Do you like cats? In the field of quantum information, we love them very much. Today, we’re going to talk about cat qubits, whose properties offer hope for building a fault-tolerant quantum computer. The goal here is to introduce the concept of cat states, their use in quantum information, and their interesting properties in quantum error correction (QEC). As a Level-2 post, a solid foundation in quantum mechanics and quantum information is essential to fully enjoy reading it.
Let it purr…
1 ~ What do you mean “a cat”?
You’re wondering. To understand why we are talking about cats here, let me define what is needed. Let’s take, for example, two opposite-phase coherent states, i.e. quantum states of the quantum harmonic oscillator1
where is the coherent amplitude. These states are not orthogonal, since , hence they cannot be distinguished. However, when increases, decreases exponentially so that and can be discriminated with a probability of failure that approaches zero exponentially with . For this reason, we say that and are quasi-orthogonal, for large enough.
Now, if we sum these two states, we obtain a superposition of two macroscopically2 distinct states. This image might remind you of something: Schrödinger’s cat thought experiment. Thus, the superposition
is, up to normalization, called a cat state. Since
i.e. it contains only even Fock states, we say that it is an even cat state. Equivalently, since
i.e. it contains only odd Fock states, we say that it is an odd cat state.
We thus denote the normalized even and odd cat states as follows:
1.1 ~ And what about “cat qubits”?
You probably know that one can encode information using continuous-variable systems, such as a harmonic oscillator realized in superconducting circuits. This is what we call bosonic encoding. That’s great because it lets us encode information using the cat states we defined above: a qubit whose code space is defined by the two coherent states and . A cat qubit is then defined as the superposition of cat states:
This is equivalent to3 and . You might find it a little easier to understand with a Bloch sphere (see Figure 1).
Such an encoding pictured in Figure 1 is called a cat code.
Another way to visualize these states is through their Wigner function. I used the qutip package to plot Wigner functions of coherent states and a cat state. Here is the Python code that was used to produce the Wigner functions plotted in Figure 2.
The standard Wigner function of a (free and time-independent) cat state can be calculated5:
for
the complex phase-space variable, and the amplitude. It consists of three pieces that we can recognize in Figure 2: two Gaussians centered at representing each state of the cat in phase-space, and an interference term between these two superposed states. These oscillatory fringes are the genuinely quantum part since they produce negative regions of the Wigner function.
1.2 ~ So what’s the point of these cat codes?
The number one enemy of quantum information is, of course, noise. A qubit can be affected by it and undergo a bit-flip (switching and ) or a phase-flip (switching and ). QEC thus aims to provide techniques designed to protect quantum information from such errors arising from decoherence and noise. Correcting both types of errors requires error-correcting codes whose complexity is proportional to the square of the number of qubits (for example, by using surface codes).
That’s where the magic of cat codes comes in! Cat qubits have intrinsic resistance to bit-flip errors. Specifically, in the regime of , the 2-component cat encoding can be exploited to simplify existing QEC protocols: the two coherent states separate for increasing values of , the probability of a physical error induces a bit-flip exponentially decreases as , i.e. exponentially. With this important property of cat qubits, we can assume bit-flip errors as of no consequence and, hence, can be disregarded completely. It then remains only to correct the remaining phase-flip errors by using a simple repetition code. This reduces the complexity of the problem to linear versus quadratic. Two dimensions vs. only one! Such a low overhead can help leading the race to building a fault-tolerant quantum computer.
The important thing to remember is that, by the very nature of cat qubits, error correction is intrinsinc for bit-flips. As if bit-flips were corrected even before a QEC protocol was used, or as if cat qubits never “experienced” such errors. But why and how does that happen? Via the process called two-photon dissipation.
2 ~ Two-photon what?
Two-photon dissipation is what stabilizes the cat qubits and gives them their resistance to bit-flips. Indeed, cat codes are encoded in an oscillator subspace that is fragile under ordinary noise. And to prevent the oscillator state from leaving the computational subspace driven by and , one needs an engineered loss process in which the oscillator loses or gains energy in pairs of photons, so the dynamics drive the mode toward a desired manifold of states rather than toward ordinary vacuum decay. That’s the idea behind two-photon dissipation. Bit-flip errors are exponentially suppressed (with the mean number of photon ) if the stabilization rate realized by the two-photon dissipation is higher than that of typical errors.
How does this work in reality? Although a complete and detailed description of this engineering process is beyond the scope of this post, I will try to present the key equations and fundamental concepts in order to better understand its importance in protecting cat qubits.
2.1 ~ Dissipation?
A qubit is never perfectly isolated: it interacts with an environment (electromagnetic modes, thermal photons, materials, etc.). Hence, the system is an open quantum system. What we need is then an equation that describes the coherent evolution, dissipation and irreversible process: this is the role of master equations. When the interaction between the system and the environment (the external quantum system, called the bath) is weak, any changes to the combined system over time can be approximated as originating from only the system in question: this is a Markovian open quantum system. A Markovian bath has no memory and information lost into the bath never comes back. One of the general forms of Markovian master equations is the Franke–Gorini–Kossakowski–Sudarshan–Lindblad master equation (often simply called a Lindbladian or a Lindblad master equation).
A Lindbladian generalizes, in a way, the Schrödinger equation to open quantum systems: it preserves probability, positivity, and Hermicity. It has two parts:
- The Hamiltonian (ordinary quantum evolution of energy) ;
- The dissipative term (describes irreversible coupling to the environment).
If we now imagine our open quantum system such that the environment removes photons two at a time, we obtain the so-called two-photon dissipation.
2.2 ~ How can Lindbladian describe these kinds of systems?
We are going to use the jump operator6 so that we can define a two-photon loss by the Lindblad operator
Indeed, you can notice that
i.e. we “lost” two photons. Notice also how the two-photon loss preserves parity, whereas single-photon loss flips parity. This is why this process was chosen for cat states stabilization, since cat states are parity eigenstates (see Section 1)!
However this Lindblad example operator is not the best to describe such a system. Pure loss alone would simply drain photons away until vacuum. We can use a more sophisticated and engineered process:
giving the Lindblad master equation
where is the time derivative of the quantum state described by density operator (the open system analogue of Schrödinger equation) and is the two-photon dissipation rate setting the speed of the engineered dissipation (in Hz).
is called the Lindblad dissipator, a superoperator defined as
It describes the effect of irreversible coupling to the environment with specifying the physical process (here, the physical process is the loss of two photons). We recognize two parts in it: which describes stochastic jumps caused by the environment, and which are the normalization and backaction terms. This more “complicared” master equation describes how the environment continuously drives the oscillator toward states satisfying
namely and . This dissipation continuously keeps the state confined near the protected manifold. That is autonomous quantum error correction, or more specifically, an hardware level bit-flip error resilience!
This is, of course, a simplifed model. In practice, you need to model the full physical device that generates the effective two-photon dissipation. If you’re interested, you can check out the innovative and pioneering work of Alice&Bob7, where they use two bosonic modes: the storage mode that stores the cat qubit, and the buffer mode intentionally very lossy. One of their significant experimental findings showed that for each added photon in the cat-qubit state, the bit-flip time is multiplied by , which is an exponential bit-flip error suppression.
3 ~ How are cat qubits a game-changer in QEC?
Achieving universal fault tolerance traditionally requires a code capable of handling both bit-flip and phase-flip errors simultaneously. That’s why surface codes currently represent the industry standard for large-scale and fault-tolerant quantum computation architectures. These are two-dimensional topological quantum error-correcting codes, meaning physical qubits are arranged in a 2D checkerboard lattice. Conventional computers, on the other hand, require only a single dimension of error correction. Correcting both types of quantum errors at the same time is therefore quite difficult.
Fortunately, as we’ve seen, by using such cat codes, we effectively eliminate one of the two types of errors (bit-flips). All that remains is to correct the second type of error, phase-flips. This effectively amounts to 1D error correction, just like with classical computers, which is much simpler and yields much more efficient results!
3.1 ~ So what about phase-flips?
Since bit-flips are insignificant in this architecture, the only issue with a significant probability of occurring is the switch between the and , namely the even and odd cat states. When we increase the number of photons to reduce the number of bit-flips, we also increase the risk of losing one, and thus of causing a phase-flip. Indeed, losing a photon from an even cat state makes it odd, and losing one from an odd cat state makes it even.
Is this a snake biting its own tail (or, in this case, a cat)? Not really, because this risk increases linearly with the number of photons added. Reducing the number of bit-flips exponentially versus increasing the number of phase-flips linearly remains a more than acceptable trade-off! And we know how to correct a single type of error that occurs with a linear probability. For example, we can use repetition codes, which are used by conventional computers.
3.2 ~ But how does repetition codes work with quantum states?
Good question. You just redundantly encode a logical qubit into several physical qubits so local errors can be detected and corrected without directly measuring the logical quantum information. Let me explain it with an example.
Imagine that you want to encode the qubit
Then, a repetition code simply involves encoding it as a logical state
Figure 3 shows the circuit used to encode the logical qubit.
Now, if you want to identify a bit-flip error, this can be done through syndrome measurements using ancillary qubits. Instead of measuring the encoded state directly, parity measurements reveal which qubit has been corrupted while preserving the amplitudes and (see Table 1).
A syndrome measure means that you determine the parity of each term in the expression. For example, is even, so it will yield ; is odd, so it will yield ; is even8, so it will yield , and so on. Do this for both the and terms, and refer to Table 1, which provides a table of the gates to apply based on the result obtained. If you get , you will apply ; if you get , you will apply ; etc.
This way, you correct bit-flip errors. But wait a minute! We said that with cat qubits, bit-flips weren’t really our problem anymore and that we needed to worry about phase-flips instead. And you’re right. Repetition codes can also be used to correct phase-flips instead of bit-flips. The repetition code in Figure 3 can be adapted to phase-flip errors by applying Hadamard gates before and after the encoding circuit. In this rotated basis, phase errors become effective bit-flip errors and can therefore be corrected using the same repetition principle, and the same syndrome measurement described in Table 1. This gives us our simple repetition-correcting code for phase-flip errors, designed for our cat qubits that are already protected against bit-flips.
3.3 ~ Can we make it even more efficient?
Yes! Cat qubits truly have interesting properties for error correction. Repetition codes are certainly interesting, but imagine if we could use this dimensionality reduction to do things that would be impossible without this property of cat qubits. This is what recent research has been focusing on, particularly the researchers at Alice&Bob. LDPC (Low-Density Parity-Check) codes, used notably in flash storage technologies, have found a use in quantum computing. While using these codes normally requires a 3D architecture, using cat qubits reduces this to just 2D. The researchers were thus able to design a 2D LDPC architecture requiring five times fewer logical qubits than their previous error-correcting codes. As a result, Shor’s famous algorithm could be implemented on a 2048-bit RSA key using just… 100,000 cat qubits. This is 200 times fewer than other traditional approaches9!
This is only the beginning: from a theoretical standpoint, but even more so from a practical one. Mastering a cat qubit in the lab and making the technology scalable enough to be useful requires a great deal of talent and effort. Only time will tell if these tiny cats, the smallest you’ll ever encounter, will enable humanity to solve complex problems, as promised by everyone in the quantum sector.
4 ~ Can I ask a question?
How to compute the Wigner function plotted in Figure 2?
I aim to provide the calculations involved in the construction of the Wigner function of a coherent state and the Wigner function of a superposition of coherent state, namely a cat state.
Coherent state Wigner function
To better understand the construction of a coherent state Wigner function, we need to recall the definition and property of this state. A coherent state is the unique eigenstate of the annihilation operator with eigenvalue , meaning it remains unchanged by the annihilation of field excitation: . From the representation of in the Fock basis (see Section 1), we recognize the power-series expansion of the exponential function, and thus
Since and satisfy the canonical commutation relation, it derives from the Baker-Campbell-Hausdorff formula that the equation above leads to the equality
where is called the displacement operator. Indeed, acts on the vacuum state by displacing it into a coherent state of amplitude . In the phase-space representation, displaces a localized state by a magnitude , which makes sense when we look at the Wigner function of : a Gaussian blob of the vacuum, shifted so its center is at . Indeed, let be the wave function of the vacuum state
then, it follows that a coherent state is a displaced vacuum state
where and are the phase-space coordinates, related to coherent amplitude , i.e. and . Since the global phase is irrelevant for the Wigner function, we keep this definition of the coherent state wave function:
Our goal is now to compute the Wigner function of a coherent state of wave function described above. First, we need to compute
and
and multiplying them:
Using
with ,
and therefore
Now
This Gaussian integral is
and taking , and , we obtain the Wigner function of a coherent state:
This is a minimum-uncertainty Gaussian in phase-space, centered at . It is also possible to express it directly in terms of amplitude , since the complex phase-space variable is
the result then becomes
Similarly,
They both are simply two Gaussian peaks in phase-space.
Cat state Wigner function
Let
be an arbitrary cat state built from two coherent states of amplitude and , and is some relative phase and the normalization constant. The density operator is
Since the Wigner transform is linear,
is the Wigner function of a coherent-state cat, where comes from the off-diagonal coherence terms. The cross-term has Wigner function
which gives
up to normalization factor. Finally, combining everything, we obtain the standard Wigner function of a (free and time-independent) cat state:
Notice also that for an even cat state or an odd cat state, defined in Section 1, and or which simplifies the equation above and clearly shows two Gaussian packets separated in and oscillatory fringes along .
Are these cat states easy to produce in practice?
Not really… In practice, production of cat states with a large mean photon number is difficult. States restricted to small values of are referred to as kitten states. How cute!
These states are useful, since their small10 amplitude can be amplified to transform them into bigger cat states. Moreover, they are “easy”11 to produce, since they can be well approximated through photon subtraction from a squeezed vacuum state. This is a way to produce bigger cat states, and that is the topic discussed in this section.
Photon subtraction is the process of removing a single photon from a light field. Following the same approach as in the question about the Wigner function (with the displacement operator), we introduce a unitary operator acting on the vacuum state , called the squeeze operator . When the squeezing operator is applied directly to the vacuum state,
rather than to a coherent state , the resulting state is called the squeezed vacuum state, where , i.e. . Let this be the initial state that is split into two modes on a beam splitter and vacuum on the other input port, i.e.
such that mode is the signal mode (the photon transmitted mode) and mode is the trigger mode (the photon reflected mode). It is known that, if a photon number detector in the trigger mode records photons, then the resulting state transmitted in the signal mode will be a squeezed vacuum state with photons subtracted. In the Fock basis, we have that
where is the magnitude of . A beam splitter operator of transmittance transforms into
Using these equations above, our initial state is thus transformed after the beam splitter into the output state
In this form, it is clear that if the photon detector in the trigger mode gets photons, then the state is projected onto , whereas the signal mode collapses into the photon-subtracted state
of normalization factor such that is the probability of detecting photons in the trigger mode (mode ), given by
From this, we notice that when an even number of photons detected in the trigger mode, the signal mode will consist of a superposition of only even Fock states, which reminds us of the definition of an even cat state (see Section 1). In the same way, an odd number of photons detected will give a state similar to an odd cat state. It is possible to decompose the state and to compute the Wigner functions of the resulting states of an even or odd photons detected number, and then showing how we can produce, using this technique, a cat state of weak amplitude, namely a kitten state.
You said that cat states are fragile because of noise. What is their decoherence?
The larger is, the more fragile a cat state becomes. Since cat states are produced during an optical experiment, the primary source of decoherence is photon absorption. Absorption is the process by which the energy of a photon ( according to Planck relation for some photon frequency ) is absorbed by another particle, such as an electron. This operation destroys the photon, which destroys the quantum information if it is encoded in cat states.
If the initial state of the cat system is
where the labels and represent the optical modes of the system, and if these two modes pass through a beam splitter of transmissivity , then the state of the system becomes
where is the normalization factor. As we can see, mode (which contained the vacuum before entering the beam splitter) is lost to the environment. Let
be the density matrix of the two-mode system. Indeed, mode is given by the partial trace
and we recognize
which is a cat state of amplitude . This shows how photon absorption affects a cat state by decreasing the amplitude of its coherent state. Using the full and normalized definition of , one can calculate the fidelity as a function of and between an initial even cat state and the resulting mode state :
where . This function gives quick drops to as decreases from , and for a large , even a small decrease in causes a significant drop in fidelity. For instance, with12 and a transmissivity of 99%, fidelity drops directly to 56%. Of course, all these results are similar if we use odd cat states.
What are your sources for this post?
I spent six months working on this topic for my master’s degree; here is the bibliography for my project report. It will provide you with useful resources to help you understand, explore further, and even verify some of my calculations.