Inference with Truth Table

This example demonstrates how to define stabilizers, encode data qubits measure syndromes, use truth table to infer error type and position, and correct the error.

We take the [[7,1,3]] Steane code as an example. We use Yao.jl to simulate a physical quantum devise and perform error correction.

Definition of Stabilizers

using TensorQEC, TensorQEC.Yao
st = stabilizers(SteaneCode())

6-element Vector{PauliString{7}}:
 XIXIXIX
 IXXIIXX
 IIIXXXX
 ZIZIZIZ
 IZZIIZZ
 IIIZZZZ

Then we can generate the encoding circuits of the stabilizers by encode_stabilizers. qc is the encoding circuit, data_qubits are the qubits that we should put initial qubtis in, and code is the structure records information of the encoding circuit.

qc, data_qubits, code = encode_stabilizers(st)
vizcircuit(qc)

Construction of the Truth Table

We can generate the truth table for the error correction by make_table. The TruthTable is a struct that records the truth table, the number of qubits, the number of stabilizers, and maximum error legth of the errors.

table = make_table(st, 1)

┌────────────────────┬────────────────┐
│            Sydrome │          Error │
├────────────────────┼────────────────┤
│             [1, 3] │         5 => Z │
│          [4, 5, 6] │         7 => X │
│                [5] │         2 => X │
│             [4, 5] │         3 => X │
│                [4] │         1 => X │
│                [1] │         1 => Z │
│                [6] │         4 => X │
│             [2, 3] │         6 => Z │
│       [1, 3, 4, 6] │ 5 => Z, 5 => X │
│             [3, 6] │ 4 => Z, 4 => X │
│             [1, 4] │ 1 => Z, 1 => X │
│             [1, 2] │         3 => Z │
│          [1, 2, 3] │         7 => Z │
│             [4, 6] │         5 => X │
│             [5, 6] │         6 => X │
│                [3] │         4 => Z │
│         ⋮          │       ⋮        │
└────────────────────┴────────────────┘
                         5 rows omitted

We can save the truth table to a file by save_table, and load the truth table from a file by load_table.

save_table(table, "test_table.txt")
table2 = load_table("test_table.txt", 9, 8, 1)
table.table == table2.table

true

Circuit Simulation with Yao.jl

Create a random qubit state to be encoded.

reg1 = rand_state(1)

ArrayReg{2, ComplexF64, Array...}
    active qubits: 1/1
    nlevel: 2

We use place_qubits to create a quantum register. data_qubits records the position of data qubits, and the rest ancilla qubits are in the $|0\rangle$ state.

reg = place_qubits(reg1, data_qubits, nqubits(qc))

ArrayReg{2, ComplexF64, Array...}
    active qubits: 7/7
    nlevel: 2

Apply the encoding circuits.

apply!(reg, qc)

ArrayReg{2, ComplexF64, Array...}
    active qubits: 7/7
    nlevel: 2

Apply an X error on the third qubit.

apply!(reg, put(7, 5 => Y))

ArrayReg{2, ComplexF64, Array...}
    active qubits: 7/7
    nlevel: 2

Measure the Syndrome and Inference the Error Type and Position

We first measure the stabilizers to get the error syndrome by measure_syndrome!. 1 means the stabilizer is not violated, and -1 means the stabilizer is violated.

measure_outcome = measure_syndrome!(reg, st)

6-element Vector{Int64}:
 -1
  1
 -1
 -1
  1
 -1

The measrue outcome shows that the stabilizer 1,3,4 and 6 are violated. According to the truth table, the error is $X_5$ and $Z_5$, which is exactly the error we applied. We can use table_inference to look up the syndromes in the truth table and infer the error type and position.

table_inference(table, measure_outcome)

2-element Vector{Any}:
 5 => Z
 5 => X

If we look up the syndrome that is not in the truth table, it will return nothing.

table_inference(table, [-1,-1,-1,-1,1,1])

No such syndrome in the truth table.

Error Correction

Then the following error correction is trivial. We apply the error correction.

apply!(reg, put(7, 5 => X))
apply!(reg, put(7, 5 => Z))

ArrayReg{2, ComplexF64, Array...}
    active qubits: 7/7
    nlevel: 2

Measure the stabilizers after error correction to check whether the error is corrected.

measure_syndrome!(reg, st)

6-element Vector{Int64}:
 1
 1
 1
 1
 1
 1

And we can calculate the fidelity after error correction to check whether the initial state is recovered.

apply!(reg, qc')
fidelity_after = fidelity(density_matrix(reg, data_qubits), density_matrix(reg1))

0.9999999999999998

This page was generated using Literate.jl.