Fully-programmable universal quantum simulator with a one-dimensional quantum processor

The dynamics of driven many-body systems has many relevant applications in quantum technologies and quantum information processing Amico et al. (2008); Cheneau et al. (2012); Nemoto et al. (2014); Lewis-Swan et al. (2019). Existing quantum devices allow us to perform tasks such as quantum metrology Degen et al. (2017), quantum communication Hasegawa et al. (2019) and quantum simulation Roushan et al. (2017a); Ma et al. (2019); Ye et al. (2019); Yan et al. (2019); Roushan et al. (2017b); Chiaro et al. (2019); O’Malley et al. (2016); Kandala et al. (2017); Arute et al. (2020), that can be carried out without error correction. Among the Noisy Intermediate-scale Quantum (NISQ) devices, superconducting Roushan et al. (2017a); Ma et al. (2019); Ye et al. (2019) and trapped ions Lanyon et al. (2011); Blatt and Roos (2012); Zhang et al. (2017) quantum processors play a fundamental role and have huge potential for near-term applications Preskill (2018). These devices are highly programable and potentially able to outperform classical computers on specific tasks Arute et al. (2019). The available processors can be controlled locally through control lines, enabling the tuning of parameters of the Hamiltonian such as on-site energies and couplings Roushan et al. (2017b); Chiaro et al. (2019); Ye et al. (2019); Yan et al. (2019); Zha et al. (2020). Therefore, given the high degree of control the current technology already allows, it is now necessary to design a pathway towards practical applications of such devices.

Quantum machines promise dramatic speed-up in computation time for the simulation of quantum chemistry and many-body problems Babbush et al. (2018); Georgescu et al. (2014). However, a major challenge lies in encoding the actual problem onto the limited physical connectivity of the quantum simulator. Ideally, a quantum simulator would be fully-programmable such that it can achieve direct connections between arbitrary qubits that can be modified at any time. Unfortunately, each device has a fixed and limited connectivity and it is quite challenging to reconfigure it after fabrication, such that it is able to perform another task. In the long term, a fault tolerant quantum computer would allow this but that option is not available in the NISQ regimePreskill (2018). In practice, due to the hardware restrictions, one simply designs and builds quantum processors tailored to solve the particular problem of interest. This current approach to quantum simulators is quite limited and time-consuming.

In this work, we enhance NISQ devices with tailored driving sequences to become fully re-programmable quantum simulators. Our approach is based on modest assumptions of local controllability of a quantum device, which are already available in many systems Roushan et al. (2017b); Chiaro et al. (2019); Ye et al. (2019); Yan et al. (2019); Zha et al. (2020); Wei et al. (2019); Rubio-Abadal et al. (2020). This allows one to apply a periodic drive to control different parameters of the system. Thus, the Hamiltonian is periodic in time with $\hat{H}(t)=\hat{H}(t+T)$. Associated with this is the evolution operator within a period of the drive $\hat{\mathcal{F}}=\hat{U}(T;0)$, known as the Floquet operator  Bastidas et al. (2018); Eckardt (2017); Restrepo et al. (2016); Bukov et al. (2015); Oka and Kitamura (2019). As the Floquet operator is unitary, it is natural to define an effective Hamiltonian such that $\hat{\mathcal{F}}=\exp(-\mathrm{i}\hat{H}_{\text{eff}}T/\hbar)$. Now if we measure the dynamics of the system at stroboscopic times $t=nT$, the device described by the instantaneous Hamiltonian $\hat{H}(t)$ can be used to simulate the effective Hamiltonian $\hat{H}_{\text{eff}}$. Within the framework of Floquet engineering, the usual approach is that for a given driving, one can analytically obtain the effective Hamiltonian in the high-frequency regime Eckardt (2017); Restrepo et al. (2016); Bukov et al. (2015); Oka and Kitamura (2019). However, the inverse problem is highly nontrivial, because it requires the design of an appropriate driving sequence to the device in order to achieve a desired target Hamiltonian $\hat{H}_{\text{target}}$. Here, we provide a solution to this problem using GRAPE, an optimization algorithm that allows us to obtain the desired driving sequence. In figure 1 a) we depict an example of our idea, where we drive a linear chain to simulate star and ring topologies. A related work Onodera et al. (2020), focuses on stroboscopic simulation of effective Hamiltonians by driving a device with all-to-all coupling to achieve a desired effective Hamiltonian. This approach, however, requires coupling to an external bus, which can propagate errors to the whole system.

Our paper is organized as follows. In Sec. II we introduce the Bose Hubbard Hamiltonian describing interacting microwave photons and discuss the hardcore boson limit. In Sec. III we discuss GRAPE, an optimization algorithm used to find an optimal driving sequence to simulate the target Hamiltonian. We also discuss the family of Hamiltonians that can be simulated using our method. In Sec. IV we show how to apply our method to simulate arbitrary topologies by driving a one dimensional quantum processor. After that, in Sec. V we discuss how to apply our method to stroboscopically simulate a 3-Sat problem and its adiabatic deformation. We also show how to use our scheme to simulate problems in quantum chemistry in Sec. VI.

Here we tailor our approach to one of the most promising NISQ devices based on a one-dimensional array of $L$ coupled superconducting qubits, described by the Bose-Hubbard Hamiltonian Roushan et al. (2017a); Chiaro et al. (2019); Ye et al. (2019); Yan et al. (2019); Zha et al. (2020)

where $\hat{n}_{l}=\hat{a}^{\dagger}_{l}\hat{a}_{l}$ is the number operator, $\hat{a}_{l}$ and $\hat{a}^{\dagger}_{l}$ are bosonic annihilation and creation operators at site $l$, respectively. With current experimental feasibilities, it is possible to use Z and XY control lines of the superconducting processor to drive the angular frequencies of the qubits $g_{l}(t)$ and the coupling strengths $J_{l}(t)$. However, the anharmonicity $U$ is kept fixed and it is determined from fabrication. The latter plays an important role in the dynamics. For example, in the hardcore-boson regime $U\gg g_{l},J_{l}$, the large anharmonicity prevents two microwave photons from being at the same site Ye et al. (2019); Yan et al. (2019). In this case, the Hamiltonian (1) can be written in terms of Pauli matrices $\sigma^{\alpha}_{l}$ with $\alpha\in\{x,y,z\}$, as follows

It is useful to mention that the Hamiltonians (1) and (2) preserve the total number $M$ of excitations. By repeating a control sequence in a periodic fashion with a period $T$, the Hamiltonians (1) and (2) become periodic in time, i.e., $\hat{H}(t)=\hat{H}(t+T)$. Therefore, we can use Floquet theory to explore the dynamics of the system.

The cornerstone of our approach is to use an optimization algorithm to determine a driving sequence that is applied to a one-dimensional quantum processor. To achieve that, we consider the decomposition of the Floquet operator as a product of $N$ unitary operators $\hat{U}_{j}$

at different time steps $t_{j}=j\tau$ with $\tau=T/N$. This decomposition can be obtained by keeping the parameters of the system constant in certain time intervals. We can fix a set of values of the frequencies $g_{l}(t)=\bar{g}^{j}_{l}$ or couplings $J_{l}(t)=\bar{J}^{j}_{l}$ for all the qubits in the time interval $t_{j-1}\leq t\leq t_{j}$. If we let the system evolve during that time interval, we obtain the evolution operator $\hat{U}_{j}=\exp(-\mathrm{i}\hat{H}(t_{j})\tau/\hbar)$. The basic idea of the control scheme is depicted in Fig. 1 b). Given a control sequence with $N$ time steps, it is not trivial a priori to know which values $\bar{g}^{j}_{l}$ and $\bar{J}^{j}_{l}$ we should choose to obtain our desired target Hamiltonian.

Here we translate the task of finding the driving sequence into an optimization problem that we solve using GRAPE Machnes et al. (2011); Johansson et al. (2013); Wu et al. (2019), which is a well-known optimization algorithm. By using this method, we can simulate effective Hamiltonians for finite $U$ and in the hardcore boson regime, as we show below.

Let us now present some specific examples that illustrate how we can simulate a nontrivial topology starting from a simple one. Let us start with the simplest example of a single excitation $M=1$ in an array of $L=9$ sites, where $D_{1,9}=9$ is the dimension of the Hilbert space. In the single-excitation subspace, we consider the basis $|1_{l}\rangle=|0,0,\ldots,1_{l},0,\dots,0\rangle$ with $l=1,\dots,L$, where $|1_{l}\rangle$ denotes an excitation at the $l$-th site. In this case, we use Hamiltonian (1), but as we are working with a single excitation, it is also possible to use Hamiltonian (2) to obtain the same results. Consider that we want to simulate a star network topology of the effective Hamiltonian (8) or (9) as depicted in Fig. 2 a). In Fig. 2 b) we illustrate the matrix representation of target Hamiltonian in the single-excitation basis $|1_{l}\rangle$. As we discussed before, it is non-trivial to obtain the right sequence of driving parameters $\bar{g}^{j}_{l}$ and $\bar{J}^{j}_{l}$ for sites $l$ and time $t_{j}$ in order to achieve the target Hamiltonian. Using GRAPE, we obtain an effective Hamiltonian as in Fig. 2 c) that is very close to the target. This can be done experimentally by driving the onsite energies $\bar{g}^{j}_{l}$ of the chain, while keeping the couplings constant $(\bar{J}^{j}_{l}=J)$. The driving protocol is shown in Fig. 2 d).

Next let us move beyond the single excitation subspace and effectively expand the computational space by considering $M=2$ excitations in $L=8$ sites. In this subspace, we consider the basis states $|1_{l},1_{m}\rangle=|0,0,\ldots,1_{l},0,\dots,1_{m},0,\ldots,0\rangle$ with $l,m=1,\dots,L$. Here $|1_{l},1_{m}\rangle$ denotes the state of two excitations at sites $l$ and $m$. Note that here we consider $U\sim g_{l},J_{l}$, which is far from the hardcore boson regime and thus, we allow for states $|2_{l}\rangle=|0,0,\ldots,2_{l},0,\ldots,0\rangle$ describing two excitations being at the same site. In this case, the dimension of the Hilbert space is $D_{2,8}=36$. Our aim here is to simulate a target Hamiltonian (1) describing a device with all-to-all connectivity, as it is depicted in Fig. 2 e). Figs. 2 f) and g) show the target and effective Hamiltonians, respectively. The driving sequence of the onsite energies that is sufficient to simulate this topology is illustrated in Figs. 2 h). In contrast to the single particle case, here we drive both the onsite energies $\bar{g}^{j}_{l}$ as well as the couplings $\bar{J}^{j}_{l}$. In appendix C we show the driving sequence of the couplings $\bar{J}^{j}_{l}$.

In the hardcore boson regime we can simulate a target Hamiltonian (9) describing $M=3$ excitations in a ring with $L=8$ sites [see Fig. 2 i)]. In this regime, however, there is a restriction on the type of Hamiltonians that can be simulated (see methods), because the doubly and triply occupied states are forbidden due to the large nonlinearity. For this reason, we work with the basis $|1_{k},1_{l},1_{m}\rangle=|0,\ldots,1_{k},0,\dots,1_{l},0\dots,1_{m},\ldots,0\rangle$ with $k\neq l\neq m$ and the dimension of the Hilbert space is $D_{3,8}=55$. Figs. 2 j) and k) show the target and effective Hamiltonians, respectively and the driving sequence is illustrated in Fig. 2 l). In appendix E we show the scaling of our method in the hardcore boson regime as we increase the number of particles $M$.

We have shown that by exploiting tools of optimization, we can simulate complex topologies at stroboscopic times by using Hamiltonians (1) and (2). Next, let us explore a practical application of our approach as a stroboscopic simulator of a SAT solver for combinatorial optimization. The term SAT itself refers to satisfiability of equations involving boolean variables and some variants of SAT problems are hard Battaglia et al. (2005). 3-SAT problems are of utmost importance because any $k$-SAT problem can be decomposed into a sequence of $3$-SAT instances. The simplest $3$-SAT problems involve $3$-body interactions Battaglia et al. (2005). To generate this kind of interactions in near-term analogue quantum devices can be quite challenging, due to the restrictions of the hardware. Therefore, we expect our method to provide a route to design the desired target Hamiltonian. Next, to demonstrate the versatility of our approach, we explore an example of a $3$-SAT problem to solve the clauses $1+a_{2}+a_{3}+a_{1}a_{3}=0$, $1+a_{1}+a_{3}+a_{1}a_{2}=0$, and $a_{1}+a_{2}+a_{2}a_{3}=0$, where $a_{1},a_{2}$ and $a_{3}$ are boolean variables. The latter is a typical example of a system of equations that arises when performing cryptanalysis of block ciphers or hash functions Gravel et al. (2019). By using algebraic tools Mullen and Panario (2013), it is possible to map this combinatorial problem to a spin Hamiltonian

with three-body interactions, where $\mu^{\alpha}_{l}$ are Pauli matrices and $\omega$ is a parameter with units of angular frequency ( see appendix D). Thus, by finding its ground state, we can obtain the solution of the optimization problem, which in this case, corresponds to the triplet $(a_{1},a_{2},a_{3})$ satisfying the 3 clauses discussed above. In the eigenbasis of $\mu^{z}_{l}$, the matrix representation of Hamiltonian (10) is a $8\times 8$ matrix, as it is depicted in Fig.3 a). It can also be represented as a graph with high-connectivity, which is difficult to implement experimentally. With our approach, we obtain a driving sequence that stroboscopically simulate the target Hamiltonian $\hat{H}_{\text{SAT}}$, as we discuss in appendix D. This can be mapped to a single-excitation in terms of Hamiltonian (8) and (9).

Next, we explore an adiabatic deformation in the space of effective Hamiltonians by slowly switching on the driving protocol Tanaka and Nemoto (2010). In this way, we can deform a trivial diagonal Hamiltonian $\hat{H}_{\text{diag}}$ into the Hamiltonian (10) by using the Hamiltonian $\hat{H}_{\text{adiabatic}}(\lambda)=(1-\lambda)\hat{H}_{\text{diag}}+\lambda\hat{H}_{\text{SAT}}$, where $0<\lambda<1$ is slowly modulated in time. Fig.3 a) depicts the adiabatic deformation of the Hamiltonian $\hat{H}_{\text{adiabatic}}(\lambda)$ and its corresponding representation in terms of graphs for different values of $\lambda$. During the adiabatic deformation, the quasienergies exhibit multiple anticrossings, as depicted in Fig.3 b), but the system remains in its lowest quasienergy state, as illustrated in Fig.3 c). By finding the ground state when $\lambda=1$, we obtain a solution $(a_{1},a_{2},a_{3})=(1,0,0)$ satisfying the three clauses discussed above. For more details, we refer the reader to the appendices. In the next section, we provide an example of how to use our approach in quantum Chemistry to simulate a LiH molecule at bond distance, where nontrivial many-body terms can directly encoded into the effective Hamiltonian.

In the previous section we discussed an application of our method to simulate optimization problems. Here, we explore yet another application to stroboscopically simulate Hamiltonians in quantum chemistry. This is one of the most relevant near term applications of the existing NISQ devices.

Understanding biological processes and designing new pharmaceutical products is of utmost importance in todays age. To accomplish task, we have to understand the inner workings of atoms, molecules and proteins better by calculating their properties such as their quantum mechanical configuration and dynamics from first principles. However, the simulation of molecules is a difficult task for classical computers as the computational resources scale exponentially with the number of electrons that have to be calculated. Quantum computers do not suffer from this scaling and thus promise to be able to calculate large molecules which are intractable with classical computation. As benchmark, small molecules can be calculated already with state-of-the-art quantum computers Kandala et al. (2017). The Hamiltonian that describes the molecule includes non-local interactions between multiple qubits that should be controlled. Most quantum computers support only two-body interactions and thus require multiple operations just to fulfill a single non-local operation acting on multiple qubits.

With Floquet engineering, all those interactions including the complex n-body qubit interactions can directly encoded into the effective Hamiltonian. We demonstrate this by simulating a LiH molecule at bond distance in Fig.4. The corresponding parameters of the Hamiltonian can be found in Kandala et al. (2017). They have been obtained by the STO-3G basis. It approximates the atomic orbitals with three Gaussians to obtain the one and two-electron integrals for the electron interactions. For the LiH molecule, the 1s orbital of H and the 1s, 2s and 2p_x and 2p_z were assumed to be occupied, all other orbitals are assumed to be empty. By including the parity symmetries, the LiH molecule can be described effectively by 4 qubits or a Hilbert space of dimension $D_{1,16}=16$. It consists of 99 coupling terms expressed as a product of different Pauli operators (see appendix F). Fig.4 a) illustrate the instantaneous Hamiltonian of the device. We use a one-dimensional chain with a single excitation $M=1$ in $L=16$ sites. Figs.4 b) and c) show the good agreement between the target and simulated Hamiltonian, respectively. The stroboscopic simulation of the LiH molecule can be efficiently achieved by driving the onsite energies, as depicted in Fig.4 d).

In this work we have demonstrated that one can simulate a target Hamiltonian with arbitrary connectivity by driving a lattice with a simple fixed topology. To achieve this, we exploited tools of optimization to engineer a periodic drive that allows us to simulate the target Hamiltonian at stroboscopic times. We demonstrate the versatility of our approach by simulating star, all-to-all and ring connectivities. Further, we propose a practical application of our methods: We simulated the 3-SAT Hamiltonian, a paradigmatic model of combinatorial optimization. In terms of Pauli matrices, the 3-SAT model involves 3-body interactions, which are difficult to realize experimentally. We envision potential applications of our results to quantum chemistry and quantum simulation in noisy intermediate-scale quantum (NISQ) devices.

We thank M. P. Estarellas, A. Sakurai, L. Wright for valuable discussions and Y. Sato for proof-reading the manuscript. This work was supported in part by the Japanese MEXT Quantum Leap Flagship Program (MEXT Q-LEAP), Grant Number JP-MXS0118069605, the MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas Science of hybrid quantum systems Grant No.15H05870 and the JSPS KAKENHI Grant No. 19H00662. The computational work for this article was partially per-formed on resources of the National Supercomputing Centre, Singapore (https://www.nscc.sg).