# Harmonizing continuous noise to build a modular photonic quantum computer

This is a Perspective on "Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer" by J. Eli Bourassa, Rafael N. Alexander, Michael Vasmer, Ashlesha Patil, Ilan Tzitrin, Takaya Matsuura, Daiqin Su, Ben Q. Baragiola, Saikat Guha, Guillaume Dauphinais, Krishna K. Sabapathy, Nicolas C. Menicucci, and Ish Dhand, published in Quantum 5, 392 (2021).

By Francesco Arzani (Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany).

## Quantum computing with light

Light has been related in many ways to the development of quantum mechanics. It is only natural that it should play a primary role in the development of quantum technologies as well, and quantum computation is no exception. Moreover, light is arguably a nearly ideal medium to encode and transmit information, also in the classical realm. For example, photons interact weakly among themselves and with their typical environments, which makes light robust to noise, information can be hosted by many different degrees of freedom (polarization, frequency, arrival time, angular momentum to name just a few) and there is literally nothing that can deliver a message faster than light. As a result, much work has been devoted to describe, produce, control and detect light. A whole arsenal of technologies is at our disposal to build something interesting. The downside is that it may become hard to tell, amid the virtually infinitely many conceivable setups, which one is the most promising for a given application, and where the most urgent gaps to be filled may lie, be they technological or theoretical. Concerning quantum computation in particular, this highlights how beneficial it can be to have a vision and a description as detailed as possible of what a complete, fault-tolerant, photonic quantum computer would look like. The authors of [1] provide precisely one such blueprint. By combining previous ideas, some of which were initially developed in widely different contexts, the authors describe a modular setup composed of components that are already commercially available, or may become so in the near future. The keystone of this new scheme, as well as the main technical contribution of the article, is a clever way to describe what the authors identify as the main sources of noise, such that they match the requirements of each single module. This is not trivial because the noise model, on the one hand, needs to capture the relevant aspects of the actual implementation and, on the other hand, needs to result in something that an error correcting code can handle at the logical level. This is achieved in the paper using the language of continuous variables.

## Computation by entangling and measuring oscillators

Most of the theory of quantum computation deals with qubits, that is two-level systems. Consequently, much of the experimental efforts are also geared to systems that naturally present two energy levels that can be sufficiently isolated and well controlled. The setup of [1] takes a different route, which is known as continuous-variable approach. The latter consists of considering the full infinite dimensional Hilbert space of an ensemble of harmonic oscillators, specifically modes of the electromagnetic field. In this setting, the most relevant and easily accessible observables have continuous spectra. To have a concrete example in mind, almost everything can be phrased in terms of position and momentum, denoted $\hat{x}$ and $\hat{p}$ in the following. As recognized early on [2], discretization must still be enforced at some point to deal with continuous unwanted perturbations. But instead of the energy levels corresponding to photon number, finite-dimensional computational subspaces are specified here by different constraints: information is encoded in the simultaneous eigenspace of some $\textit{displacement operators}$. In a phase-space picture of harmonic oscillators, treating positions and momenta on equal footing, displacements are simply translation operators. The codespace is then identified by the states that are invariant under certain phase-space translations.

In the simplest example, a single qubit is encoded in one oscillator. The computational basis states are, in the ideal case, superpositions of position eigenstates

\begin{split}
\left|0\right>_\mathrm{GKP} &= \sum_{j=-\infty}^{\infty} \left|x = 2j\sqrt{\pi}\right> \\
\left|1\right>_\mathrm{GKP} &= \sum_{j=-\infty}^{\infty} \left|x = (2j+1)\sqrt{\pi}\right>
\end{split}\label{eq:GKPbasis}

which are simultaneous eigenstates of the stabilizer operators $\hat{S}_x = \exp\left(2i\sqrt{\pi}\hat{x}\right)$ and $\hat{S}_p = \exp\left(2i\sqrt{\pi}\hat{p}\right)$. The operators $\hat{X}_\mathrm{GKP} =\sqrt{\hat{S}_x}$ and $\hat{Z}_\mathrm{GKP} =\sqrt{\hat{S}_p}$ play the role of the logical $X$ and $Z$ Pauli operators. These encodings were initially introduced by Gottesman, Kitaev and Preskill [3] and are thus called GKP codes. They received much attention after recent experiments [4,5] have shown that the code-states are not as hard to realize as it was initially suspected. Interest in these codes stems from the fact that their symmetries make the encoded information inherently robust to small displacements in phase space. Since displacements form an operator basis, much like Pauli matrices for qubits, the robustness of GKP codes extends to other relevant noise sources, such as losses [6]. Moreover, imposing the GKP stabilizer conditions discretizes the state space, so that when GKP error correction fails, continuous displacements are converted into qubit-level errors. These can in turn be dealt with by imposing another error correcting code at the logical level, in an instance of a procedure called $\textit{code concatenation}$.

The code chosen by the authors for the higher concatenation level is tailored to a $\textit{measurement-based}$ [7], or one-way, approach to quantum computation (MBQC). Rather than applying one logic gate after the other to static systems, MBQC proceeds by embedding the information to be processed in a large entangled state and then measuring some parts of it. One can imagine the entangled state (called cluster state) as a strip that is consumed by the measurements. At the beginning, the encoded information sits at one end of the strip, then measurements are performed, consuming some of the strip and teleporting the information one step ahead. In the process, some logical operation is attached to the encoded state, which depends on the choice of the measurement. The beauty of it is that the whole strip does not need to exist at the same time: it can be dynamically produced on one end, while it is consumed on the other, the encoded information always sitting in between the two stages (see the Figure). This implies that the photons populating any of the modes that physically carry the logical information only traverse a constant number of optical components before they are destructively measured. The number of components does not depend on the length of the computation, which eases the coherence time requirements. For this reason MBQC has been extensively studied also in other photonic setups (see for example [8,9,10]).

## Code states, non-Gaussian operations and noise

For CV-MBQC linear combinations of position and momentum need to be measured, which is implemented fairly easily through homodyne detection. Entanglement is also fairly easy to generate compared to other setups. The reason is that all the operations involved in the tasks above only require so-called $\textit{Gaussian}$ resources. These are a class of operations that can be realized deterministically and with high accuracy in most continuous-variable settings. In contrast, their counterpart, $\textit{non-Gaussian}$ operations, require higher nonlinearities, which are either very difficult to realize, or can only be achieved non-deterministically. A tight bottleneck in optical GKP-based approaches comes from the fact that the resource states are in fact non-Gaussian, a hurdle that is shared by other photonic architectures as well. So what is the best way to cope with this? Well, while it is possible to produce GKP states deterministically in trapped ions [4] or as the steady state of a superconducting system [5], the most viable approaches in photonics involve post-selection. This is also the case in [1]: the authors consider a setup initially developed for Gaussian boson sampling (GBS, another light-based computational paradigm), in which a “simple” Gaussian state is produced, then all modes except one are measured with photon counters. If the right number of photons is detected in each mode the state is projected onto an approximate GKP code state.

A small aside is in order concerning this approximation. First of all, note that $\textit{some}$ approximation is necessary since the ideal states in Eq. \eqref{eq:GKPbasis} cannot be normalized. Usually each position eigenstate is replaced by a Gaussian packet and an overall envelope is added to capture the shape of physically realizable approximations. The quality of the approximate states is measured in terms of $\textit{squeezing}$, which is inversely related to the width of the Gaussian peaks replacing the Dirac deltas in the wave functions. The physical code states are thus called $\textit{finitely squeezed}$.

Returning to the preparation strategy, clearly the probability of projecting on the desired state is less than one. As a matter of fact, it decreases if a better approximation is demanded. It can, however, be boosted by multiplexing. In particular, the authors propose to run many Gaussian boson samplers in parallel and in case none of them produces the desired state just output a squeezed state, which is a simple Gaussian state. Behind this seemingly innocent choice is the realization that the entanglement structure of the cluster state for MBQC stays essentially the same if some modes are in a squeezed state instead of a GKP state. Moreover, both the errors coming from the approximate nature of the code states and from the probabilistic replacement can be treated within a single noise model. To give an idea, the authors start from a common model for finitely squeezed GKP states as those obtained by applying random displacements $\mathcal{D}\left(\alpha\right)$, where the shift $\alpha$ is drawn from a Gaussian probability distribution $G_{\Sigma}(\alpha)$, with covariance matrix $\Sigma\propto \mathrm{Id}$, to the ideal states
\rho_\mathrm{GKP}^\mathrm{physical} = \int \mathrm{d}^{2n}\alpha\, G_{\Sigma}\left( \alpha\right) \mathcal{D}\left(\alpha \right) \rho_\mathrm{GKP}^\mathrm{ideal}\mathcal{D}^\dagger\left(\alpha \right)
with $n$ the number of modes. To prepare the cluster states, one has to start from logical $\left| + \right>_\mathrm{GKP} \propto \left| 0 \right>_\mathrm{GKP} + \left| 1 \right>_\mathrm{GKP}$ states and apply (Gaussian) entangling gates. The above Gaussian noise with a modified covariance matrix turns these states into something similar to $\textit{a mixture of squeezed states}$. While this is not a perfectly accurate description of the actual physical process, it captures some of its most important features and allows one to easily simulate the effect of the (heralded) replacement of some code states with squeezed states in the cluster. Finally, as the authors show, this noise model is well suited for the error correction capabilities of the chosen MBQC model. Hence the logical-level error probabilities can be rendered arbitrarily small as long as the approximate code words are sufficiently good and the replacement probability sufficiently low.

## Conclusions and perspectives

The proposed setup has a number of appealing properties. First, it operates almost entirely at room temperature, with the exception of the photon counters at the state preparation (GBS) stage. The latter requires cooling because the photon counters rely on superconducting detectors. Second, it has a clear path towards scalability. Most of the setup can fit on integrated photonics circuits. The recent developments towards more integration, including the work of the Canadian company Xanadu, to which several of the authors are affiliated, give hope that it will soon be possible to also integrate the missing parts (such as good sources of highly squeezed states). Third, its modular structure allows to clearly separate the requirements of each stage. For example, only the state preparation module needs to be cooled, and only the measurement module needs to be tunable on the fly, contrasting with the entangling gates of the computational module that can be performed by a static device operating at room temperature.

Some of the technical requirements are, however, still out of reach of current technology. But, as mentioned earlier, the indication as to $\textit{what}$ needs to be improved is one of the most important contributions of the paper. The most pressing matters seem to concern the production of high quality squeezed states in integrated circuits on the technological side, and the development of more realistic theoretical analysis, that will inevitably have to include the impact of losses. It is unclear how long it will take to solve these issues, despite the considerable efforts already underway. Notwithstanding these uncertainties, by describing a complete fault-tolerant, modular quantum computer in such detail, this paper shows how much closer we are to building a working device than we used to be. This also helps putting into much better focus the holes and cracks that are yet to be filled, a merit that should not be underestimated at this stage.

## Acknowledgments

I would like to thank Jonathan Conrad and Bohan Lu for valuable feedback on an earlier version of this perspective article and Johannes Jakob Meyer for providing help in preparing the manuscript.

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