Universal quantum computation with optical four-component cat qubits

Introduction.–Quantum computers promise to solve some problems faster than classical computers by utilizing quantum entanglement and coherence Shor (1994); Grover (1996); Nielsen and Chuang (2002); Ladd et al. (2010). However, noise can easily destroy the entanglement and coherence of a quantum system, dramatically degrading the performance of a quantum computer Unruh (1995). The standard procedure to deal with noise is to introduce quantum error correcting codes, in particular, to encode quantum information in the subspace of a multi-qubit system and correct the errors by using the redundant subspace  Shor (1995); Steane (1996); Bennett et al. (1996); Laflamme et al. (1996); Ekert and Macchiavello (1996); Knill and Laflamme (1997); Kitaev (2003); Fowler et al. (2012). An alternative is to encode the quantum information in an infinite-dimensional bosonic system, e.g., a single harmonic oscillator. These sorts of codes are called bosonic codes. Examples include the Gottesman-Kitaev-Preskill (GKP) codes Gottesman et al. (2001), cat codes Cochrane et al. (1999); Ralph et al. (2003); Leghtas et al. (2013); Mirrahimi et al. (2014), binomial codes Michael et al. (2016) and rotation-symmetric codes Grimsmo et al. (2020).

A bosonic code is usually introduced to correct a specific kind of error, e.g., the GKP code corrects random phase and amplitude displacements Gottesman et al. (2001). However, a bosonic code itself is not sufficient for fault-tolerant quantum computation. The strategy is to concatenate the bosonic codes with qubit codes, which potentially reduces the resource overhead. It has been shown that a finite-squeezing GKP code can allow for fault tolerance Menicucci (2014), and the squeezing threshold can be reduced to around 10 dB Fukui et al. (2018); Noh and Chamberland (2020); Vuillot et al. (2019). Another important bosonic code is the two-component cat code, which can protect itself from phase rotation error but not photon loss, leading to a biased error model. The photon loss can be corrected by concatenating the cat code with some simple qubit codes, e.g., repetition code, giving more feasible protocols for fault-tolerant quantum computing Guillaud and Mirrahimi (2019); Puri et al. (2020). In optical platforms where gates are probabilistic, additional located erasure errors need to be addressed Lund et al. (2008), which can also be handled by concatenating with qubit codes. Along this line, the photon loss threshold for the two-component optical cat codes has been estimated Lund et al. (2008); Lee and Jeong (2013).

Another way to mitigate the photon loss is to introduce a cat code with more components Bergmann and van Loock (2016), e.g., the four-component cat code Mirrahimi et al. (2014). The four-component cat code is introduced to correct the single photon loss and its performance has been studied in Ref. Albert et al. (2018). The preparation and manipulation of the four-component cat code have been proposed for platforms with strong nonlinearity Mirrahimi et al. (2014). However for optical platforms, techneques to implement the universal set of gates and to correct the single photon loss have not been explored.

In this work we propose a scheme to implement the universal set of gates for the four-component cat code via teleportation, with appropriate entangled resource states and photon-number-resolving (PNR) detection. This is particularly appealing due to the rapid development of PNR detectors in the optical platform Rosenberg et al. (2005); Jönsson and Björk (2019); Morais et al. (2020). The advantage is twofold: the PNR detectors not only can be used for teleportation, but also can be used to generate the required entangled resource states Sabapathy et al. (2019); Su et al. (2019); Magana-Loaiza et al. (2019). We remark that this scheme is also promising for other platforms provided PNR detection and entangled resource states are available. Also based on teleportation, we then develop a concrete procedure to correct the single photon loss, which accomplishes the syndrome measurement and recovery operation simultaneously. By further concatenating with qubit codes, we estimate the photon loss threshold for fault-tolerant quantum computation.

Encoding.–In the four-component cat code, the state of a qubit is encoded in the superposition of four coherent states $|\pm\alpha\rangle$ and $|\pm i\alpha\rangle$, where $\alpha$ is assumed to be real. Specifically, the encoded Pauli $Z$ eigenstates are defined as $|0_{L}\rangle=|\mathcal{C}^{+}_{\alpha}\rangle=\mathcal{N}_{+}(|\alpha\rangle+|-\alpha\rangle)$ and $|1_{L}\rangle=|\mathcal{C}^{+}_{i\alpha}\rangle=\mathcal{N}_{+}(|i\alpha\rangle+|-i\alpha\rangle)$, where $\mathcal{N}_{+}=1/\sqrt{2(1+e^{-2\alpha^{2}})}$ is the normalization factor. Note that $|0_{L}\rangle$ and $|1_{L}\rangle$ are not orthogonal, $\langle 0_{L}|1_{L}\rangle=\langle\mathcal{C}^{+}_{\alpha}|\mathcal{C}^{+}_{i\alpha}\rangle=\cos\alpha^{2}/\cosh\alpha^{2}$, however the overlap is small for large $\alpha$. Since $|\mathcal{C}^{+}_{\alpha}\rangle$ and $|\mathcal{C}^{+}_{i\alpha}\rangle$ are related by a $\pi/2$ phase rotation, the encoded Pauli $X$ operation can thus be implemented deterministically. This is similar to the situation for the two-component cat code where the logical states $|\alpha\rangle$ and $|-\alpha\rangle$ are non-orthogonal and are related by a $\pi$ phase rotation.

Teleportation and universal set of gates.–We choose the universal set of quantum gates as Pauli $X$, arbitrary rotation about $Z$, Hadamard gate and controlled-$Z$ gate. The Pauli $X$ gate can be implemented deterministically while other gates are implemented probabilistically via gate teleportation. Before studying the gate teleportation, we first consider the state teleportation which simply propagates the input state to the output, and can be considered as an identity gate or a memory.

The state teleportation requires an entangled resource state and the Bell measurement. Here we choose the entangled resource state as the encoded Bell state,

where $\mathcal{N}=\cosh\alpha^{2}/\sqrt{2(\cosh^{2}\alpha^{2}+\cos^{2}\alpha^{2})}$ is the normalization factor. The Bell measurement is performed by appropriately arranging four $50:50$ beam splitters, one phase shifter and four PNR detectors, as shown in Fig. 1. A similar circuit has been used to amplify coherent states and teleport four-component cat states Neergaard-Nielsen et al. (2013). The novelty of our circuit is that we use PNR detectors and a different entangled resource state, which are essential for implementing quantum gates and correcting single photon loss.

The four PNR detectors count the photon number at each output, giving click patterns $[n_{A},n_{C},n_{A^{\prime}},n_{C^{\prime}}]$. If all components in the circuit are ideal, i.e., no photon loss, then the total number of detected photons $N=n_{A}+n_{C}+n_{A^{\prime}}+n_{C^{\prime}}$ is even. This is because both the encoded states and the entangled resource state consist of even numbers of photons. The click patterns are divided into two categories, one for success of teleportation and the other for failure of teleportation. When the teleportation was successful, one needs to perform appropriate encoded Pauli corrections, depending on the click patterns, to recover the input state. These are summarized in Table 1.

To implement the gates via teleportation, the entangled resource states need to be modified accordingly. For rotation about the $Z$ direction, one needs an entangled state $|\Phi_{R}\rangle\sim e^{i\theta/2}|0_{L}\rangle|0_{L}\rangle+e^{-i\theta/2}|1_{L}\rangle|1_{L}\rangle$ with $\theta$ the rotation angle; for the Hadamard gate, one needs an entangled state $|\Phi_{H}\rangle\sim|0_{L}\rangle|0_{L}\rangle+|0_{L}\rangle|1_{L}\rangle+|1_{L}\rangle|0_{L}\rangle-|1_{L}\rangle|1_{L}\rangle$; and for the controlled-$Z$ gate, one needs a four-qubit entangled state $|\Phi_{CZ}\rangle\sim|0_{L}\rangle|0_{L}\rangle|0_{L}\rangle|0_{L}\rangle+|0_{L}\rangle|0_{L}\rangle|1_{L}\rangle|1_{L}\rangle+|1_{L}\rangle|1_{L}\rangle|0_{L}\rangle|0_{L}\rangle-|1_{L}\rangle|1_{L}\rangle|1_{L}\rangle|1_{L}\rangle$, which is used as the shared entanglement of two teleportation circuits. Depending on the gate to implement, the Pauli correction needs to be adjusted accordingly, for example, for the Hadamard gate the $Z$ correction is converted to $X$ correction and the $X$ correction is converted to $Z$ correction Lund et al. (2008).

Generating entangled resource states.–We now consider a strategy for creating the entangled resource state required for the teleportation. We take the single mode encoded superposition state, $|+_{L}\rangle\sim|0_{L}\rangle+|1_{L}\rangle$, with amplitude $\sqrt{2}\alpha$ as our “free” resource state and seek a non-deterministic, heralded, linear optics method for producing the teleportation resource state given in Eq. (1).

Consider an optical circuit depicted in Fig. 2 which mixes two superposition states and two vacuum inputs together on a series of 50:50 beam splitters. When we obtain equal counts on the two detectors, $n_{{\mathrm{A}}^{\prime}}=n_{{\mathrm{C}}^{\prime}}$, and $n_{{\mathrm{A}}^{\prime}}$ is even then the projected state is the teleportation resource state given in Eq. (1). If $n_{{\mathrm{A}}^{\prime}}=n_{{\mathrm{C}}^{\prime}}$, but $n_{{\mathrm{A}}^{\prime}}$ is odd we obtain the phase flipped Bell state $\mathcal{N}\big{(}|0_{L}\rangle|0_{L}\rangle-|1_{L}\rangle|1_{L}\rangle\big{)}$, which is still useful for teleportation, just requiring a different Pauli operator correction.

The entangled resource states used to implement the universal set of gates for the four-component cat code are non-Gaussian states. For optical platforms, these non-Gaussian states can be probabilistically generated by measuring multimode Gaussian states using PNR detectors and post selecting certain measurement outcomes Sabapathy et al. (2019); Su et al. (2019); Magana-Loaiza et al. (2019). For platforms where strong nonlinearity is available, these non-Gaussian states can be generated deterministically.

Correcting single photon loss.–The standard error correction procedure consists of two steps: the first is to detect whether an error has occurred without disturbing the state, known as the syndrome measurement; the second is to recover the original state according to the results of the syndrome measurement, known as the recovery operation. The performance of the four-component cat code in correcting single photon loss has been explored in Ref. Albert et al. (2018), however a concrete error correcting procedure is still absence. Here we propose a concrete procedure to correct single photon loss via teleportation, which realizes the syndrome measurement and recovery operation simultaneously.

The four-component cat code is designed to correct single photon loss using the redundancy in the Fock basis states. The states of a qubit are encoded in the subspace with even numbers of photons. If there is single photon loss, the states are shifted to the subspace with odd numbers of photons. Therefore, a syndrome measurement is basically a parity measurement. When detecting an even number of photon, one concludes that no photons have been lost and no further operation is needed; when detecting an odd number of photon, one concludes that a photon has been lost and a photon needs to be added to the output state to recover the original state. If more than one photons have been lost, the above procedure results in uncorrectable errors.

Define two new cat states $|\mathcal{C}^{-}_{\alpha}\rangle=\mathcal{N}_{-}(|\alpha\rangle-|-\alpha\rangle)$ and $|\mathcal{C}^{-}_{i\alpha}\rangle=\mathcal{N}_{-}(|i\alpha\rangle-|-i\alpha\rangle)$, with $\mathcal{N}_{-}=1/\sqrt{2(1-e^{-2\alpha^{2}})}$ the normalization factor. The states $|\mathcal{C}^{-}_{\alpha}\rangle$ and $|\mathcal{C}^{-}_{i\alpha}\rangle$ consist of odd numbers of photons, therefore are not in the code subspace. We refer to them as being in the loss subspace. Together with $|\mathcal{C}^{+}_{\alpha}\rangle$ and $|\mathcal{C}^{+}_{i\alpha}\rangle$, one can form four states that are superposition of four coherent states: $|\psi_{\alpha}^{(0)}\rangle=c_{0}|\mathcal{C}^{+}_{\alpha}\rangle+c_{1}|\mathcal{C}^{+}_{i\alpha}\rangle$, $|\psi_{\alpha}^{(1)}\rangle=c_{0}|\mathcal{C}^{-}_{\alpha}\rangle+ic_{1}|\mathcal{C}^{-}_{i\alpha}\rangle$, $|\psi_{\alpha}^{(2)}\rangle=c_{0}|\mathcal{C}^{+}_{\alpha}\rangle-c_{1}|\mathcal{C}^{+}_{i\alpha}\rangle$, and $|\psi_{\alpha}^{(3)}\rangle=c_{0}|\mathcal{C}^{-}_{\alpha}\rangle-ic_{1}|\mathcal{C}^{-}_{i\alpha}\rangle$, where $c_{0}$ and $c_{1}$ are complex coefficients. Note that $|\psi_{\alpha}^{(0)}\rangle$ and $|\psi_{\alpha}^{(2)}\rangle$ are in the code subspace and are related by a Pauli $Z$ transformation, while $|\psi_{\alpha}^{(1)}\rangle$ and $|\psi_{\alpha}^{(3)}\rangle$ are not in the code subspace. When a photon is lost, these states are transformed as $|\psi_{\alpha}^{(k)}\rangle\rightarrow|\psi_{\alpha}^{[(k+1)\,{\rm mod}\,4]}\rangle$, up to a normalization constant.

Consider an encoded cat qubit with initial state $|\psi_{\alpha}^{(0)}\rangle$ going through a lossy channel which is modelled by a beam splitter with transmission $\eta=1-\epsilon$. If there is no photon loss, the state becomes $|\psi_{\alpha^{\prime}}^{(0)}\rangle$ with $\alpha^{\prime}=\sqrt{\eta}\alpha$; if there is single photon loss, the state becomes $|\psi_{\alpha^{\prime}}^{(1)}\rangle$ (see Appendix for details). The state after the lossy channel is then fed into the teleportation circuit shown in Fig. 1, namely, it acts as the input state. Note that the amplitude of the cat code decreases from $\alpha$ to $\sqrt{\eta}\alpha$ regardless of whether a photon is lost or not, therefore the amplitude of the entangled resource state has to be adjusted such that the amplitude of arm $C$ is $\alpha^{\prime}$, whilst that of arm $B$ remains $\alpha$. In the case where there is no photon loss, the state is simply teleported to the output and the total number of detected photon is even. The teleportation corrects the amplitude: $\alpha^{\prime}\rightarrow\alpha$. In the case where a photon is lost, the total number of detected photon is odd. This is because the input couples with one of the modes of the encoded Bell state Eq. (1), which consists of even numbers of photon. Therefore, the parity of the total number of detected photon tells us whether there is single photon loss or not when the encoded cat qubit goes through the lossy channel. This accomplishes the syndrome measurement.

Interestingly, the recovery operation can also be implemented via teleportation. From Eq. (1) and Fig. 1 it is evident that the output state of the teleportation (in the other mode of the encoded Bell state) consists of even numbers of photon and is in the code subspace. In order to recover the initial state of the encoded cat qubit, one only needs to apply appropriate encoded unitary transformations. In the case of state teleportation and no photon loss, these required unitary transformations are encoded Pauli operators, as shown in Table 1. We find that one can still recover the encoded initial state by applying appropriate encoded Pauli operators when there is single photon loss in the encoded cat qubit. The encoded Pauli operator that needs to be applied depends on the click pattern and the total number of detected photon (see Supplementary Material for details), as summarized in Table 2. The Pauli correction in the teleportation thus plays the role as recovery operation.

Other than photon loss in the encoded cat qubit, there are various sources of photon loss in the teleportation circuit, including the encoded Bell state, beam splitters, phase shifter and PNR detectors. An important question is whether the proposed error correction procedure can also correct single photon loss within the teleportation circuit. The answer is affirmative. In the situation where a single photon is lost in one of the modes of the entangled Bell state, we show that the initial state can be recovered (see Supplementary Material for details). The output states after photon number detection are the same as those where a single photon is lost in the encoded cat qubit, except for a possible global phase that is irrelevant. This implies the Pauli corrections that need to be applied are also given by Table 2.

Single photon loss in the beam splitters, phase shifter and photon number detectors can also be mitigated. Assume that the optical elements and the PNR detectors are not perfect, but the rate of photon loss along each path (the modes $A,C,A^{\prime}$ and $C^{\prime}$) is the same. Under this balanced loss assumption, one can combine all lossy channels along each path into one effective lossy channel. Since the loss rate in each path is the same, the effective lossy channels commute with the optical elements. One can therefore move them to the front of the teleportation circuit, giving a lossy channel in the input mode $A$, which may cause single photon loss in the encoded input state, and another lossy channel in one of the arms of the Bell state, which may cause singe photon loss in the Bell state. As a result, the photon loss from the optical elements and PNR detectors can be mitigated via the proposed error correction procedure.

Estimating loss threshold.–Since the four-component cat code can protect the encoded qubit from single photon loss, it is expected that the loss threshold will be higher if it is used for fault-tolerant quantum computation, as compared to codes which cannot correct single photon loss like the two-component cat code. Here we estimate the loss threshold based on the results of Ref. Lund et al. (2008), where concatenation of the two-component cat code with the Steane code or Golay code is used to estimate the loss threshold. The purpose of this preliminary estimation is to show that the four-component cat code is a promising candidate for fault-tolerant computation. To obtain an accurate loss threshold value one needs to perform a numerical simulation. By concatenating the cat codes with some more advanced qubit codes, e.g., surface code, which promises a much higher threshold, one expects the loss threshold would be higher. We leave this for future work.

There are two kinds of error one needs to deal with: the located error due to the failure of the teleportation and the phase flip error coming from the photon loss. If the teleportation failed, the gate implementation was not successful and one simply removes the relevant qubits. For the two-component cat code, the teleportation fails when two PNR detectors detect no photons. For the four-component cat code, there are more click patterns that result in failure of teleportation, as shown in Table 1. The failure probability is obtained by summing the probabilities of detecting all those failure click patterns. We find that for a fixed cat amplitude, the failure probability for the two-component cat code is lower than that for the four-component cat code. If the failure probability is fixed, the amplitude of the four-component cat code is approximately twice as big as that of the two-component cat code (see supplementary materials).

The phase flip error rate for the two-component cat code is the probability of losing an odd number of photons and is dominated by the probability of losing one photon. Since the four-component cat code can correct a single photon loss, so the phase flip error rate is dominated by the probability of losing two and three photons. The phase flip error rate depends on the cat amplitude and the photon loss rate. In order to obtain a fair comparison of loss threshold between the two cat codes, we adjust the amplitudes of the two codes so as to set the located error rate of the two codes equal. By fixing the phase flip error rate, we can compare the loss value between the two cat codes.

It is shown in Ref. Lund et al. (2008) that if the amplitude of the two-component cat code $\alpha_{2}>1.2$, the loss threshold is around $\epsilon=5\times 10^{-4}$. As an example, we choose the amplitude of the two-component cat code as $\alpha_{2}=1.5$, so a four-component cat code with amplitude $\alpha_{4}=3.0$ would produce approximately the same located error rate. Once the cat amplitudes are given, the relation between the phase flip error rate and the loss is known, as plotted in Fig. 3. It is evident that a significant improvement in the loss threshold can be achieved by employing the four-component cat code. For a phase flip error rate that corresponds to a loss value around $5\times 10^{-4}$ for the two-component cat code, the corresponding loss value for the four-component cat code is about $5\times 10^{-3}$ with perfect teleportation circuit and $3\times 10^{-3}$ with lossy teleportation circuit.

Conclusion.–We developed a scheme to implement the universal set of quantum gates for the four-component cat code via gate teleportation. This is the first proposal for optical implementation and is particularly promising due to the fast development of PNR detectors. It is also suitable for other platforms provided the PNR detection is available. Also based on the teleportation, a concrete procedure to correct the single photon loss was proposed. Furthermore, we estimated the loss threshold for the four-component cat code when it is concatenated with qubit codes, and found a significant improvement as compared to the two-component cat code. This shows that the four-component cat code is a promising candidate for fault-tolerant quantum computation.

Note: In the preparation of this work, Ref. Hastrup and Andersen (2021) was posted, which like this current work addresses the task of all-optical error correction with a cat code albeit using a different and complementary approach.

We thank Austin Lund for providing some valuable references.