferable gate sequences. Given a specific ansatz, such as the high-frequency expansion in Eq. (17), the truncated series can be determined explicitly by the following algorithm. Going through these steps for k_max = 1 gives Eqs. (4) and (5).

Algorithm

for deriving $\tilde{K}(t)$ and ${\tilde{H}}_{{\rm{eff}}}$

Simulation methods

The first order QHiFFS ansatz yielded a constant depth circuit to simulate the BNNNI model on quantum hardware as both $\exp (-it{\tilde{H}}_{{\rm{eff}}})$ and $\exp (\pm i\tilde{K}(t))$ were exactly implementable. Indeed, in the case of Quantinuum’s H1-1 trapped ion device, with native single qubit rotations and two-qubit gates $\exp (-i\frac{\theta }{2}\hat{Z}\hat{Z})$, the simulation was implemented exactly with a fixed circuit using the native gate set. This meant that the QHiFFS circuit, for any simulation time, had the same depth as a single standard low-order Trotter-Suzuki step. Thus, the final simulation time t affected the error from the high-frequency approximation, but not the error from hardware imperfections.

Initial state preparation for a physically motivated quantum simulation is a non-trivial task, in general. In our case, preparing the ground states exactly would require deep circuits but using classical compilation techniques we found an approximation of the ground state that still captured the same underlying physics and required a much shorter depth circuit to prepare. Our final circuits used 40 2-qubit gates to prepare an approximation of the ground state for different h and κ with an average overlap of 0.964 relative to the true ground state.

The complete quantum circuit for the full simulation consisted of 110 2-qubit gates and 120 single qubit gates (see Fig. 5) for all non-half-integer time steps. At half-integer time steps the circuit implementing the time evolution compiled to the identity and so the circuit consisted of only the 40 2-qubit gates required for state preparation. As an example, we provide the precise hardware quantum circuit for h = 2, κ = 0.25 and t = 22.25 T (compare Fig. 3) as well as the code to run it via Quantinuum’s API in the Supplementary Information.

Fig. 5: Quantum hardware circuit.

Native universal one-qubit gates are colored in green and native RZZ two-qubit gates are depicted in yellow. The first 4 two-qubit layers (40 gates) are used for approximate preparation of the BNNNI model ground-state $\left\vert {\psi }_{0}\right\rangle$ (average overlap ≈ 96.4 %). The remaining 7 two-qubit layers (70 gates) are used to implement $\exp (-i(t-{t}_{0}){\hat{H}}_{{\rm{eff}}})\approx \exp (-i(t-{t}_{0}){\hat{H}}_{0})$. Both $\hat{K}(t)$ and ${\hat{H}}_{0}$ can be implemented exactly up to ${\mathcal{O}}\left(\frac{1}{{\omega }^{2}}\right)$ (compare Eqs. (4) and (5)), which allows for constant depth approximate fast-forwarding.

Classical numerical simulations in Figs. 2–4 were performed using a state vector simulator. To simulate the exact time evolution, we trotterized it and decomposed the resulting evolution operator to single- and two-qubit unitaries enabling the usage of the state vector simulator. The Trotter step was determined using analytic bounds on the Trotterization error discussed in section Analytical methods.

Analytical methods

Upper bounds for QHiFFS and standard Trotter decomposition are derived using the estimate:

$$\left|\left| \mathop{\prod }\limits_{j=1}^{{n}_{j}}{e}^{i{\hat{A}}_{j}}-\mathop{\prod }\limits_{j=1}^{{n}_{j}}{e}^{i{\hat{B}}_{j}}\right|\right|\le \mathop{\sum }\limits_{j=1}^{{n}_{j}}\left|\left| {\hat{A}_{j}-{\hat{B}}_{j}}\right|\right|.$$

(18)

For the QHiFFS approximation (n_j = 3) this allows to derive a general upper bound by comparison of different orders (details see Supplementary Note IV):

$$\begin{array}{ll}\left\Vert {\hat{U}}_{{\rm{exact}}}({t}_{0},t)-{\tilde{U}}_{{\rm{QHiFFS}}}({t}_{0},t)\right\Vert &\le \left\Vert \hat{K}(t)-\tilde{K}(t)\right\Vert +| t-{t}_{0}| \left\Vert {\hat{H}}_{{\rm{eff}}}-{\tilde{H}}_{{\rm{eff}}}\right\Vert \\&+\,\left\Vert \hat{K}({t}_{0})-\tilde{K}({t}_{0})\right\Vert\\& ={\mathcal{O}}\left(\frac{| t-{t}_{0}| }{{\omega }^{k+1}}+\frac{1}{{\omega }^{k+1}}\right),\end{array}$$

(19)

where $\hat{\square }$ indicates exact expansions and $\tilde{\square }$ labels the truncation at 1/ω^k. This means we can upper-bound the QHiFFS error by simply adding the individual errors in both kick operators and the effective Hamiltonian. To derive an accurate estimate of the QHiFFS error, the norm $\left\vert \left\vert {\hat{U}}_{{\rm{exact}}}(t)-{\tilde{U}}_{{\rm{QHiFFS}}}(t)\right\vert \right\vert$ can be expanded in its trace form ${\text{Tr}}\,({\hat{U}}_{{\rm{exact}}}{(t)}^{\dagger }{\tilde{U}}_{{\rm{QHiFFS}}}(t))$ and in orders of $\frac{1}{\omega }$ instead of bounding it. To do this, the truncated (e.g., ${e}^{-i{\tilde{K}}(t)}$) and the non-truncated exponential (e.g., ${e}^{-i{{\hat{K}}(t)}}$) are merged via the identity

$${e}^{\hat{A}}{e}^{-\hat{A}+\chi }-{{1}}=\mathop{\sum }\limits_{j=0}^{\infty }\frac{1}{(j+1)!}\left[{\hat{A}}^{(j)},\chi \right]+{\mathcal{O}}({\chi }^{2}),$$

(20)

which is a combination of the Hadamard lemma, the Baker-Campbell-Hausdorff lemma and a general formula for the derivative of the exponential map. χ denotes the tail of the high-frequency expansion and is small compared to $\hat{A}$. The leading error term is hence determined by the linear contribution in χ (see Supplementary Note I for details). Explicitly doing this calculation for the BNNNI model gives Eq. (10).

To derive the standard Trotter error bound and estimate respectively, we can use Eq. (18) to upper bound the standard Trotterization error

$$\begin{array}{ll}\left\Vert {\hat{U}}_{{\rm{exact}}}({t}_{0},t)-{\hat{U}}_{{\rm{Trotter}}}({t}_{0},t)\right\Vert &\le \left\Vert {\mathcal{T}}{e}^{-i\mathop{\int}\nolimits_{{t}_{0}}^{t}\hat{H}(s)ds}-{e}^{-i\mathop{\int}\nolimits_{{t}_{0}}^{t}\hat{H}(s)ds}\right\Vert \\ &+\,\left\Vert {e}^{-i\mathop{\int}\nolimits_{{t}_{0}}^{t}\hat{H}(s)ds}-{\hat{U}}_{{\rm{discrete}}}({t}_{0},t)\right\Vert \\&+\,\left\Vert {\hat{U}}_{{\rm{discrete}}}({t}_{0},t)-{\hat{U}}_{{\rm{Trotter}}}({t}_{0},t)\right\Vert \end{array}$$

(21)

with ${\hat{U}}_{{\rm{discrete}}}({t}_{0},t)=\mathop{\prod }\nolimits_{r = 1}^{m}{e}^{i\delta t\hat{H}({t}_{0}+r\delta t)}$ and ${\hat{U}}_{{\rm{Trotter}}}({t}_{0},t)=\mathop{\prod }\nolimits_{r = 1}^{m}\mathop{\prod }\nolimits_{j = 1}^{{n}_{j}}{e}^{i\delta t{\hat{H}}_{j}({t}_{0}+r\delta t)}$. The first term describes errors due to non-commutativity of $\hat{H}(t)$ at different times, the second term gives the integral discretization error and the third term is standard non-time-dependent Trotter error.

For the BNNNI model the first and third term scale as ${\mathcal{O}}\left(\frac{{t}^{2}}{m}J(1+\kappa )h\right)$, which is why the dominant error term arises from the Riemann discretization of the time integral, which can be bounded by another application of Eq. (18),

$$\left\Vert {e}^{-i\mathop{\int}\nolimits_{{t}_{0}}^{t}\hat{H}(s)ds}-{\hat{U}}_{{\rm{discrete}}}({t}_{0},t)\right\Vert \le \left\Vert \mathop{\int}\nolimits_{{t}_{0}}^{t}\hat{H}(s)ds-\delta t\mathop{\sum }\limits_{r=1}^{m}\hat{H}({t}_{0}+r\delta t)\right\Vert\,.$$

(22)

Explicitly calculating this then gives the error estimate in Eq. (9) (compare Supplementary Note V).