Abstract
The preparation of initial superposition states of discretetime quantum walks (DTQWs) is necessary for the study and applications of DTQWs. Based on an encoding method, here, we propose a DTQW protocol in linear optics, which enables the preparation of arbitrary initial superposition states of the walker and the coin and enables to obtain the states of the DTQW in addition to the probability distribution of the walker. With this protocol, we experimentally demonstrate the DTQW in the polarization space with both the walker and the coin initially in superposition states, by using only passive linearoptical elements. The effects of the walker’s different initial superposition states on the spread speed of the DTQW and on the entanglement between the coin and the walker are also experimentally investigated, which have not been reported before. When the walker starts with superposition states, we show that the properties of DTQWs are very different from those of DTQWs starting with a single position. Our findings reveal different properties of DTQWs and pave an avenue to study DTQWs with arbitrary initial states. Moreover, this encoding method enables one to encode an arbitrary highdimensional quantum state, using a single physical qubit, and may be adopted to implement other quantum information tasks.
Introduction
Quantum walks (QWs)^{1} are extensions of the classical random walk and have wide applications in quantum algorithms,^{2,3,4} quantum simulations,^{5,6,7,8,9,10} quantum computation,^{11,12,13} and so on.^{14,15} In standard onedimensional (1D) discretetime quantum walks (DTQWs),^{16,17} the walker’s position can be denoted as x〉 (x is an integer) and the coin can be described with the basis states 0〉_{c} and 1〉_{c}. The evolutions of the walker and the coin are usually characterized by a timeindependent unitary operator U = TS_{c}(ψ). In each step, the coin is tossed by
with ψ ∈ (0°, 90°), and the walker is shifted by \(T =\mathop {\sum}\nolimits_x x + 1\rangle \langle x \otimes 1\rangle _{\mathrm {c}}\langle 1 + \mathop {\sum}\nolimits_x x  1\rangle \langle x \otimes 0\rangle _{\mathrm {c}}\langle 0\). In general, the result of the DTQW with a finite number of steps is determined by the initial states of the coin and the walker, as well as the operator U. The systematic investigation of DTQW properties and the effect of the initial states on DTQW is important and necessary for applications of DTQWs.
It is worth noting that DTQW with the walker initially in a superposition state has rarely been demonstrated in experiments,^{18} though there were several theoretical works.^{19,20,21,22,23,24,25} DTQWs have been experimentally realized in several systems, such as linear optics,^{4,5,6,26,27,28} ion traps,^{29,30} cavity quantum electrodynamics,^{31} and neutral atom traps.^{32} In most DTQW experiments, the walker starts from the original position 0〉 and it is difficult to prepare an arbitrary initial quantum superposition state of the walker. Moreover, initial superposition states are usually required for applications of DTQWs. For example, an initial uniform superposition state of the walker’s position is required in the Grover walk.^{2} DTQWs with initial superposition states of the walker are quite different from the standard DTQW (i.e., the walker starts from a single position x〉), because the final state of the former is a coherent superposition of the final states of the standard DTQW starting from different single positions and thus contains more quantum interference.
Note that the effects of the coin’s initial state on DTQW have been well studied,^{16,17,26,27,28,33} but the effects of the walker’s initial superposition states on DTQW have rarely been investigated before in experiments. In previous linearoptical DTQW implementations,^{34,35,36} the coin was always encoded in the twodimensional polarization space and the position of the walker must be encoded in highdimensional spaces. In ref. ^{34} the coin was encoded using the polarization state and the walker’s positions were encoded using the paths of the photons, in which an additional calcite beam displacer was needed in each step to perform the conditional shift of the walker, and DTQWs with up to six steps were measured in experiments. In ref. ^{18} the coin was encoded using the polarization state and the walker’s positions were encoded using the orbital angular momentum of light, in which the whole process develops in a single light beam and optical resources scaling linearly with the number of steps are required. In ref. ^{35} the coin was also encoded using the polarization state, and the walker’s positions were encoded using the temporal information of single photons, in which the coin operator was adjustable and fivestep DTQWs were studied experimentally. In linear optics, the preparation of arbitrary initial superposition states of the coin is not difficult, because it is easy to obtain arbitrary polarization states using halfwave plates (HWPs) and quarterwave plates (QWPs). Each DTQW implementation scheme has advantages and limitations, but the preparation of arbitrary initial superposition states of the walker is difficult in most DTQW schemes.
To our knowledge, how to encode the walker’s position using the polarization state has not been reported in both theory and experiment. It is difficult to encode a highdimensional state using the twodimensional state. Moreover, the higher the dimension of a quantum state is, it becomes more complex to measure the state or the corresponding density matrix. In DTQW experiments, the measured quantity is always the probability distribution of the walker. In optical experiments, the probability distribution of each position is measured one by one, i.e., a multistep DTQW should be performed and measured many times, unless all the walker’s positions can be separated and measured simultaneously, which is challenging in most experiments. If the walker’s position can be encoded using the polarization state, arbitrary superposition states of the walker can be easily prepared and measured in linearoptical experiments.
In this work, we propose an encoding method to encode and read out highdimensional states using only a 2D physical qubit. With this method, we report a DTQW protocol with the walker’s position encoded in the polarization space of single photons. The main advantages of this protocol are as follows: (i) the DTQW protocol together with the design of a linearoptical setup, in which the coin is encoded by two paths, allows us to arbitrarily set the input state of the walker and coin; (ii) the whole state of the multistep DTQW can be obtained, rather than the probability distribution, which can help to study deep properties of DTQWs and extend the possible applications of DTQWs; (iii) the devices needed are simple and common, optical resources do not increase when increasing DTQW steps; (iv) the mature technique for operations of polarization of photons makes the implementation of this protocol easy and efficient. Then we demonstrate the experimental implementation of DTQW with the walker initially in superposition states. In this case, the effect of the walker’s different initial superposition states on the spread speed of the walker and on the entanglement between the coin and the walker (i.e., the entropy of the reduced density matrix of the coin) are also studied experimentally, which has not been studied in experiments before.
Results
DTQW protocol in linear optics
To implement DTQWs in polarization space, we propose an encoding method in the “Methods” section. Here, we make the specific choice of encoding the walker’s positions using nonorthogonal states \(k\rangle _p = {\mathrm{cos}}{\kern 1pt} \theta _kH\rangle + e^{i\phi _k}{\mathrm{sin}}{\kern 1pt} \theta _kV\rangle\) (with integer k) in the polarization space of single photons, by setting θ_{k} = kΔθ and ϕ_{k} = 0. The coin is encoded by two paths 0〉 and 1〉 of the single photons. As shown in Fig. 1a, the operator for the conditional translation of k〉_{p} should be expressed as
In this case, the coin operator S_{c} can be implemented by a cos^{2} ψ/sin^{2} ψ beam splitter (BS) and the operator T_{p} can be implemented via two HWPs oriented at 0° and two HWPs oriented at Δθ/2 (step forward) and −Δθ/2 (step back), as shown in Fig. 1b. Note that each of the HWPs oriented at 0° can be replaced by a mirror.
In each step of a DTQW in polarization space, the state of the system can be expressed as Ψ_{0}〉_{p}0〉 + Ψ_{1}〉_{p}1〉, with
where \(a^{\prime}_k = a_k/C_a\), \(b^{\prime}_k = b_k/C_b\), \(C_a = \sqrt {{\sum} a_k^2}\), \(C_b = \sqrt {{\sum} b_k^2}\), N_{a} = N_{p}/C_{a}, N_{b} = N_{p}/C_{b}, and N_{p} is the normalization coefficient, which guarantees \(\mathop {\sum}\nolimits_k a_k^2 + \mathop {\sum}\nolimits_k b_k^2 = 1\). If one derives all values of a_{k} and b_{k}, the probability distribution of the DTQW can be obtained as \(a_k^2 + b_k^2\).
The values of a_{k} (b_{k}) can be obtained by using \(a_k = C_{a}{a_{k}^\prime}\) (\(b_k = C_{b}{b_{k}^\prime}\)). Note that Ψ_{0}〉_{p} (Ψ_{1}〉_{p}) has the same form as that of Ψ〉_{q} in Eq. (4) and the condition \(\mathop {\sum}\nolimits_k a^{\prime}_k^2 = 1\) \(\left(\mathop {\sum}\nolimits_k b^{\prime}_k^2 = 1\right)\) is satisfied. By using the encoding method introduced in the “Methods” section, \(a^{\prime}_k\) and \(b^{\prime}_k\) can be obtained one at a time by implementing the DTQW with different angles Δθ and measuring Ψ_{0}〉_{p} and Ψ_{1}〉_{p} in path 0〉 and path 1〉, respectively. Next, C_{a} and C_{b} can be obtained by solving two equations
where \(r = \mathop {\sum}\nolimits_k a_k{\mathrm{cos}}(k\theta )^2/\mathop {\sum}\nolimits_k b_k{\mathrm{sin}}(k\theta )^2\) is the ratio between the number of photons counted in path 0〉 and those counted in path 1〉 at the same time period for an arbitrary θ ≠ 0. With this protocol, DTQWs with the walker initially in an arbitrary superposition state can be easily implemented and the entire DTQW state can be obtained rather than just the probability distribution. It seems that DTQWs must be performed for at least N times, to obtain an Ndimensional state of the walker. Actually, in other DTQW experiments, the probability of each walker position is measured one by one, i.e., the DTQW also needs to be performed and measured for at least N times to obtain the probability distribution of N positions of the walker, unless the walker state is encoded by the paths of photons and multiple detectors are used simultaneously.
Experimental demonstrations
A feasible linearoptical setup for a DTQW in the photon polarization space is shown in Fig. 2. A pair of photons are generated by the typeI spontaneous parametric downconversion in a 3mmthick nonlinear betabarium borate (BBO) crystal pumped by a 100mW diode laser (centered at 405.8 nm). One photon is directly detected by detector D_{0} as the trigger and the other photon is adopted to make a DTQW. Then an arbitrary coin state can be initialized by a combination of one polarizing beam splitter (PBS), two QWPs, and one HWP. An arbitrary state of the walker’s position is represented as a polarization state, which can be easily initialized by using two HWPs on the two paths of the DTQW photon after the PBS.
In Fig. 3, we show the results of the 1D Hadamard (i.e., setting ψ = π/4) DTQW with an initial state of (0.8−1〉_{p} + 0.61〉_{p})0〉. Besides the probability distribution of the walker’s position a_{k}^{2} + b_{k}^{2}, the coefficients a_{k} and b_{k} of the states Ψ_{0}〉_{p} and Ψ_{1}〉_{p} for a twostep DTQW, a fourstep DTQW, and a sixstep DTQW are obtained and compared with the theoretical results. Note that the error bars only indicate statistical uncertainty, which are obtained by numerical simulations. Our results indicate that the protocol introduced here is realizable in experiments, and DTQW with small errors can be implemented by using the setup shown in Fig. 2.
Since the walker is initially in a superposition state of positions −1〉_{p} and 1〉_{p}, the outermost positions of the walker after an nstep DTQW are denoted by − n− 1〉_{p} and n + 1〉_{p}. To simplify the calculation of a_{k} and b_{k}, we have used a_{n+1} = 0 and b_{−n−1} = 0 for this type of DTQW. In this case, only n different Δθ are needed to obtain a_{k} and b_{k} for an nstep DTQW.
The initial state of the walker’s position can be easily prepared by rotating the angles of the HWPs in our experiment. We have also experimentally studied the DTQW with the initial state \((0.6  2\rangle _p + 0\rangle _p + 0.82\rangle _p)0\rangle /\sqrt 2\). The results of the reconstructed a_{k}, b_{k}, and a_{k}^{2} + b_{k}^{2} after a sixstep DTQW are shown in Fig. 4, which also fit well with the theoretical results.
By means of this DTQW experiment, we also study the relation between the spread speed s and the initial state of the walker’s position. The spread speed s is defined as s(n) = [σ(n) − σ(0)]/n, where σ(n) is the variance of the position for an nstep DTQW. The results for a fourstep DTQW are obtained and plotted in Fig. 5a, in which the initial state is set as \({\mathrm{\Phi }}_0\rangle = \left( {\alpha   1\rangle _p + \sqrt {1  \alpha ^2} 1\rangle _p} \right)\left( {0\rangle + i1\rangle } \right)/\sqrt 2\), with α ∈ (−1, 1). It is shown that different initial states of the walker’s position will lead to different spread speeds of DTQW. Especially, for the DTQWs with α and −α (α ≠ 0), they have different σ(n) and s(n) (n > 0) though starting with the same σ(0), which is quite different from the classical random walk as shown in Fig. 5a, because of the quantum interference. In Fig. 5a, we also give the simulated relation of s ~ α for a DTQW with n = 16 and n → ∞. It is shown that the properties of the DTQW starting from superposition states of the walker are very different from that starting from a single position of the walker. For example, the parameter s for α ≠ 0 approaches a constant more slowly than that for α = 0, as the number of DTQW steps increases.
A significant advantage of our DTQW protocol is that the whole final state of the coin and the walker’s position can be obtained, rather than the position distributions only in other DTQW experimental schemes. In this case, we can investigate the entanglement between the coin and the walker in experiments by calculating the entropy of the reduced density matrix of the coin. The entropy is defined as E = −Tr(ρ_{c}log_{2}ρ_{c}), where ρ_{c} = Tr_{x}(ρ_{cx}) is obtained by tracing out the walker’s position.
The E ~ α relation of DTQW with n = 4 from the experiment and of a DTQW with n = 4, n = 16, and n → ∞ from the theory is plotted in Fig. 5b. The properties of entanglement (entropy) of a DTQW with nonlocal initial states have been studied theoretically in refs. ^{19,25} As the number of DTQW steps increases, the entropy E with the initial state Φ_{0}〉 will start from 0, then oscillate, and approach to the asymptotic entanglement^{19,37}
As shown in Fig. 5b, the E ~ α relation for n = 4, n = 16, and n → ∞ have very similar forms, so most properties of the entanglement can be obtained by a DTQW with a low step, such as a DTQW with four steps. It indicates that different initial states of the walker will lead to different behaviors of entanglement E. The maximum (minimum) E is obtained for \(\alpha \sim  \sqrt 2 /2\) (\(\alpha \sim \sqrt 2 /2\)). This also indicates that the rate at which the asymptotic value for entanglement is approached is faster for higher entanglement, which fits with the result in ref. ^{19} For a local initial state (i.e., α = 0, ±1), E with n = 16 is close to the asymptotic \(\bar E\), while E with n = 4 is not. For the nonlocal initial state with \(\alpha =  \sqrt 2 /2\) (\(\alpha = \sqrt 2 /2\)), E with both n = 16 and n = 4 are (are not) close to the asymptotic \(\bar E\). Comparing Fig. 5a with Fig. 5b, it is clear that in general, a larger spread speed of the DTQW corresponds to a larger entanglement between the coin and the walker.
Discussion
To implement DTQWs for larger steps, the errors of the experiment should be analyzed and improved. The errors are related to the following factors:

(i)
The number of coincidence counts: Because of the leakage of photons at BS 1, the number of coincidence counts will decrease rapidly with increasing the DTQW steps. The ratio of reflection to transmission of all BSs adopted in our experiment is 50:50. The loss of detected photons after every two steps of DTQW is about 70%. It is an advantage of two optical loops adopted in the setup: photons are leaked and detected after every two steps rather than after each step.
The errors in the experiment can be reduced by increasing the number of coincident counts. One method is to replace the 50:50 coupler (i.e., BS 1) by a coupler with better ratios. The other method is to increase the number of input photons. This can be achieved in several ways. For example, the BBO can be replaced by a highefficiency periodically poled KTiOPO_{4} (PPKTP), the power of the laser can be improved, or weak laser pulses can be directly used as the input photons of DTQW.

(ii)
The quality of the interference in the optical loops: QWs are different from classical random walks because of quantum interference. There are two optical loops in the setup. To guarantee good interference of DTQW, anticlockwise and clockwise optical paths must be adjusted to be the same in each loop, and two loops must be adjusted to match each other. In general, the quality of the interference will become worse when increasing the walk steps. An important factor leading to interference degradation is the imperfections of devices, such as the reflection dependence of the polarization and nonplanar optical surfaces.^{34,35}
To show the scalability of this DTQW protocol and the setup, we estimate the maximum number of steps that are in principle achievable. We assume the use of perfect devices (including detectors). In addition, we assume that the loss of detected photons after every two steps of a DTQW is 70%. By replacing the 50:50 coupler with a 99:1 coupler and replacing the input single photons with laser pulses of 1.4W power (5 MHz), there should be ~10,000 photons detected in 1 second after 150 steps, provided that the signaltonoise ratio can be improved by adding an active switch to couple the photons out of the loops.
In summary, we have experimentally demonstrated a 1D DTQW with the walker moving in the polarization space of single photons. This work differs from previous works in that the walker is initially in a superposition state and the walker’s position is encoded using the photon polarization state. The ability to operate with arbitrary initial superposition states of DTQWs and to implement DTQWs in a lowdimensional space opens an avenue not only for discovering more properties of DTQWs but also more DTQW applications. This DTQW protocol is quite general and can be applied to other quantum systems (e.g., cavity or circuit QED systems). This work may be extended to realize multidimensional DTQWs, which are of importance for largescale quantum computing based on QWs. Moreover, the proposed encoding method enables one to encode an arbitrary highdimensional quantum state, using a single physical qubit, and may be adopted to implement other quantum information tasks.
Methods
The encoding method
Suppose there is an ndimensional state
where k〉 (k = 1, 2, …, n) are orthogonal states. In the following, we propose a method to encode this state with a twodimensional qubit and to read out all a_{k} from the qubit by measurement. The obtained a_{k} can be used to reconstruct the ndimensional state Ψ〉 of Eq. (3).
The state Ψ〉 can be encoded by using n nonorthogonal states \(k\rangle _q = {\mathrm{cos}}\theta _k0\rangle + {\mathrm {e}}^{i\phi _k}{\mathrm{sin}}\theta _k1\rangle\) (k = 1, 2, …, n) of a qubit, which correspond to n different points in the Bloch sphere of the qubit. With k〉 replaced by k〉_{q}, the state of the qubit in the encoded state can be expressed as
where \(C_0 = \mathop {\sum}\nolimits_{k = 1}^n a_k{\mathrm{cos}}\theta _k\), \(C_1 = \mathop {\sum}\nolimits_{k = 1}^n a_ke^{i\phi _k}{\mathrm{sin}}\theta _k\), and \(N_q = \sqrt {C_0^2 + C_1^2}\). Note that the same state Ψ〉_{q} can be constructed by performing suitable operations on an initial state of the qubit, which is just the case considered in the DTQW protocol above.
We now show how to read out a_{k} from Ψ〉_{q}. The density matrix of the encoded state Ψ〉_{q} can be expressed as
which can be easily obtained by using tomographic measurement on the qubit. Then one homogeneous equation of a_{k} can be obtained as
where R = C_{0}/C_{1} is obtained from the measured ρ_{q} as
In this way, (n − 1) homogeneous equations for a_{k} can be obtained by repeatedly encoding the state Ψ〉 onto a qubit with (n − 1) different sets \(\left\{ {1\rangle _q^j,2\rangle _q^j, \ldots ,n\rangle _q^j} \right\}\) (where j = 1, 2, …, n − 1, and \(k\rangle _q^j = {\mathrm{cos}}\theta _k^j0\rangle + {\mathrm {e}}^{i\phi _k^j}{\mathrm{sin}}\theta _k^j1\rangle\)), and by measuring the corresponding R^{j} one at a time. Then a_{k} (k = 1, …, n) can be solved from the (n − 1) homogeneous equations
and the normalization condition \(\mathop {\sum}\nolimits_{k = 1}^n {\kern 1pt} a_k^2 = 1\). Now the ndimensional encoded state is reconstructed. Note that the ndimensional state is actually encoded in (n − 1) qubits rather than one qubit, and all information is reconstructed from the (n − 1) qubits. Of course, one can adopt only one physical qubit which will be repeatedly used.
To study the generality and scalability of this encoding method, we calculated the fidelity between random 50dimensional encoded states and their reconstructed states from numerical simulations of the encoding process, in which random errors of measured R^{j} were assumed to be within ±10% and a specific choice of the nonorthogonal states
was used (i.e., \(\theta _k^j = jk{\mathrm{\Delta }}\theta\) and \(\phi _k^j = k{\mathrm{\Delta }}\phi\)). The averaged fidelity over 1000 random simulations was obtained and its dependence on Δθ and Δϕ is plotted in Fig. 6a. The averaged fidelity is larger than 90% in the ~41% area of Δθ ∈ (0, 180°) and Δφ ∈ (0, 360°). It is not hard to find suitable θ_{k} and ϕ_{k} to implement the encoding method, and high fidelity can be achieved for encoding largedimensional states if measurement errors are not very large.
Moreover, we experimentally demonstrated the encoding and reading of a 16dimensional state in a linear optics system. At first, we encode a uniform state \({\mathrm{\Phi }}_{16} = \frac{1}{4}\mathop {\sum}\nolimits_{k = 1}^{16} k\rangle\) in the twodimensional polarization space of single photons by using Δθ = 22° and Δϕ = 12° in Eq. (9). This encoding can be easily implemented by injecting photons into an adjusted setup consisting of two QWPs and one HWP, which can be used to construct an arbitrary onequbit gate in the polarization space. Then density matrices of the encoding photons can be measured and the initial uniform state can be reconstructed. The experimental result of the reconstructed state is shown in Fig. 6b, and a high fidelity of 99.1% is achieved.
The encoding method introduced here is feasible for the present technology of linear optics and can be easily applied to other quantum systems. One key advantage for the use of this encoding method is the simplicity of the setup, because only one physical qubit is needed.
The implementation of a DTQW
With the setup shown in Fig. 2, the steps of a DTQW are as follows:

(i)
Preparation of the initial states of the coin and the walker: An arbitrary coin state can be initialized by a combination of one PBS, two QWPs, and one HWP. An arbitrary state of the walker’s position is represented as a polarization state, which can be easily initialized by using two HWPs on the two paths of the DTQW photon after the PBS. Note that the lengths of the two paths from the PBS to BS 2 should be the same by adjusting the prism.

(ii)
Steps for implementing the DTQW: The DTQW starts as the photon enters BS 2 for the first time and is implemented in loop 1 and loop 2. In each loop, the anticlockwise and clockwise photon paths are adjusted to overlap with each other and the loop length is designed to be ~30 cm. Here, the loop’s interference visibility is measured as ~98.5%. The steps for the DTQW realization are described below:
Step 1: As the photon passes through BS 2, the coin state is tossed by the operator S_{c}. The photon after BS 2 will rotate anticlockwise (or clockwise) in loop 2, and the walker achieves a rotation of Δθ (or −Δθ) in the polarization space via HWP 2, just like that shown in Fig. 1a.
Step 2: Then the photon goes through BS 2 a second time and the coin is tossed once again. After BS 2, one possibility is that the photon will rotate anticlockwise (or clockwise) in loop 1 and the walker’s state achieves a rotation of Δθ (or −Δθ) in the polarization space via HWP 1. The photon will enter BS 2 one more time, and the next step of the DTQW will start. The other possibility is that the photon goes through BS 1 and then is detected by D_{1} (D_{2}) to get a_{k} (b_{k}) for a twostep DTQW.

(iii)
Extracting the final state of a 2nstep DTQW: The final state of a 2nstep DTQW can be obtained by measuring the photons which are detected after repeating steps 1 and 2 for n times. Because the total length of loop 1 and loop 2 is about 60 cm, the time gap between the photon detection after a (2n−2)step DTQW and the photon detection after a 2nstep DTQW is ~2 ns. In this case, the time window of the coincidence detections between D_{0} and D_{1} (D_{2}) is set as 0.49 ns.
Data availability
All data supporting the findings of this study are available within the article or from the corresponding author upon reasonable request.
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Acknowledgements
This work is supported in part by the NKRDP of China (Grant No. 2016YFA0301802) and the National Natural Science Foundation of China under Grant Nos. [11504075, 11374083, 11774076, 11375003, and 11775065]. This work is partially supported by MURI Center for Dynamic MagnetoOptics via the Air Force Office of Scientific Research (AFOSR) (FA95501410040), Army Research Office (ARO) (Grant No. W911NF1810358), Asian Office of Aerospace Research and Development (AOARD) (Grant No. FA23861814045), Japan Science and Technology Agency (JST) (via the QLEAP program, the ImPACT program, and the CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (JSPSRFBR Grant No. 175250023 and JSPSFWO Grant No. VS.059.18N), the RIKENAIST Challenge Research Fund, and the John Templeton Foundation.
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Q.P.S. devised various aspects of the project and designed the experimental methodology. Q.P.S., Y.Z., J.Q.Z. and S.J.X. carried out the experiment and analyzed the data. Q.P.S., L.Y., C.P.Y., J.S.J. and X.Q.X. developed the theoretical aspects. Q.P.S., C.P.Y. and F.N. wrote the paper, with contributions from K.F.C., Z.S. and Q.J.X. All authors discussed the results and contributed to refining the paper.
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Correspondence to Franco Nori or ChuiPing Yang.
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