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Zhang Zijing, Wang Feng, Song Jie, Zhao Yuan. The effect of phase fluctuation and beam splitter fluctuation on two-photon quantum random walk*. Chinese Physics B, 2020, 29(2): 020503  
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The effect of phase fluctuation and beam splitter fluctuation on two-photon quantum random walk*
Zhang Zijing1, †, Wang Feng2, Song Jie1, Zhao Yuan1
Department of Physics, Harbin Institute of Technology, Harbin 150001, China
Tianjin Jinhang Institute of Physical Technique, Tianjin 300192, China

 

† Corresponding author. E-mail: zhangzijing@hit.edu.cn

Abstract

In the optical quantum random walk system, phase fluctuation and beam splitter fluctuation are two unavoidable decoherence factors. These two factors degrade the performance of quantum random walk by destroying coherence, and even degrade it into a classical one. We propose a scheme for the simulation of quantum random walk using phase shifters, tunable beam splitters, and photodetectors. This proposed scheme enables us to analyze the effect of phase fluctuation and beam splitter fluctuation on two-photon quantum random walk. Furthermore, it is helpful to guide the control of phase fluctuation and beam splitter fluctuation in the experiment.

PACS: 05.40.Fb;05.40.-a;03.65.Yz
Keyword:quantum random walk;decoherence;random walk
1. Introduction

Classical random walk is widely used in classical algorithms and plays an important role in many areas, such as polymer chemistry, biology, computer science, economics, and so on.[1–3] By endowing the walker with quantum properties, quantum interference leads to a new type of walk, named quantum random walk, which diffuses much faster than its corresponding classical counterpart.[4] Quantum random walk quickly becomes a research hotspot with two reasons: on the one hand, quantum random walk has richer dynamical properties than the classical one for an interest in basic science;[5–10] on the other hand, quantum random walk may provide a similar insight for quantum algorithms, which is equivalent to classical random walk for classical algorithms.[11,12] It has been proven that quantum random walk allows the realization of universal quantum computation[13] and the speed-up of search algorithms.[14] There have been several suggestions for a practical implementation of quantum random walk, using waveguide structures,[15] single photon systems,[16] two-photon systems,[17,18] trapped atoms,[19] trapped ions,[20–22] and nuclear magnetic resonance (NMR) systems.[23,24] More recent works on the implementation of quantum walk are reported, based on a photonic chip,[25] cross-Kerr nonlinearity,[26] dense coding of coin operators,[27] spin–orbital angular momentum space of photons,[28] and a lattice with twisted photons,[29] even quantum walk with one variable absorbing boundary.[30]

Project supported by the National Natural Science Foundation of China (Grant No. 61701139).

All practical implementations of various quantum random walk might be destroyed by decoherence in a real environment. Coherence is the essence of the difference between quantum random walk and classical random walk. With the increase of the degree of decoherence, the distribution of a totally quantum random walk can change into a classical Gaussian distribution.[31,32] Recently, decoherence in quantum random walk has been discussed by many researchers. Brun et al. studied quantum random walk with decoherent coins, through increasing the number of coins which drive the walk and using a new coin at each step.[33,34] Broome et al. controlled a relative angle between two beam displacers to adjust photon’s temporal and spatial mode overlap, in order to discuss tunable decoherence in quantum random walk.[35] Romanelli et al. investigated the quantum walk when decoherence was introduced through random failures in the links between neighboring sites.[36] Quantum random walk is highly sensitive to decoherence with the increase of the number of steps.[37] This is because the effects of decoherence will be accumulated, and therefore decoherence research is more important for the application of large-scale quantum random walk system in the future. Photons are well known to be extremely powerful for carrying and manipulating quantum information. In this paper, phase shifters and tunable beam splitters are used to simulate two main kinds of decoherence factors, phase fluctuation and beam splitter fluctuation (erratic transmittivity, i.e., coin fluctuation). We in detail discuss the effect of these two kinds of decoherence factors on two-photon quantum random walk. This is of great significance to both basic theory and experimental research. On the one hand, the role of two kinds of decoherence factors in two-photon quantum random walk is clarified theoretically; on the other hand, it is helpful to preserve quantum properties in the experiment of two-photon quantum random walk by controlling two main kinds of decoherence factors.

2. Theoretical analysis

Tunable decoherence two-photon quantum random walk system scheme is composed of a series of phase shifters, tunable beam splitters and detectors, as shown in Fig. 1. Two entangled photons are used as input signals and enter into this system from the above two ports. Beam splitter implements the quantum coin operation, deciding whether the walker moves left or right in each step. Each beam splitter is randomly adjustable with erratic transmittivity, in order to simulate beam splitter fluctuation T that can be described as Gaussian distribution (μBS, σBS). Besides, phase fluctuation is introduced by phase shifters placed at each beam splitter’s input port, and phase fluctuation Φ can be described as Gaussian distribution (μphase, σphase).

Fig. 1. Two-photon quantum random walk system scheme with tunable beam splitters and phase shifters. TBS: tunable beam splitter; PS: phase shifter; D1–D8: detectors.

As is well known, a perfect beam splitter (BS) can be represented as a 2 × 2 matrix . Here considering two main kinds of decoherence factors (beam splitter fluctuation T and phase fluctuation Φ), the imperfect BS 2 × 2 matrix is

After the first beam splitter, the output results of the first step is obtained as |ψ1⟩ = U|ψin⟩. Then the output results of the n-th step can be got through the iterating method The first term of this formula is the tensor product of imperfect BS 2 × 2 matrix U and n-order unit matrix. The imperfect BS 2 × 2 matrix U is expanded into a 2n × 2n matrix, which represents the total effect of n imperfect BS on the n-th step; The second term is the input states of the n-th step. When n > 1, considering two outermost beam splitters of each step of two-photon quantum random walk system, only one port has input and the other port is idle. Two idle ports on either side can be treated as the vacuum state |0⟩. Thus according to |ψn⟩, the probability distribution of the n-th step is obtained P(n) = ⟨ψn|ψn⟩. Next, the effects of phase fluctuation and beam splitter fluctuation on two-photon quantum random walk are in detail analyzed.

3. Simulation results and analysis

First, we control each phase shifter to generate independent phase fluctuation Φ which satisfies Gaussian distribution (μphase, σphase). The mean value is μphase = 0, and the standard deviation σphase are 0, 0.1π, 0.2π, 0.3π, respectively. Their coincidence probability distribution results are shown in Figs. 2(a), 2(c), 2(e), 2(g). Here coincidence probability of any point (x, y) is the probability of one photon at position x and one photon at position y. Figures 2(b), 2(d), 2(f), 2(h) are photon distributions of panels (a), (c), (e), (g), respectively. When σphase = 0, two-photon quantum random walk shows typical quantum characteristics. At this time, photons diffuse rapidly, and two discrete peaks appear. With the increase of phase fluctuation, it can be seen that phase fluctuation gradually weakens quantum characteristics, and the coincidence probability distribution and photon distribution are gradually close to the center. When σphase = 0.3π, it is almost completely degenerated to Gaussian distribution of the classical random walk. This shows that the fundamental reason why quantum random walk is different from classical random walk is interference.[38] The interference requires a stable phase environment, and however random phase fluctuation will degrade the interference effect rapidly, making quantum random walk degenerate into classical random walk.

Fig. 2. Coincidence probability distribution of 10-step two-photon quantum random walk with phase fluctuation of (a) σphase = 0, (c) σphase = 0.1π, (e) σphase = 0.2π, (g) σphase = 0.3π. Panels (b), (d), (f), (h) are photon distributions of panels (a), (c), (e), (g), respectively.

In order to quantitatively analyze the effect of phase fluctuation on two-photon quantum random walk, we calculate the photon distribution variance and the coincidence probability distribution similarity, as shown in Figs. 3(a) and 3(b), respectively.

Fig. 3. (a) Photon distribution variance and (b) similarity of two-photon quantum random walk with different phase fluctuations.

Photon distribution variance ,[33,34,39] where x is the position of the triggered detector, and is the mean position, which is is photon distribution probability of the i-th position (detector). Photon distribution variance directly shows the diffusion rates of ‘walker’ photons. In Fig. 3(a), photon distribution variance decreases gradually with the increase of phase fluctuation, that is to say the diffusion rates of ‘walker’ photons decrease gradually. When σphase ⩾ 0.3π, photon distribution variance remains almost unchanged, and this is because it basically approaches classical random walk. Besides, figures 3(b) shows coincidence probability distribution similarity with different phase fluctuation. Coincidence probability distribution similarity ,[40,41] where Pj,k is coincidence probability distribution with no phase fluctuation σphase = 0 and is coincidence probability distribution with phase fluctuation σphase ≠ 0. Coincidence probability distribution similarity directly shows the effect of phase fluctuation on coincidence probability distribution of quantum random walk. It can be seen that coincidence probability distribution similarity decreases gradually with the increase of phase fluctuation. When σphase ⩾ 0.3π, coincidence probability distribution similarity also remains unchanged. This change point σphase = 0.3π is in accordance with the one of photon distribution variance, which shows it basically approaches classical random walk with σphase = 0.3π. In addition, from Figs. 3(a) and 3(b), it can be seen that the larger the step of the quantum random walk is, the greater the effect of the phase fluctuation will be.

Secondly, tunable beam splitters (TBS) are used to generate independent random beam splitter fluctuation T which satisfies Gaussian distribution (μBS, σBS). The mean value is μBS = 0.5, and the standard deviation σBS are 0, 0.1, 0.2, 0.3, 0.4, respectively. Their coincidence probability distributions are shown in Figs. 4(a), 4(c), 4(e), 4(g), 4(i). Figures 4(b), 4(d), 4(f), 4(h), 4(j) are photon distributions of panels (a), (c), (e), (g), (i), respectively. As the standard deviation of beam splitter fluctuation increases, coincidence probability distribution and photon distribution get close to the center gradually, and degenerate into a classical one. This indicates that beam splitter fluctuation is also able to affect the interference effect, so that quantum random walk gradually degenerates into the classical random walk.

Fig. 4. Coincidence probability distribution of 10-step two-photon quantum random walk with tunable beam splitters of (a) σBS = 0, (c) σBS = 0.1, (e) σBS = 0.2, (g) σBS = 0.3, (i) σBS = 0.4. Panels (b), (d), (f), (h), (j) are photon distributions of panels (a), (c), (e), (g), (i), respectively.

We calculate the photon distribution variance and the coincidence probability distribution similarity under different beam splitter fluctuations, as shown in Figs. 5(a) and 5(b), respectively. In Fig. 5(a), photon distribution variance decreases gradually with the increase of beam splitter fluctuation, that is to say the diffusion rates of ‘walker’ photons decreases gradually. Figure 5(b) shows coincidence probability distribution similarity decreases gradually with the increase of beam splitter fluctuation. Besides, figures 5(a) and 5(b) show that the larger the step of quantum random walk is, the greater the effect of beam splitter fluctuation will be.

Fig. 5. (a) Photon distribution variance and (b) similarity of two-photon quantum random walk with beam splitter fluctuation.

Finally, we synthetically consider the effect of two factors (phase fluctuation and beam splitter fluctuation). Figures 6(a) and (b) show that photon distribution variance and coincidence probability distribution similarity of 10-step two-photon quantum random walk with phase fluctuation and beam splitter fluctuation. The results of Fig. 6 are helpful to guide us implement a better quantum random walk through the control of phase fluctuation and beam splitter fluctuation in the experiment. For example, coincidence probability distribution similarity of more than 90% requires that phase fluctuation standard deviation should be controlled within σphase ⩽ 0.05π and beam splitter fluctuation standard deviation should be controlled within σBS ⩽ 0.1. Controlling σBS ⩽ 0.1 is easy to achieve, while the main difficulty that needs our attention is to control phase fluctuation within σphase ⩽ 0.05π in the experiment.

Fig. 6. (a) Photon distribution variance and (b) coincidence probability distribution similarity of 10-step two-photon quantum random walk with phase fluctuation and beam splitter fluctuation.
4. Conclusion

In this paper, we introduce phase shifters and tunable beam splitters into a two-photon quantum random walk system. Phase shifters and tunable beam splitters are used to simulate two unavoidable decoherence factors, phase fluctuation, and beam splitter fluctuation. We find that both coincidence probability distribution and photon distribution gradually tend to the center and degenerate into a classical Gaussian distribution with the increase of phase fluctuation and beam splitter fluctuation. The photon distribution variance and coincidence probability distribution similarity are used to quantitatively analyze the effect of phase fluctuation and beam splitter fluctuation on the two-photon quantum random walk. Our work shows that the controlling phase fluctuation is the key point for realizing a better result of quantum random walk. When phase fluctuation σphase ⩾ 0.3π, the two-photon quantum random walk basically approaches classical random walk. If similarity of more than 90% for a typical quantum result is required, phase fluctuation within σphase ⩽ 0.05π is a key point and must be satisfied.

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