# The additional complexity of quantum mechanics: how will quantum computers correct errors in the future?

Sina science and technology news Beijing time on November 29, according to foreign media reports, the topic of quantum error correction is far less than “quantum hegemony”. For the application of quantum computers, quantum error correction is far more important than quantum hegemony. So, what error correction methods will be used in practical quantum computers? Recently, researchers demonstrated it on a real machine.

In 1994, Peter Sauer, a mathematician who was still working at Bell Laboratories in New Jersey, proved that quantum computers can solve some problems much faster than classical machines, even exponentially. The question is, can we build quantum computers? Skeptics believe that quantum states are so fragile that the environment will inevitably confuse the information in quantum computers so that they are not quantum states at all.

A year later, Peter Sauer responded. The classical error correction scheme is to correct errors by measuring a single bit, but this method is not suitable for quantum bits, because any measurement will destroy quantum states and interfere with quantum computing. Sauer found a way to detect whether an error occurred without measuring the state of the qubit itself. This method marks the beginning of the field of quantum error correction.

With the vigorous development of this field, most physicists began to regard Sauer’s algorithm as the only way to build a practical quantum computer. Without this method, the performance of quantum computer can not be improved to solve really difficult problems.

As with normal quantum computing, developing error correction code is one thing, but implementing it in a working machine is another. However, in early October 2021, the research team led by physicist Chris Monroe of the University of Maryland reported that they successfully demonstrated many elements necessary for Sauer’s error correction loop to run.

So how did Peter Sauer solve this problem? In short, he took advantage of the additional complexity of quantum mechanics.

Repeated comparison

When designing error correction coding, Peter Sauer modeled the coding of classical repeaters, including copying each bit of information, and then checking these copies regularly. If one bit is different from the others, the computer can correct the error and continue the calculation.

In the quantum error correction coding designed by Sauer, he uses three independent “physical” qubits to encode a single qubit of information, namely “logical” qubits. Of course, his quantum relay coding cannot be exactly the same as the classical version. The advantage of quantum computing essentially comes from the fact that qubits can exist in the “superposition” of 0 and 1 at the same time. Since measuring quantum states will destroy this superposition, there is no direct way to check whether an error has occurred.

Instead, Sauer found a way to judge whether the three physical qubits are in the same state. If one of the qubits is different, an error has occurred.

Checking for errors in qubits is almost the same as solving a simple logic puzzle. If you give three as like as two peas, one of them may have different weights. Give you a simple balance. What measurement method will you use to determine whether there are balls with different masses? If so, which ball?

The answer is to pick out two balls and compare their weights, then replace one with the remaining ball and check it again. If the balance is balanced twice, all the balls are the same weight; If the balance is balanced only once, the weight of the replaced ball or the ball used for replacement is different; If the balance is unbalanced twice, the stationary ball is the one with different mass.

The Sauer code replaces the balance with two additional “auxiliary” qubits. First, the first auxiliary qubit is compared with the first and second physical qubits; Then, another auxiliary qubit is compared with the second and third physical qubits. By measuring the states of these auxiliary qubits, we can know whether they are in the same state without interfering with the three physical qubits containing information.

This code prevents the so-called “bit flip” – the only error that can occur in classical computing. However, qubits have another potential source of error.

Superposition is the key to quantum computing, but what is important is not only the value of qubits, but also the relative “phase” between qubits. You can think of the phase as a wave, which describes the positions of the peaks and troughs. When two waves are in phase, their ripples are synchronous. If two waves collide, constructive interference will occur and merge into a wave twice as large; However, if the two waves are reversed, when one wave is at the peak and the other wave is at the lowest point, they will cancel each other, that is, destructive interference.

The quantum algorithm makes use of the phase relationship between quantum bits to set up a situation in which the correct result of the calculation is long interference, which is amplified, and the wrong result is eliminated by destructive interference. However, if an error causes a phase reversal, the destructive interference will become a long interference, and the quantum computer will begin to amplify the result of the error.

Peter Sauer found that he could correct phase errors using a principle similar to bit flipping. Each logical qubit is encoded into three qubits, and the auxiliary qubit checks whether one of the phases is reversed. Then, Sauer combined the two codes to obtain the code, which can convert one logical qubit into nine physical qubits, so as to correct bit reversal and phase error.

Fault tolerance

Sauer’s coding can protect a single logical qubit from error in principle. But what if the error measurement itself is wrong? In this case, when you try to correct a non-existent error, a bit flip may occur, unknowingly introducing a real error. In some cases, this can cause errors to cascade into the coding.

Sauer’s coding also does not consider how to operate a quantum computer constructed by logical qubits. In 1996, after three years of pioneering research, Peter Sauer put forward the concept of fault tolerance. Fault tolerant coding can deal with the errors introduced by the environment, the imperfect operation of these qubits, and even the errors introduced by the error correction step itself – provided that the frequency of these errors is lower than a certain threshold.

In October 2021, a research team announced that they had successfully used bacon shor code, a fail safe version of Shor code, to demonstrate almost all the tools needed for a fully fault-tolerant quantum computer. They encode a logical qubit into a quantum state of nine ions, and then use four auxiliary qubits to prove that all single qubit operations required for quantum computing can be performed under fault-tolerant conditions. The results show that the fault-tolerant quantum computer is feasible.

However, it is still far from achieving this goal. We will see the advantages of error correction only when the quantum computer reaches about 100 logical qubits. Such a machine needs about 1300 physical qubits, because each logical qubit needs 9 physical qubits plus 4 auxiliary qubits (at present, the largest quantum processor is eagle newly released by IBM, with 127 physical qubits). Only then can we start to build a qubit factory and introduce error correction coding. (Ren Tian)