Shortcuts to Adiabaticity in Digitized Adiabatic Quantum Computing

Quantum computing is known to have significant advantages in solving certain computational tasks, such as simulating quantum systems Feynman (1999); Georgescu et al. (2014); Houck et al. (2012); Lanyon et al. (2011); Gerritsma et al. (2010), machine learning Biamonte et al. (2017); Lloyd et al. (2013); Dunjko et al. (2016); Schuld and Killoran (2019), solving optimization problems Moll et al. (2018); Farhi and Harrow (2016); Arute et al. (2020), cryptography Gisin et al. (2002); Bennett and Brassard (2014), and several others. Recent advancements in quantum technologies have already shown that quantum computers can outperform currently existing classical computers Arute et al. (2019).

Quantum adiabatic algorithms (QAA) Farhi et al. (2001); Ambainis and Regev (2004); Young et al. (2010); Peng et al. (2008) are one of the leading candidates for solving optimization problems Reichardt (2004); Steffen et al. (2003); Bapst et al. (2013). In adiabatic quantum computation (AdQC), we start with a simple Hamiltonian whose ground state can be easily prepared and evolve the system adiabatically to the ground state of the final Hamiltonian, which encodes the solution of the optimization problem. This is embodied by the well-known method of quantum annealing Das and Chakrabarti (2008). Quantum annealers, such as the D-Wave machine Johnson et al. (2011), provide the test-bed for adiabatic algorithms Boixo et al. (2013). Despite its applications, quantum annealers have certain limitations, such as difficulty in implementing non-stoquastic Hamiltonian, limited qubit connectivity and noise. Although AdQC is equivalent to the standard circuit model Aharonov et al. (2008), the advantage of digital quantum computation (DQC) over quantum annealers is that the circuit model offers more flexibility to construct arbitrary interactions, and it is consistent with error correction. The recent work by R. Barends et al. Barends et al. (2016) combines the advantage of AdQC and the circuit model, termed as digitized adiabatic quantum computation (DAdQC), has been presented and implemented on a superconducting system.

QAAs are generally governed by the quantum adiabatic theorem that restricts a system to evolve along a specific eigenstate, i.e., from the ground state of an initial Hamiltonian $\hat{H}_{i}$ to the ground state of a final Hamiltonian $\hat{H}_{f}$, while the evolution is considerably slow. The computation time for the QAA depends on the minimum energy gap between the successive eigenstates during the evolution. When the system size increases, this poses a significant disadvantage for the implementation of QAA as the energy gap decreases with increasing system size, which ends up in transition between various instantaneous eigenstates. One has to increase the adiabatic evolution time to circumvent such an issue. However, in practice, evolution time for QAA is significantly larger than the coherence time of the current quantum computers, leading to the loss of fidelity of the evolution.

The techniques of “Shortcut to adiabaticity” (STA) Torrontegui et al. (2013); Guéry-Odelin et al. (2019) have been developed during the past decade and proved to be extremely useful for accelerating quantum adiabatic processes in general Takahashi (2019). Various techniques like counter diabatic (CD) driving (equivalently transitionless quantum algorithm) Demirplak and Rice (2003, 2005); Berry (2009), invariant based inverse engineering Chen et al. (2010, 2011), fast-forward approach Masuda and Nakamura (2008, 2010) are rigorously explored and implemented in several studies Schaff et al. (2010); An et al. (2016); Du et al. (2016). Among these works, studies related to quantum spin systems such as Ising and Heisenberg spin models are of particular interest due to their relevance to the applicability and ease of implementation in the development of modern-day quantum algorithms Lucas (2014). In particular, the CD driving has been useful for studying fast dynamics del Campo et al. (2012); Özgüler et al. (2018); Takahashi (2013); Hartmann and Lechner (2019); Petiziol et al. (2018), preparation of entangled states Opatrnỳ and Mølmer (2014); Petiziol et al. (2019); Ji et al. (2019); Zhou et al. (2020), quantum annealing Vinci and Lidar (2017); Takahashi (2017); Passarelli et al. (2020) and, others.

In this work, we explore the STA techniques to enhance the performance of the DAdQC using Ising spin systems. Starting from a single spin, we extend our study to many spins with nearest-neighbor interactions using the CD driving. We find out that the CD interactions in the QAA improve the fidelity remarkably compared to the previous studies. Due to the difficulty in the implementation of the exact CD term for a many-body system, we opt to find local CD terms that can drive smaller systems precisely and able to achieve the target state in very few time steps. By considering approximate CD terms using the nested commutator, we study the non-integrable Ising-model and extend this idea for the preparation of Bell state and GHZ state in larger systems.

The paper is organized as follows. In the next Sec. II, we give a detailed insight into the implementation of STA in a single spin, where the CD interaction can be exactly calculated using Berry’s formula. Sec. III explains the application of the approximate local CD driving and its limitations when applied to strongly interacting many-spin systems. In Sec. IV, we show the improvement in fidelity when the approximate CD driving is calculated using the nested commutator through the adiabatic gauge potential using the variational approach. This is followed by an example of entangled state preparation, where we show the preparation of Bell state and GHZ state for few spin systems with high fidelity. In the final Sec. V, we summarize our findings and discuss the scope for future research.

We begin our heuristic discussions with a single spin in the presence of time-dependent external magnetic-field $\boldsymbol{h}(t)$, represented by a two-level Hamiltonian, given by

where $\boldsymbol{\hat{\sigma}}$ represents the Pauli matrices, and the superscript $1$ represents the number of spin. Following the general method for AdQC, also that of quantum annealing, we express this Hamiltonian as a combination of two time-independent parts.

$\hat{H}_{i}$ and $\hat{H}_{f}$ are time-independent with ground states $\ket{\psi_{i}}$ and $\ket{\psi_{f}}$, respectively. The time dependence of the system is introduced through the parameter $\lambda(t)$. The initial Hamiltonian is chosen as $\hat{H_{i}}=h_{x}\sigma_{x}$, and the final Hamiltonian as $\hat{H_{f}}=h_{z}\sigma_{z}$, where $h_{x}$ and $h_{z}$ being the magnetic field strength along respective directions. Such a choice leads to express the magnetic field effectively as $\boldsymbol{h}(t)=[h_{x}(1-\lambda(t))\quad 0\quad h_{z}\lambda(t)]^{T}$. AdQC, in its rudimentary approach, allows $\lambda(t)$ to be any function that varies from $0$ to $1$ and drives the system from $\ket{\psi_{i}}$ to $\ket{\psi_{f}}$. Although, the most general way to choose it as a linear function, here to begin with, it is considered as $\lambda(t)=\sin^{2}{(\omega t)}$, where $\omega=\pi/2T$ with $T$ being the total evolution time. Although $\hat{H}_{0}(t)$ is extremely elementary and can easily be implemented in the circuit model Sieberer et al. (2019), there are hints that the evolution can be improved significantly using the STA Özgüler et al. (2018). In this case, one should be tempted to find out the CD term, which is somewhat straightforward to calculate using Berry (2009),

yielding the explicit form as following,

Therefore, the total Hamiltonian assisted with CD term becomes

Note that the introduction of the CD term should not affect the initial and final states, since $F^{1}(t)$ should always satisfy the boundary conditions, $F^{(1)}(t=0)=F^{(1)}(t=T)=0$. Also, the STA methods generally follow the inverse engineering approach of quantum control, i.e., designing the interaction for achieving the desired eigenstates. Therefore the notion of the eigenstate, although not that essential in traditional AdQC, turns out to be extremely important in the present case. Here, the initial state of $\hat{H}_{i}$ is chosen in the computational basis, i.e., $\{\ket{0},\ket{1}\}$ , as $\ket{\psi_{i}}=\ket{+}$ and the final state is, $\ket{\psi_{f}}=\ket{1}$. It should be noted that $\ket{+}=(\ket{0}+\ket{1})/\sqrt{2}$ and $\ket{1}$ are the natural ground states of $\hat{H}_{i}$ and $\hat{H}_{f}$, respectively, but these choices are not restricted to the ground states only. However, such a choice restraints the qubit in the ground state, minimizing the effect of the decoherences.

To implement the evolution using one qubit, we used first-order Trotter-Suzuki formula. The time evolution is digitized with $n$ number of small time steps $\Delta t$ (see Eq (18)). Ideally, the discretized version of AdQC approaches the actual adiabatic evolution for $n=T/\Delta t\to\infty$, ($\Delta t\to 0$). Although, in real situations, $n$ is finite and it has to be a relatively small number since each trotter step is being implemented by three rotation gates (see Appendix A). The error associated with the first-order trotterization is $\mathcal{O}({\Delta t}^{2})$ Suzuki (1976).

To perform the simulation, we used publicly available five qubit superconducting quantum computer of IBM Quantum Experience ibm . For the single spin experiment we use qubit $Q_{0}$ on $ibmq\_essex$. Since the single qubit gate error of the device is of the order of $10^{-4}$, the initial state is prepared with very high fidelity. Also a significant error in this simulation comes from the readout error $(\sim 4\%)$, for that we used the error mitigation technique using matrix inversion method described in Appendix C.

In the simulation, time evolution takes place between $\ket{+}$ to $\ket{1}$ and measured in the computational basis, which restricts the variation of the probability between $0.5$ to $1$, see Fig. 2. Since the trotter error is of the order $\mathcal{O}(\Delta t^{2})$, we choose, $\Delta t=0.2$, and $T=1$ for the comparison of the evolution using DAdQC and STA assisted DAdQC. To study the probability of final ground state $P_{gs}=|\braket{\psi(t)}{1}|^{2}$, where $\ket{\psi(t)}$ denotes the time evolved state of $\hat{H}_{0}(t)$, measurement is performed at each progressing time step. Fig. 2(b) shows a single evolution of the qubit governed by both Eq. (2) and Eq. (5), using both simulation and experiment on a real quantum device. With the application of CD driving $F^{(1)}(t)$, the probability of getting the state $\ket{1}$ from the experiment comes out to be around $0.997$. Whereas, when the CD driving is zero, i.e., for the adiabatic evolution, the final state probability is around $0.561$ only. However, if the evolution is extended for a larger $T$, we could obtain a much higher probability, even without $F^{(1)}(t)$, a signature of the typical adiabatic process. This is evident in Fig. 2(a), where the fidelity of the evolution (in the computational basis) for different $T$ is shown. Even when $T=1$ ($\Delta t=0.02$) using the STA method, the final ground state is reached with nearly unit fidelity. We observe that fidelity for the STA method for large $T$ maintains its value at around $0.978$. However, for the adiabatic case, fidelity gradually increases with increasing simulation time and the average fidelity will be around $0.927$ for $T=5$. Notice that in Fig. 2(a), the experimental values differ slightly from the exact simulation values, and the difference is slightly larger for the STA assisted case. As $T$ increases, the circuit depth becomes larger, which results in ramping up the gate errors, affecting STA more than the adiabatic case as it requires more gates for implementation.

The results in Sec. II establishes the fact that STA assisted DAdQC shows significant improvement over the DAdQC, at least when a single qubit is considered. However, such implementation becomes far more interesting when multiple qubits are considered. The simplest choice is a system of $N$ interacting spins in one-dimensional lattice, coupled by a time-dependent exchange interaction $J(t)$ with a rotating magnetic field is acting upon it. Here we consider $J(t)$ to constitute $\sigma_{z}\sigma_{z}$ type interaction with $J_{0}$ being the coupling amplitude. The spins are initially aligned along the transverse magnetic field, $h_{x}$, while an Ising Hamiltonian represents the system’s final state. The total Hamiltonian is represented as,

The scheduling $\lambda(t)$ is chosen similarly, as in Eq. (2). The traditional approaches to finding the CD driving are predominantly limited to two and three-level systems and become more complex for higher dimensional many-body systems. However, for interacting many spin systems, as in the preceding section, a local CD driving could be more useful. Instead of acting on the whole system, a set of approximated interactions could be designed to control the spins individually. Such type of local CD driving is more general and can be extended to a larger number of spins.

Following the discussion in the preceding section, when complex many-body systems are considered, the calculation of the exact CD term becomes difficult. Also, the form of the CD term can be severely complicated with different non-local and many-body interaction terms. Besides, it becomes rather difficult to implement systems with such interactions on current quantum computers. Although, the local terms, see Eq. (9), from variational approach gives an optimal solution, it is not that useful for preparing entangled states, especially when $h_{z}^{(j)}=0$. The nature of the CD term from variational calculation depends on the choice of appropriate adiabatic gauge potential $\hat{A}_{\lambda}^{*}$. A recently proposed method gives a more general way to choose the gauge potential by using the nested commutator (NC) Claeys et al. (2019),

where $l$ determines the order of the expansion. Depending on the required accuracy, we can keep the number of variational coefficients small. If we consider only the first-order term, our ansatz will be $\hat{A}_{\lambda}^{(1)}=i\alpha_{1}(t)[\hat{H},\partial_{\lambda}\hat{H}]$, and the effective Hamiltonian can be written as,

where $\dot{\lambda}{\hat{A}}_{\lambda}^{(1)}$ is the relevant CD term.

First of all, we apply this technique to non-integrable Ising spin model, described by the Hamiltonian in Eq. (6). Considering the two-qubit system ($N=2$), we approximate CD term using first-order nested commutator,

with,

The second-order term ($l=2$) can give the exact gauge potential Claeys et al. (2019). However, for the experimental demonstration, we only consider the first-order term and implement the time evolution on a quantum processor. The circuit implementation for the CD driving is shown in Appendix A. Using this method, the final ground state is achieved with very few trotter steps compared to digitized adiabatic evolution, which drastically reduces the number of gates required as well as the total simulation time. In Fig. 5, we depicted the fidelity as a function of evolution time using first-order nested commutator method when the final ground state is degenerate and compared the result with the local CD term from Eq. (9). The fidelity is much better compared to the local CD case for degenerate state and thereby it justifies our argument in the preceding section.

Secondly, we shall check the reliability and validity as well as the extent of the variational approach in the many body regime. To this aim, we apply this technique to prepare the GHZ state in Ising spin chain with many spins, described by the Hamiltonian,

with $N$ being the number of spins. Here, the periodic boundary condition $\sigma^{N+1}=\sigma^{0}$ is assumed. Following the procedure described in Sec. IV, by considering only the first-order expansion, we calculate the approximate gauge potential as,

The variational coefficient, $\alpha^{N}_{1}(t)$ is calculated by minimizing the action $S$. For the experimental demonstration on a quantum processor, we choose a small system with two and three qubits to prepare a Bell state and GHZ state. For the bell state, $(\ket{00}+\ket{11})/\sqrt{2}$, governed by the Hamiltonian in Eq. (14), the variational coefficient is calculated as $\alpha_{1}(t)=-J_{0}h_{x}/2[J_{0}^{2}\lambda^{2}+4(1-\lambda)^{2}h_{x}^{2}]$. Here we have noticed that, for two spins, the first order commutator is proportional to the higher-order terms. The resulting CD driving from the approximate gauge is exact and produces unit fidelity in ideal situations Claeys et al. (2019). The same procedure can be followed in case of more qubits to prepare a GHZ state $\ket{GHZ}=(\ket{0}^{\otimes N}+\ket{1}^{\otimes N})/\sqrt{2}$ starting from the $N$ qubit ground state $\ket{+}^{\otimes N}$. Specifically, the variational coefficient for a three-qubit case is given by ${\alpha}_{1}(t)=-J_{0}h_{x}/[5J_{0}^{2}\lambda^{2}+8(1-\lambda)^{2}h_{x}^{2}]$.

The simulation was performed on a five qubit quantum processor ibmq_ourense. A similar trotterization, as in Eq. (20), is used to study the evolution with digitized time step $dt=0.01$. Using the CD driving, with only three trotter steps, the desired bell state is obtained with experimental fidelity $0.984$. The ideal digital simulation gives almost unit fidelity ($F=0.999$). The fidelity is calculated as, $F\left(\rho_{1},\rho_{2}\right)=\braket{\psi_{1}\left missing}{\rho_{2}}{\psi_{1}}$, where the exact bell state is represented by, $\rho_{1}=\left|\psi_{1}\right\rangle\left\langle\psi_{1}\right|$. Similarly, for the three-qubit system, the ideal digital evolution gives the fidelity $0.935$ with the exact GHZ state, and the corresponding experimental fidelity is $0.819$. The density matrix representation of the final state ($\rho_{2}$) is obtained by performing quantum state tomography for both Bell and GHZ states and is depicted in Fig. 6. Whereas, Fig. 7, shows how the fidelity varies with increasing system size on a ideal digital simulator with six trotter steps. The first-order approximation of the CD term provides high fidelity for small system size. As we increase $N$, the probability of obtaining the final ground state decreases gradually. This can be overcome by considering the higher-order commutators while calculating the approximate CD term. As the variational method tends to provide exact CD driving for larger $l$-values, which in principle, can give a better fidelity in many-body systems.

In conclusion, we have demonstrated the implementation of digitalized STA on a superconducting quantum processor. The problem Hamiltonian, chosen for the simulation, emulates the one-dimensional Ising spin chain. The STA is realized by means of the local and approximate CD driving, which is obtained using mainly two methods: the long-established Berry’s algorithm and the newly proposed variational approach. The CD term in our simulation is non-stoquastic in nature, therefore it can’t be simulated efficiently on a classical computer. The effective Hamiltonian is implemented using the available quantum gates, and the time evolution of the system is studied to achieve the ground state of the problem Hamiltonian. We have showed that the time steps required to reach the target state are minimal compared to the DAdQC method, leading to minimal loss due to the decoherences and accumulated gate errors. The local CD driving proved to be very effective for weakly interacting spin chains; however, is not useful in case of strong interaction and fails to produce degenerate target states. To remedy this situation, the approximate CD driving is useful, which can be calculated using the nested commutator method. We applied the first-order approximate CD term to prepare Bell and GHZ state with high fidelity for few qubit systems. Furthermore, for the many qubit systems, the fidelity can be further improved with higher-order terms at the cost of gate numbers.

This work provides the evidence that significant enhancement of the DAdQC approach can be achieved using the STA methods by decreasing the total computational cost and hence achieving the desired results within the coherence time of the device. To our knowledge, this work is the first to successfully realize the STA methods in a contemporary superconducting circuit-based quantum computer. Due to the decoherence and gate errors, the time evolution studies are challenging to perform in such devices. Our result provides a beginning to such studies on the speed up of AdQC algorithms.

The authors are grateful to Kazutaka Takahashi for useful discussions. We acknowledge support from National Natural Science Foundation of China (NSFC) (11474193), STCSM (2019SHZDZX01-ZX04, 18010500400 and 18ZR1415500), Program for Eastern Scholar, Ramón y Cajal program of the Spanish MCIU (RYC-2017-22482), QMiCS (820505) and OpenSuperQ (820363) of the EU Flagship on Quantum Technologies, Spanish Government PGC2018-095113-B-I00 (MCIU/AEI/FEDER, UE), Basque Government IT986-16, as well as the and EU FET Open Grant Quromorphic. This work is also partially supported from Huawei HiQ funding for developing QAOA&STA (Grant No. YBN2019115204).