Implementation of a two-dimensional quantum walk using cross-Kerr nonlinearity

Wei-Chao Gao; Cong Cao; Xiao-Fei Liu; Tie-Jun Wang; Chuan Wang

doi:10.1364/OSAC.2.001667

1. Introduction

Quantum walk (QW) [1] is the quantum analogues of classical random walk which has exponential superiority over its classical counterpart in virtue of the coherent superposition and quantum interference. It provides a versatile tool for the realization of quantum algorithms and quantum simulation [2–5]. At a certain computational errand, QW provides an polynomial speedup counterpointing to the classical computation which could be employed as an effective tool to simulate the artificial networks [6], energy transport in the photosynthetic process [7,8], graph isomorphism [9], and topological phase [10–13]. Both the variants, discrete-time and continuous-time QW, have been shown to be universally quantum computation primitive which have great potential for realizing the vast majority of quantum computation tasks [14,15]. Inspired by the prospects of QW, controlled evolution of QW has been demonstrated in various ways experimentally, including the nuclear magnetic resonance [16,17], trapped neutral atoms [18], trapped ions [19,20], cold atoms [21,22], and photonic systems [23–27].

Among the various experimental implementations, photonic systems, especially the free-space linear optical systems, possess some extraordinary features. Benefit from its properties of fast transmission, spatial flexibility and low decoherence rate, photonic systems are considered as the ideal platform and several important progresses have been done [28–32] which are related to the problems based on the QW system exploiting the spatial mode of photons created by the beam displacer (BD) as the information carrier. Recently, an important method in quantum information processing using the weak cross-Kerr nonlinearity is discovered, that the weak nonlinearity could be compensated by using a strong coherent probe field. The phase information of the coherent state is generated as a special degree of freedom which can expected to perform infinite step QW system. The proposed method not only fulfills the peer-previous works but also extirpats the evident limitation: the evolving scale of 2D QW remains in small scale. Because the simple demonstration of finite-step two-dimensional QD has been unable to meet the growing demand for quantum simulations that could be further accelerated or more complex with quantum search algorithms [33,34].

Here in this study, we propose a scheme to realize the 2D QW with the assistance of auxiliary coherent state for unlimited steps. The dynamics of the discrete-time variables are defined on two Hilbert spaces, one contains the spatial positions (orbital angular momentum (OAM) state and phase information state), and the other is the Hilbert space of the particles in which the unitary quantum coin operation could be operated followed by conditional shift operators. Exploiting the spatial light modulator (SLM) and $\textbf {X}$ homodyne detection, the position of the quantum walk can be readout independently. The numerical simulation of the feasibility shows that the success probability of the deterministic measurement remains a high value which is not affected by the steps of the quantum walk. Moreover, the proposed scheme can be further used to demonstrate more complex quantum algorithms using photonic systems.

2. Two-dimensional quantum walks using cross-Kerr nonlinearity

Recently, the study of the weak cross-Kerr nonlinearity attracts much attention in quantum information science. It has been used in various tasks of quantum information processing, such as the quantum gates [35,36], entanglement generation [37], entanglement purification and concentration [38,39], and Bell state analysis [40]. Here the cross-Kerr media plays a key role in creating the spatial mode information of the photons during the quantum nondemolition measurement and parity check. In detail, suppose there is a nonlinear weak cross-Kerr interaction between a single photon state (photonic qubit) $|\phi \rangle =\mu |0\rangle +\nu |1\rangle$ ($\mu ^2+\nu ^2=1$) and a coherent sate $|\alpha \rangle$. The evolution of the system can be described as:

(1)$$U_{cross-Kerr}|\phi\rangle= e^{i\chi n_{a} n_{c}}(\mu|0\rangle+\nu|1\rangle)|\alpha\rangle=\mu|0\rangle|\alpha\rangle+\nu|1\rangle|\alpha e^{i\theta}\rangle,$$

where $\theta$ is induced by the nonlinearity, $n_{a}(n_{c})$ denotes the number operator for mode $a(c)$. And $\chi$ denotes the coupling strength of the nonlinearity. $|0\rangle$ and $|1\rangle$ represent the vacuum state and single-photon state, respectively. Traditionally, through a general homodyne-heterodyne measurement on the phase of the coherent state, the single qubit state $|\phi \rangle$ can be projected onto the Fock state with definite number, or the superposition of the Fock state. In other words, the characteristics of the photon number state decisively influence the phase distribution of the coherent state.

Besides the novel features of the weak cross-Kerr nonlinearity mentioned above, another important merit is that orbital angular momentum (OAM) state of light is associated with the spatial distribution of the light wave, which implies a useful degree of freedom that has no limitation on the dimensions. It has been widely studied in the fields of both classical and quantum optics since it was first introduced [41]. It has demonstrated that beams with OAM have helical phase fronts and possess an azimuthal phase dependence of $e^{i l \vartheta }$, where $l$ (azimuthal phase index of integer value) indicates the number of azimuthal phase rotations in one full cycle. The OAM-carried light also has been widely used in many fields, such as optical manipulation [42,43] optical trapping [44], optical vortex knots [45], imaging [46], free-space information transfer and communications [47], and quantum information processing [48–50].

A discrete-time QW can be described by the repeated application of the unitary evolution operator $U$, which acts on a Hilbert space $H^{C_{1, 2}}\otimes H^{S_{x,y}}$, where the Hilbert space $H^{C_{1, 2}}$ and $H^{S_{x,y}}$ are associated with a quantum cion (coin space) and the nodes of the two-dimensional (2D) graph, respectively. And the operator $U$ in 2D QW can be specified as:

(2)$$U=S_{y}C_{2}S_{x}C_{1},$$

where $C_{1, 2}$ is the unitary matrix which corresponds to “tossing” the quantum coin, visualizing to analogy to a classical random walk, to generate the quantum coin state. While $S_{x}$ and $S_{y}$ are permutation matrixes that are performed a controlled shift based on the state of quantum coin.

Here we propose a scheme of infinite steps 2D quantum walk in which the OAM state of single photon and phase information of the auxiliary coherent state induced by cross-Kerr nonlinearity serve as two distinct degrees of freedom in the position space of 2D QW, and illustrated in Fig. 1. The coin state of 2D QW is encoded by the spatial mode ($|U\rangle$ and $|D\rangle$ ) of the Fock state, and the 2D lattice is mapped to a grid implemented by the OAM states ($|l\rangle$) and phase information ($|\alpha e^{ i\theta }\rangle$) of the auxiliary coherent state ($|\alpha \rangle$). The evolution of each step in the 2D QW consists of four processes:

Fig. 1. Schematic diagram showing the principle of implementing 2D QW. The coin state of 2D QW is encoded by the path modes $|U\rangle$ and $|D\rangle$, and the 2D lattice is mapped to a grid with the help of the phase information of the coherent state $|\alpha \rangle$ indeced by cross-Kerr media and the OAM states of the photon number state resorting to q-plate. SLM (spatial light modulator) and $\textbf {X}$ (homodyne-heterodyne measurement) are employed to read the site $x$ and site $y$, respectively.

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(i) At first, a single photon is transmitted through a beam splitter (BS) which has two accessible path modes (denoted as the coin state $|U\rangle$ and $|D\rangle$) with the same probability. It represents the process of tossing a quantum coin, and the operation can be described by a Hadamard transformation as follows:

(3)$$H\left[ \begin{array}{c} |U\rangle \\ |D\rangle \end{array}\right] = \frac{1}{\sqrt{2}}\left[ \begin{array}{c} |U\rangle +|D\rangle\\ |U\rangle -|D\rangle\end{array}\right] :=C_{1}\left[ \begin{array}{c} |U\rangle \\ |D\rangle \end{array}\right],$$

where the two paths are specified by the coin states labeled $|U\rangle$ and $|D\rangle$.

(ii) The different sites $x$ in the quantum walk is denoted by different OAM states ($|l\rangle$) of the single photons, here the movement of the position along the $x$ direction to the left or to the right is controlled by the path modes described with the conditional shift operator:

(4)$$\sum_{l}|l+1\rangle \langle l|\otimes|U\rangle \langle U|+|l-1\rangle \langle l|\otimes|D\rangle \langle D|,$$

where $|l\rangle$ represents the OAM state of the photon. For clarity, we formally replaced the $|l\rangle$ with the site $|x\rangle$ and rewrite the conditional shift operator as:

(5)$$\sum_{x}|x+1\rangle \langle x|\otimes|U\rangle \langle U|+|x-1\rangle \langle x|\otimes|D\rangle \langle D|:=S_{x}.$$

(iii) Subsequently, the operator $C_{2}$ is similar to the $C_{1}$ operation which has been manipulated to generate quantum coin state,

(6)$$C_{2}\left[ \begin{array}{c} |U\rangle \\ |D\rangle \end{array}\right] =\frac{1}{\sqrt{2}}\left[ \begin{array}{c} |U\rangle +|D\rangle\\ |U\rangle -|D\rangle\end{array}\right].$$

(iv) The phase information ($|\alpha e^{ i\theta }\rangle$) of the coherent state ($|\alpha \rangle$) is defined as the other degree of freedom in the 2D quantum walk. Owing to the interaction of the cross-Kerr nonlinearity, the path modes ($|U\rangle$ and $|D\rangle$) of the Fock state (the walker) possess an absolute dominant relationship with the phase information ($|\alpha e^{ i\theta }\rangle$, sites $y$) of the coherent state, which decides the movement of the position along the $y$ direction to the left or to the right. The conditional shift operator can be expressed as:

(7)$$\sum_{\theta}|\alpha e^{i\theta}\rangle \langle \alpha|\otimes|U\rangle \langle U|+|\alpha e^{-i\theta}\rangle \langle \alpha|\otimes|D\rangle \langle D|,$$

here the phase information $e^{\pm i \theta }$ is induced by the cross-Kerr nonlinearity. Similarly, the $|\alpha \rangle$ is replaced with the site $|y\rangle$ and the conditional shift operator can be rewritten as:

(8)$$\sum_{y}|y+1\rangle \langle y|\otimes|U\rangle \langle U|+|y-1\rangle \langle y|\otimes|D\rangle \langle D|:=S_{y}.$$

So, for a given single Fork state walking on a 2D lattice, the unitary operator of a single step is:

(9)$$U^{QW}=S_{y}C_{2}S_{x}C_{1}.$$

As illustrated in Fig. 1, suppose an auxiliary coherent light and a single photon in the up mode are injected into the input ports of the optical circuits. The initial state of the composite system is prepared in $|\psi \rangle ^{0}=|U\rangle |x\rangle |y\rangle$ (when $x=y=0$, which means that the single photon with the Laguerre-Gaussian (LG) modes and the coherent state with the zero phase information). The 2D QW dynamics is governed by the unitary evolution operator $U^{QW}$. For example, the evolution of the state after the walking of one step satisfying

(10)$$\begin{aligned}U^{QW}|\psi\rangle^{0}\Rightarrow & \frac{1}{2} |U\rangle |x+1\rangle |y+1\rangle +\frac{1}{2} |D\rangle |x+1\rangle |y-1\rangle \\ & +\frac{1}{2} |U\rangle |x-1\rangle |y+1\rangle -\frac{1}{2} |D\rangle |x-1\rangle |y-1\rangle. \end{aligned}$$

Obviously, by measuring the OAM state and phase information state, the probability of distribution of the walker at the site $x$ and $y$ on the 2D lattice are: 1/4 at position (1, 1), 1/4 at position (1, −1), 1/4 at position (−1, 1), and 1/4 at position (−1, −1), respectively. In Figs. 2(a)-(c), we numerically simulated the probability distribution after the twentieth step 2D QW based on the feasibility analysis measurement. To further manifest the accuracy and effectiveness of the scheme, we introduce the quantity of similarity which is defined as $S_{[P_{th}(x,y),P_{nu}(x,y)]}=(\Sigma ^{n}_{x,y=-n}\sqrt {P_{th}(x,y)P_{nu}(x,y)})^{2}$ [31] to evaluate the quality of the QW. Here $P_{th}(x,y)$ represents the theoretical probability distribution in the 2D QW while $P_{nu}(x,y)$ is the numerical simulation of probability distribution using X Homodyne measurement. It is apparently that the value of $S$ is approaching to unity when the measurement results are same with the theoretic results. The similarity of the QW with different initial states after 20 steps is discussed, the values are always close to 1. Based on the evolution in Fig. 2(d), the success rate are quantitatively studied. It keeps very high within 400 steps [black dash line, top figure] (200 steps [black dash line, bottom figure]) of the QW, but it will decrease slightly beyond 400 steps (200 steps). Additionally, there are two methods to obtain a high success rate for the large-scale evolution 2D QW: One relies on employing a large step length for QW, and the other is to adopt the large-amplitude coherent state, both of which can be implemented under current technology and independent of the measuring site $x$.

Fig. 2. Probability distribution after 20 steps of the 2D QW with auxiliary coherent state and corresponding deterministic measurement success probability as a function of the evolution scale. In (a), (b), and (c) we have simulated the evolution pattern of 2D QW based on $\textbf {X}$ homodyne analysis and the walker starts from the lattice site (0, 0). The initial input state are (a) $|U\rangle$, (b) $|D\rangle$, (c) $1/\sqrt {2}(|U\rangle +|D\rangle )$. All phases caused by cross-Kerr media are set to $\theta ' = 1^\circ$. (d) When the step length is set to $\theta ' = 1^\circ$ and $\theta ' = 0.5^\circ$, the success probability of deterministic measurements as the number $\textbf {n}$ of walking steps increases for different $\alpha$ is shown.

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3. Feasibility of the quantum walk scheme

In a realistic system, the quantum cion state $C_{1}$ and $C_{2}$ can be easily complied by using the 50:50 beam splitter (BS). One of the conditional shift operators $S_{x}$ along the $x$ direction could be implemented by using two q-plates while measured with the spatial light modulator (SLM) [51–54]. Moreover, employing the cross-Kerr nonlinearity to generate phase information as another degree of freedom, the conditional shift operator $S_{y}$ along the $y$ direction could be realized and the 2D QW can be constructed deterministically. In order to ensure that each operator $S_{y}$ can produce a valid site $y$, one must guarantee that there is a sufficient direct interaction for cross-Kerr nonlinearity between photon and the coherent light. However, such strong nonlinearity is not easily prepared directly in practice. And yet for all that, the Ref [55] and some related works [56–59] optimized the feature by using intense coherent states of light as a ‘bus’ to mediate interactions between photons, in which the strength of the coherent states can offset the weakness of the nonlinearities. Practically, those approaches can ‘amplify’ the effect of the rather weaker nonlinearities that are available (with $\theta$ < 1), to enable QW.

However, the scheme needs to make a distinction among the phases related to the angle values of the phase shifts $(0, \pm \theta ', \pm 2\theta ',\ldots \pm n\theta ', n\subset Z$ ($\theta '$ represents the step length corresponding to the phase information for each walk.)) occurred on the coherent state in the measurement module. To reach the demands above mentioned, $\textbf {X}$ Homodyne measurement is performed on the coherent state and the corresponding measurement outcomes map to the sites $y$ $(y=0, y=\pm 1, y=\pm 2,\ldots y=\pm n, n\subset Z )$. Considering a scenario of site $y$ using $\textbf {X}$ Homodyne measurement, the $nth$ step of QW can be calculated with the formula $\langle \textbf {x}|\boldsymbol {\alpha } \rangle =(2\pi )^{-\frac {1}{4}}e^{-(\textbf {Im} \alpha )^{2}-\frac {(\textbf {x}-2\boldsymbol {\alpha })^{2}}{4}}$ [60] presented as (Benefit from the independence of the two degrees of freedom, we assume that the success probability of measuring OAM states is 1, and only the phase information states are analyzed):

(11)$$\begin{aligned}\langle \textbf{X}|\boldsymbol{\alpha} e^{\pm i \theta}\rangle = & \Sigma^{n}_{x,y=-n}P_{nu}(x,y)f(\textbf{x}, \boldsymbol{\alpha} cos y\theta')|x\rangle|U\rangle \\&+ \Sigma^{n}_{x,y=-n}P_{nu}(x,y)f(\textbf{x}, \boldsymbol{\alpha} cos y\theta')|x\rangle|D\rangle \end{aligned}$$

and

(12)$$f(\textbf{x}, \boldsymbol{\alpha} cos y\theta')= \boldsymbol{\alpha} sin y\theta'(\textbf{x}-2\boldsymbol{\alpha} cosy\theta')$$

where $| \textbf {X}\rangle$ is the eigenstate of the $\hat {\textbf {X}}$ with the eigenvalue $\textbf {x}$, $|\boldsymbol {\alpha }\rangle$ denotes the coherent state and $P_{nu}(x,y)$ represents the probability of distribution at site ($x,y$). Note here the significant of the protocol is to read the phases of the coherent state, that is to measure the values of $f(\textbf {x}, \alpha cos\theta ')$, $f(\textbf {x}, \alpha cos2\theta ')$, $f(\textbf {x}, \alpha cos3\theta ')$,…, $f(\textbf {x}, \alpha cos y\theta ')$ deterministically. It had been indicated that a near-accurate condition is $\alpha \theta >0$ which can satisfy the above requirement [61]. Even with the weak nonlinearity (small $\theta$), one can deterministically read the phases as long as the existence of the coherent state with large amplitude. However, due to the effects of decoherence, there are inevitable existing error during the implementation of the cross-kerr effect with $\textbf {X}$ Homodyne measurement given by $\textbf {erfc}[\boldsymbol {\alpha }/{n\sqrt {2}}(1-cos\theta ')]/2$. And the total success probability of the measurement for the n$th$ step reads

(13)$$\mathcal{P}_{success}=1-\frac{1}{2}\textbf{erfc}[\frac{\boldsymbol{\alpha}}{\sqrt{2}n}(1-cos\theta')].$$

The success probability of deterministically measuring phases as a function of $\theta$ and $\alpha$ is shown in Fig. 3(a), from which one can clearly acquire the appropriate parameter settings to fulfil the experimental requirements. Specially, in Fig. 3(b), the success rate of 2D QW is calculated with three different step length ($\theta ' = 0.2^\circ$, $\theta ' = 1^\circ$, $\theta ' = 2^\circ$) for $\textbf {X}$ Homodyne measurement related to the value of $\alpha$. And the phases $e^{\pm i \theta }$ can be effectively distinguished with the assistance of a large $\alpha$ no matter how small the step length is.

Fig. 3. The success probability of 2D QW using $\textbf {X}$ Homodyne analysis. (a) Deterministic success probability of resolution measurement against the site y (the value of $\theta$) and amplitude $\alpha$ for 2D QW. (b) Three different $\theta '$s were chosen as the walking step length to simulate the success rate against the amplitude $\alpha$, inset, small $\theta '$ for big growth $\alpha$.

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4. Summary

In summary, we have proposed the 2D quantum walk scheme with arbitrary steps exploiting the coherent state and cross-Kerr nonlinearity. The quantum coin state in the quantum walk is represented by the spatial modes of the single photons which can be operated by the BS. On the other hand, the OAM states of the single photon and the phase information of the auxiliary coherent light are defined as the walker’s position site $x$ and $y$, respectively.

Compared with the previous methods using the beam displacer, the current scheme possesses the following features: in merit of two independent degrees of freedom and the success rate of measurement that do not affect each other. The flexibility in structure is easy to generalize to infinite steps walk in experimentally as long as one possesses a relatively large $\alpha$. We believe the current scheme is feasible and efficient with current technologies and may be useful for the further studies in quantum information science.

Funding

National Natural Science Foundation of China (NSFC) (61622103 and 61671083); Ministry of Science and Technology of the People's Republic of China (MOST) (2016YFA0301304); Fok Ying Tong Education Foundation (151063).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Implementation of a two-dimensional quantum walk using cross-Kerr nonlinearity

Abstract

1. Introduction

2. Two-dimensional quantum walks using cross-Kerr nonlinearity

3. Feasibility of the quantum walk scheme

4. Summary

Funding

Disclosures

References

Cited By

Figures (3)

Equations (13)

OSA Continuum