Measurement-Free QEC

This example demonstrates how to use tensor network to simulate a error correction process. We use the $[[7,1,3]]$ steane code and the measurement-free QEC^[Heußen] as an example. There are non-clifford gates in the quantum circuit, so we use tensor network to simulate the process.

Background Knowledge

Traditional quantum error correction requires measurment and feed back into quantum circuit. Limited by laws of physics, measurements are doomed to be costly timewise, e.g. in the NMR computing schme^[Boykin]. Decoherence may very well happen during measurement. Measurement-free quantum error correction was a scheme to circumvent this problem.

Measurement-free quantum error correction was first proposed by Aharonov et. al ^[Aharonov] in the 90s. Without using measurements, a universal set of fault-tolerant gates on encoded qubits was constructed in section 4 of the paper ^[Aharonov]. They used CSS code and assumed noise is local and un-correlated in time, i.e Markovian. They obtained a threshold of $\eta_c \approx 10^{-6}$, which is the considerably lower than that of the conventional method^[DiVincenzo]. This threshold was later improved to be "only about an order of magnitude worse than conventional schemes" ^[Ercan] with the Bacon-Shor code^[Paz].

Experimental realizations include

• Realization on trapped ion platform ^[Schindler]

• Realization on bosonic code qubits. ^[Gertler]

Implementation

Traditional quantum error correction involves several key procedures. First, the encoding procedure involves mapping the logical quantum information into a larger, redundant quantum state using QEC codes such as the Shor code or the surface code. Usually, those quantum code are defined by stabilizers. Then, the syndrome extraction is the process that we extract the value of the stabilizers into ancilla qubits. And we measure the ancilla qubits to detect the value of stabilizers. After detection, the error syndromes are identified, which indicates the presence and location of errors in the quantum state. Next, the error correction procedure uses quantum gates to apply operations that reverse the effects of errors, effectively restoring the quantum state to its original form.

In the measurement-free QEC protocol, we embed the classical truth table of the error correction into the quantum circuit directly. For example, if we measure the stabilizer 1 and 2 to ancilla qubit 1 and 2, and if they are both in state $|1\rangle$, we know that there is an X error on the first qubit. We can encode this information into the quantum circuit directly by a multi-controlled-X gate.

Since such multi-controlled gate is non-clifford, we can't simulate it with clifford circuit simulator.

Definition of Stabilizers and Encoding Circuits

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

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

Generate the encoding circuits of the stabilizers by encode_stabilizers. qcen 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.

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

Syndrome Extraction and Measurement-Free Error Correction

First, we generate the steane measurement circuit by measure_circuit_steane and st_pos records the ancilla qubits that store the measurement results of the stabilizers.

qcm,st_pos  = measure_circuit_steane(data_qubits[1],st;qcen)
vizcircuit(qcm)

Then we generate truth table for the error correction by make_table. For more detials on truth table, please check Inference with Truth Table.

table = make_table(st, 1;error_type = "Z")

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

Now we use correct_circuit to generate the measurement-free correction circuit by encoding the truth table on the quantum circuit directly.

num_qubits = nqubits(qcm)
qccr = correct_circuit(table, 22:24, num_qubits)
vizcircuit(qccr)

Circuit Simulation with Tensor Networks

We connect the encoding circuit, the measurement circuit, and the correction circuit to form a full circuit. And we apply a Y error on the third qubit after encoding.

qcf=chain(subroutine(num_qubits, qcen, 1:7),put(27,3=>Z),qcm,qccr,subroutine(num_qubits, qcen', 1:7))
vizcircuit(qcf)

Then we transform the circuit to a tensor network and optimize its contraction order. QCInfo records the information of the quantum circuit, including the data qubits and the number of qubits. fidelity_tensornetwork constructs the tensor network to calculate the fidelity after error correction.

tn = fidelity_tensornetwork(qcf, QCInfo(data_qubits, 27))

TensorNetwork
Time complexity: 2^244.0
Space complexity: 2^0.0
Read-write complexity: 2^10.817783121774532

Finally, we optimize the contraction order and contract the tensor network to get the infidelity after error correction.

optnet = optimize_code(tn, TreeSA(; ntrials=1, niters=5), OMEinsum.MergeVectors())
infidelity = 1 - abs(contract(optnet)[1])

0.6250000000000017

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