# Quantum computing Floquet band structures

Benedikt Fauseweh<EMAIL_ADDRESS>Institute for Software Technology,
German Aerospace Center (DLR), Linder Höhe, 51147 Cologne, Germany  Jian-Xin
Zhu<EMAIL_ADDRESS>Theoretical Division, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA  Center for Integrated Nanotechnologies, Los
Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

###### Abstract

Quantum systems can be dynamically controlled using time-periodic external
fields, leading to the concept of Floquet engineering, with promising
technological applications. Computing Floquet band structures is harder than
only computing ground state properties or single time-dependent trajectories,
and scales exponentially with the Hilbert space dimension. Especially for
strongly correlated systems in the low frequency limit, classical approaches
based on truncation break down. Here, we present two quantum algorithms to
determine effective Floquet modes and band structures. We combine the defining
properties of Floquet modes in time and frequency domains with the
expressiveness of parameterized quantum circuits to overcome the limitations
of classical approaches. We benchmark our algorithms and provide an analysis
of the key properties relevant for near-term quantum hardware.

##  1 Introduction

Figure 1:  Graphical representation of both algorithms for the determination
of Floquet eigenstates and energies. a) Algorithm Fauseweh-Zhu-1. The
parameterized ansatz circuit $U_{\mathbf{\Theta}}\left|0\right\rangle$ is time
evolved over a full period and the overlap to its starting state is
calculated. Once the overlap is maximized, we compute the Floquet quasi-energy
using iterative quantum phase estimation. b) Algorithm Fauseweh-Zhu-2. The
Hilbert space is extended by Fourier expansion of the Hamiltonian. A
parameterized quantum circuit approximates the combined Floquet state
described by both physical and Fourier quantum numbers. Excited state VQE is
used to determine Floquet states and energies.

The interaction between the electromagnetic field and matter is one of the
basic principles, which is used to probe and manipulate electronic and
magnetic degrees of freedom. Quantum systems that are subjected to time-
periodic irradiation show intriguing phenomena such as light-induced surface
states [1], topological phases of matter [2, 3], high harmonic generation [4]
and light-induced superconductivity [5]. With advances in high-power ultra-
fast spectroscopy, non-linear phenomena, including multi-photon processes,
have opened up new prospects of dynamically controlling quantum dynamics,
engineering atomic [6], molecular [7, 8, 9] and solid-state [10, 11]
properties on demand. On the theoretical side the study of non-equilibrium
dynamics in highly entangled many-body systems is at the forefront of current
research [12, 13, 14, 15, 16, 17, 18, 19, 20]. Light-driven quantum systems
are described by time-dependent Hamiltonians $H(t)$, which include the
interaction between the external electromagnetic field and the electronic or
magnetic degrees of freedom.
If the field is periodic in time, i.e., $H(t)=H(t+T)$ for some period $T$, one
can use the concept of Floquet engineering to control the microscopic degrees
of freedom [21]. Floquet engineering is based on the Floquet theorem [22, 23],
which states that there exists a complete and orthogonal set of solutions
$\left|\Psi(t)\right\rangle$ to the time-dependent Schrödinger equation

$\displaystyle
i\frac{\mathrm{d}}{\mathrm{d}t}\left|\Psi(t)\right\rangle=H(t)\left|\Psi(t)\right\rangle,$
(1)

which have the form

$\displaystyle\left|\Psi_{\alpha}(t)\right\rangle=\left|\Phi_{\alpha}(t)\right\rangle
e^{-i\epsilon_{\alpha}t},\,\,\left|\Phi_{\alpha}(t+T)\right\rangle=\left|\Phi_{\alpha}(t)\right\rangle\;.$
(2)

Here we have chosen the reduced Planck constant $\hbar=1$,
$|\Phi_{\alpha}(t)\rangle$ is the Bloch amplitude periodic in time, and
$\epsilon_{\alpha}$ denotes the Floquet quasi-energy. The quasi-energies
depend on the eigenstates of the non-driven system as well as the specific
form of the driving field. They are uniquely defined up to the fundamental
frequency $\Omega=2\pi/T$. Changing the modulation of the driving field, such
as shape, intensity and frequency allows to modify the $\epsilon_{\alpha}$’s
and thereby the quasi-energy or Floquet band structure, inducing novel
topological phases not present in equilibrium [24].
While the Floquet theorem is a strong statement, computing the quasi-energy
band structure explicitly, depending on the microscopic degrees of freedom and
the time-dependence of the external field, remains a difficult problem due to
the exponentially scaling Hilbert space. Various classical techniques, such as
time-dependent dynamical mean-field theory (t-DMFT) [25, 26, 27, 28], time-
dependent density matrix renormalization group (t-DMRG) [29], kinetic
equations [30, 31], perturbative high-frequency expansions [32, 33, 34, 35,
36, 37, 38] and exact diagonalization [39, 40, 41] have been employed, but
unfortunately most are either not universally applicable or scale
exponentially in system size. Especially the computation of the whole quasi-
energy band structure requires more than a simple time-evolution.
Another problem in Floquet-engineering comes from the limited theoretical
modeling, in which only a few band model is considered and the perturbative
infinite frequency expansion can be applied. In real materials however the
situation is more difficult, as higher bands become important once the
frequency is sufficiently large to induce optical transitions.
In this work, we propose to use quantum computers to overcome the limitations
of previous classical approaches. We present two algorithms that in principle
can calculate all Floquet eigenstates and quasi-energies. Our quantum-
classical hybrid approach takes the limitations of modern Noisy Intermediate-
Scale Quantum (NISQ) hardware [42] into account and allows to either improve
accuracy by increasing the number of auxiliary qubits or by deepening the
quantum circuits. Both algorithms make use of the enhanced expressiveness [43]
of parameterized quantum circuits to find approximate Floquet eigenstates and
compute the corresponding quasi-energies. While the first algorithm works in
the original Hilbert space of the time-dependent problem, the second algorithm
makes use of the extended Floquet Hilbert space in frequency domain. After a
detailed explanation of the basic principles, we evaluate the performance of
both algorithms by investigating the simplest system that is not analytically
solvable: a spin-$\frac{1}{2}$ in a linear polarized periodic magnetic field.

##  2 First Algorithm

In the first algorithm, Fauseweh-Zhu-1, we use a parameterized quantum circuit
$U_{\mathbf{\Theta}}$, with variational parameters $\mathbf{\Theta}$, in
combination with the defining properties of the time evolution operator

$\displaystyle\left(H(t)-i\frac{\mathrm{d}}{\mathrm{d}t}\right)U(t,t_{0})=0,\quad
U(t_{0},t_{0})=\mathbb{1},$ (3)

to determine a good approximation for Floquet eigenstates. Without loss of
generality, we set $t_{0}=0$ and denote $U_{t}=U(t,0)$. The Floquet theorem
implies that for a full time period $t=T$ the time evolution operator has
complex eigenvalues of the form

$\displaystyle
U_{T}\left|\Psi_{\alpha}(0)\right\rangle=\left|\Psi_{\alpha}(T)\right\rangle=e^{-i\epsilon_{\alpha}T}\left|\Psi_{\alpha}(0)\right\rangle$
(4)

We represent the Floquet eigenstate $\left|\Psi_{\alpha}(0)\right\rangle$
using a parameterized quantum circuit

$\displaystyle\left|\Psi_{\alpha}(0)\right\rangle\approx
U_{\mathbf{\Theta}}\left|0\right\rangle,$ (5)

where $\left|0\right\rangle$ is the initial state of the quantum computer. If
$U_{\mathbf{\Theta}}\left|0\right\rangle$ is a Floquet eigenstate, then

$\displaystyle\left|\left\langle
0\right|U_{\mathbf{\Theta}}^{\dagger}U_{T}U_{\mathbf{\Theta}}\left|0\right\rangle\right|^{2}=1$
(6)

holds for the overlap between the time-evolved state and the initial state.
Note that overlaps of the form $|\langle 0|U^{\dagger}V|0\rangle|^{2}$ can be
efficiently computed on a quantum computer by interpreting them as the
measurement of the computational initial state projector

$\displaystyle|\langle 0|U^{\dagger}V|0\rangle|^{2}=\langle
0|V^{\dagger}U|0\rangle\langle 0|U^{\dagger}V|0\rangle.$ (7)

Thus we devise a hybrid algorithm that maximizes the overlap to determine a
Floquet eigenstate. An optimal solution $\mathbf{\Theta}_{\alpha}$ has a
maximal overlap of $1$. To obtain the complete set of eigenstates, we modify
the target function using a Lagrange multiplier $\lambda>0$ to project out
solutions $\mathbf{\Theta}_{\beta}$ that have previously been computed:

$\displaystyle\mathcal{L}({\mathbf{\Theta}})=\left|\left\langle
0\right|U_{\mathbf{\Theta}}^{\dagger}U_{T}U_{\mathbf{\Theta}}\left|0\right\rangle\right|^{2}-\lambda\sum\limits_{\beta}\left|\left\langle
0\right|U_{\mathbf{\Theta}_{\beta}}^{\dagger}U_{\mathbf{\Theta}}\left|0\right\rangle\right|^{2}.$
(8)

Here $\lambda$ has to be choosen sufficiently large for the algorithm to find
a new solution, see also [44]. Once a Floquet eigenstate has been determined,
we use iterative quantum phase estimation (IQPE) [45, 46, 47] to compute the
complex phase $e^{-i\epsilon_{\alpha}T}$ of $U_{T}$ upon application to the
eigenstate. The algorithm is sketched in Fig. 1 a).

Algorithm 1 Fauseweh-Zhu-1

1:

2:parameterized quantum circuit $U_{\mathbf{\Theta}}$

3:previous solutions $\mathbf{\Theta}_{\beta}$

4:periodic Hamiltonian $\mathcal{H}(t)$

5:procedure Optimize($U_{\mathbf{\Theta}}$)

6: Choose initial parameters $\mathbf{\Theta}$

7: while Maximize $\mathcal{L}(\mathbf{\Theta})=\left|\left\langle
0\right|U_{\mathbf{\Theta}}^{\dagger}\left.\right|U_{T}^{\phantom{\dagger}}U_{\mathbf{\Theta}}^{\phantom{\dagger}}\left|0\right\rangle\right|^{2}-\lambda\sum_{\beta}\left|\left\langle
0\right|U_{\mathbf{\Theta}_{\beta}}^{\dagger}\left.\right|U_{\mathbf{\Theta}}\left|0\right\rangle\right|^{2}$
do

8: Evaluate circuit
$U_{\mathbf{\Theta}}^{\dagger}U_{T}U_{\mathbf{\Theta}}^{\phantom{\dagger}}\left|0\right\rangle$

9: Evaluate circuits
$U_{\mathbf{\Theta}_{\beta}}^{\dagger}U_{\mathbf{\Theta}}^{\phantom{\dagger}}\left|0\right\rangle$

10: Update parameters $\mathbf{\Theta}$ to increase target
$\mathcal{L}(\mathbf{\Theta})$

11: Iterative quantum phase estimation on optimized state
$U_{\mathbf{\Theta_{\alpha}}}\left|0\right\rangle$ for

12: return $\epsilon_{\alpha},\,\mathbf{\Theta_{\alpha}}$

We turn to a high-level analysis of the algorithm and its requirements. The
basis for the algorithm is the optimization of a parameterized quantum
circuit. Thus one of the fundamental requirements is that the manifold spanned
by all the possible quantum circuits within the ansatz [48] contains the
Floquet eigenstates or is at least close to it. Thus carefully choosing a
circuit ansatz is important for the applicability of our algorithm, and is
currently a subject of intensive research [49, 50, 51, 52, 53, 54]. Note that
the target function $\mathcal{L}({\mathbf{\Theta}})$ allows for a gradient-
based optimization using parameter shift rules [55].

The algorithm optimizes a global observable and is therefore, in principle
subject to the problem of barren plateaus [56]. This can be avoided by
replacing the global observable with a local observable with identical
extremum, see also [57].

We do not specify how the time evolution in the first part of the algorithm is
performed. Trotterization is in principle applicable to $k$-local Hamiltonians
[58] and is the most accurate. However, it is disadvantageous with respect to
circuit depth [59, 60, 61]. Hence it is best applied if $T$ is the smallest
time scale, i.e., in the large frequency limit. Various other methods have
been developed [62, 63, 64, 65, 66] that make use of shallower circuits,
including approaches that work directly within the variational manifold [67,
57, 68]. All of these methods can be combined with our algorithm, as long as
they keep track of the global phase alignment.

The final part of the algorithm uses a single ancillary qubit to perform IQPE.
It applies controlled $(U_{T})^{2^{n}}$ gates to compute the $n$-th bit of the
phase $2\pi\phi=\epsilon_{\alpha}T$. The iteration depths $n_{\mathrm{max}}$
of IQPE determines the phase resolution and thereby desired precision of the
quasi-energies. Unless powers of $U_{T}^{2}$ can be obtained within the same
circuit depth, e.g., with variational time evolution, IQPE leads to an
increasing circuit depth requirement for increasing precision.

##  3 Second algorithm

The second algorithm, Fauseweh-Zhu-2, uses Fourier analysis to map the problem
onto an extended Hilbert space that is then solved using variational quantum
eigensolver approaches. From the definition of the Floquet modes in (2) one
can derive that the Hermitian operator
$\mathcal{H}(t)=H(t)-i\frac{\mathrm{d}}{\mathrm{d}t}$ has eigenvalues

$\displaystyle\mathcal{H}(t)\left|\Phi_{\alpha}(t)\right\rangle=\epsilon_{\alpha}\left|\Phi_{\alpha}(t)\right\rangle.$
(9)

Applying a Fourier expansion we can map the time-dependent evolution problem
to a time-independent eigenvalue problem

$\displaystyle H(t)=\sum\limits_{j}e^{-ij\Omega
t}H_{j},\quad\left|\Phi_{\alpha}(t)\right\rangle=\sum\limits_{j}e^{-ij\Omega
t}\left|\Phi_{\alpha}^{j}\right\rangle$ (10)
$\displaystyle\Rightarrow\sum\limits_{j}\left(H_{j-k}-j\Omega\delta_{j,k}\right)\left|\Phi_{\alpha}^{j}\right\rangle=\epsilon_{\alpha}\left|\Phi_{\alpha}\right\rangle.$
(11)

Note that the state $\left|\Phi_{\alpha}^{k}\right\rangle$ is part of an
extended Hilbert space $\mathcal{R}\otimes\mathcal{T}$ containing the original
Hilbert space $\mathcal{R}$ onto which $H(t)$ acts and the Hilbert space
$\mathcal{T}$ of square-integrable periodic functions of period $T$. This
extended Hilbert space is not physical, but it has a clear connection to the
original Hilbert space: the eigenvalues of the time-independent problem are
identical to the original time-dependent problem, up to multiples of
$j\Omega$. A graphical interpretation of this procedure is shown in Fig. 1 b),
as the original system is expanded into infinitely many copies. The
Hamiltonian $H_{0}$ acts within the plane, while $H_{\pm\gamma},\gamma>0$
introduces hopping in the $j$ direction. In the literature, this is understood
as extending the system in a new dimension [21]. This is certainly true for
non-interacting systems, but not in the strict many-body sense, as with each
additional layer in the quantum number $j$ the Hilbert space dimension
increases by dim($\mathcal{R}$), but in a many-body system it would be
multiplied by dim($\mathcal{R}$).

The eigenvalue problem in (11) is the starting point for our second algorithm.
We parameterize the Floquet eigenstates $\left|\Phi_{\alpha}^{k}\right\rangle$
using a parameterized quantum circuit

$\displaystyle\left|\Phi_{\alpha}^{k}\right\rangle\approx
U_{\mathbf{\Theta}}\left|0\right\rangle_{\mathcal{T}}\left|0\right\rangle_{\mathcal{R}}.$
(12)

Notice that we used the notation
$\left|0\right\rangle_{\mathcal{T}}\left|0\right\rangle_{\mathcal{R}}$ to mark
the extended Hilbert space computational basis. Here the first number marks
the quantum number $j$ of the $\mathcal{T}$ Hilbert space part, while the
second number refers to the original Hilbert space $\mathcal{R}$. In the
following we neglect the indices $\mathcal{T}$ and $\mathcal{R}$ of the
states. We define the effective Hamiltonian

$\displaystyle H_{\mathrm{eff}}^{j,k}=H_{j-k}-j\Omega\delta_{j,k}.$ (13)

Now we want to compute the eigenstates of $H_{\mathrm{eff}}$ to obtain the
Floquet band structure. We use the variational principle to optimize the
parameters $\mathbf{\Theta}$. As the extended Hilbert space is infinite
dimensional, we truncate $H_{\mathrm{eff}}$ at $\pm j_{\mathrm{max}}$. This
truncation introduces finite size errors in our calculation. We expect that
finite size errors are smallest in the center of the band structure. We
therefore minimize $H_{\mathrm{eff}}^{2}$ instead of $H_{\mathrm{eff}}$,
giving us the squared eigenvalues $\epsilon_{\alpha}^{2}$. The sign of the
eigenvalues can then be determined after optimization by measuring the
expectation value of $H_{\mathrm{eff}}$. As in the first algorithm we define
our target function using a Lagrange multiplier $\lambda>0$ to project out
solutions $\mathbf{\Theta}_{\beta}$ that have previously been computed

$\displaystyle\mathcal{L}(\mathbf{\Theta})=\left\langle 0\right|\left\langle
0\right|U_{\mathbf{\Theta}}^{\dagger}H_{\mathrm{eff}}^{2}U_{\mathbf{\Theta}}^{\phantom{\dagger}}\left|0\right\rangle\left|0\right\rangle-\lambda\sum_{\beta}\left|\left\langle
0\right|\left\langle
0\right|U_{\mathbf{\Theta}_{\beta}}^{\dagger}U_{\mathbf{\Theta}}\left|0\right\rangle\left|0\right\rangle\right|^{2},$
(14)

which corresponds to an excited state variational quantum eigensolver [44] for
the squared effective Hamiltonian. An overview of the Algorithm is sketched in
Fig. 1 b).

Algorithm 2 Fauseweh-Zhu-2

1:

2:parameterized quantum circuit $U_{\mathbf{\Theta}}$ in extended Hilbert
space $\mathcal{R}\otimes\mathcal{T}$

3:previous solutions $\mathbf{\Theta}_{\beta}$

4:Fourier expansion of Hamiltonian
$H_{\mathrm{eff}}=H_{j-k}-j\Omega\delta_{j,k}$

5:Truncation value $j_{\mathrm{max}}$

6:procedure Optimize($U_{\mathbf{\Theta}}$)

7: Choose initial parameters $\mathbf{\Theta}$

8: while Maximize $\mathcal{L}(\mathbf{\Theta})=\left\langle
0\right|\left\langle
0\right|U_{\mathbf{\Theta}}^{\dagger}H_{\mathrm{eff}}^{2}U_{\mathbf{\Theta}}^{\phantom{\dagger}}\left|0\right\rangle\left|0\right\rangle-\lambda\sum_{\beta}\left|\left\langle
0\right|\left\langle
0\right|U_{\mathbf{\Theta}_{\beta}}^{\dagger}\left.\right|U_{\mathbf{\Theta}}\left|0\right\rangle\left|0\right\rangle\right|^{2}$
do

9: Evaluate observable $\left\langle 0\right|\left\langle
0\right|U_{\mathbf{\Theta}}^{\dagger}H_{\mathrm{eff}}^{2}U_{\mathbf{\Theta}}^{\phantom{\dagger}}\left|0\right\rangle\left|0\right\rangle$

10: Evaluate circuits
$U_{\mathbf{\Theta}_{\beta}}^{\dagger}U_{\mathbf{\Theta}}^{\phantom{\dagger}}\left|0\right\rangle\left|0\right\rangle$

11: Update parameters $\mathbf{\Theta}$ to increase target
$\mathcal{L}(\mathbf{\Theta})$

12: Compute $\epsilon_{\alpha}\pm j\Omega=\left\langle 0\right|\left\langle
0\right|U_{\mathbf{\Theta}}^{\dagger}H_{\mathrm{eff}}U_{\mathbf{\Theta}}^{\phantom{\dagger}}\left|0\right\rangle\left|0\right\rangle$

13: return $\epsilon_{\alpha}\pm j\Omega,\,\mathbf{\Theta_{\alpha}}$

By analyzing the second algorithm in comparison to the first algorithm, we
immediately identify the increased auxiliary qubits that are required due to
the extension of the Hilbert space. While this increases the width of the
quantum circuit, its depth now depends purely on the depth of the ansatz. This
is in striking contrast to the first algorithm, where the maximum depth is
determined by the IQPE part of the algorithm, and hence by the required quasi-
energy resolution. The second algorithm also has the property of benefiting
from a mixed qudit-qubit architecture, as the truncated $T$ part of the
Hilbert space naturally leads to states such as $\left|\pm
j\right\rangle\left|\phi\right\rangle$, with $\left|\phi\right\rangle$ being a
state in the $\mathcal{R}$ Hilbert space. This could be useful for quantum
computer architectures that have access to more than two states per
fundamental quantum building block. The accuracy of the approach also directly
depends on the maximum truncation $j_{\mathrm{max}}$, that needs to be
determined, depending on the driving and the original Hamiltonian.

##  4 Benchmark

To evaluate the performance of both algorithms, we investigate one of the
simplest systems that remains not analytically solvable, the linear driven
spin-$\frac{1}{2}$,

$\displaystyle H(t)=-\frac{\Delta}{2}\sigma_{z}+\frac{A}{2}\cos(\Omega
t)\sigma_{x}\;.$ (15)

Here $\sigma_{i}$ are the Pauli matrices. We fix the energy spacing $\Delta=1$
and the driving frequency $\Omega=2.5$, and investigate the Floquet band
structure as a function of the amplitude $A$.

###  4.1 Algorithm Fauseweh-Zhu-1

We use the $U3$ gate as a generic single qubit rotation for the parameterized
quantum circuit:

$\displaystyle
U3(\theta,\phi,\nu)=\begin{pmatrix}\cos(\frac{\theta}{2})&-e^{i\nu}\sin(\frac{\theta}{2})\\\
e^{i\phi}\sin(\frac{\theta}{2})&e^{i(\phi+\nu)}\cos(\frac{\theta}{2})\end{pmatrix}$
(16)

The time evolution $U_{T}=\mathcal{T}\exp(-i\int\mathrm{d}tH(t))$ is performed
using Trotterization with $100$ time steps. Optimization is performed using
conjugate gradient descent on a quantum computer simulator using the IBM
qiskit package [69]. The eigenstates are optimized with $10^{4}$ samples,
while the IQPE uses $100$ samples with $5$ iterations. Results for the Floquet
band structure are shown in Fig. 2. We see a very good agreement between the
simulated results and the exact quasi-energy band structure. The error bars
are the result of the limited iteration depth in the IQPE.

Figure 2:  Simulation results for the Floquet energies of the driven
spin-$\frac{1}{2}$ system using algorithm Fauseweh-Zhu-1 with $\lambda=5$.
Straight lines show the exact result. Errorbars from sampling show the
circular standard-deviation.

###  4.2 Algorithm Fauseweh-Zhu-2

As the second algorithm uses the extended Hilbert basis we transfer the
problem in (15) to frequency space with a truncation value of
$j_{\mathrm{max}}=\pm 1$

$\displaystyle
H_{\mathrm{eff}}=\frac{\Delta}{2}\sigma_{z}\otimes\mathbb{1}+\frac{A}{2}\sigma_{x}\otimes
S_{x}\;,$ (17)

where the matrix $S_{x}$ is defined in the appendix A. To account for this, we
must also enlarge the ansatz in the parameterized quantum circuit and cannot
use a single qubit gate. We use a variational Hamiltonian approach [50] to
effectively reach the target states,

$\displaystyle U_{\mathbf{\Theta}}=\prod\limits_{l}\exp(-iK_{l}\Theta_{l}).$
(18)

We specify the hermitian generators $K_{l}$ in the appendix A. The parameters
are optimized using the conjugate gradient descent method. We used $10^{4}$
samples for optimization and evaluation of observables on a quantum computing
simulator. For comparison we also computed the exact eigenvalues of the
Floquet matrix. The results are shown in Fig. 3. We see a very good agreement
between the simulated results and the exact quasi-energy structure of the
truncated Hamiltonian. Naturally the truncation leads to an increasing error
with amplitude $A$ and only qualitatively captures the exact quasi-energy band
structure to a certain point.

Figure 3:  Simulation results for the Floquet energies of the driven
spin-$\frac{1}{2}$ system using algorithm Fauseweh-Zhu-2 with $\lambda=5$ and
$j_{\mathrm{max}}=1$. Straigt lines show the exact result, dashed lines show
the exact eigenstates of the truncated Hamiltonian in the extended Hilbert
space. Errorbars are from $10^{4}$ samples.

##  5 Discussion

In this paper we have presented two quantum algorithms, Fauseweh-Zhu-1 and
Fauseweh-Zhu-2, to compute Floquet eigenstates and quasi-energies. Both
algorithms use parameterized quantum circuits in combination with a quantum-
classical hybrid approach to find solutions to the defining Floquet properties
in time- and frequency-domain, respectively. While the precision of the first
algorithm depends on the depth of the quantum circuit due to the IQPE
application, the precision of the second algorithm mainly depends on the
frequency truncation $j_{\mathrm{max}}$ and thereby on the width of the
parameterized quantum circuit. There are no fundamental hurdles in applying
both algorithms on NISQ devices, with the advantage of complementary
requirements in circuit depth and width for increasing system size. We tested
both algorithms on a quantum computer simulator for the linear driven
spin-$\frac{1}{2}$ problem and showed the principal applicability of our
approach. As with other variational hybrid algorithms, the ansatz for the
parameterized quantum circuit is fundamental for the success of the approach.
In this context, the qudit-qubit structure of the second algorithm calls for
novel schemes that have so far not been explored. It will be interesting to
explore more complicated driving schemes with our approach and to test the
performance on real devices using advanced error mitigation methods [59] in
the near future.

## Acknowledgments

We thank Andrew Sornborger, Zoe Holmes, Michael Epping, Tim Bode, Peter
Schuhmacher, Elisabeth Lobe and Tobias Stollenwerk for useful discussions on
related problems. This work was carried out under the auspices of the U.S.
Department of Energy (DOE) National Nuclear Security Administration under
Contract No. 89233218CNA000001. It was supported by the LANL LDRD Program.
This research used computing resources provided by the LANL Institutional
Computing Program.

## References

  * Wang  _et al._ [2013] Y. H. Wang, H. Steinberg, P. Jarillo-Herrero, and N. Gedik, Science 342, 453 (2013).
  * Jotzu  _et al._ [2014] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, 237 (2014).
  * Rechtsman  _et al._ [2013] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
  * Ghimire and Reis [2019] S. Ghimire and D. A. Reis, Nat. Phys. 15, 10 (2019).
  * Cavalleri [2018] A. Cavalleri, Contemp. Phys. 59, 31 (2018).
  * Krausz and Ivanov [2009] F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 (2009).
  * Rabitz  _et al._ [2000] H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, Science 288, 824 (2000).
  * Levis  _et al._ [2001] R. J. Levis, G. M. Menkir, and H. Rabitz, Science 292, 709 (2001).
  * Levin  _et al._ [2015] L. Levin, W. Skomorowski, L. Rybak, R. Kosloff, C. P. Koch, and Z. Amitay, Phys. Rev. Lett. 114, 233003 (2015).
  * Fausti  _et al._ [2011] D. Fausti, R. I. Tobey, N. Dean, S. Kaiser, A. Dienst, M. C. Hoffmann, S. Pyon, T. Takayama, H. Takagi, and A. Cavalleri, Science 331, 189 (2011).
  * Basov  _et al._ [2011] D. N. Basov, R. D. Averitt, D. van der Marel, M. Dressel, and K. Haule, Rev. Mod. Phys. 83, 471 (2011).
  * Polkovnikov  _et al._ [2011] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011).
  * Abanin  _et al._ [2019] D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019).
  * Harper  _et al._ [2020] F. Harper, R. Roy, M. S. Rudner, and S. Sondhi, Annu. Rev. Condens. Matter Phys. 11, 345 (2020).
  * Schwarz  _et al._ [2020a] L. Schwarz, B. Fauseweh, N. Tsuji, N. Cheng, N. Bittner, H. Krull, M. Berciu, G. S. Uhrig, A. P. Schnyder, S. Kaiser, and D. Manske, Nat. Comm. 11, 287 (2020a).
  * Fauseweh and Zhu [2020] B. Fauseweh and J.-X. Zhu, Phys. Rev. B 102, 165128 (2020).
  * Schwarz  _et al._ [2020b] L. Schwarz, B. Fauseweh, and D. Manske, Phys. Rev. B 101, 224510 (2020b).
  * Yarmohammadi  _et al._ [2021] M. Yarmohammadi, C. Meyer, B. Fauseweh, B. Normand, and G. S. Uhrig, Phys. Rev. B 103, 045132 (2021).
  * Zhu  _et al._ [2021] W. Zhu, B. Fauseweh, A. Chacon, and J.-X. Zhu, Phys. Rev. B 103, 224305 (2021).
  * Paeckel  _et al._ [2020] S. Paeckel, B. Fauseweh, A. Osterkorn, T. Köhler, D. Manske, and S. R. Manmana, Phys. Rev. B 101, 180507 (2020).
  * Oka and Kitamura [2019] T. Oka and S. Kitamura, Annu. Rev. Condens. Matter Phys. 10, 387 (2019).
  * Shirley [1965] J. H. Shirley, Phys. Rev. 138, B979 (1965).
  * Sambe [1973] H. Sambe, Phys. Rev. A 7, 2203 (1973).
  * Potter  _et al._ [2016] A. C. Potter, T. Morimoto, and A. Vishwanath, Phys. Rev. X 6, 041001 (2016).
  * Tsuji  _et al._ [2008] N. Tsuji, T. Oka, and H. Aoki, Phys. Rev. B 78, 235124 (2008).
  * Tsuji  _et al._ [2009] N. Tsuji, T. Oka, and H. Aoki, Phys. Rev. Lett. 103, 047403 (2009).
  * Lee and Park [2014] W.-R. Lee and K. Park, Phys. Rev. B 89, 205126 (2014).
  * Murakami  _et al._ [2017] Y. Murakami, N. Tsuji, M. Eckstein, and P. Werner, Phys. Rev. B 96, 045125 (2017).
  * Heidrich-Meisner  _et al._ [2010] F. Heidrich-Meisner, I. González, K. A. Al-Hassanieh, A. E. Feiguin, M. J. Rozenberg, and E. Dagotto, Phys. Rev. B 82, 205110 (2010).
  * Seetharam  _et al._ [2015] K. I. Seetharam, C.-E. Bardyn, N. H. Lindner, M. S. Rudner, and G. Refael, Phys. Rev. X 5, 041050 (2015).
  * Dehghani  _et al._ [2014] H. Dehghani, T. Oka, and A. Mitra, Phys. Rev. B 90, 195429 (2014).
  * Bukov  _et al._ [2015] M. Bukov, L. D’Alessio, and A. Polkovnikov, Adv. Phys. 64, 139 (2015).
  * Mentink  _et al._ [2015] J. H. Mentink, K. Balzer, and M. Eckstein, Nat. Comm. 6, 6708 (2015).
  * Eckardt [2017] A. Eckardt, Rev. Mod. Phys. 89, 011004 (2017).
  * Abanin  _et al._ [2017] D. A. Abanin, W. De Roeck, W. W. Ho, and F. m. c. Huveneers, Phys. Rev. B 95, 014112 (2017).
  * Eckardt and Holthaus [2008] A. Eckardt and M. Holthaus, Phys. Rev. Lett. 101, 245302 (2008).
  * Eckardt and Anisimovas [2015] A. Eckardt and E. Anisimovas, New J. Phys. 17, 093039 (2015).
  * Rodriguez-Vega  _et al._ [2018] M. Rodriguez-Vega, M. Lentz, and B. Seradjeh, New J. Phys. 20, 093022 (2018).
  * Moskalets and Büttiker [2002] M. Moskalets and M. Büttiker, Phys. Rev. B 66, 205320 (2002).
  * Rakcheev and Läuchli [2020] A. Rakcheev and A. M. Läuchli, (2020), arXiv:2011.06017 [cond-mat.stat-mech] .
  * Bukov  _et al._ [2016] M. Bukov, M. Heyl, D. A. Huse, and A. Polkovnikov, Phys. Rev. B 93, 155132 (2016).
  * Preskill [2018] J. Preskill, Quantum 2, 79 (2018).
  * Du  _et al._ [2020] Y. Du, M.-H. Hsieh, T. Liu, and D. Tao, Phys. Rev. Research 2, 033125 (2020).
  * Higgott  _et al._ [2019] O. Higgott, D. Wang, and S. Brierley, Quantum 3, 156 (2019).
  * Griffiths and Niu [1996] R. B. Griffiths and C.-S. Niu, Phys. Rev. Lett. 76, 3228 (1996).
  * van Dam  _et al._ [2007] W. van Dam, G. M. D’Ariano, A. Ekert, C. Macchiavello, and M. Mosca, Phys. Rev. Lett. 98, 090501 (2007).
  * Dobšíček  _et al._ [2007] M. Dobšíček, G. Johansson, V. Shumeiko, and G. Wendin, Phys. Rev. A 76, 030306 (2007).
  * Funcke  _et al._ [2021] L. Funcke, T. Hartung, K. Jansen, S. Kühn, and P. Stornati, Quantum 5, 422 (2021).
  * Kandala  _et al._ [2017] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Nature 549, 242 (2017).
  * Wecker  _et al._ [2015] D. Wecker, M. B. Hastings, and M. Troyer, Phys. Rev. A 92, 042303 (2015).
  * Choquette  _et al._ [2021] A. Choquette, A. Di Paolo, P. K. Barkoutsos, D. Sénéchal, I. Tavernelli, and A. Blais, Phys. Rev. Research 3, 023092 (2021).
  * Gard  _et al._ [2020] B. T. Gard, L. Zhu, G. S. Barron, N. J. Mayhall, S. E. Economou, and E. Barnes, Npj Quantum Inf. 6, 10 (2020).
  * Cerezo  _et al._ [2021a] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Nat. Rev. Phys. 3, 625 (2021a).
  * Ganzhorn  _et al._ [2019] M. Ganzhorn, D. Egger, P. Barkoutsos, P. Ollitrault, G. Salis, N. Moll, M. Roth, A. Fuhrer, P. Mueller, S. Woerner, I. Tavernelli, and S. Filipp, Phys. Rev. Applied 11, 044092 (2019).
  * Li  _et al._ [2017] J. Li, X. Yang, X. Peng, and C.-P. Sun, Phys. Rev. Lett. 118, 150503 (2017).
  * Cerezo  _et al._ [2021b] M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Nat. Comm. 12, 1791 (2021b).
  * Barison  _et al._ [2021] S. Barison, F. Vicentini, and G. Carleo, Quantum 5, 512 (2021).
  * Lloyd [1996] S. Lloyd, Science 273, 1073 (1996).
  * Fauseweh and Zhu [2021] B. Fauseweh and J.-X. Zhu, Quantum Inf. Process. 20, 138 (2021).
  * Smith  _et al._ [2019] A. Smith, M. S. Kim, F. Pollmann, and J. Knolle, Npj Quantum Inf. 5, 106 (2019).
  * Lamm and Lawrence [2018] H. Lamm and S. Lawrence, Phys. Rev. Lett. 121, 170501 (2018).
  * Berry  _et al._ [2015] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Phys. Rev. Lett. 114, 090502 (2015).
  * Campbell [2019] E. Campbell, Phys. Rev. Lett. 123, 070503 (2019).
  * Childs  _et al._ [2019] A. M. Childs, A. Ostrander, and Y. Su, Quantum 3, 182 (2019).
  * Low and Chuang [2019] G. H. Low and I. L. Chuang, Quantum 3, 163 (2019).
  * Cîrstoiu  _et al._ [2020] C. Cîrstoiu, Z. Holmes, J. Iosue, L. Cincio, P. J. Coles, and A. Sornborger, Npj Quantum Inf. 6, 82 (2020).
  * Yuan  _et al._ [2019] X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. C. Benjamin, Quantum 3, 191 (2019).
  * Otten  _et al._ [2019] M. Otten, C. L. Cortes, and S. K. Gray, (2019), arXiv:1910.06284 [quant-ph] .
  * Andersson  _et al._ [2020] S. Andersson, A. Asfaw, A. Corcoles, L. Bello, Y. Ben-Haim, M. Bozzo-Rey, S. Bravyi, N. Bronn, L. Capelluto, A. C. Vazquez, J. Ceroni, R. Chen, A. Frisch, J. Gambetta, S. Garion, L. Gil, S. D. L. P. Gonzalez, F. Harkins, T. Imamichi, H. Kang, A. h. Karamlou, R. Loredo, D. McKay, A. Mezzacapo, Z. Minev, R. Movassagh, G. Nannicini, P. Nation, A. Phan, M. Pistoia, A. Rattew, J. Schaefer, J. Shabani, J. Smolin, J. Stenger, K. Temme, M. Tod, E. Wanzambi, S. Wood, and J. Wootton., “Learn quantum computation using qiskit,”  (2020).

##  Appendix A Generators for variational Hamiltonian ansatz

To find an efficient variational ansatz for the Fauseweh-Zhu-2 algorithm we
first separate the gates required for the $\mathcal{T}$ and the $\mathcal{R}$
part of the Hilbert space. For $j_{\mathrm{max}}=\pm 1$ the qudit space is $3$
dimensional. A general unitary matrix in SU($3$) can be parameterized using
the Gell-Mann matrices, which span the corresponding Lie-Algebra. For our
particular case we find, that a much more limited set is sufficient to find
all Floquet eigenstates. For $\mathcal{R}$ a simple Ry gate is sufficient.
Naturally for nonzero $A$ the Floquet eigenstates do not separate into the two
Hilbert spaces anymore and we need an entangling gate to account for this. We
use the matrices

$\displaystyle S_{4}=\begin{pmatrix}0\,0\,0\\\ 0\,0\,1\\\ 0\,1\,0\\\
\end{pmatrix}\quad S_{5}=\begin{pmatrix}0\,1\,0\\\ 1\,0\,0\\\ 0\,0\,0\\\
\end{pmatrix}\quad S_{x}=\frac{1}{2}\left(S_{5}+S_{4}\right),$ (19)

to define our variational Hamiltonian ansatz

$\displaystyle
U_{\mathbf{\Theta}}=e^{i\Theta_{1}\sigma_{y}}e^{i\Theta_{2}S_{4}}e^{i\Theta_{3}S_{5}}e^{i\Theta_{4}S_{x}\sigma_{x}}e^{i\Theta_{5}\sigma_{y}}e^{i\Theta_{6}S_{4}}e^{i\Theta_{7}S_{5}},$
(20)

where $e^{i\Theta_{4}S_{x}\sigma_{x}}$ is the aforementioned entangling gate.