Measurement-based quantum error correction

Measurement-based (or one-way) quantum error correction (MBQEC) is a method with the capability to detect and correct any errors present in a measurement-based quantum computation (MBQC) setup [1]. There are a variety of methods and protocols we can use to perform QEC, although few that have been successfully implemented experimentally [2, 3]. An MBQC protocol requires a resource state and will inevitably experience errors, which may be caused by a wide variety of factors, including coherent, systematic control errors, environmental decoherence, channel loss and measurement errors [4]. This essay will discuss the theoretical methods utilised for MBQEC and how these may be implemented. Our long-term aim is to create a quantum computation setup that successfully corrects against a reasonable threshold percentage of errors. Such a thing is said to be fault tolerant. Before leaping straight into the process of MBQC, it seems prudent to first discuss the finer details of QEC. We know that all forms of quantum computation will experience errors, leaving scientists with the crucial, yet difficult task of developing an implementable method of rectifying this. Previous research into quantum information has shown that there are two main types of error, a bit-flip error and a phase-flip error. A bit-flip error occurs when two bits in a state are swapped. For instance, in the single qubit state defined by |ψ〉in = α |0〉+ β |1〉 a bit-flip error will output |ψ〉out = α |1〉+ β |0〉. This is equivalent to performing an X operation on the state, meaning that this may also be referred to as an X error. A phase-flip error incurs a change in sign, giving |ψ〉out = α |0〉 − β |1〉, also known as a Z error. Classically, although we still experience errors in our states, the system is easily correctable, as we can simply send many copies of the same state and take the outcome that occurs with the highest probability to be correct [5]. For a QEC procedure, we may not simply reproduce the classical configuration in a quantum setting as this relies on our ability to copy states, a well-known caveat of quantum mechanics [6]. Instead, we must consider alternative correction methods. One well-established method is the Shor code, which corrects one error on a system of nine qubits (or three logical qubits) [7]. However, this method is only useful in a quantum circuit setup, so we must instead discuss possible correction methods for a resource state.