Alright, fault tolerant detection of a quantum error! Let’s dive right into it :)

At the core of this paper is a simple concept. All error correction schemes rely on ancilla qubits to detect errors on qubits containing quantum information. Those ancilla qubits need to be interacting with those info qubits, making quantum nondemolition measurements to detect error syndromes, or results that indicate the existence of errors. This is very different from trying to make a direct measurement on info qubits, as that would collapse their state, or try to otherwise directly gain information on errors.

This paper uses the circuit quantum electrodynamics model that is commonly found in RSL and Qlabs, where a microwave photon cavity is coupled to a transmon qubit. This coupling, as described in the Wallraff 2004 paper, is significant because the transmon qubit can be easily controlled and read, while the microwave photon cavity has a very long lifetime. Therefore, the qubit is able to be “stored” in the cavity, while read and controlled using the circuit. Note that the cavity is primarily considered by numbers of photons, while the transmon qubit can be described as different energy levels. The three lowest energy levels in the transmon are |g>, |e>, and |f>.

In this kind of system, a basic form of error correction uses a “cat” state, where the cavity has two different coherent states superposed on each other. The main significance of this is that the two logical states, 0 and 1, both correspond to an even number of photons in the cavity. Therefore, since the dominant kind of error in the cavity is single photon loss, if there was a detection of an odd number of photons in the cavity, then the experimenters can correct for such an error. The measurement of the parity of photons in the cavity is therefore the error syndrome for the loss of a photon. In fact, there is a coupling between the cavity and the transmon, such that if there is an odd number of photons in the cavity, the transmon will experience a rotation.

To prepare this, start with a qubit in a ground state of |g> and rotate it to |g> + |e>. Then, if there is an odd number of photons in the cavity, the transmon will experience a pi rotation to go from |g> + |e> to |g> – |e>. Otherwise, if there is an even number of photons in the cavity, the transmon will remain in the |g> + |e>. After the period of error syndrome measurement is complete, the transmon qubit is rotated, such that |g> + |e> is rotated to |e>, and |g> – |e> is rotated to |g>. Finally, a measurement of the transmon qubit will be able to determine the parity of the cavity, and then be used to reset the transmon for the next iteration.

The process sounds pretty great, except for one small problem: the cavity isn’t the only place where there could be a quantum error! While single photon losses can be found in the cavity when the cavity a photon “leaks” out of the cavity, errors can also happen in the excited states of the transmon qubit. And when errors occur on the transmon, bad things can happen in the cavity. See, the interaction that leads to the detection of cavity errors can flow the opposite way: a different state in the transmon qubit can also affect the cavity. In fact, the photons in the cavity will oscillate at different frequencies depending on the state of the transmon. We can refer to each of these frequency of oscillations as f(|g>), f(|e>), and f(|f>).

Typically, qubit errors can occur in two forms: A dephasing error, where the sign on the qubit gets turned around, from |g> + |e> to |g> – |e>, or a relaxation error, where the excited state falls down into the ground state. One can tell that the dephasing error will actually not affect the cavity state. That is, the frequency in the cavity will not change between the |g> + |e> state and the |g> – |e> states, which is great! It means that this dephasing error would essentially be “transparent”, and the cavity does not change because of a dephasing error.

However, the relaxation error is a greater concern. When that happens, you can see that the |g> + |e> state will fall into the |g> state. This induces a frequency change in the cavity. Furthermore, since this decay can happen at any point of time during the error correction process, it introduces a “random” period of time where this error has occurred, completely scrambling the information contained within the cavity! How might we solve this?

The crucial idea introduced in this paper is to exploit the higher states of the transmon qubit. See, the transmon qubit is often used because it acts as a two level system, with different energy transitions between each of the energy states. However, just because we commonly use two levels, doesn’t mean we don’t have access to the higher levels. In fact, if we use the third excited level, the |f> level, we can do some interesting things…

By the laws of quantum mechanics, there is no relaxation that allows the transmon qubit to go from |f> to |g>. This is referred to as a forbidden transition. The only kind of relaxation that could be present here would be a relaxation from |f> to |e>.

But wait, wouldn’t that be just as bad? After all, the cavity would view the transmon state at |f> as different from |e>, right? And herein lies the second crucial insight of the paper: you can change how the cavity qubit is affected by the transmon state! In other words, there is a way to introduce an additional drive such that f(|f>) = f(|e>). Then, such a relaxation error in the transmon is once again “transparent” relative to the cavity!

How does such a change occur? The primary idea is that an additional off-resonant sideband drive is introduced that causes an additional phase to be picked up by the cavity. This phase/frequency that is picked up is like a geometric phase that is dependent on two main factors: the detuning strength of the interaction and the dephasing frequency.

Given that the detuning strength is fixed, the experimenters were able to choose a dephasing frequency that introduced the exact amount of additional phase such that f(|e>) = f(|f>)!

Alright. So what does that mean?

The key is that with the new system, no longer would relaxation errors in the ancilla directly affect the cavity. Is this perfect? Not exactly – those relaxation and dephasing errors still mess up the cavity state, and could cause incorrect error correction to occur. However, it does mean that the cavity will be protected from simple errors in the transmon. This directly translates to a 5 times increase in lifetime of the transmon!

Eventually, given enough error correction improvements, we would hope to see fault-tolerant quantum computing, so that we can implement a full quantum computer! My words are getting more and more fuzzy, so I’m going to just go ahead and post this up :)