2.1.

Quantum Walks with Arbitrary Hamiltonian Design

Quantum walks, the quantum analog of classical random walks,^51,52 have unique features of interference and superposition, which bring quantum walks remarkably different behaviors and normally faster transport performances compared with classical random walks. Therefore, quantum walks have become a powerful approach to quantum information algorithms,^53–55 and quantum simulation for various systems.^5,8,56 To apply quantum walks to real applications, a two-dimensional evolution space of a large scale and a flexible design of the Hamiltonian for quantum walks are two preliminary requirements. A few examples of such large-scale two-dimensional quantum walks have now been experimentally realized on the integrated photonic chip.^10,11 This module of QW provides a platform to allow for more Hamiltonian designs to help researchers explore a rich diversity of quantum walk applications.

As concerned in this module (QW), single photons are injected into the photonic lattice consisting of straight waveguides to form two-dimensional quantum walks. Photons propagating through these coupled waveguides are described by the Hamiltonian

Fig. 2

Theoretical calculation for quantum walks on a photonic chip. The coupling coefficient $C$ as a function of the center-to-center waveguide spacing and the flowchart for the procedures of calculating the evolution result for quantum walks on a photonic chip.

For a quantum walk that evolves along the waveguide, the propagation length $z$ is proportional to the propagation time by $z=c_{w}t$, where $c_w$ is the speed of light in the waveguide, so we use $z$ instead of $t$ for simplicity. The wavefunction that evolves from an initial wavefunction satisfies

In the experiment, we observe such probability distribution by injecting a laser beam or single-photon source into one waveguide and measuring the evolution patterns using an intensified charge-coupled device camera. The normalized light intensity of each waveguide corresponds to the probability at this waveguide. As the facula at each waveguide is normally in the shape of the two-dimensional Gaussian distribution, we also plot the Gaussian-shaped facula for presenting the theoretically obtained probability distribution in the software modules.

2.2.

GUI and Functions for the Module of QW

The workflow in this module for theoretically calculating quantum walks on a photonic chip includes the following four steps: generate Hamiltonian, evolve (exponentiate Hamiltonian), obtain probability distribution, and apply Gaussian facula (See Fig. 2). A GUI of this module is further given in Fig. 3, which shows a case of a two-dimensional quantum walk with an arbitrary Hamiltonian design from the panel “design your own chip pattern.” Users can then define the propagation distance and input nodes on the GUI. The Hamiltonian matrix corresponding to the pattern and the calculated probability distribution are displayed and can be exported as image and data files.

Fig. 3

The GUI for the module of QW. An array forming the shape of $\mathrm{\Psi }$ has been set by manually plotting the waveguide in the interactive board, and its evolution pattern with Gaussian-shaped facula for a certain propagation distance and input node is plotted. Different panels on the GUI are marked in the figure, and the functions of all buttons in these panels are explained in Table 1.

Table 1

Functions of all panels for the module of QW.

2.3.

Educational Use for the Module of QW

The highlight of this module is the flexible setting of Hamiltonian, which is fun and straightforwardly shows the possible Hamiltonian manipulation on the photonic lattice. Teachers can then go deeper by showing that the specifically designed Hamiltonians can indeed be applied to different quantum simulation tasks. For instance, the Hamiltonians that are required in many topological photonics models such as the Aubry–Andŕe–Harper model⁵⁷ and Su–Schrieffer–Heeger model⁵⁸ can be implemented on the photonic lattice.

2.3.1.

Case study

In a course that we deliver on analog quantum computing,⁵⁹ we use this module to demonstrate a case of quantum fast hitting. This is a representative algorithm to show the advantages of quantum walks, originally proposed by Childs et al.,⁵² and it was recently adapted to a photonic experiment.¹¹ Students are asked to plot a hexagonal glued tree structure (see Fig. 4, which is reproduced from Ref. 11) and input light in an entry site, the left-most node as the vertex of one tree. Students then observe the hitting efficiency, the probability at the exit site, and the right-most node as the vertex of another tree. They see that the probability changes dynamically over time, which reveals the dynamic nature of quantum walks, and they see that the hitting efficiency can be very high, usually above 90%. The teacher demonstrates running the hitting task with classical random walks, which only gets a low hitting efficiency. Through this example, students see quantum advantages in fast-hitting tasks.⁵²

Fig. 4

Quantum fast-hitting on hexagonal graphs. (a) Mapping the hexagonal glued trees onto the photonic chip. (b) The hitting efficiencies for quantum and classical hitting. Inset of (b) is the experimental pattern with the highest hitting efficiency. The figures are reproduced from Ref. 11.

3.1.

Quantum Stochastic Walks on the Photonic Chip

In real systems, environmental noise easily causes some dephasing and decoherence of the original quantum system. This leads to the issue of an open quantum system,^3–5 which can be addressed with a useful approach named the quantum stochastic walk.⁶⁰

The Lindblad master equation, the most common formulation of quantum stochastic walks, is a differential equation that incorporates both quantum and classical walks in the continuous-time evolution:^36,60

The first part right to Eq. (3) represents the quantum walk evolution, where $H$ is the Hamiltonian operator. The second part, which contains the Lindblad operators $L_{ij}$, describes the classical random walk evolution. The parameter $\omega$ interpolates the weight of quantum walk and classical walk, that is, a higher value of $\omega$ suggests a larger portion of the classical walk in the mixture of the quantum stochastic walk.

There are a few solvers for Lindblad equations, such as Qutip^34,35 and QSWalk.³⁶ However, the parameters in the Lindblad equation do not have a clear correspondence to the physical parameters of the photonic quantum systems. In the photonic chip, a $\mathrm{\Delta }\beta$ photonic model was raised⁶¹ instead to achieve a continuous-time quantum stochastic walk

3.2.

GUI and Functions for the Module of QSW

The GUI for this module of QSW is shown in Fig. 5. It is similar to that for the module of QW, in that they both follow the same four-step workflow to get the theoretical result. Still, QSW has its special panels to facilitate the study of the $\mathrm{\Delta }\beta$ approach for quantum stochastic walks. Users can define how frequently $\mathrm{\Delta }\beta$ varies along the waveguide (by filling “$z$ interval”), and the strength $\mathrm{\Delta }\beta$ variation (by filling “noise amplitude $\mathrm{\Delta }\beta$”). In panel 5, users can view the evolution probability against the evolution length for each specified waveguide [see Fig. 5(b)].

Fig. 5

The GUI for the module of QSW. (a) Different panels on the GUI are marked in the figure, and the functions of all buttons in these panels are explained in Table 2. (b) In panel 5, users can also view the probability of each specified waveguide.

Table 2

Functions of all panels for the module of QSW.

3.3.

Educational Use for the Module of QSW

By adding various $\mathrm{\Delta }\beta$ detunings into the photonic lattice, users are able to create quantum stochastic walks that meet wider scenarios in quantum simulation. Via this module, users will learn the rich capabilities of photonic lattice and explore the difference between quantum stochastic walks and quantum walks.

3.3.1.

Case study

Students can try an exercise using quantum stochastic walks to form the Haar measure, as has been recently conducted in an experiment⁶² (see Fig. 6, which is reproduced from Ref. 62). By averaging the evolution probabilities for many quantum stochastic walk patterns, students observe that the average probabilities converge to a uniform distribution, which meets the criteria for a Haar measure.^62,63 They see that this feasible and scalable experimental approach can make the Haar randomness useful as it can be applied to many quantum information processing tasks. Students are then asked to turn the “noise amplitude $\mathrm{\Delta }\beta$” into zero, which reduces the model to a pure quantum walk, and they see that the average probabilities always change dynamically and never converge. This exercise shows the difference between quantum walks and quantum stochastic walks.

Fig. 6

Generating Haar randomness via quantum stochastic walks on photonic chips. (a) Illustration of generating quantum stochastic walks via $\mathrm{\Delta }\beta$ detunings on the photonic lattice. (b) The average of experimental results converges to uniform distribution as the propagation length increases. The figures are reproduced from Ref. 62.

4.1.

Multiparticle Quantum Walks

In the above two modules, QW and QSW, we have focused on a single particle evolving in a two-dimensional quantum system. It is natural to wonder about the case of injecting more than one particle into the system, i.e., two or more indistinguishable particles. In this case, two main features of quantum optics, quantum interference and quantum correlation play central roles. The genuine quantum interference [also known as Hong–Ou–Mandel (HOM) effect⁶⁴], which cannot be classically simulated, gives rise to nonclassical quantum correlations.⁶⁵ With two particles populating a two-dimensional lattice, the corresponding state space can be easily extended to a greater dimension.⁶⁶ Such high-dimensional graphs demonstrate quantum speedup for the continuous-time search algorithm.⁵⁵

Another striking departure of quantum mechanics from classical physics is that particle statistics, either bosonic or fermionic, can influence the quantum correlation, exhibiting either bunching or antibunching behaviors. These phenomena can be physically simulated with bipartite entanglement in the form of

Phase $\varphi$ controls the states’ symmetry, thus determining the particle statistics.

With an even greater number of $N$ particles involved in the evolution through a system of $M$ waveguides, the process can be regarded as a generalized HOM effect and a photon scattering model. The outcome also depends on whether the particles are distinguishable or indistinguishable and if they are indistinguishable, whether they are bosons or fermions. This distinguishability depends on whether the particles are identical, e.g., whether they have temporal delay or the same frequency when they enter the waveguide array.

If the particles are indistinguishable bosons, then given a Hamiltonian $H$ and its corresponding unitary evolution $U=e^{-\frac{iHt}{\hslash }}$, we can calculate the probabilities of various states occurring. If we define a configuration $S_i$ to be the number of photons injected in waveguide $i$ and $T_j$ to be the number of photons exiting waveguide $j$, then the probability of an initial state $S=|S_1\cdots S_M⟩$ evolving into a final state $T=|T_1\cdots T_M⟩$ is given as¹⁵

With these formulas incorporated into the module of the multiparticle, we are able to use this software to calculate the multiparticle evolution.

4.2.

GUI and Functions for the Module of MultiParticle

The GUI for multiparticle is shown in Fig. 7. Users can define the “number of nodes” and “interval between nodes” for a one-dimensional array. They then choose input nodes for the “initial state.” In particular, users can choose the “particle type,” which can be bosons, fermions, or distinguishable. The above panel shows the breakdown of probability distribution of all states, and a scroll bar can be used for the very long list. Users can specify an output state and observe its probability at different evolution lengths. They can also view the evolution dynamics for inputting only a single particle in a specified input node.

Fig. 7

The GUI for the module of MultiParticle. (a) Different panels on the GUI are marked in the figure, and the functions of all buttons in these panels are explained in Table 3. (b) The probability of a specified quantum state at different propagation lengths and (c) the probability of a specified waveguide at different propagation lengths are also provided in panel 5 of the GUI.

Table 3

Functions of all panels for the module of MultiParticle.

4.3.

Educational Use for the Module of MultiParticle

This module can be a good tool for teaching the bunching and antibunching effects of different types of particles.^67,68 This is a key issue in quantum optics and can be very naturally linked to the up-to-date research in quantum optics and quantum information processing.

4.3.1.

Case study

Students are asked to generate any initial input state and then pick an output state, e.g., $|00002000⟩$, where two photons output from the same waveguide. Students try the three different particle types and write down the output probability of that output state (e.g., $|00002000⟩$) for each particle type. They see that the probability goes to zero for fermions because of the antibunching effects, while the value for bosons doubles the value for distinguishable particles because of photon bunching. Students are then impressed to see the same effect in real experiments. In a work of topological protection of two-photon quantum correlation on a photonic chip,⁶⁹ the authors show the measured two-photon coincidence count from the output end of a waveguide in the photonic lattice. The count has a clear peak that doubles the bottom value when the delay between the two photons reduces to zero (see Fig. 8, which is reproduced from Ref. 69). This module of multiparticle can well explain those experimental phenomena.

Fig. 8

The photon bunching effect in the experiment. (a) A one-dimensional photonic lattice with photons injected into the boundary waveguide. (b) The two-photon coincidence counts against different delay lines between two photons, measured from the output end of the boundary waveguide. The figures are reproduced from Ref. 69.

4.4.

Optimization Algorithms for the Permanent Calculation

It is also worth mentioning that we manage to use effective optimization algorithms for the key part of the theoretical model of this module: the permanent calculation. Computing the permanent has been proven to be of $\#P$ complexity, even harder than NP-complete, and is widely believed to be classically intractable. The permanent of an $n\times n$ matrix per $M$ is defined as

Two better algorithms have been discovered, the first of which is due to Ryser⁷⁰

These two formulas are both related to polarisation identities of symmetric tensors and in fact part of a larger family of permanent algorithms.⁷²

These formulas give algorithms of complexity $O(2^n{n}^2)$ and can be further reduced to $O(2^{n}n)$ using a gray code.⁷³ We explain this using Ryser’s formula, and the same logic applies to Glynn’s. If we say that ${\lambda }_k=\sum _{j=1}^n{ϵ}_j{M}_{jk}$, then if $ϵ$ and ${ϵ}^{\prime }$ differ by a single ${ϵ}_m$ (i.e., the Hamming distance is 1), then ${\lambda }_m^{\prime }={\lambda }_m+({ϵ}_m^{\prime }-{ϵ}_m)M_{mk}$, and we only need to calculate one element of the sum instead of all $n$ elements.

Comparing the two formulas, it seems that Glynn’s formula is faster than Ryser’s by a factor of 2, as the number of possible $\delta$ is half the number of possible $ϵ$. This is indeed true as $n\to \infty$, but Ryser’s formula seems to be better optimized for small matrices where $N<6$ in our implementations (see Fig. 9).

Fig. 9

Comparison of the calculation time using different permanent algorithms.

Therefore, in this software, we use “$\text{Ryser}+\text{gray}$ and $\text{Glynn}+\text{gray}$” mixed algorithm to calculate the permanent, i.e., “$\text{Ryser}+\text{gray}$” for $N<6$ and “$\text{Glynn}+\text{gray}$” for large $N$. This ensures that we always have an optimized permanent calculation efficiency for different scenarios of photon counts.

5.1.

Boson Sampling

Boson sampling, an analog quantum computing model first raised by Aaronson and Arkhipov,⁷⁴ has been an emerging field in the quantum research community. Unlike universal quantum computation scheme that usually requires active elements, boson sampling schemes only require a passive linear optics interferometer, single-photon source, and photodetection (photon number resolving is not necessary). Results show that these experimentally friendly devices may offer the first evidence that refutes the extended church-turing (ECT) thesis and set a key milestone in the quantum computing field called quantum supremacy.¹²

The boson sampling problem can be modeled as a multiphoton scattering process. We first prepare an input state consisting of $N$ single photons in an $M$ modes passive linear optics interferometer. The process is very similar to that in the multiparticle quantum walks in which $N$ single photons are injected into $M$ waveguides, but a key difference lies in there not being a unitary matrix in boson sampling schemes that can be effectively described by the Hamiltonian, similar to $U=e^{-\frac{iHt}{\hslash }}$ in the case of multiphoton quantum walks. In boson sampling, one normally defines $U$, a unitary map on the creation operators, directly. The injected photons in the boson sampling scheme interfere and scatter in the linear optics interferometers. The interferometer implements a Haar-random $M$-mode transformation $U$ on these $N$ indistinguishable bosons:

The hardness of this problem is rooted in evaluating the value of $P(T|S)$, related to the permanent of the scattering amplitudes. The calculation can follow the same equation as Eq. (6).

5.2.

GUI and Functions for the Module of BosonSampling

In the GUI of BosonSampling, users need to import an $M\times M$ unitary scattering matrix as $U$ and are informed of an error if the imported matrix does not satisfy the requirement for a unitary matrix. The users also need to design their interferometers. Generally, there are two types of decomposition, namely, the Reck’s type⁷⁵ and the Clements’s type⁷⁶ (see Fig. 10).

Fig. 10

Schematic diagram of the decomposition. An example of (a) 10 modes in the Reck decomposition and (b) 10 modes in the Clements decomposition.

After choosing the unitary scattering matrix $U$ and the decomposition style, users need to set the parameters for beam splitters and phase shifts that accomplish pairwise transformations between channels by satisfying $U=D(\prod T_{m,n})$, where $D$ is a diagonal matrix and $T_{m,n}$ is the transformation from channel $m$ to $n$ ($m=n-1$) through a lossless beam splitter with the reflectivity $\cos \theta$ and through a phase shift $\varphi$ at the input $m$ side.⁷⁶ $T_{m,n}(\theta ,\varphi )$ reads as

Users can manually set the parameters for all $T_{m,n}\mathrm{s}$. They can also ask the software to randomly set the parameters by choosing “reset” for $\theta$ and $\varphi$. A GUI for BosonSampling is shown in Fig. 11, where the decomposition with the defined type, number of modes, and injected photons is shown in the interactive board. The interferometer parameters can be shown by placing the mouse on the corresponding white dot that represents an interferometer. The obtained probability distribution data can be exported to facilitate further analysis for boson sampling.

Fig. 11

The GUI for the module of BosonSampling. Different panels on the GUI are marked in the figure, and the functions of all buttons in these panels are explained in Table 4.

Table 4

Functions of all panels for the module of BosonSampling.

5.3.

Educational Use for the Module of BosonSampling

Users can get familiar with the framework of boson sampling via this module. When they increase the injected photon numbers and mode numbers, they experience the increase of complexity of the output states and the computational time on classical computers. This deepens their understanding of the quantum advantage for boson sampling in real quantum hardware.