† Corresponding author. E-mail:
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.
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.
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.
![]() | 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
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
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.
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.
![]() | Fig. 3. (a) Photon distribution variance and (b) similarity of two-photon quantum random walk with different phase fluctuations. |
Photon distribution variance
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.
We calculate the photon distribution variance and the coincidence probability distribution similarity under different beam splitter fluctuations, as shown in Figs.
![]() | 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
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.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] | |
[36] | |
[37] | |
[38] | |
[39] | |
[40] | |
[41] |