A important set of methods in the area of quantum computing, quantum error correction (QEC) is designed to protect quantum information against defects in quantum gate operations as well as from errors resulting from the delicate character of qubits and their interactions with the environment.
The Need of Quantum Error Correction (QEC): Because “quantum hardware” is far more delicate than in classical computers, quantum computers are well more susceptible to errors. Physical systems highly sensitive to perturbations are quantum bits (qubits). These disturbances can cause decoherence, the loss of quantum information resulting from entanglement with the environment, and faults in the application of quantum gates. Any quantum computing beyond a few steps will most certainly be unreliable without sufficient error correction. For a functional quantum computer, the quality of present qubits and logic operations calls for QEC.
Challenges in Quantum Error Correction Compared to Classical Error Correction
Implementing error correction in the quantum realm is much harder than in the traditional realm for a number of important reasons:
- No-Cloning Theorem: The no-cloning theorem, which asserts that unknown quantum states cannot be accurately replicated, means that the classical method of merely repeating a bit to identify and fix faults is typically not applicable for quantum states.
- Continuous mistakes: Bit-flip mistakes, in which 0 becomes 1 or vice versa, are the main problem for classical systems. On the other hand, because the quantum world is continuous, qubits can encounter an endless number of various types of errors, such as bit flips (X errors), phase flips (Z errors), and combinations of these (Y errors). Qubits need to be protected since they have both phase and amplitude information.
- Measurement Issue: A qubit’s superposition state will collapse and the quantum information it contains will be destroyed if it is directly measured to check for faults. Without measuring the state itself, error rectification must extract information about the error.
Quantum Error-Correcting Codes (QECC):
- Three-Qubit Codes: Among the first quantum codes were the three-qubit codes. Bit-flip and phase-flip errors can be corrected using different three-qubit codes.
- Nine-Qubit Shor Code: This code combines the bit-flip and phase-flip correction codes to fix arbitrary single-qubit faults (bit flip, phase flip, or both).
- Steane Code (code): Any single qubit fault can be fixed by this more complex code. These error correcting systems are based on entanglement. The Steane code has a well-established stabilizer description.
- Surface Codes: These codes show promise for fault-tolerant quantum computing because of their local connectivity needs and relatively high fault-tolerance threshold, which may make hardware implementation simpler.
- Quantum CSS code : One type of quantum code that is derived from conventional error-correcting codes is called a quantum CSS code.
- The particular noise properties of the quantum hardware determine which quantum error-correcting code is used.
How does work Quantum Error-Correcting Codes (QECC)
In order to shield quantum information from the damaging effects of noise, quantum error correction employs a number of meticulously planned procedures. This is a detailed explanation.
1. Encoding the Quantum Information: Converting one or more logical qubits into a greater number of physical qubits is the first stage. The no-cloning theorem, which forbids merely copying an unknown quantum state, introduces redundancy in this process in a fundamentally different way than conventional repetition. Rather, the logical qubit is represented by entanglement between the physical qubits in quantum error-correcting codes (QECC). For instance, a logical qubit in the three-qubit bit-flip algorithm.
jψi = aj0i + bj1i encode it to be jψci= aj000i+bj111i
In order to accomplish this encoding, a unitary operator is usually used. The coding space is the collection of encoded states.
2. Errors Occur on Physical Qubits: Because of interactions with the environment (decoherence) or flaws in the quantum gates, the physical qubits are prone to a variety of errors during storage or manipulation. Bit-flips (X errors), phase-flips (Z errors), or a mix of the two (Y errors) are examples of these faults. Notably, the Pauli operators (I, X, Y, Z) can be linearly combined to express any arbitrary single-qubit error. Because of the linearity of quantum mechanics, QECC are frequently made to correct these basic Pauli errors, which enables them to correct more general faults.
3. Measurement of Syndrome (Error Detection): A procedure known as syndrome measurement is used to determine whether an error has happened and maybe which one. Through the use of particular quantum gates (such as CNOT or CZ), ancilla qubits—auxiliary qubits prepared in a known state, often $|0\rangle$—interact with the encoded qubits. The purpose of these interactions is to extract information about the parity of the encoded qubits, which indicates whether a qubit or qubits have experienced an error of a certain type. The important thing about measuring syndrome is that it avoids the breakdown of the encoded logical state by not measuring it directly. The type and location of the error are indicated by the error syndrome, a classical bit string that is the result of the syndrome measurement. The defective state is projected into one of a limited number of error instances in this process. Using parity checks, the fundamental idea is taken from traditional error-correcting codes.
4. Error Correction (Recovery): A particular recovery operation (a unitary gate or a series of gates) is applied to the physical qubits in order to reverse the detected error, based on the derived error syndrome. For instance, an X gate is applied to the second qubit to return it to its initial state if the syndrome of the three-qubit bit-flip code indicates that it has flipped. The incorrect state is successfully mapped back to the code space via the recovery process. The ancilla qubits are usually returned to their initial state for the subsequent error detection cycle after the correction.
5. Logical Qubit Remains Protected: The logical qubit is represented by the encoded data, which is currently housed in the many physical qubits’ rectified state. The logical qubit is shielded from noise as long as the amount of errors that occur within a certain time frame or operation sequence is less than the code’s error-correcting capabilities (connected to its code distance). Significantly greater resistance to noise is offered by the logical qubit.
6. Iterative Error Correction: To continuously monitor and fix errors that may build, quantum error correction is frequently carried out repeatedly during the computation. Repeated cycles of syndrome measurement and repair are necessary for this.
The quantum gates themselves must be implemented on the encoded qubits in a fault-tolerant way in order for a quantum computer to do intricate calculations. This makes sure that errors brought to by the gates don’t spread so much that the code’s error-correction capabilities are overloaded.
Even lower error rates can be attained by using more sophisticated strategies, such as concatenated codes. This entails recursively encoding a logical qubit with one QECC, followed by encoding each of the code’s physical qubits with another QECC, and so forth. If the physical error rate falls below a specific threshold, this process can exponentially suppress the logical error rate (quantum threshold theorem).
Read more about the Examples of QECC
Read more about the Quantum Computing Algorithms