Historical research guide · 0587 · 1996
Perfect Quantum Error Correcting Code
Raymond Laflamme · Cesar Miquel · Juan Pablo Paz · Wojciech Hubert Zurek
In one minute
Raymond Laflamme, Cesar Miquel, Juan Pablo Paz and Wojciech Zurek, working at Los Alamos and in Buenos Aires, settled a sizing question at the heart of quantum computing. A quantum bit is fragile: let the outside world touch it and it picks up one of three kinds of damage — a flip, a sign change, or both at once. Classical machines fix errors by keeping copies, but a quantum state cannot be copied, so you spread it out instead. Peter Shor had protected one bit using nine; Andrew Steane had used seven. This paper shows that five is enough, and that five is the floor. The counting is simple: sixteen outcomes have to be told apart — no error, plus the fifteen ways a single one of five qubits can go wrong — and only a five-qubit space is large enough. The team writes down the encoding, gives a short circuit that builds it, and notes that the same circuit run backwards both names the error and sets up the repair.
Connection
This one is a shelf-neighbour rather than a pillar. It says nothing about the vacuum, gravity, propulsion or fusion, and nobody should read it as though it did. It sits in chapter 1, the evidence ladder, as a clean example of what the top rung looks like: a result stated, proved minimal, checked by others within months, and now the working foundation of every quantum machine built since — including the hardware on which later vacuum-energy experiments are run.
From the source
One qubit of information can be protected against a general one-qubit error by distributing it over five qubits, and five is the minimal number for which this is possible.Abstract; text following Eq. 4
The counting argument fixes the minimum. Every qubit can suffer three distinguishable errors plus the unperturbed case, giving three times the qubit count plus one subspaces, doubled to hold both logical states — so two times three n plus one must not exceed two to the n. Shor’s nine-qubit and Steane’s seven-qubit codes satisfy it; five is the smallest that does.Equation 4 and surrounding text
Any interaction between one qubit and its environment resolves into exactly four operations: leave it alone, flip its sign, flip the bit, or do both. Correcting all four for any single qubit is therefore enough to correct a general one-qubit error.Equations 2 and 3
The encoding circuit is also the decoder. Run it backwards and the four extra qubits read out a syndrome that identifies which of the sixteen alternatives occurred; a simple unitary transformation then restores the original state. Earlier schemes needed a separate correction circuit.Abstract; Figure 1b; Table 1
The protection is quadratic. If the probability of an error on a single qubit is p, the encoded qubit’s fidelity behaves as one minus a constant times p squared, against one minus p for an unprotected qubit.Closing paragraphs, following Table 1
The code is not a classical linear code. Classical coding theory is built on the Hamming distance, which is too restrictive here; the five-qubit encoding is a genuinely quantum object built from three-particle Bell states, and its sign pattern — two minus signs in one logical state, four in the other — is the only remaining freedom.Introduction; Equations 5 and 6
The way in
https://doi.org/10.1103/PhysRevLett.77.198How to cite it
Raymond Laflamme, Cesar Miquel, Juan Pablo Paz, Wojciech Hubert Zurek (1996) Perfect Quantum Error Correcting Code doi:10.1103/PhysRevLett.77.198