Chief Mentor, Acadinnet
Part III – Measurement and interpretation
In Part I we laid the foundation on which quantum algorithms are built. In part II we harnessed such exotic aspects of quantum mechanics as superposition, entanglement and collapse of quantum states to show how powerful quantum algorithms can be constructed for efficient computation. In Part III (the concluding part) we discuss two aspects of quantum computation: (1) the problem of correcting errors that inevitably plague physical quantum computers during computations, by algorithmic means; and (2) a possible underlying mechanism for the collapse of the wave function during measurement.
12. Quantum error corrections
In designing quantum algorithms, we assumed an ideal, non-interfering, execution environment. This is far from reality. Superposition and entanglement are very fragile quantum states because of the difficulty of insulating quantum computers from a variety of causes — decoherence, cosmic radiation, and spontaneous emission. Maintaining the state of a qubit for prolonged periods is difficult enough leave alone preserving the states of entangled qubits. Inevitably the computer and the environment couple to vitiate the computer’s quantum state. Attempts to eliminate or minimize this problem by hardware and software means is an on-going research area. Here we restrict ourselves to software means, i.e., build error correction algorithms by creating information redundancies in an enlarged Hilbert space for error detection and correction.
A classical bit is in either state 0 or 1. Its physical state is defined by a large number of electrons, so a few electrons going astray will not affect its state identity.1 A physical qubit is often represented by a single or a few quantum particles (e.g., electrons), hence inadvertent errors are serious. Classical digital computers have inbuilt hardware bit-parity checks and self-correcting steps to restore a bit’s state if it inadvertently flips. A qubit, on the other hand, has a continuum of quantum states and it is not obvious that similar corrective measures are generally possible. For example, a likely source of error in setting a qubit’s state is over-rotation by an imperfect unitary gate (unitary gates rotate state vectors without changing the vector’s length). Thus, a state α|0〉 + β|1〉 instead of becoming α|0〉 +βeiψ|1〉, may become α|0〉 + βei(ψ+δ)|1〉. While δ may be very small, it will still be wrong, and if left uncorrected, will amplify to larger errors. Thus, not only must bit flips be corrected but also its phase must be correct, else, quantum parallelism, which depends on coherent interference between the superpositions, will be lost. Another source of error – inadvertent ‘measurement’ – occurs due to unexpected computer-environment interactions where the environment interferes with the Hilbert space reserved for computation in a manner that collapses the evolving state vector. This is generally seen as a hardware design problem.
1One may view this as a form of repetition code, where each bit is encoded in many electrons (the repetition), and after each machine cycle, it is returned to the value held by the majority of the electrons (the error correction).