Introduction

The field of quantum computing grew out of the need for efficient quantum simulations1,2. As Feynman pointed out3, the possibility of mapping one quantum system into another opens up new windows for efficiently exploring the properties and dynamics of general quantum systems4,5. Large-scale, programmable quantum computers could offer exactly this type of possibility of mapping and simulating complex quantum systems6,7,8. While such large-scale quantum computers do not yet exist, it is worth exploring the potential for quantum simulations using currently available noisy intermediate-scale quantum (NISQ) computers9,10. One such potential area of NISQ application is the topic of quantum chaos—the study of quantum systems that exhibit chaos in some classical limit. The question of how classical chaos emerges from quantum dynamics remains one of the open fundamental questions in quantum theory11,12,13. On the experimental side, the quantum control and precision needed to explore quantum chaotic dynamics over a wide range of parameters and long time scales remains quite challenging. So far, relatively few experiments in limited parameter regimes and for short times have been performed14,15,16,17,18.

Classical chaos is characterized by exponential sensitivity to initial conditions quantified by the Lyapunov exponent, that is a measure of the rate of divergence of neighbouring trajectories19,20,21. A corresponding quantum measure of chaos is challenging to define due to the uncertainty principle and the linearity of quantum evolution. In recent years, the question of quantum-classical correspondence in classically chaotic systems has been explored in the context of quantum information processing. The connection between fundamentally quantum phenomena such as entanglement and classical chaos has puzzled physicists for decades, and has gained new relevance for quantum computing applications. It has been shown that classical chaos can affect the implementation of quantum computing algorithms22,23. Chaos can also affect the generation of dynamical entanglement, an important resource for quantum computing.

To understand chaos in the quantum context, it is important to explore signatures of classical chaos in the deep quantum regime, where the standard Bohr correspondence principle cannot be invoked24. A textbook model for studying quantum chaos is the quantum kicked top (QKT)12, which is a finite-dimensional spin system that displays chaotic dynamics in the classical limit. The quantum kicked top has been extensively studied theoretically25,26,27,28,29,30,31,32. The system can be described as a collection of indistinguishable qubits, which makes it attractive to explore in the framework of quantum information processing and NISQ devices. In the deep quantum regime, periodicities and symmetries in the two- and three-qubit QKT model were theoretically studied in31,32. A few experimental studies of the QKT have also been performed16,17,18. In18, a 3-qubit model of the QKT was shown to exhibit ergodic dynamics and a resemblance between entanglement entropy and classical phase space dynamics was noted. Temporal periodicity and symmetries of the 2-qubit QKT were explored using NMR techniques in16. These experiments are limited in the number of kicks due to decoherence times of the physical qubits. Furthermore, in these implementations the chaoticity parameter \(\kappa\) is determined by the strength and duration of interaction between the qubits, making it difficult to tune. To experimentally study the long term dynamics and dependence on \(\kappa\) rigorously, one needs to explore longer time scales and a wider range of \(\kappa\).

In this work, we show that mapping the QKT on to a programmable quantum circuit in a quantum computer allows simulations of the QKT that overcome previous experimental limitations. We construct and demonstrate an exact simulation of the 2-qubit quantum kicked top using a universal set of quantum logic gates. Our quantum circuit-based simulation is programmable and enables flexible initial state preparation and evolution. Using IBM’s publicly accessible 5-qubit processor, we can prepare initial states and implement the dynamics of the QKT for an arbitrary number of kicks and a wide range of \(\kappa\). The number of gates required for this simulation is independent of the number of kicks and value of \(\kappa\). Therefore, our model does not suffer any systematic loss in fidelity with increasing number of kicks or \(\kappa\) values. Finally, quantum state tomography enables us to explore signatures of chaos in 2-qubit entanglement. The ability to vary \(\kappa\) and the number of kicks allows us to observe the periodic nature of the dynamics with respect to \(\kappa\) as well as kick number. Additionally, the temporal periodicity of the QKT can be used to obtain highly accurate time averages of relevant physical quantities. In particular, we explore the time-averaged entanglement for different initial spin coherent states (SCS). We find that a contour plot of the time average entanglement shows clear signatures of the classical phase space structures of regular islands in a chaotic sea, even in a deep quantum regime. We also show that the states initialized in chaotic regions of the phase space show intermediate values of average concurrence, whereas, the fixed points and the period-4 orbit correspond to the minimum and maximum values respectively. This behaviour is related to the degree of delocalization of the states and thus demonstrates a connection between delocalization and entanglement24. Our work shows that current quantum computers are useful for flexible quantum simulations of chaotic systems. Mapping the system onto a tunable quantum circuit lets us probe different aspects of the QKT dynamics without the need for building sophisticated customized hardware or being constrained by fixed system parameters. This method combines the ease of numerical simulation with the built-in quantum evolution of a physical system.

Results

The model

The quantum kicked top model was first introduced by Haake, Kus and, Scharf in 198712. It is now a widely studied model for exploring quantum chaos. The QKT is a time-dependent periodic spin system governed by the Hamiltonian

$$\begin{aligned} H = \hbar \frac{p J_{y}}{\tau } + \hbar \frac{\kappa J_{z}^2}{2j}\Bigg ( \sum _{n=-\infty }^{n=\infty } \delta (t-n\tau ) \Bigg ). \end{aligned}$$
(1)

where \(J_{x},J_{y}\) and \(J_{z}\) obey the standard angular momentum operator commutation relations. The operator \(J^{2}=J_x^2 + J_y^2 + J_z^2\) commutes with the Hamiltonian and the magnitude of the angular momentum is a constant of motion. The Hamiltonian consists of a series of rotations described by the \(J_{y}\) term alternating with kicks (torsion) due to the non-linear \(J^{2}_{z}\) term. \(\tau\) is the duration between consecutive kicks, p is the angle of linear rotation in the y-direction, and \(\kappa\) is the strength of twist in the z-direction.

The kick-to-kick Floquet time evolution operator can be written as

$$\begin{aligned} U= \exp \Bigg (-i\frac{\kappa }{2j}J_{z}^2\Bigg )\exp \Bigg (-ipJ_{y}\Bigg ). \end{aligned}$$
(2)

The classical map for the kicked top can be obtained by writing the Heisenberg equations of motion for the angular momentum operators and then taking the limit \({ j \rightarrow \infty }\). By defining the normalized variables \(X=J_{x}/j\), \(Y=J_{y}/j\) and \(Z=J_{z}/j\), the classical equations of motions for p=\(\pi /2\) are24

$$\begin{aligned} X_{n+1}= & Z_n\cos (\kappa X_n)+Y_n\sin (\kappa X_n), \nonumber \\ Y_{n+1}= & Y_n\cos (\kappa X_n)-Z_n\sin (\kappa X_n), \nonumber \\ Z_{n+1}= & -X_n, \end{aligned}$$
(3)

where \([X_{n+1},Y_{n+1},Z_{n+1}] = \lim _{\epsilon \rightarrow 0}[X(\tau +\epsilon ), Y(\tau +\epsilon ), Z(\tau +\epsilon )]\). Here the dynamical variables (X,Y,Z) satisfy the constraint \(X^{2} + Y^{2} + Z^{2} =1\), i.e., they are restricted to be on the unit sphere \(S ^2\). Thus, the variables can be parameterized into spherical polar coordinates as \((X,Y,Z)=(\sin (\theta )\cos (\phi ), \sin (\theta )\sin (\phi ),\cos (\theta ))\).

As the chaoticity parameter \(\kappa\) is varied, the classical dynamics ranges from regular motion (\(\kappa\)\(\le\) 2.1), to a mixture of regular and chaotic behaviour for different initial conditions (2.1 \(\le \kappa \le 4.4\)), to fully chaotic motion (\(\kappa > 4.4\)). The classical stroboscopic maps (in spherical co-ordinates) for a range of initial conditions with \(\kappa = 2.5\) and \(\kappa =3\) is plotted in Fig. 1a and 1b respectively.

Fig. 1
figure 1

Stroboscopic map showing the classical time evolution over 150 kicks for (a) \(\kappa = 2.5\) and (b) \(\kappa =3.0\) for 289 initial points in phase space.

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Performance of quantum simulations

Starting with various initial points for two different values of \(\kappa\), we run the quantum circuit for implementing N kicks on IBM vigo (see Methods section for circuit details). We reconstruct the resulting final state by performing quantum state tomography and use the fidelity of the reconstructed state as a measure of simulation accuracy. For the theoretically predicted state \(\rho _{\text {th}}\) and the reconstructed state \(\rho _{{vigo}}\), fidelity is given by \(F(\rho _{{vigo}},\rho _{\text {th}})=(\text {tr}(\sqrt{\sqrt{\rho _{\text {th}}}\rho _{{vigo}}\sqrt{\rho _{\text {th}}}} ))^2\). We observe that there is no systematic loss in fidelity with the number of kicks for different initial states and values of \(\kappa\) (Fig. 2). We compute the average fidelity over different \(\kappa\) values in the range [0.5,6.5], and for different initial states as a function of time. As seen in Fig. 3, the average fidelities remain around 0.87.

Fig. 2
figure 2

Fidelity of the tomographically reconstructed 2-qubit state for different initial states and different \(\kappa\) values on IBM vigo.

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Fig. 3
figure 3

Fidelity of the tomographically reconstructed 2-qubit state averaged over initial states with \((\theta ,\phi )\in \{(2.25,0),(\pi /2,\pi /2),(\pi /2,0)\}\) and \(\kappa \in \{0.5,2.5,4.5,6.5\}\) on IBM vigo. The error bars indicate standard deviation.

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Existing experimental implementations of QKT have reported either monotonically decreasing fidelity or a significant drop in fidelity after some particular number of kicks16,18. In this work, the non-decreasing trend in fidelity for higher number of kicks can be attributed to the fixed number of gates for arbitrary number of kicks. By decomposing the unitary into elementary, programmable quantum gates, we effectively remove any constraints on the parameters of the physical system (QKT) that we can implement. In the IBM-Q systems, the error varies only with the number of single qubit physical rotations and CNOT gates33 acting on each qubit. Since the number of gates in the circuit remains constant irrespective of the value of \(\kappa\), we observe that the fidelity values do not depend on \(\kappa\). Average fidelity plots for different initial states as a function of \(\kappa\) are included in the supplementary information (Supplementary Fig. 1).

Observation of periodicity in \(\kappa\)

Fig. 4
figure 4

Time-averaged concurrence as function of \(\kappa\). The initial state is an SCS with \(\theta\) = 2.25 and \(\phi\) = 2.0. The average was taken over 200 steps for the simulated plot and over 50 steps on IBM vigo. Average concurrence shows a periodicity of \(2\pi\).

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The ability to simulate a wide range of values of the chaoticity parameter \(\kappa\) allows us to explore how chaos affects the dynamics. In particular, we study the effect of chaos (a classical phenomenon), on entanglement—a fundamentally quantum phenomenon.

Fig. 5
figure 5

Contour plot of concurrence. (a) Over 50 kicks for different values of \(\kappa\) on IBMQ vigo (b) Over 200 kicks for different values of kappa on IBMQ simulator. The initial state is an SCS with \(\theta\) = 2.25 and \(\phi\) = 2.0.

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Figure 4 shows the time averaged 2-qubit concurrence34 as a function of \(\kappa\) for an initial SCS centered at \(\theta =2.25, \phi =2.0\). The experimental results from the IBMQ hardware are in good agreement with that of the IBM quantum circuit simulator, demonstrating the periodic behaviour of \(\kappa\). The observed period of \(2\pi\) agrees with the theoretical prediction of \(2\pi j\)32. To confirm the general periodicity of \(\kappa\), we mapped the time evolution of the concurrence between the two qubits for 25 different values of \(\kappa\) ranging from 0 to 12 to create a contour plot (Fig. 5a, b). The periodic nature of the dynamics can be observed in the concurrence as we scan over either the number of kicks or \(\kappa\), while holding the other variable constant. The periodicity of quantum dynamics in the 2-qubit QKT model was explored experimentally in a previous work using NMR qubits16. Our work greatly expands the range of \(\kappa\) and the number of kicks explored, and directly connects the periodicity to the observed entanglement dynamics.

Relationship between entanglement and delocalization

The periodicity in concurrence, combined with the ability to implement a high number of kicks, can be exploited to generate detailed average concurrence plots. By averaging over multiple periods of concurrence in the number of kicks we can reduce the error in the value of average concurrence. This allows more detailed observations of signatures of chaos in entanglement dynamics.

Fig. 6
figure 6

Comparison between classical phase space and contour plot of average concurrence. (a) Stroboscopic classical map, (b) average concurrence over 200 kicks on IBMQ simulator and (c) average concurrence over 50 kicks on IBMQ vigo for 289 initial points and \(\kappa\) = 2.5 .

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A contour plot of time-averaged concurrence as function of \(\theta\) and \(\phi\) for \(\kappa = 2.5\) reflects the structures of the stroboscopic classical map as shown in Fig. 6. Furthermore, our plots have enough resolution to observe that the chaotic regions of the classical phase space show intermediate concurrence values. The four prominently visible islands of low concurrence correspond to fixed points of the classical dynamics. These islands are clearly distinguishable on the hardware plot and the left-right symmetry is maintained as shown in Fig. 6c. Points \((J_x/j,J_y/j,J_z/j) = (1,0,0), (0,0,-1),(-1,0,0)\) and (0, 0, 1), which constitute a period-4 orbit present in the classical dynamics of the system, show the highest values of average concurrence16,18,26,31. The contour plots of average concurrence for various values of \(\kappa\) obtained on the IBMQ simulator are included in Supplementary Note 4 (Supplementary Fig. 3).

We note correspondence between the average concurrence and the degree of delocalization of various initial states after evolution according to the Floquet unitary. This degree of delocalization24 can be quantified by calculating the maximum overlap with respect to the set of minimum uncertainty spin coherent states.

$$\begin{aligned} O_{\textrm{SCS}}(|\psi (t)\rangle )=\max _{\textrm{SCS}}|\langle \textrm{SCS} \mid \psi (t)\rangle |. \end{aligned}$$
(4)
Fig. 7
figure 7

Evolution of \(O_{\textrm{SCS}}\) values for two different initial states for 50 kicks on IBMQ quito for\(\kappa =2.5\). The evolution of initial state leading to higher concurrence (\((\theta ,\phi )=(\pi /2,0)\)) is more delocalized, i.e., has lower average \(O_{\textrm{SCS}}\) than that corresponding to lower concurrence (\((\theta ,\phi )=(2.25,1)\)). The horizontal dotted lines represent the average \(O_{\textrm{SCS}}\) values for the two cases.

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Large values of \(O_{\textrm{SCS}}\) correspond to more localized states, as they indicate high overlap with spin coherent states. Delocalized states show low \(O_{\textrm{SCS}}\) values. The values of \(O_{\textrm{SCS}}\) for two different initial states, one in the high concurrence region and one in the low concurrence region, are plotted against number of steps in Fig. 7, and show the connection between entanglement and delocalization. The corresponding Husimi distributions for these points are shown in Supplementary Note 3 (Supplementary Fig. 2).

This approach of quantifying delocalization is equivalent to the definition of geometric entanglement given in35. For a symmetric state \(|\psi \rangle\), the geometric entanglement is given by

$$\begin{aligned} E_{G}(|\psi \rangle ) = 1-G(\psi )^2 \end{aligned}$$
(5)

where the quantity \(G(\psi )\) is the overlap of \(|\psi \rangle\) with the closest product state

$$\begin{aligned} G(\psi ) = \max _{|\phi \rangle = |a\rangle |b\rangle |c\rangle ...} |\left\langle \psi |\phi \right\rangle |. \end{aligned}$$
(6)

Since all the spin coherent states are separable, and every separable state of this Hilbert space is a spin coherent state, for our case the quantity \(G(\psi )\) is exactly equal to the previously defined \(O_{SCS}(|\psi \rangle )\). Hence, this definition implies that highly delocalized states have higher geometric entanglement and more localized states have lower geometric entanglement. This behaviour, in the case of the 2-qubit QKT, agrees with the trend in concurrence. This observation of the connection between entanglement and delocalization in the deep quantum regime confirms previous theoretical predictions24,36.

Discussion

In this work, we have proposed a quantum circuit-based approach to simulate and explore quantum chaos, and demonstrated its advantages over existing methods. The proposed method can be applied in general to any periodically driven finite dimensional quantum system. In our study, IBM’s 5-qubit open access quantum chip (vigo) was used as the experimental platform to implement the proposed approach for the 2-qubit quantum kicked top (QKT). The Hamiltonian of the QKT can be exactly expressed in terms of qubits since it is a finite-dimensional quantum system. Therefore, its evolution operator can be decomposed into quantum gates. Traditionally, experimental studies of quantum chaos have applied the same set of operations n times to explore time evolution. Here, we decomposed the unitary evolution operator for n kicks, \(U^n\), into elementary quantum gates. This results in a fixed number of operations implementing the QKT evolution for any number of kicks. This hybrid combination of classical processing and quantum computing opens up the ability to perform high fidelity experimental studies of quantum chaos in new parameter regimes.

Since the value of the chaoticity parameter \(\kappa\) only determines the parameters of unitary rotations in the quantum circuit, and since the single qubit rotation errors are independent of the parameters, we were able to experimentally study chaotic dynamics over a wider range of \(\kappa\) and kick number previously inaccessible to experimental studies. By taking advantage of the high fidelity obtained for both a large number of kicks and arbitrary \(\kappa\) values, we experimentally demonstrated the periodicity of entanglement with time and \(\kappa\) with high accuracy. Our studies also clearly showed signature of chaos in the contour plot of average 2-qubit concurrence despite being in the deeply quantum regime. Furthermore, we reported the first observation of the correspondence between average entanglement and delocalization in the 2-qubit QKT.

Our results demonstrate the advantages of circuit-based NISQ devices for exploring fundamental questions in quantum information and quantum chaos despite their noise and scale limitations. The method can be extended to the case of more qubits. The symmetries in the Hamiltonian may also enable polynomial scaling of the number of elementary gates required to simulate unitary evolution of an N-qubit QKT. With the rapid improvement in qubit error rates it is foreseeable that we will eventually be able to simulate the evolution of a large N-qubit chaotic system at a scale that is intractable on classical computers. Our work provides the first stepping stone towards this eventual quantum advantage. Furthermore, this programmable computing approach to simulating increasingly larger systems would also allow detailed studies of the quantum-classical transition in chaotic systems - a long-standing open problem. Previous studies22,23 have noted that chaos could influence the efficient and stable operation of quantum computers. In37, it was shown that chaos affects the balance between the disorder that maintains the stability of qubits and nonlinear resonator couplings that is used to manipulate interactions. This plays an integral role in future transmon device engineering. The gate-based circuit model of QKT can be used as an efficient tool for studying these effects in different situations.

Methods

Floquet operator

The QKT Hamiltonian commutes with the total angular momentum operator \(J^2\), \([H,J^2] = 0\). Hence, it can be considered as an \(N = 2j\) qubit system24. The spin-j operators are written in terms of the single qubit Pauli rotation operators as:

$$\begin{aligned} J_{\alpha }=\frac{1}{2} \sum _{i=1}^{2 j} \sigma _{i \alpha }, \quad \alpha \in \{x, y, z\} \end{aligned}$$
(7)

where \(\sigma _{i\alpha }\) denotes \(\sigma _{\alpha }\) acting on the \(i^{th}\) qubit. This allows us to rewrite the QKT Hamiltonian in 2j-qubit space:

$$\begin{aligned} H=\hbar \frac{\kappa }{8 j}\left( 2 j \mathbb {I}+\sum _{\begin{array}{c} {i, k=1} \\ {i < k} \end{array}}^{2 j} \sigma _{i z} \otimes \sigma _{k z}\right) \sum _{n=-\infty }^{\infty } \delta (t-n \tau )+\hbar \frac{p}{2 \tau } \sum _{i=1}^{2 j} \sigma _{i y}. \end{aligned}$$
(8)

In particular, the \(j=1\) QKT is described by the 2-qubit Hamiltonian

$$\begin{aligned} H = \frac{\hbar k}{4}\left( \mathbb {I} + \sigma _z \otimes \sigma _z\right) \sum _{n}\delta (t-n\tau ) + \frac{\hbar p}{2\tau }(\sigma _y\otimes \mathbb {I}+\mathbb {I}\otimes \sigma _y) \end{aligned}$$
(9)

and the corresponding single-kick time evolution unitary is

$$\begin{aligned} U = \exp \bigg (-i\frac{\kappa }{4}(\mathbb {I}+\sigma _z \otimes \sigma _z)\bigg )\exp \bigg (-i\frac{p}{2}(\sigma _y\otimes \mathbb {I}+\mathbb {I}\otimes \sigma _y)\bigg ). \end{aligned}$$
(10)

Initial states

Minimum uncertainty states in spin systems are spin coherent states (SCS)38 that satisfy the uncertainty relation

$$\begin{aligned} \Delta J_i\Delta J_k = \frac{\hbar }{2}|\Delta J_l|, \end{aligned}$$
(11)

where i, k and l are permutations of x,y and z. The uncertainty for these states is distributed symmetrically over the two operators. For larger j values, the SCS becomes highly localized around the point \((\theta ,\phi )\) in the phase space. Hence in the classical limit of \(j\rightarrow \infty\), the SCS approximates the classical angular momentum state located at \((\theta ,\phi )\)12.

Spin squeezed states, which have asymmetric distribution of uncertainty, can display entanglement in the corresponding multi-qubit representation39. Since we are interested in studying entanglement that arises from the dynamics of the system, we choose SCSs as our initial states, thus ensuring there is no initial entanglement between the qubits. Given any point \((\theta ,\phi )\) in the classical phase space, we construct the corresponding SCS \(|j;\theta ,\phi \rangle\)

$$\begin{aligned} |j;\theta ,\phi \rangle = \exp [i\theta (J_x\sin \phi -J_y\cos \phi )]|j,j\rangle . \end{aligned}$$
(12)

In the 2j-qubit space, we define our initial states as the SCSs

$$\begin{aligned} |j;\theta ,\phi \rangle = |\theta ,\phi \rangle ^{\otimes 2j}, \end{aligned}$$
(13)

where \(|\theta ,\phi \rangle\) are points on the Bloch sphere.

Implementation of unitaries with a quantum circuit

Any n-qubit unitary can be decomposed into \(2^n(2^n-1)/2\) single-qubit gates with controls. We follow the prescription in40 to decompose our 2-qubit Floquet operator as

$$\begin{aligned} U=U_{1}\times U_{2}\times U_{3}\times U_{4}\times U_{5}\times U_{6}, \end{aligned}$$
(14)

where \(U_i\) are either single-qubit unitaries or controlled unitaries of the form

In the circuit above, the two wires represent the two qubits, the solid dot represents the control qubit, and the box indicates the single qubit operation \(V_i\) on the target qubit. The exact decomposition for a 2-qubit gate in this scheme is

(15)

Implementation of these gates on a quantum computer requires further decomposition into rotations and CNOT gates. Given a 1-qubit unitary W\(\in\) SU(2), a controlled gate of the from (\(|0\rangle \langle 0|\otimes \mathbb {I}+|1\rangle \langle 1|\otimes W)\) can be decomposed as41

(16)

where \(R_x\), \(R_y\) and \(R_z\) describe rotations on the Bloch sphere and \(\alpha\), \(\beta\) and \(\theta\) are such that

\(R_z(\alpha )R_y(\theta )R_z(\beta ) = W\).

A general 1-qubit unitary V is of the form \(\exp (i\delta )\times W\) where \(W\in\) SU(2). A 2-qubit gate of the form (\(|0\rangle \langle 0|\otimes \mathbb {I}+|1\rangle \langle 1|\otimes V)\) can be written as

(17)

where \(U_{\delta } = \exp (-i\delta )\times \mathbb {I}\). This controlled phase gate can be further simplified by moving the phase over to the other qubit:

$$\begin{aligned} (|0\rangle \langle 0|\otimes \mathbb {I} + |1\rangle \langle 1|\otimes U_{\delta })= & (|0\rangle \langle 0|\otimes \mathbb {I}+ \exp (i\delta )|1\rangle \langle 1|\otimes \mathbb {I}) \nonumber \\= & (|0\rangle \langle 0| + \exp (i\delta )|1\rangle \langle 1|)\otimes \mathbb {I} \nonumber \\= & R_z(\delta )\otimes \mathbb {I}. \end{aligned}$$
(18)

Here, we ignore a global phase factor of \(\exp (i\delta /2)\) in the final step. Finally we are left with

(19)

Similar analysis follows when the control and target qubits are exchanged. For two-qubit unitaries of the type \(\mathbb {I}\otimes V\) and \(V\otimes \mathbb {I}\), the phase factors appearing on V are global and can be ignored. These gates have a similar decomposition as the one in Eq. 16 with the CNOT gates replaced by X gates. For example, a unitary of the form \(\mathbb {I}\otimes V\) can be decomposed as

(20)

In this scheme, our 2-qubit Floquet operator can be constructed from 46 total gates, with 8 two-qubit CNOT gates and 38 single qubit rotations. Consecutive rotations have been counted as separate single qubit gates. Depending on the universal gate set for the particular quantum computer, the actual number of gates needed to simulate the Floquet unitary may be reduced.

Decomposition into \(U_1\) and \(U_3\) gates on IBMQ

We implement our quantum circuits on the quantum hardware and simulator back-end of the IBM Quantum Experience42. The interfacing with the quantum hardware is done using Qiskit43. Qiskit allows us to implement 1-parameter and 3-parameter single-qubit unitary operators of the form

$$\begin{aligned} U_3(\theta ,\phi ,\lambda )= & \left( \begin{array}{cc} \cos (\theta / 2) & -e^{i \lambda } \sin (\theta / 2) \\ e^{i \phi } \sin (\theta / 2) & e^{i \lambda +i \phi } \cos (\theta / 2) \end{array}\right) \nonumber \\ U_1(\lambda )= & \left( \begin{array}{cc} 1 & 0 \\ 0 & e^{i \lambda } \end{array}\right) . \end{aligned}$$
(21)

We decompose the gates in Eq. 15 into a combination of \(U_1\) and \(U_3\) gates. Gates of the form \(V\otimes \mathbb {I}\) and \(\mathbb {I}\otimes V\) are implemented directly as \(U_3\) gates. For controlled gates, the decomposition given in Eq. 17 is used where W\(\in\) SU(2) is implemented as a \(U_3\) gate and \(U_{\delta }\) is implemented as \(U_1(\delta )\). Hence, we obtain the final circuit decomposition for our 2-qubit Floquet operator on IBMQ:

(22)

with \(W_i = U_3(\theta _i,\phi _i, \lambda _i)\).

Time evolution after multiple kicks is calculated by applying the Floquet unitary on the initial state repeatedly. This could be achieved by applying the set of gates given in Eq. 22 consecutively N times to simulate evolution by N time steps. However, to mitigate the errors which may arise from the increasing number of gates, in our approach, we decompose the effective N-step unitary \(U^N\) using the same procedure as mentioned above. This means that the state after any arbitrary number of steps can be obtained by applying the same set of gates given in Eq. 22 with appropriate parameters. The advantage of this approach is that the number of gates is fixed and does not grow with the number of kicks. Similarly, time evolution for different values of \(\kappa\) can be implemented by computing the relevant parameters for the set of gates corresponding to the unitary \(U^N(\kappa )\). This affords us a more fine-grained and flexible control over this parameter compared to other qubit-based realizations where the value of \(\kappa\) is set by tuning the time duration of interactions between the physical qubits.

After applying the appropriate set of gates to the initial states, the final states density matrix is constructed using state-tomography circuits built into Qiskit Ignis44. Physical quantities of interest can be calculated from this density matrix.