Measurement-free preparation of grid states

Quantum computing offers exponential speeds-ups in solving certain computational problems, with wide-ranging consequences for information processing, information security, fundamental physics and chemistry and more. Impressive progress has been achieved towards realising quantum computing, including recent experimental demonstration of a quantum advantage over classical computation Arute et al. (2019). However, real devices are subject to noise and imperfections. As computations grow in size and complexity, errors accumulate and eventually destroy any quantum advantage unless mitigated. Achieving fault tolerance, where errors are corrected sufficiently fast to allow scalable computation, is a central challenge to making universal quantum computing practical.

Quantum error correction (QEC) enables large-scale quantum computing in the presence of noise by redundantly encoding logical qubits into a larger Hilbert space. In traditional, discrete-variable QEC many physical qubits make up a single logical qubit. However, in 2001 Gottesman, Kitaev and Preskill (GKP) proposed encoding a logical qubit into the infinite-dimensional Hilbert space spanned by the continuous variables of a single bosonic mode Gottesman et al. (2001). With this encoding, small displacement errors of the bosonic mode can be detected and corrected using only simple Gaussian operations. Furthermore, recent results have shown that the GKP code also performs very well against boson loss Albert et al. (2018), in many cases outperforming other bosonic codes designed specifically against loss such as cat codes Leghtas et al. (2013); Ofek et al. (2016) and binomial codes Michael et al. (2016); Hu et al. (2019). In fact, numerical optimization suggests that the hexagonal GKP code might be the optimal loss-resistant code among all bosonic codes Noh et al. (2018). Additionally, GKP codes have recently been shown to have applications within continuous-variable QEC Noh et al. (2019) and quantum metrology Duivenvoorden et al. (2017).

An ideal GKP code is embedded in an idealized grid state which forms a lattice structure, consisting of an infinite superposition of position eigenstates. Such states require infinite energy and are hence unphysical. Importantly, however, it is possible to use approximate grid states with finite energy, composed of finitely squeezed states to achieve fault tolerance by concatenating the GKP code with discrete-variable error-correcting codes, provided that the grid states are sufficiently quadrature squeezed. In 2014, a conservative threshold for fault tolerance of 20.5 dB squeezing was derived for a measurement-based quantum computing approach Menicucci (2014). Later this threshold was significantly reduced to less than 10 dB squeezing by exploiting the analog information contained in the syndrome measurements Fukui et al. (2018); Fukui (2019). Other approaches such as concatenating the GKP code with the surface code Noh and Chamberland (2019), the toric code Wang (2019); Vuillot et al. (2019) and Knill’s C₄/C₆ code Fukui et al. (2017) have recently been proposed. For any of these proposals the squeezing threshold will depend not only on the involved codes, but also on the type and magnitude of the noise and experimental errors of the given system. It is therefore crucial to test the feasibility of these approaches with high-quality grid states experimentally. Additionally, as with any quantum error-correcting code, one would ideally use grid states with squeezing levels well above the threshold to avoid impractical resource overheads associated with the repeated concatenation of the codes.

The preparation of grid states have, however, proven to be highly challenging. Recently, such states were prepared for the first time in ground-breaking experiments in the motional state of a trapped ion Flühmann et al. (2019) and in a microwave cavity field coupled to a superconducting circuit Campagne-Ibarcq et al. (2019). The states realised in these experiments clearly exhibit the required grid structure in phase space. However, the quality of the states needs to be improved for implementation with fault tolerant schemes. The main experimental limitation is that during the preparation protocol, the states accumulate noise e.g. from boson dephasing and losses, rendering the produced grid states noisy. To minimize this noise one has to increase the speed of the preparation protocol. The state-preparation protocols currently implemented in experiment use oscillator-qubit couplings and rely on repeated measurements of the ancilla qubit. These measurements and their associated processing times constitute about half of the total preparation time. Therefore, to improve the quality of the GKP codes, it is crucial to replace the slow measurement-based approach with a faster approach.

In this work, we present a new, measurement-free grid-state preparation protocol, which is significantly faster than known methods, without introducing additional resources. The key interaction of our protocol is the Rabi interaction Hamiltonian between an oscillator and a two-level system Kockum et al. (2019); Forn-Díaz et al. (2019), which can be effectively simulated in trapped-ion and microwave systems. This interaction is also used in the experiments of Refs. Flühmann et al. (2019); Campagne-Ibarcq et al. (2019). Such interactions were recently shown to enable deterministic, non-Gaussian operations by using many weak interactions Park et al. (2017, 2018). Here, we instead use only a few, but stronger interactions, to generate the highly non-Gaussian grid states. Our work thus provides further demonstration of Rabi interactions as a powerful and versatile non-Gaussian resource in trapped ion and superconducting circuit platforms.

The speed-up obtained with our approach is large enough to prepare grid states with more than $10$ dB of effective squeezing in practical systems that are readily available in both trapped-ion and microwave cavity platforms. With a further reduction of noise levels in future experiments, our protocol enables the generation of grid states with squeezing levels well above the fault-tolerance threshold levels, thus facilitating scalable quantum computing.

In this section we review the basic structure of grid states and the figures of merit used in this article. For a more extensive review, see e.g. Ref. Tzitrin et al. (2019).

Bosonic modes of harmonic oscillators are associated with the creation and annihilation operators $\hat{a}$ and $\hat{a}^{\dagger}$ and the corresponding dimensionless quadrature operators $\hat{X}=\frac{1}{\sqrt{2}}(\hat{a}+\hat{a}^{\dagger})$ and $\hat{P}=\frac{1}{\sqrt{2}i}(\hat{a}-\hat{a}^{\dagger})$ satisfying $[\hat{X},\hat{P}]=i$. The 2-dimensional code space of the GKP-code is defined in the common +1 eigenspace of the stabilizer operators

Here $\hat{D}(x)=e^{x\hat{a}^{\dagger}-x^{*}\hat{a}}=e^{i\sqrt{2}(-\textrm{Re}(x)\hat{P}+\textrm{Im}(x)\hat{X})}$ is the displacement operator with displacement amplitude $x$, satisfying the commutation relation

By choosing $\operatorname{Im}(\alpha\beta^{*})=2\pi$ we ensure that the stabilizers commute, which enables the existence of simultaneous eigenstates. Furthermore we can define logical operators

which commute with the stabilizers and anti-commute with each other. The logical GKP qubit states, $\lvert 0\rangle_{\textrm{GKP}}$ and $\lvert 1\rangle_{\textrm{GKP}}$, are then defined as the $\pm 1$ eigenstates of $\hat{Z}_{L}$. These satisfy the expected logic $\hat{X}_{L}\lvert 0\rangle_{\textrm{GKP}}=\lvert 1\rangle_{\textrm{GKP}}$ and $\hat{X}_{L}\lvert 1\rangle_{\textrm{GKP}}=\lvert 0\rangle_{\textrm{GKP}}$.

The relative directions and magnitude of $\alpha$ and $\beta$ determine the lattice of the corresponding grid states. For example, rectangular grid states are generated by $\alpha=i2\pi/\beta^{*}$. Further choosing $\beta=\sqrt{2\pi}$ yields the square grid states for which the code space is symmetric with respect to $X$ and $P$. Alternatively, choosing $\alpha=i\sqrt{\frac{4}{\sqrt{3}}\pi}$ and $\beta=e^{-i\frac{\pi}{3}}\alpha$ yields the hexagonal grid states. In the following we will consider only the square grid, returning to the case of rectangular and hexagonals grid in Section C. The (unnormalizable) ideal square grid states can be written as:

where $\lvert X=0\rangle$ denotes the eigenstate of $\hat{X}$ with eigenvalue $0$ and $\mathbb{Z}$ denotes the set of integers. The ideal grid states are thus infinite superpositions of equidistant position eigenstates and their Wigner functions are an infinite grid of 2-dimensional delta-functions (see Fig. 2(b)). Ideal grid states can be approximated by finite-energy states in several ways. The most commonly used representation for deriving fault tolerance thresholds is a superposition of finitely squeezed states of width $e^{-r}$ under a Gaussian envelope of width $\kappa^{-1}$:

where $\hat{S}_{r}=e^{-\frac{1}{2}r(\hat{a}^{2}-\hat{a}^{\dagger 2})}$ is the squeezing operator (not to be confused with the stabilizers $\hat{S}_{x}$ and $\hat{S}_{z}$). The squeezing parameter $r$ and envelope $\kappa$ characterises the quality of the states in the $X$- and $P$-quadratures respectively and in the limit $(e^{-r},\kappa)\rightarrow(0,0)$ the approximate states converge to the exact states of equation (4). For $\kappa=e^{-r}$ the states can correct noise equally well in $X$ and $P$.

However, physical grid states will never exactly be of the form given in equation (5). First, physical states are not pure and are generally described by a density matrix $\hat{\rho}$. Secondly, the exact Gaussian envelope can be difficult to obtain and most preparation protocols yield a finite sum of squeezed states. Therefore, the parameters $r$ and $\kappa$ are not well-defined for practically realizable states. Instead, more generic figures of merit, the effective squeezing parameters, have been suggested in Ref. Duivenvoorden et al. (2017). They quantify the effective degree of squeezing in each quadrature of the peaks constituting the grid state and are defined as:

where the effective squeezing levels in units of dB are given by $\Delta_{\textrm{dB}}=-10\log_{10}(\Delta^{2})$. The expectation values in these definitions are exactly the expectation values of the stabilizers $\hat{S}_{z}$ and $\hat{S}_{x}$ for square GKP states. High quality grid states should therefore have $|\langle\hat{S}_{z/x}\rangle|\approx 1$, in which case $\Delta_{X/P}\rightarrow\infty$ dB. These definitions also have the nice property that for squeezed states they reproduce the squeezing parameter, i.e. $\Delta_{X}(\hat{S}_{r}\lvert\textrm{vac}\rangle)=e^{-r}$ and $\Delta_{P}(\hat{S}_{r}\lvert\textrm{vac}\rangle)=e^{r}$. Furthermore, for approximate square lattice grid states of Eq. (5) we extract the parameters $\Delta_{X}(\lvert\tilde{0}\rangle_{\textrm{GKP}})=e^{-r}$ and $\Delta_{P}(\lvert\tilde{0}\rangle_{\textrm{GKP}})\approx\kappa$. The last approximation is very accurate for $e^{-r},\kappa>10$ dB. Thus, if the state resembles a grid state consisting of a grid of squeezed peaks in phase-space, the effective squeezing parameters approximately quantify the squeezing of these peaks in each quadrature direction.

However, it is important to note that the effective squeezing is not directly related to the fault-tolerance thresholds. Most GKP-based fault-tolerance thresholds are derived based on the specific, approximate states given in Eq. (5) and refer to $r$ and $\kappa$. Any other state can therefore in general not be guaranteed to enable fault tolerant computations, even when the effective squeezing parameters are both above these thresholds. Moreover, the effective squeezing parameters say nothing about the logic state of the GKP qubit, e.g. a mixed code state might be strongly squeezed, but might not necessarily be useful for quantum computing.

Nevertheless, it is reasonable to assume that states with a high degree of effective squeezing can be used for fault tolerance if they otherwise closely resemble the approximate pure grid states of Eq. (5), e.g. in terms of their fidelity with the approximate states. In the further analysis we therefore compliment the effective squeezing parameters with the fidelity to verify the appropriateness of using the effective squeezing parameters as quantifiers of the protocol performance. Moreover, we also verify that the produced states have the expected grid structure in terms of their Wigner function. Another alternative figure of merit is the effective shift error Glancy and Knill (2006) which is discussed and calculated in Appendix A.

Several proposals exist for the preparation of approximate grid states Gottesman et al. (2001); Travaglione and Milburn (2002); Campagne-Ibarcq et al. (2019); Terhal and Weigand (2016); Shi et al. (2019); Su et al. (2019); Eaton et al. (2019); Vasconcelos et al. (2010); Weigand and Terhal (2018). The original GKP paper Gottesman et al. (2001) includes a proposal based on an radiation-pressure-like interaction between two bosonic modes under the Hamiltonian $\hat{X}_{1}\hat{a}_{2}^{\dagger}\hat{a}_{2}$ in the quantum non-linear regime. However, experimental realization of the required strongly nonlinear coupling has proven highly challenging and has not yet been achieved.

In Travaglione and Milburn (2002), a preparation protocol based on the Rabi interaction Hamiltonian $\hat{P}\hat{\sigma}_{x}$ (where $\hat{\sigma}_{x}$ is the Pauli-$x$ matrix), between the bosonic mode and a two-level system was proposed. Such an interaction can be realized in trapped ions Haljan et al. (2005) and microwave cavities Campagne-Ibarcq et al. (2019). This protocol, however, has three main drawbacks: First it is probabilistic, with a success probability inversely proportional to the mean photon number of the generated state. Secondly, the output states have a box-shaped envelope rather than the Gaussian envelope of equation (5). This means that the effective squeezing parameters are suboptimal given the number steps required to prepare the states. Hence, excessively large states need to be generated to obtain useful effective squeezing. Finally, the protocol requires qubit measurements, which in realistic systems will constitute a significant contribution to the total preparation time during which the state decoheres.

The two first issues were solved by Terhal and Weigand in Ref. Terhal and Weigand (2016): By adding a single measurement-based feed-forward displacement operation as well as suitable qubit rotations, the protocol is made deterministic. Furthermore, by using a different strength of the Rabi interactions, the envelope of the output state is made nearly Gaussian, making the protocol much more efficient. However, their protocol still relies on qubit measurements, which limits the quality of the states that can be realistically generated in the laboratory today.

Our protocol addresses all the above mentioned problems by adding additional short Rabi interactions of the form $\hat{X}\hat{\sigma}_{y}$, which effectively act as “deterministic measurements” by disentangling the bosonic mode and the qubit and further enables us to shape the envelope of the state. This interaction can be obtained from the $\hat{P}\hat{\sigma}_{x}$ Hamiltonian by simple rotations of the qubit and the bosonic mode, Similarly, both interaction types can be obtained from the more commonly considered Rabi Hamiltonian $\hat{X}\hat{\sigma}_{x}$. Fig. 1(a) shows a circuit diagram of the protocol. It consist of $N$ groups of interactions, each consisting of 3 gates:

• •

  a preparation gate, $\hat{U}_{k}=e^{iu_{k}\hat{X}\hat{\sigma}_{y}}$.

• •

  a displacement gate, $\hat{V}_{k}=e^{iv_{k}\hat{P}\hat{\sigma}_{x}}$.

• •

  a disentangling gate, $\hat{W}_{k}=e^{iw_{k}\hat{X}\hat{\sigma}_{y}}$.

These interactions can be interpreted as either conditional displacements of the bosonic mode or conditional rotations of the qubit, as illustrated in Fig. 1(b). Since the preparation and disentangling gates are of the same type, i.e. $\hat{X}\hat{\sigma}_{y}$, the preparation gate of round $k$ can be combined with the disentangling gate of round $k-1$ into a single gate. The interaction strengths of the displacement and disentangling gates are given by

while the interaction strengths of the preparation gates, $u_{k}$, are found numerically (see Appendix B). In the first round the optimal preparation gate strength is $u_{1}=0$ i.e. $\hat{U}_{1}=\mathbb{1}$ so $\hat{U}_{1}$ is thus ignored in Fig. 1(a). The input state is a squeezed vacuum state $\hat{S}_{r}\lvert\textrm{vac}\rangle$ and the output state is an approximation to the state $\lvert 1\rangle_{\textrm{GKP}}$, which can subsequently be transformed into an arbitrary grid state, as will be discussed later. Note that all gates commute with $\hat{D}(i\sqrt{2\pi})=e^{i2\sqrt{\pi}\hat{X}}$. Therefore, $\Delta_{X}$ is left invariant under the protocol, i.e. the effective squeezing of the output state in the $X$-quadrature is $\Delta_{X}=e^{-r}$. The effect of the protocol is thus to create a superposition of $2^{N}$ squeezed states and thereby improve $\Delta_{P}$. The effect of each gate is illustrated in Fig. 1(c) for the case of $N=3$, but the procedure can be extended for arbitrary $N$.

We now consider the effect of relevant noise sources on our protocol. To include noise effects in our model, we consider each gate as being implemented with a specific Hamiltonian for a set duration, e.g. the gate $e^{ic\hat{X}\hat{\sigma}_{y}}$ is implemented via the Hamiltonian $\hat{H}=\frac{1}{T}\hat{X}\hat{\sigma}_{y}$ within the time $t=cT$. To simulate the added noise, we use a master equation approach in which noise is included in the Lindblad terms $\hat{L}$:

where $\rho$ is the density matrix of the composite boson-qubit system. We consider four common noise channels:

• •

  Boson loss: $\hat{L}=\sqrt{\gamma}\hat{a}$

• •

  Boson dephasing: $\hat{L}=\sqrt{\gamma}(\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a})$

• •

  Qubit dephasing: $\hat{L}=\sqrt{\gamma}\hat{\sigma}_{z}$
  

• •

  Qubit decay: $\hat{L}=\sqrt{\gamma}(\hat{\sigma}_{x}+i\hat{\sigma}_{y})/2$

The effect of these noise sources on the effective squeezing of the output states is shown by the solid lines in Fig. 4. For each noise source we consider $N=2$ and $N=3$ rounds with $11.5$ dB and $16.6$ dB squeezed input states respectively. It is clear that our protocol is sensitive to all types of noise. By increasing $N$ we also increase the implementation time of the protocol, thus increasing the effect of noise. Therefore, there exists an optimal number of rounds that depends on the magnitude and type of noise. E.g. for large noise contributions, two rounds ($N=2$) of the scheme produces states with higher effective squeezing degrees than three rounds ($N=3$), and this is simply a result of the extended time over which noise can accumulate. This clearly illustrates the importance of a fast preparation protocol.

Even though the quality of the generated states is limited by qubit and bosonic errors, the effect of qubit errors can be significantly suppressed by adding a few qubit measurements, after each of the disentangling gates $\hat{W}$. In the noiseless case, the qubit should be in a known state, disentangled from the bosonic mode at these points, as illustrated in the rightmost windows of Fig. 1(c). Therefore, if we measure the qubit in a different state, we know that an error has occurred, and the realization should be discarded and the protocol restarted. The result of such a postselection strategy is shown by the dashed lines in Fig. 4, demonstrating that we can improve the effective squeezing of the output state by several dB. Bosonic errors, on the other hand, are largely unaffected by the postselection strategy. Thus, when these errors are dominating the only way to improve the output states is to increase the interaction speed or reduce the rate of the noise. For the calculation of Fig. 4 we assumed instantaneous measurements to isolate the effect of qubit projections. In real systems the measurements will take time, during which noise accumulates thus resulting in lower effective squeezing parameters. However, compared to the measurement-based schemes, e.g. phase-estimation Terhal and Weigand (2016), we require exponentially fewer measurements and therefore still attain a significant speed-up.

Using realistic noise parameters and operation speeds from recent experiments with trapped ions Flühmann et al. (2019) and microwave cavities Campagne-Ibarcq et al. (2019), we find that grid states with effective squeezing parameters above 10 dB in both quadrature can be realistically generated in both platforms using input states squeezed by 11 dB (see Appendix D). Squeezing levels of 12.6 dB in trapped ions Kienzler et al. (2015) and 10 dB in microwave cavities Castellanos-Beltran et al. (2008) have been experimentally generated.

In conclusion, we have presented a measurement-free protocol to deterministically prepare GKP states using only few interactions of the type $\hat{X}\hat{\sigma}_{y}$ and $\hat{P}\hat{\sigma}_{x}$, which are readily available in trapped-ion and microwave-cavity platforms. Our protocol requires no measurements, resulting in a speed-up over previous methods, which enables the generation of grid states with high effective squeezing levels. Furthermore, by adding a few measurements we can partly detect qubit errors, thus making the protocol robust against qubit noise. Although the exact requirements for general CV states (i.e. states not exactly on the form of Eq. (5)) to enable fault-tolerance with the GKP encoding are yet unknown, it seems reasonable that states generated using this protocol suffices, due to their high fidelity with the commonly considered approximate grid states of Eq. (5).

Finally, our protocol exemplifies the versatility of sequential applications of non-commuting Rabi Hamiltonians, e.g. $\hat{P}\hat{\sigma}_{x}$ and $\hat{X}\hat{\sigma}_{y}$, demonstrating that highly non-Gaussian states can be deterministically engineered with only a few of these interactions. The full power of such repeated combination of Rabi interactions remains still relatively unexplored, but we expect that many other interesting applications are possible using this technique.