Demonstration of topologically path-independent anyonic braiding in a nine-qubit planar code: supplementary material

S1. The


A. Preparation of the ground state
In the experiment, the ground state is created in 5 steps as follows: Step 1. Create 3 pairs of entangled-states |H 1 H 3 + |V 1 V 3 , |H 5 H 6 + |V 5 V 6 , |H 7 H 9 + |V 7 V 9 and two qubits |H 4 , |V 8 using 3 sandwiched-BBOs and a single-BBO (and some half wave plates) respectively.
Step 2. Apply the H gate on photons 3 and 4, and let photons 3 and 4 interfere on the PBS 1 , and post-select the events where there is exactly one photon exiting each output. This creates the GHZ-state |H 1 H 3 H 4 + |V 1 V 3 V 4 . Using a similar method, we could also create the GHZ-state |H 5 H 6 H 8 + |V 5 V 6 V 8 .
Step 3. Perform rotations on two GHZ-states to obtain |+ 1 + 3 + 4 + |− 1 − 3 − 4 and |+ 5 + 6 + 8 + |− 5 − 6 − 8 . Then we let photon 4 and 5 interfere on the PBS 3 and post-select the state Step 4. Let photons 7 and 8 interfere on the PBS 4 to create the state: Step 5. By using a Mach-Zehnder interferometer (MZI), we encode the spatial mode of photon 1 as qubit-2, to obtain the state where U and D are the upper and lower path of the MZI. Encoding H and D as qubit 0, and encode V and U as qubit 1, we obtain the ground state

B. Preparation of the excited state
By performing different operations on the ground state, we can create an excited state or a superposition of the state with and without the e anyons. The excited state is created by performing the Z 3 operator on photon 3 and the X 4 operator on photon 4, which can be realized using a HWP of 0 • and 45 • , respectively. The superposition in equation (6) of the main text is created by performing the √ Z 3 operator on photon 3 and the X 4 operator on photon 4, which can be realized using QWP of 0 • and HWP of 45 • , respectively.

C. Measurement of photonic qubit
In our experiment, two kinds of photonic qubit, polarization qubit and spatial qubit, are used. The measurement methods for these two kinds of photonic qubit are as follows.
Polarization qubit measurement: The experimental device for measuring polarization qubits consists of HWP, PBS and singlephoton detectors, as shown in Fig. S3(a). The Pauli-X measurement is implemented when the angle of HWP is 22.5 • , and the Pauli-Z measurement is implemented when the angle of HWP is 0 • . Spatial qubit measurement: The experimental device for measuring spatial qubits consists is shown in Fig. S3(b). The Pauli-X measurement is implemented via placing the Double-BS and setting the angle of HWP to 22.5 • . The Pauli-Z measurement is implemented via removing the Double-BS and setting the angle of HWP to 0 • .

OBSERVABLE FOR READING OUT THE ANYONIC PHASE
In this section discuss the observable P(θ) = | + +θ 123 + + θ| 123 (S5) which is used to read out the anyonic phase φ in the main text. In general we desire an observable O(θ) that takes the form where the expectation value is taken with respect to the state Such a quantity is convenient since the phase can be read out easily by a displacement of the function f . The operator (S5) is a simple quantity that satisfies the above requirements as can be shown below. To evaluate the expectation value it is more convenient to write the projection operator in terms of Pauli operators. Let us first define which is the Pauli operator that has as its positive eigenvector Ω(θ)|θ = |θ . Then the projection operator (S5) can be written We may now evaluate the state obtained after each of the terms in (S10) on the state (S7). We obtain cos θ|m 1 , e 1 , e 2 + e iφ cos θ|m 1 + i sin θ|m 2 , e 1 , e 2 − ie iφ sin θ|m 2 X 2 Ω 3 (θ)|χ = 1 √ 2 (X 2 Z 3 cos θ + iX 2 Z 3 X 3 sin θ) (1 + e iφ Z 3 )|ϕ = 1 √ 2 cos θ|m 2 , e 1 , e 2 + e iφ cos θ|m 2 − i sin θ|m 1 , e 1 , e 2 − ie iφ sin θ|m 1 .