Qualitative quantum simulation of resonant tunneling and localization with the shallow quantum circuits

Quantum computing can be used to investigate quantum systems as a universal simulator [1, 2, 3]. In quantum mechanics, the time evolution of quantum states is driven by a Hamiltonian and described by a unitary operator. In a digital quantum computer, the computing is carried out by using a set of basic quantum gates, and usually each gate is a single-qubit or two-qubit unitary operator. The combination of these basic gates allows us to implement the evolution operator of a multi-qubit system. A specific approach is Trotter-Suzuki decomposition [4, 5, 6, 7], in which we approximate the continuous-time evolution with a discrete-time evolution. For a general local-interaction Hamiltonian, we can explicitly construct the evolution operator for a short time, i.e. one time step, from quantum gates. By repetitive gates of one time step for $N_{T}$ times, we realize the target time evolution. With a smaller step size, the discrete-time evolution is closer to continuous-time evolution, however, this requires a larger $N_{T}$, i.e. more quantum gates. Considering a practical device [8], quantum computing is inaccurate due to decoherence and imperfect control, and usually the error increases with the gate number [9, 10, 11]. Fault-tolerant quantum computing using quantum error correction is able to remove the error but impractical using today’s technologies, because of the large qubit overhand for encoding [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. A family of practical methods have been developed to mitigate errors, however the gate number is usually limited due to the finite error rate on the physical level [26, 27, 35, 29, 30, 31, 32, 33, 34, 35, 36]. Therefore we can only realize the discrete-time Trotter evolution with a small number of Trotter steps. This motivates researches on the effect of large step size, i.e. few Trotter steps. Trotter errors induced by large step sizes in digital quantum simulation have received extensive attention[37, 38, 39, 40, 41]. Some studies show that the Trotter step sizes can separate quantum chaotic phase from localized phase and comparatively large Trotter steps can retain controlled errors for local observables[42, 43].

In this paper, we are interested in, when the step size is large (or equivalently the steps are few), whether some typical physical effects in the limit of continuous-time can still be observed. The physical effects that we focus on are the resonant tunneling phenomenon and the localization in disordered systems [44, 45]. We find that, in a large-step-size Trotter circuit, the resonant tunneling with $n$ resonant peaks can be observed in circuits with $n+1$ Trotter steps. Experiments on an IBM quantum computer are implemented to demonstrate the resonant tunneling with up to three peaks. We also study the spin transport with the disordered configurations of the $R_{z}$ gates (we will specify these gates later) using the large step size. The numerical simulation of circuits with $15$ qubits and tens of Trotter steps exhibits the localization in the disordered configuration. The results indicate that shallow quantum circuits on near-term quantum computers are sufficient to qualitatively simulate some significant physical phenomena. The localization phenomenon of the spin excitation distribution implies that the bit-flip error does not affect the measurement on distant qubits if the configuration of the $R_{z}$ gates is disordered. These conclusions can be generalized if we replace XY gates with controlled-$R_{x}$ gates, which can transform one spin excitation into multiple spin excitations.

This paper is organized as follows. In Sec. 2, we discuss the quantum transverse-field XY model and corresponding quantum circuits, the map between the Hamiltonian of model and corresponding circuit is established in the limit of small Trotter step size. In Sec. 3, we discuss the propagation of the spin excitation, and compare resonant tunneling effects in circuits in the small-step-size limit and the large-step-size limit. In Sec. 4, we investigate the transport of the spin excitation in the ordered and disordered configurations of single-qubit $R_{z}$ gates. Conclusion is given at the end of the paper.

In this paper, we study the particle transport in the discrete-time evolution in the quantum transverse-field XY model. The purpose of this study is to investigate whether some typical quantum phenomena occurring in the continuous-time limit can be qualitatively observed in discrete-time evolution when the step size is large. The typical quantum phenomena we concerned here including resonant tunneling and localization effect, which caused by interference during the particle transport. Generally, the quantum transverse-field XY model can be used to describe the propagation of spin excitations, or particle transport when spin excitations can be treated as particles [46]. Additionally, in the quantum transverse-field XY model, the corresponding discrete-time evolution operator can be mapped into a quantum circuit according to the Trotter-Suzuki decomposition. Therefore, we investigate the particle transport with the quantum transverse-field XY model

where $j$ is the lattice site, $N$ is the system size, $J_{j}$ is the interaction strength and $V_{j}$ is the transverse-field strength. $H_{j,j+1}^{XY}=(\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y})/4$ and $H_{j}^{Z}=\sigma_{j}^{z}/2$, where $\sigma_{j}^{i}$ $(i=x,y,z)$ represents the Pauli matrix at the $j$th site. In the case of $J_{j}<0$ and $V_{j}<0$, the ground state of the transverse-field XY chain is $\left|00...0\right\rangle$. In this work, we consider the time evolution of the initial state $\left|\psi(0)\right\rangle=\left|10...0\right\rangle$ which represents a spin excitation on the first site. The time evolution is in the subspace of single spin excitation.

The time evolution operator $U(t)$ of quantum transverse-field XY model can be approximated with a quantum circuit. According to Trotter-Suzuki decomposition, $U(t)$ can be expanded approximately

where $N_{T}$ is the number of Trotter steps, $\tau=t/N_{T}$ is the size of each Trotter step. $U_{j,j+1}^{XY}(\theta_{j})=e^{-iH_{j,j+1}^{XY}\theta_{j}}$ and $U_{j}^{Z}\left(\phi_{j}\right)=e^{-iH_{j}^{Z}\phi_{j}}$, where $\theta_{j}=J_{j}\tau$ and $\phi_{j}=V_{j}\tau$.

The corresponding discrete-time evolution is realized with the quantum circuit as shown in Fig. 1(a). The time evolution of each term, i.e. $U^{XY}$ and $U^{Z}$, are two-qubit XY gate and single-qubit $R_{z}$ gate, respectively. The circuit has $N$ qubits $\{q_{1},q_{2},...,q_{N}\}$ and $N_{T}$ Trotter steps, and every Trotter step contains one layer of two-qubit XY gates and one layer of single-qubit $R_{z}$ gates (We neglect $R_{z}$ gates in the last Trotter step, because these gates does not have any effect on the distribution of the spin excitation). In a quantum circuit, a NOT gate (i.e. the Pauli $\sigma_{x}$ matrix) can flip the qubit $\left|0\right\rangle$ to $\left|1\right\rangle$, therefore we prepare the initial state $\left|\psi(0)\right\rangle$ by applying a NOT gate on the first qubit, i.e. $\left|\psi(0)\right\rangle=\sigma_{1}^{x}\left|0\right\rangle^{N}=\left|10...0\right\rangle$ [see the dashed rectangle in Fig. 1(a)]. The matrix representation of XY gate is shown in Fig. 1(c).

The behavior of a spin excitation under discrete-time evolution depends on step size. When the Trotter step size is sufficiently small, the propagation of a spin excitation in quantum circuit is equivalent to the particle transport in continuous-time evolution. In this case, the propagation of spin excitation can exhibit some typical physical phenomena in particle transport. A natural question to ask is, in shallow quantum circuits with a large Trotter step size, whether some physical phenomena during continuous-time evolution nevertheless remains, so that we can observe these phenomena using fewer quantum gates. In the following text, we show that we observe the resonant tunneling and localization effect in shallow circuits even if the Trotter step size is large.

In this section, we study the resonance phenomenon related to the transport of the spin excitation during the discrete-time evolution with a large step size. We study the situation where the size of the circuit system is $N=2,3,4,5$ respectively, where the spin excitation is created by a NOT gate and transmitted through the XY gate. In each one case, we qualitatively find the resonance tunneling in the limit of continuous-time evolution and give the corresponding minimum number of Trotter steps.

We also study the transport of a spin excitation through the controlled-$R_{x}$ gate. In quantum computing, computational errors occur due to the imperfect control and decoherence always error. The typical one is bit-flip error corresponding to an unwanted NOT gate. So studying the transmission of the spin excitation can help us understand the propagating of bit-flip errors. In actual quantum circuits, errors are not only transmitted, but also replicated. For example, a single-qubit error will become a multi-qubit error after passing through the controlled-$R_{x}$ gate. This effect potentially has a greater impact in quantum computing, because multi-qubit errors can lead to the failure of quantum error correction. So in this section, we also study the behavior of a spin excitation propagating through the controlled-$R_{x}$ gate and find the resonance phenomenon similar to that propagating through the XY gate.

In this section, we study whether the localization can be observed in the discrete-time evolution when the Trotter step size is large. We still study the transverse field XY model, but the parameters of a layer of $U^{Z}$ (i.e. a layer of single-qubit $R_{z}$ gates) are random. We compare the probability distribution of the spin excitation in different configurations of the parameters of a layer of $R_{z}$ gates. The stronger the randomness of the parameters, the higher the localization of the distribution of the spin excitation, which means higher the probability of observing the spin excitation near a specific qubit. In this study, the propagation of the bit-flip error (i.e. an unwanted NOT gate) is similar to the transport of the spin excitation. Therefore, the localization indicates that the bit-flip error may be localized near a specific qubit and may not affect the measurement on distant qubits.

The localization phenomenon is studied[52, 53, 54, 55, 56] during the discrete-time evolution with a large step size. We begin with the Hamiltonian in Eq. (1) with $J_{j}=J$. For convenience the parameters are defined as $J\tau=\theta,V_{j}\tau=\phi_{j}.$ For the purpose of investigating localization, one layer of parameters for $U^{Z}$ in Eq. (2) is {$\phi_{j}$}$\equiv${$\phi_{1},-\phi_{2},\phi_{3},-\phi_{4},\cdots$}, where $\phi_{j}=\phi+r_{j}$, $r_{j}\in[-R$, $R]$ is a random number. {$\phi_{j}$} is the same for each layer of $U^{Z}$. {$\phi_{j}$} is ordered (periodic) configuration when $R=0$ and disordered configuration when $R>0$. The inverse participation ratio [57] (IPR) is a measure of localization and defined as

where

$\left|\psi(\eta)\right\rangle$ represents the quantum state at the $\eta$th Trotter step, $P_{i}(\eta)$ represents the corresponding amplitude at the $i$th qubit. In general, IPR_η varies from $1/N$ (system size) to $1$ and a large value of IPR_η means a stronger localization effect. The localization of $\left|\psi(\eta)\right\rangle$ changes with $\eta$, thus the average IPR is introduced to character the average level of localization during the whole discrete-time evolution[42],

In Fig. 4(a), we plot $IPR_{\eta}$ (the solid lines) as function of $\eta$ for the ordered ($R=0$) and disordered ($R=\pi/2$) configuration respectively. The drawing parameters are $N=N_{T}=80,\theta=\phi=\pi/2$. $IPR_{\eta}$ for both the ordered (the red lines) and disordered (the red lines) configuration show a periodic-like behavior and is larger than $1/N$, which means $\left|\psi(\eta)\right\rangle$ exhibits localization effect in both cases. However, the average value (the horizontal line), i.e. $IPR_{ave}$, of the blue line is larger than the red line, and the peaks of the blue line are closer to $1$. This indicates stronger localization in the disordered case. Furthermore, in Fig. 4(b), we show the $IPR_{ave}$ varying with the degree $R$ of the randomness. With the increase of disorder, the localization becomes stronger.

In this study, the propagation of the bit-flip error (i.e. an unwanted NOT gate) is similar to the transport of the spin excitation, thus the localization implies that single bit-flip error propagated by disordered {$U_{j}^{Z}(\phi_{j})$} does not affect the measurement on a distant qubit. To illustrate this point, we propose a physical quantity $P_{\mathrm{t}}\equiv\sum_{q_{i}=2N/3}^{N}p_{q_{i}}$ which is the average probability of finding the spin excitation on the last third of the qubits, where $p_{q_{i}}$ denotes the probability of finding the error on the $i$th qubit. The smaller the $P_{\mathrm{t}}$, the shorter the distance the spin excitation travels. As shown in Fig. 4(c), $P_{\mathrm{t}}$ is lower when $r_{i}\neq 0$, which demonstrates that only a little probability is propagated to the last several qubits. $P_{\mathrm{t}}$ is almost vanishing as the degree of randomness keeps increasing. In Fig 4(d), we plot $p_{q_{i}}$ at $\eta=10$. As we can see, more probabilities are propagated to the last few qubits for $r_{i}=0$ and are localized at the first few qubits for $r_{i}\neq 0$. Figure 4(d) also show that the greater the degree of randomness, the stronger the localization phenomenon. Above results indicates that the measurement is almost unaffected by the bit-flip error on the first qubit for $r_{i}\neq 0$.

The above conclusion still holds if XY gates in the circuit are replaced by controlled-$R_{x}$. As shown in Fig. 3(c), $P_{t}$ grows with the increasing Trotter steps. However, $P_{\mathrm{t}}$ becomes lower when the random perturbation is applied to a layer of $R_{z}$ gates, which means less probabilities are propagated to the end of the circuit. With the increasing degree of disorder, the inhibiting effect is more significant. In Fig. 3(d), we plot the probability distribution of spin excitation when $\eta=10,\theta=\phi=\pi/2$. These results show that, with a strong randomness of {$\phi_{j}$}, the spin excitation will not be propagated to the farther qubits. At the same time, this also shows that the randomness of the parameters will inhibit the propagation of the bit-flip error and protect the measurement on the distant qubits is not affected.

We study the transport of the spin excitation in the discrete-time evolution using Trotter circuits with a large step size, and qualitatively observe the quantum phenomena in the continuous-time limit, i.e. the resonant tunneling and localization. We observe the resonance phenomenon during the transport of the spin excitation in systems of sizes $N=2,3,4,5$ respectively. The probability distribution of spin excitations propagating through several Trotter steps agree qualitatively with that in the continuous-time limit. The corresponding minimum number of Trotter steps is given for each system size. In a Trotter circuit with random parameters of $R_{z}$ gates, we can observe the localization phenomenon of the spin excitation distribution even with a large step size. We study the spin excitation propagating through the XY gates and also through the controlled-$R_{x}$ gates. Our research indicates that a discrete-time quantum simulator with a large step size can qualitatively demonstrate some physical phenomena in the continuous-time limit. Qualitative observations require fewer quantum gates than quantitative calculations and therefore are a promising application on near-term quantum computers. In quantum computing, some errors, such as bit-flip errors, behave like spin excitations, thus, our finding can be used to understand the propagating of these errors in the quantum circuits.