Correcting biased noise using Gottesman-Kitaev-Preskill repetition code with noisy ancilla

We briefly review the notations used throughout this paper Gerry and Knight (2005); Weedbrook et al. (2012). The annihilation and creation operators of a single bosonic mode are denoted as $\hat{a}$ and $\hat{a}^{\dagger}$, respectively, and satisfy the commutation relation $[\hat{a},\hat{a}^{\dagger}]=1$. The position quadrature $\hat{q}$ and momentum quadrature $\hat{p}$ are defined as

$\displaystyle\hat{q}=\frac{1}{\sqrt{2}}(\hat{a}+\hat{a}^{\dagger}),~{}~{}~{}~{}~{}~{}\hat{p}=\frac{i}{\sqrt{2}}(\hat{a}^{\dagger}-\hat{a}),$

(1)

and they satisfy the commutation relation $[\hat{q},\hat{p}]=i$, where we have used the unit $\hbar=1$. This definition implies that the variance of the vacuum state is normalized to $1/2$.

The displacement operator $\hat{D}(\alpha)=e^{\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}}$, with $\alpha$ a complex number, represents a displacement of the quantum state in phase space. It is also useful to define $\hat{X}(u)$ and $\hat{Z}(v)$ as

$\displaystyle\hat{X}(u)=e^{-iu\hat{p}},~{}~{}~{}~{}~{}~{}\hat{Z}(v)=e^{iv\hat{q}},$

(2)

which represent a displacement of the position quadrature with $u$ and a displacement of the momentum quadrature with $v$, namely,

$\displaystyle\hat{X}^{\dagger}(u)\hat{q}\hat{X}(u)=\hat{q}+u,~{}~{}~{}~{}~{}~{}\hat{Z}^{\dagger}(v)\hat{p}\hat{Z}(v)=\hat{p}+v.$

(3)

If $\ket{s}_{q}$ and $\ket{s}_{p}$ are the position and momentum eigenstates, respectively, then

$\displaystyle\hat{X}(u)\ket{s}_{q}=\ket{s+u}_{q},~{}~{}~{}~{}~{}~{}\hat{Z}(v)\ket{s}_{p}=\ket{s+v}_{p}.$

(4)

A general displacement operator can be written as

$\displaystyle\hat{D}(u,v)=e^{-iu\hat{p}+iv\hat{q}}=e^{iuv/2}\hat{X}(u)\hat{Z}(v).$

(5)

The squeezing operator $\hat{S}(s)$ is defined as

$\displaystyle\hat{S}(s)=e^{i\ln(s)(\hat{q}\hat{p}+\hat{p}\hat{q})/2},$

(6)

where $s>0$ is the squeezing factor, and its action on the position and momentum quadratures is

$\displaystyle\hat{S}^{\dagger}(s)\hat{q}\hat{S}(s)=\frac{1}{s}\,\hat{q},~{}~{}~{}~{}~{}~{}\hat{S}^{\dagger}(s)\hat{p}\hat{S}(s)=s\,\hat{p}.$

(7)

It is evident that if $s>1$ then the position is squeezed and the momentum is anti-squeezed, while if $s<1$ then the position is anti-squeezed and the momentum is squeezed. A squeezed vacuum state is obtained by applying the squeezing operator to the vacuum state,

$\displaystyle\ket{s}_{\rm sq}=\hat{S}(s)\ket{0}.$

(8)

The variance of any quadrature in the vacuum state is the same, $\sigma_{\rm vac}^{2}=1/2$; while in the squeezed vacuum state, the variances of the position and momentum are not the same and are given by

$\displaystyle\sigma_{\rm q}^{2}=\frac{1}{2s},~{}~{}~{}~{}~{}~{}\sigma_{\rm p}^{2}=\frac{1}{2}s,$

(9)

respectively.

A two-mode unitary operator that is frequently used throughout this paper is

$\displaystyle\hat{U}_{2}=e^{-i\hat{q}_{1}\hat{p}_{2}},$

(10)

which is known as the SUM gate Gottesman et al. (2001). The action of the SUM gate on the position and momentum quadratures is given by

			$\displaystyle\hat{q}_{1}\rightarrow\hat{q}_{1},\quad\hat{p}_{1}\rightarrow\hat{p}_{1}-\hat{p}_{2},$		(11)
			$\displaystyle\hat{q}_{2}\rightarrow\hat{q}_{2}+\hat{q}_{1},\quad\hat{p}_{2}\rightarrow\hat{p}_{2}.$

The state of a continuous-variable (CV) quantum system can be represented by the Wigner function, which is defined in the $q$-$p$ phase space as

$\displaystyle W(q,p)=\frac{1}{\pi}\int{\rm d}y\,e^{-2ipy}{}_{q}\bra{q-y}\hat{\rho}\ket{q+y}_{q},$

(12)

where $\hat{\rho}$ is the density matrix.

The ideal GKP states are common eigenstates of two commuting operators, $\hat{S}_{q}=\hat{X}(2\sqrt{\pi})$ and $\hat{S}_{p}=\hat{Z}(2\sqrt{\pi})$, known as the stabilizers Gottesman et al. (2001). Therefore, they form a two-dimensional code subspace of an infinite-dimensional Hilbert space of a bosonic system, the computational bases of which can be chosen as

$\displaystyle\ket{\bar{j}}=\sum_{n=-\infty}^{+\infty}\ket{(2n+j)\sqrt{\pi}}_{q},$

(13)

where $j=0,1$, and the subscript “$q$” indicates a position eigenstate. An arbitrary ideal GKP state is a linear superposition of $\ket{\bar{0}}$ and $\ket{\bar{1}}$. The wave functions of the basis states $\ket{\bar{0}}$ and $\ket{\bar{1}}$ in position space are given by

$\displaystyle\bar{\psi}_{j}(q)=\sum_{n=-\infty}^{+\infty}\delta\big{(}q-(2n+j)\sqrt{\pi}\,\big{)}.$

(14)

It is evident that the wave function of an ideal GKP state is a superposition of a sequence of Dirac delta functions with a fixed period $2\sqrt{\pi}$.

Ideal GKP states are unphysical and cannot be prepared in experiments, though sometimes it is easier to analyze them and some instructive results can be derived. A physical GKP state is a linear superposition of squeezed coherent states with finite squeezing. Mathematically, the physical GKP state can be obtained through coherently superposing randomly displaced ideal GKP states that follow a given probability distribution Gottesman et al. (2001); Grimsmo and Puri (2021), namely,

$$\ket{\tilde{\psi}}=N\int{\rm d}u{\rm d}v\,\eta(u,v)e^{-iu\hat{p}+iv\hat{q}}\ket{\bar{\xi}},$$

(15)

where $\ket{\bar{\xi}}$ is an ideal GKP state, $N$ is a normalization factor and $\eta(u,v)$ is the probability amplitude describing the quantum diffusion process, which is chosen as a two-dimensional Gaussian distribution throughout this paper,

$$\eta(u,v)=\frac{1}{\sqrt{\pi\kappa\Delta}}\exp\left[-\frac{1}{2}\left(\frac{u^{2}}{\Delta^{2}}+\frac{v^{2}}{\kappa^{2}}\right)\right],$$

(16)

with $\Delta$ and $\kappa$ the standard deviations of the Gaussian distribution. It can be shown that the physical GKP state defined in Eq. (15) is normalizable (see Appendix A for details) and therefore contains a finite amount of energy. In the limit of $\Delta\rightarrow 0$ and $\kappa\rightarrow 0$, the physical computational basis states can be approximated as

	$\displaystyle\tilde{\psi}_{j}(q)=\bigg{(}\frac{4\kappa^{2}}{\pi\Delta^{2}}\bigg{)}^{1/4}\sum_{n=-\infty}^{+\infty}e^{-\pi(2n+j)^{2}\kappa^{2}}$
	$\displaystyle\times\exp\bigg{\{}-\frac{\big{[}q-(2n+j)\sqrt{\pi}\,\big{]}^{2}}{2\Delta^{2}}\bigg{\}}.$		(17)

The noise model that we consider is an anisotropic Gaussian displacement channel (GDC), namely, the noise in one quadrature and its conjugate quadrature may not be the same. Since we use the square-lattice GKP code, a biased logical error will be induced due to the anisotropic GDC, which is further corrected by concatenating with repetition code. This is equivalent to the scheme where an isotropic GDC is considered while the GKP code is biased Stafford and Menicucci (2022).

Suppose $\hat{\rho}_{0}$ is an input density matrix, then the output density matrix $\hat{\rho}$ after the anisotropic GDC is

$\displaystyle\hat{\rho}=\mathcal{N}_{f}(\hat{\rho}_{0})=\int{\rm d}u{\rm d}vf(u,v)\hat{D}(u,v)\hat{\rho}_{0}\hat{D}^{\dagger}(u,v),$

(18)

where $\hat{D}(u,v)$ is the general displacement operator defined in Eq. (5), and $f(u,v)$ is a bivariate Gaussian distribution,

$$f(u,v)=\frac{1}{\pi\delta_{q}\delta_{p}}\exp\left(-\frac{u^{2}}{\delta_{q}^{2}}-\frac{v^{2}}{\delta_{p}^{2}}\right),$$

(19)

with $\delta_{q}$ and $\delta_{p}$ the standard deviation of the position and momentum quadratures, respectively. According to the definition of the GDC, the input state is imposed a random displacement $\hat{D}(u,v)$ each time, and the output state is an incoherent mixture of all possible displacements. This implies that one can focus on a specific displacement each time when analyzing the error correction.

Under the GDC, the Wigner function is transformed in a simple way,

$\displaystyle W(q,p;\hat{\rho})=\int{\rm d}u{\rm d}vf(u-q,v-p)W_{0}(u,v;\hat{\rho}_{0}).$

(20)

This shows that the output Wigner function is a convolution of the input Wigner function and the error distribution function. Although the physical GKP state is not Gaussian, its Wigner function is a weighted sum of a sequence of Gaussian functions. Since the convolution of two Gaussian functions gives also a Gaussian function, the Wigner function of a physical GKP state after the GDC is still a weighted sum of a sequence of Gaussian functions. In addition, the variances of the output Gaussian functions are the sum of the variances of the input Gaussian functions and error distribution function. Therefore, the Wigner function is blurred after the GDC.

Note that the GDC is essentially different from the coherent superposition of random displacements that involved in defining physical GKP states in Eq. (15), in particular, a physical GKP state cannot be generated by simply passing an ideal GKP state through a GDC (see Appendix A for details). The physical GKP states have finite energy while the states obtained by passing ideal GKP states through GDC are unphysical and have infinite energy. However, these two sets of state have exactly the same noise property if we set

$$f(u,v)=\left|\eta(u,v)\right|^{2}.$$

(21)

This is clear when comparing their Wigner functions: the Wigner function of the former is a weighted sum of a sequence of Gaussian functions, with the weight decreases exponentially for large integers; the Wigner function of the latter is a sum of the same Gaussian functions but with equal weight for all integers. Therefore, we can treat the noise in the physical GKP state in the same way as we treat the noise from the GDC when the envelope is irrelevant.

Since the displacement errors in position and momentum spaces are independent, the error distribution of the physical GKP state given by Eq. (21) can be factored into the position and momentum parts,

$$f(u,v)=f_{q}(u)f_{p}(v),$$

(22)

where the error distribution in position and momentum space is respectively given by

$$f_{q}(u)=\frac{1}{\sqrt{\pi}\Delta}e^{-\frac{u^{2}}{\Delta^{2}}},\quad f_{p}(v)=\frac{1}{\sqrt{\pi}\kappa}e^{-\frac{v^{2}}{\kappa^{2}}}.$$

(23)

The displacement noise we usually consider is unbiased, i.e., $\Delta=\kappa$. In this paper, we consider GKP code with biased noise, where the noise in one quadrature is suppressed while that in the conjugate quadrature is amplified. The error distribution of biased noise can be parameterized as

$$f_{q}(u)=\frac{1}{\sqrt{\pi}\,r\Delta}e^{-\frac{u^{2}}{(r\Delta)^{2}}},\quad f_{p}(v)=\frac{r}{\sqrt{\pi}\,\Delta}e^{-\frac{v^{2}}{(\Delta/r)^{2}}},$$

(24)

where $r$ is a real positive number and represents the bias level. By choosing $r>1$, the noise in momentum space is suppressed while that in position space is amplified. In this case, we can concatenate GKP code with repetition code to suppress the logical Pauli error induced by the large displacement error in position space. The Wigner function of GKP code with unbiased and biased noise is shown in Fig. 1, where we choose $\Delta=0.25,r=1$ and $\Delta=0.25,r=\sqrt{2}$ for comparison.

The quantum error correction circuit using SUM gate is shown in Fig. 2. The ancillary qubit couples with the data qubit via the SUM gate, then its position quadrature is measured and the measurement outcome is fed forward to the data qubit Gottesman et al. (2001). Suppose the data qubit is prepared in a physical GKP state $\ket{\tilde{\psi}}$ with $\eta(u,v)$ given by Eqs. (15) and (16). Now we only consider correcting the displacement in position space and rewrite $\ket{\tilde{\psi}}$ as

$$\ket{\tilde{\psi}}=\int{\rm d}v\,\eta(v)e^{iv\hat{q}}\cdot\int{\rm d}u\,\eta(u)e^{-iuv/2}\ket{\psi(u)},$$

(25)

where $\ket{\psi(u)}$ is an ideal GKP state with position shifted by $u$,

$$\left|\psi(u)\right\rangle=\alpha\sum_{s}\left|2s\sqrt{\pi}+u\right\rangle_{q_{1}}+\beta\sum_{s}\left|(2s+1)\sqrt{\pi}+u\right\rangle_{q_{1}}.$$

The ancillary qubit is assumed to be in the ideal GKP $\ket{\bar{+}}$ state,

$$\ket{\bar{+}}=\sum_{k}\left|k\sqrt{\pi}\right\rangle_{q_{2}}.$$

(26)

According to the transformation rule of the SUM gate given by Eq. (11), the state after the SUM gate is given by

	$\displaystyle\int{\rm d}v\,\eta(v)e^{iv\hat{q}}\cdot\int{\rm d}u\,\eta(u)e^{-iuv/2}$
	$\displaystyle\times\bigg{[}\alpha\sum_{s,k}\ket{2s\sqrt{\pi}+u}_{q_{1}}\ket{(2s+k)\sqrt{\pi}+u}_{q_{2}}$
	$\displaystyle+\beta\sum_{s,k}\ket{(2s+1)\sqrt{\pi}+u}_{q_{1}}\ket{(2s+k+1)\sqrt{\pi}+u}_{q_{2}}\bigg{]}$
	$\displaystyle=\int{\rm d}v\,\eta(v)e^{iv\hat{q}}\cdot\int{\rm d}u\,\eta(u)e^{-iuv/2}$
	$\displaystyle\times\ket{\psi(u)}\bigg{(}\sum_{k}\ket{k\sqrt{\pi}+u}_{q_{2}}\bigg{)}.$		(27)

The homodyne measurement of the ancillary qubit gives a fixed value for $\hat{q}_{2}$,

$$q_{2}=k\sqrt{\pi}+u,$$

(28)

with $k$ an integer. This implies that the superposition of different displacements is destroyed and the state in Eq. (III.1) collapses to a component with a fixed $u$. However, the superposition between GKP states $\ket{\bar{0}}$ and $\ket{\bar{1}}$ (shifted by $u$) is preserved, since they cannot be distinguished by the measurement outcome.

Since we consider small displacement errors, so with a high probability $q_{2}$ deviates from $k\sqrt{\pi}$ in a small amount. Therefore, we infer the true value of $u$ by subtracting from $q_{2}$ the nearest $k\sqrt{\pi}$. Define a function $g(x)$, which gives the distance between $x$ and its nearest $k\sqrt{\pi}$,

$$g(x)=x-k\sqrt{\pi},~{}~{}~{}\text{ for }(k-\frac{1}{2})\sqrt{\pi}\leq x<(k+\frac{1}{2})\sqrt{\pi}.$$

(29)

So our guess for the value of $u$ is $g(q_{2})$ and we apply a displacement $-g(q_{2})$ to the data qubit in order to correct the error. With a high probability the displacement error can be corrected successfully, while sometimes the error correction procedure can introduce a large displacement error and therefore result in a logical Pauli error. Define the residual displacement of the GKP state after the SUM gate and feed forward as

$$u^{\prime}=u-g(q_{2})=u-g(u).$$

(30)

If $\left|u-2k\sqrt{\pi}\right|<\sqrt{\pi}/2$, then $g(u)=u-2k\sqrt{\pi}$ and $u^{\prime}=u-(u-2k\sqrt{\pi})=2k\sqrt{\pi}$, which means a stabilizer is applied to the GKP state and no error occurs. If $\left|u-(2k+1)\sqrt{\pi}\right|<\sqrt{\pi}/2$, then $g(u)=u-(2k+1)\sqrt{\pi}$ and $u^{\prime}=u-[u-(2k+1)\sqrt{\pi}]=(2k+1)\sqrt{\pi}$, which means a stabilizer and a logical Pauli operator $\bar{X}$ that flips the computational basis states are applied to the GKP state and a logical Pauli error occurs. We divide the displacement error in position space into two different zones, denoted as Pauli error zone (PZ) and no Pauli error zone (NPZ), according to whether they lead to a logical Pauli error or not,

	$\displaystyle\text{PZ}_{~{}}$	$\displaystyle=$	$\displaystyle\left\{u:|u-(2k+1)\sqrt{\pi}|<\frac{\sqrt{\pi}}{2},~{}k\in\mathbb{Z}\right\},$
	NPZ	$\displaystyle=$	$\displaystyle\left\{u:|u-2k\sqrt{\pi}|<\frac{\sqrt{\pi}}{2},~{}k\in\mathbb{Z}\right\}.$		(31)

For narrative convenience we define a serial number for PZ and NPZ,

	$\displaystyle\text{PZ}_{m~{}}$	$\displaystyle=$	$\displaystyle\left[(2m-\frac{m}{\left|m\right|}-\frac{1}{2})\sqrt{\pi},(2m-\frac{m}{\left|m\right|}+\frac{1}{2})\sqrt{\pi}\right),$
	$\displaystyle\text{NPZ}_{m}$	$\displaystyle=$	$\displaystyle\left[2m\sqrt{\pi}-\frac{\sqrt{\pi}}{2},2m\sqrt{\pi}+\frac{\sqrt{\pi}}{2}\right),$		(32)

with $m\in\mathbb{Z}$. Note that $\text{PZ}_{0}$ is not defined for the sake of symmetry. The location of NPZ_m and PZ_m is shown in Fig. 3.

With these definitions the error correction procedure can be summarized as follows:

	$\displaystyle u$	$\displaystyle\in$	$\displaystyle\text{NPZ}\Rightarrow u^{\prime}(\text{mod}\sqrt{\pi})=0~{}\Rightarrow~{}\text{perfect correction},$
	$\displaystyle u$	$\displaystyle\in$	${}_{~{}}\text{PZ}_{~{}}\Rightarrow u^{\prime}(\text{mod}\sqrt{\pi})=\sqrt{\pi}~{}\Rightarrow~{}\text{Pauli}~{}\bar{X}~{}\text{error}.$

The failure probability of error correction, $P_{\bar{X}}$, which is also known as the logical Pauli $\bar{X}$ error rate, is the probability that $u$ falls in the PZ.

	$\displaystyle P_{\bar{X}}$	$\displaystyle=$	$\displaystyle\int_{\text{PZ}}f_{q}(u){\rm d}u=\sum_{n=-\infty}^{+\infty}\int_{\sqrt{\pi}/2+2n\sqrt{\pi}}^{3\sqrt{\pi}/2+2n\sqrt{\pi}}f_{q}(u){\rm d}u$
		$\displaystyle=$	$\displaystyle\frac{1}{2}\sum_{n=-\infty}^{+\infty}\left[\text{erf}\left(\frac{4n+3}{2\Delta}\sqrt{\pi}\right)-\text{erf}\left(\frac{4n+1}{2\Delta}\sqrt{\pi}\right)\right].$

The relation between $P_{\bar{X}}$ and $\Delta$ is plotted in Fig. 4. We can see that a smaller $\Delta$, which corresponds to a higher degree of squeezing, leads to a lower logical Pauli $\bar{X}$ error rate.

We now consider a more realistic error correction procedure with physical ancillary GKP qubits. Suppose the variances of the data and ancillary qubit of the position quadrature are $\Delta^{2}$ and $\tilde{\Delta}^{2}$, and the displacement errors in position space are $u_{1}$ and $u_{2}$, respectively. The probability distribution of $u_{1}$ and $u_{2}$ are given by

$$f_{q_{1}}(u_{1})=\frac{1}{\sqrt{\pi}\Delta}e^{-\frac{u_{1}^{2}}{\Delta^{2}}},\quad f_{q_{2}}(u_{2})=\frac{1}{\sqrt{\pi}\tilde{\Delta}}e^{-\frac{u_{2}^{2}}{\tilde{\Delta}^{2}}}.$$

(34)

According to the transformation rule of the SUM gate, one can show that the measurement outcome of the ancillary qubit is

$$q_{2}=k\sqrt{\pi}+u_{1}+u_{2}.$$

(35)

However, both $u_{1}$ and $u_{2}$ are unknown. By using the same procedure as before, we infer the true value of $u_{1}$ by subtracting from $q_{2}$ the nearest $k\sqrt{\pi}$. This means our guess for the error in the data qubit is $g(u_{1}+u_{2})$. This is not exactly the same as $u_{1}$ except that $u_{2}=m\sqrt{\pi}$. However, this procedure is acceptable when $u_{2}$ is sufficiently small. We then apply a displacement $-g(u_{1}+u_{2})$ to the data qubit in order to correct its displacement error. The residual displacement in the data qubit is

	$\displaystyle u^{\prime}$	$\displaystyle=$	$\displaystyle u_{1}-g(u_{1}+u_{2})=k\sqrt{\pi}-u_{2},$		(36)
			$\displaystyle\text{for }(k-\frac{1}{2})\sqrt{\pi}\leq u_{1}+u_{2}<(k+\frac{1}{2})\sqrt{\pi}.$

It is evident that the residual displacement $u^{\prime}$ is continuous, in contrary to the ideal case where $u^{\prime}$ is discrete. However, one can still define whether a logical Pauli error occurs or not. When $u^{\prime}$ is close to $2k\sqrt{\pi}$, no logical Pauli error occurs; when $u^{\prime}$ is close to $(2k+1)\sqrt{\pi}$, a Pauli $\bar{X}$ error occurs.

In order to understand the error correcting property with physical ancillary qubit and to evaluate the logical Pauli error rate, one needs to compute the probability distribution of $u^{\prime}$, which is given by (see Appendix B for details)

	$\displaystyle F(u^{\prime})$	$\displaystyle=$	$\displaystyle\frac{1}{2\sqrt{\pi}\tilde{\Delta}}\left[{\rm erf}\left(\frac{u^{\prime}+\frac{\sqrt{\pi}}{2}}{\Delta}\right)-{\rm erf}\left(\frac{u^{\prime}-\frac{\sqrt{\pi}}{2}}{\Delta}\right)\right]$		(37)
			$\displaystyle\times\sum_{t}{\rm exp}\left[-\frac{(u^{\prime}-t\sqrt{\pi})^{2}}{\tilde{\Delta}^{2}}\right].$

We can see that $F(u^{\prime})$ is determined by a modulating term ${\rm erf}[(u^{\prime}+\sqrt{\pi}/2)/\Delta]-{\rm erf}[(u^{\prime}-\sqrt{\pi}/2)/\Delta]$ and a wave packet term $\sum_{t}\exp\left[-\frac{(u^{\prime}-t\sqrt{\pi})^{2}}{\tilde{\Delta}^{2}}\right]$. The former is determined by the degree of squeezing of the data qubit, while the latter is determined by the degree of squeezing of the ancillary qubit.

To have an intuitive feeling of the probability distribution, we plot several examples of $F(u^{\prime})$ in Fig. 5. It can be seen that the distribution has a high peak at $u^{\prime}=0$ and two low peaks that are located symmetrically with respect to $u^{\prime}=0$. The peaks outside the ${\rm PZ}_{\pm 1}$ are strongly suppressed by the modulating term, so the residual displacement outside the ${\rm PZ}_{\pm 1}$ can be neglected. Additionally, a smaller $\tilde{\Delta}$ leads to a narrower distribution of $u^{\prime}$ in the ${\rm NPZ}_{0}$, showing a better performance of error correction. This can be understood in an intuitive way: error correction using the SUM gate is basically to substituting the error of the data qubit by the error of the ancillary qubit, hence an ancillary qubit with higher quality naturally leads to a better performance of error correction. If the ancillary qubit is ideal, i.e., $\tilde{\Delta}=0$, then the distribution of $u^{\prime}$ approaches to a delta function, which means the state can be perfectly corrected.

The error correction is successful when $u^{\prime}$ is in NPZ and fails when $u^{\prime}$ is in PZ. Therefore, the failure probability of error correction, namely, the logical Pauli error rate is given by

	$\displaystyle P_{F}(\Delta,\tilde{\Delta})$	$\displaystyle=$	$\displaystyle\sum_{n=-\infty}^{+\infty}\int_{\sqrt{\pi}/2+2n\sqrt{\pi}}^{3\sqrt{\pi}/2+2n\sqrt{\pi}}F(u^{\prime}){\rm d}u^{\prime}$		(38)
		$\displaystyle\approx$	$\displaystyle 2\int_{\sqrt{\pi}/2}^{3\sqrt{\pi}/2}F(u^{\prime}){\rm d}u^{\prime}.$

The relation between $P_{F}(\Delta,\tilde{\Delta})$ and $\tilde{\Delta}$ for a fixed $\Delta$ is plotted in Fig. 6. We can see that $P_{F}$ monotonically decreases as $\tilde{\Delta}$ decreases, showing that an ancillary GKP qubit with higher quality naturally leads to lower logical Pauli error rate. In addition, it can be shown that

$$P_{F}(\Delta,\tilde{\Delta}\to 0)=P_{\bar{X}}(\Delta).$$

(39)

It is an important property that error correction by SUM gate with a physical ancillary qubit always increases logical Pauli error rate as compared to that with an ideal ancillary qubit. In the case of ideal ancillary qubit, a logical Pauli error occurs when $u_{1}\in{\rm PZ}$ and no error occurs when $u_{1}\in{\rm NPZ}$. While in the case of physical ancillary qubit, $u_{1}\in{\rm NPZ}$ may lead to a logical Pauli error because of the presence of an additional displacement $u_{2}$. Although $u_{1}\in{\rm PZ}$ may not lead to a logical Pauli error due to the same reason, its probability is much less than the former.

In the 3-qubit bit-flip repetition code Nielsen and Chuang (2010); Devitt et al. (2013), three physical qubits are introduced to encode one logical qubit, in particular, the single-qubit state $\alpha\ket{0}+\beta\ket{1}$ is encoded as follows:

$$\ket{\psi}=\alpha\ket{0}+\beta\ket{1}\to\ket{\bar{\psi}}=\alpha\ket{000}+\beta\ket{111}.$$

(40)

If one of the three qubits was flipped, the flipped qubit can be detected by comparing any two of the three qubits and then applying the majority rule, which is known as the syndrome measurement. Once the flipped qubit is identified, it can be corrected by applying a Pauli $X$ operator. The 3-qubit repetition code can only correct single-qubit bit flip, and bit flip on two or more qubits result in logical error. Denote the bit-flip error rate of a single physical qubit as $p$, then the probability of failure is given by

$$p_{f,\text{3-rep}}^{{\rm class}}=3p^{2}(1-p)+p^{3}.$$

(41)

To realize the comparison between physical qubits without collapsing the encoded state, one needs to introduce ancillary qubits to perform the syndrome measurement. Since one has to identify no error and three single-qubit bit-flip errors, there are $C_{3}^{0}+C_{3}^{1}=4$ syndromes and therefore two ancillary qubits are needed. The comparison of states of two qubits is implemented by the CNOT gate.

To concatenate the GKP code with repetition code, one needs to replace the standard qubits by the GKP qubits and find a CV gate that corresponds to the CNOT gate. It turns out that the SUM gate we used to perform GKP error correction plays the role as a CNOT gate. The quantum circuit of encoding is shown in Fig. 7, which is a generalization of the encoding circuit for repetition code. Before encoding, the first GKP qubit is prepared in the state $\ket{\xi}=\alpha\ket{\bar{0}}+\beta\ket{\bar{1}}$, and the other two data qubits are prepared in the same state $\ket{\xi_{1}}=\ket{\xi_{2}}=\ket{\bar{0}}$. After the encoding procedure, namely, the application of two SUM gates, three GKP states become entangled with each other,

$$\ket{\xi,\xi_{1},\xi_{2}}=(\alpha\ket{\bar{0}}+\beta\ket{\bar{1}})\ket{\bar{0}}\ket{\bar{0}}\to\ket{\bar{\psi}_{3}}=\alpha\ket{\bar{0}\bar{0}\bar{0}}+\beta\ket{\bar{1}\bar{1}\bar{1}}.$$

(42)

The state $\ket{\bar{\psi}_{3}}$ as defined is an ideal GKP repetition code state. One way to construct a physical GKP repetition code state is to coherently superpose the randomly displaced ideal GKP repetition code states, namely,

$$\begin{split}&\ket{\tilde{\Psi}_{3}}=\int{\rm d}u_{1}{\rm d}v_{1}{\rm d}u_{2}{\rm d}v_{2}{\rm d}u_{3}{\rm d}v_{3}\,\eta(u_{1},v_{1})\eta(u_{2},v_{2})\\ &\eta(u_{3},v_{3})e^{i(-u_{1}\hat{p}_{1}+v_{1}\hat{q}_{1})}e^{i(-u_{2}\hat{p}_{2}+v_{2}\hat{q}_{2})}e^{i(-u_{3}\hat{p}_{3}+v_{3}\hat{q}_{3})}\ket{\bar{\psi}_{3}}.\end{split}$$

(43)

Here we assume that the displacement in each ideal GKP qubit is independent and follows the same probability distribution. This definition of the physical code state is similar to the definition of a single-qubit physical GKP state (15). The GKP repetition code state defined in Eq. (43) is different from the state generated by applying two SUM gates to three single-qubit physical GKP states, since the latter would generate correlated noise between different GKP qubits. We use the GKP repetition code state (43) only to seek convenience for calculation, and we will leave the discussion on its experimental preparation for future work.

Error correction is performed after the encoding, which is implemented by the quantum circuit shown in Fig. 8. The full process of error correction consists of three steps: one round of GKP error correction, syndrome measurement and feed forward based on the measurement outcome. The three data GKP qubits, denoted as $D_{1}$, $D_{2}$, $D_{3}$, are physical and their noise variances of the position quadrature are the same, which is assumed to be $\Delta^{2}$. Three ancillary GKP qubits, denoted as $A_{1},A_{2}$ and $A_{3}$, are introduced to perform the GKP error correction, and they are prepared in the GKP $\ket{\tilde{+}}$ state. Another two ancillary GKP qubits, denoted as $A_{1}^{\prime}$ and $A_{2}^{\prime}$, are introduced to perform the syndrome measurement, and they are prepared in the GKP $\ket{\tilde{0}}$ state. The noise variances of the position quadrature of all ancillary GKP qubits are assumed to be the same and is $\tilde{\Delta}^{2}$. The residual displacements of the three data qubits after the GKP error correction are denoted as $u_{1}^{\prime}$, $u_{2}^{\prime}$, $u_{3}^{\prime}$, respectively. Their probability distribution is given by Eq. (37). Denote the displacement errors of the ancillary qubits $A_{1}^{\prime}$ and $A_{2}^{\prime}$ as $\alpha_{1}$ and $\alpha_{2}$, respectively, whose probability distribution is given by

$$f_{q_{i}^{\prime}}(\alpha_{i})=\frac{1}{\sqrt{\pi}\tilde{\Delta}}e^{-\frac{\alpha_{i}^{2}}{\tilde{\Delta}^{2}}},~{}~{}~{}~{}~{}~{}i=1,2.$$

(44)

The purpose of the syndrome measurement is to compare the states of three data GKP qubits, which is implemented by applying four SUM gates that act on the data qubits and the ancillary qubits in an appropriate way, as shown in Fig. 8. After these SUM gates, the displacement errors of the ancillary qubits $A_{1}^{\prime}$ and $A_{2}^{\prime}$ become $u_{1}^{\prime}+u_{2}^{\prime}+\alpha_{1}$ and $u_{1}^{\prime}+u_{3}^{\prime}+\alpha_{2}$, respectively. Then measurement of the ancillary qubits $A_{1}^{\prime}$ and $A_{2}^{\prime}$ gives $M_{1}=2k_{1}\sqrt{\pi}+u_{1}^{\prime}+u_{2}^{\prime}+\alpha_{1}$ and $M_{2}=2k_{2}\sqrt{\pi}+u_{1}^{\prime}+u_{3}^{\prime}+\alpha_{2}$, with $k_{1}\in\mathbb{Z}$ and $k_{2}\in\mathbb{Z}$.

For the classical 3-qubit repetition code, the qubit with bit-flip error is identified through the measurement outcome of the ancillary qubits, known as the syndrome. The one-to-one correspondence between the syndrome and the single-qubit bit-flip error Nielsen and Chuang (2010); Devitt et al. (2013) is summarized in Tab. 1. As an example, if the ancillary qubit $A_{1}^{\prime}$ is flipped while $A_{2}^{\prime}$ is not, then the states of the data qubits $D_{1}$ and $D_{2}$ are different, while the states of the data qubits $D_{1}$ and $D_{3}$ are the same. This implies that the data qubit $D_{2}$ is flipped.

For the GKP repetition code, the way to identify the bit-flip error through the syndrome is similar. The correspondence between measurement outcomes $\{M_{1},M_{2}\}$ and the logical Pauli $\bar{X}$ error on different GKP qubits is summarized in Tab. 2. However, this decoding procedure has a subtle difference from that of the classical repetition code: sometimes a single-qubit Pauli $\bar{X}$ error could be misidentified. As an example, if $u_{1}^{\prime}=\sqrt{\pi}\in\text{PZ}$, $u_{2}^{\prime}=\sqrt{\pi}/3\in\text{NPZ}$, $u_{3}^{\prime}=\sqrt{\pi}/3\in\text{NPZ}$, $\alpha_{1}=\alpha_{2}=\sqrt{\pi}/3$, then $M_{1}=2k_{1}\sqrt{\pi}+u_{1}^{\prime}+u_{2}^{\prime}+\alpha_{1}=2k_{1}\sqrt{\pi}+5\sqrt{\pi}/3\in\text{NPZ}$, $M_{2}=2k_{2}\sqrt{\pi}+u_{1}^{\prime}+u_{3}^{\prime}+\alpha_{2}=2k_{2}\sqrt{\pi}+5\sqrt{\pi}/3\in\text{NPZ}$, from which we infer that no Pauli $\bar{X}$ error occurs but in fact a Pauli $\bar{X}$ error did occur in the qubit $D_{1}$. Continuity of the phase space is what makes GKP repetition code different from the classical repetition code Fukui et al. (2017).

Our objective is to calculate the failure probability of 3-qubit GKP repetition code. In contrary to the classical repetition code, we need to reverse the decoding process and impose some conditions to be satisfied instead. For example, if no error occurs, we need $M_{1}\in\text{NPZ}$ and $M_{2}\in\text{NPZ}$ to give the correct identification, and the area outside $M_{1}\in\text{NPZ}$ and $M_{2}\in\text{NPZ}$ must lead to failure. Similar rules apply for other cases. All possible circumstances are summarized as follows:

• •

  Case 1: If no error occurs $\Rightarrow$we require $M_{1}\in\text{NPZ},M_{2}\in\text{NPZ}$;

• •

  Case 2: If $\bar{X}$ applies on data qubit $D_{1}$ $\Rightarrow$ we require $M_{1}\in\text{PZ},M_{2}\in\text{PZ}$;

• •

  Case 3: If $\bar{X}$ applies on data qubit $D_{2}$ $\Rightarrow$ we require $M_{1}\in\text{PZ},M_{2}\in\text{NPZ}$;

• •

  Case 4: If $\bar{X}$ applies on data qubit $D_{3}$ $\Rightarrow$ we require $M_{1}\in\text{NPZ},M_{2}\in\text{PZ}$;

• •

  Case 5: If errors occur on more than one data qubit, with probability $3P_{F}^{2}(1-P_{F})+P_{F}^{3}$ $\Rightarrow$ error correction fails.

Now we calculate the failure probability for the above five cases, the sum of which gives the total failure probability. Consider case 1, there are five constraints needed to be satisfied simultaneously,

	No Pauli $\bar{X}$ error	$\displaystyle\Rightarrow$	$\displaystyle\left|u_{1}^{\prime}-2m_{1}\sqrt{\pi}\right|<\frac{\sqrt{\pi}}{2},\left|u_{2}^{\prime}-2m_{2}\sqrt{\pi}\right|<\frac{\sqrt{\pi}}{2},\left|u_{3}^{\prime}-2m_{3}\sqrt{\pi}\right|<\frac{\sqrt{\pi}}{2},$
	$\displaystyle M_{1}\in\text{NPZ},M_{2}\in\text{NPZ}$	$\displaystyle\Rightarrow$	$\displaystyle\left|u_{1}^{\prime}+u_{2}^{\prime}+\alpha_{1}-2n_{1}\sqrt{\pi}\right|<\frac{\sqrt{\pi}}{2},\left|u_{1}^{\prime}+u_{3}^{\prime}+\alpha_{2}-2n_{2}\sqrt{\pi}\right|<\frac{\sqrt{\pi}}{2},$		(45)

where $m_{i}\in\mathbb{Z}$ and $n_{i}\in\mathbb{Z}$. The probability of success is obtained by integrating the probability distribution of five variables in the domain defined by these five inequalities. However, it is challenging to derive an analytic expression for the success probability, which requires a five-dimensional linear programming. Therefore, a numerical integration method is used instead, which proceeds in two steps. We first fix a point $(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})$ in the domain defined by the first three inequalities in Eq. (IV.1), then the success probability at this given point is

	$\displaystyle P_{\alpha}^{1}(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})=\left(\sum_{n_{1}}\int_{-\sqrt{\pi}/2+2n_{1}\sqrt{\pi}-u_{1}^{\prime}-u_{2}^{\prime}}^{\sqrt{\pi}/2+2n_{1}\sqrt{\pi}-u_{1}^{\prime}-u_{2}^{\prime}}f_{q_{1}^{\prime}}(\alpha_{1}){\rm d}\alpha_{1}\right)\left(\sum_{n_{2}}\int_{-\sqrt{\pi}/2+2n_{2}\sqrt{\pi}-u_{1}^{\prime}-u_{3}^{\prime}}^{\sqrt{\pi}/2+2n_{2}\sqrt{\pi}-u_{1}^{\prime}-u_{3}^{\prime}}f_{q_{2}^{\prime}}(\alpha_{2}){\rm d}\alpha_{2}\right)$
	$\displaystyle\approx\frac{1}{4}\left[{\rm erf}\left(\frac{\frac{\sqrt{\pi}}{2}-u_{1}^{\prime}-u_{2}^{\prime}}{\tilde{\Delta}}\right)-{\rm erf}\left(\frac{-\frac{\sqrt{\pi}}{2}-u_{1}^{\prime}-u_{2}^{\prime}}{\tilde{\Delta}}\right)\right]\left[{\rm erf}\left(\frac{\frac{\sqrt{\pi}}{2}-u_{1}^{\prime}-u_{3}^{\prime}}{\tilde{\Delta}}\right)-{\rm erf}\left(\frac{-\frac{\sqrt{\pi}}{2}-u_{1}^{\prime}-u_{3}^{\prime}}{\tilde{\Delta}}\right)\right],$		(46)

where we have only kept one term with $n_{1}=n_{2}=0$ in the summation because the contribution from other terms is negligible. Then the failure probability of case 1 is given by integrating the failure probability $1-P_{\alpha}^{1}(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})$ over all points satisfying the first three constraints in Eq. (IV.1), weighted by the probability distribution $F(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})=F(u_{1}^{\prime})F(u_{2}^{\prime})F(u_{3}^{\prime})$,

	$\displaystyle P_{f,\text{3-rep}}^{1}$	$\displaystyle=$	$\displaystyle\int_{u_{1}^{\prime}\in{\rm NPZ}}\int_{u_{2}^{\prime}\in{\rm NPZ}}\int_{u_{3}^{\prime}\in{\rm NPZ}}F(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})\left[1-P_{\alpha}^{1}(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})\right]{\rm d}u_{1}^{\prime}{\rm d}u_{2}^{\prime}{\rm d}u_{3}^{\prime}$		(47)
		$\displaystyle\approx$	$\displaystyle\int_{u_{1}^{\prime}=-\sqrt{\pi}/2}^{\sqrt{\pi}/2}\int_{u_{2}^{\prime}=-\sqrt{\pi}/2}^{\sqrt{\pi}/2}\int_{u_{3}^{\prime}=-\sqrt{\pi}/2}^{\sqrt{\pi}/2}F(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})\left[1-P_{\alpha}^{1}(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})\right]{\rm d}u_{1}^{\prime}{\rm d}u_{2}^{\prime}{\rm d}u_{3}^{\prime},$

where we have only kept one term with $m_{1}=m_{2}=m_{3}=0$ in the summation because the contribution from other terms is negligible.

In a similar way, we can derive the failure probability for case 2 by taking into account the condition that $u_{1}^{\prime}\in{\rm PZ},u_{2}^{\prime}\in{\rm NPZ}$ and $u_{3}^{\prime}\in{\rm NPZ}$,

	$\displaystyle P_{f,\text{3-rep}}^{2}$	$\displaystyle=$	$\displaystyle\int_{u_{1}^{\prime}\in{\rm PZ}}\int_{u_{2}^{\prime}\in{\rm NPZ}}\int_{u_{3}^{\prime}\in{\rm NPZ}}F(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})\left[1-P_{\alpha}^{2}(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})\right]{\rm d}u_{1}^{\prime}{\rm d}u_{2}^{\prime}{\rm d}u_{3}^{\prime}$		(48)
		$\displaystyle\approx$	$\displaystyle 2\int_{u_{1}^{\prime}=\sqrt{\pi}/2}^{3\sqrt{\pi}/2}\int_{u_{2}^{\prime}=-\sqrt{\pi}/2}^{\sqrt{\pi}/2}\int_{u_{3}^{\prime}=-\sqrt{\pi}/2}^{\sqrt{\pi}/2}F(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})\left[1-P_{\alpha}^{2}(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})\right]{\rm d}u_{1}^{\prime}{\rm d}u_{2}^{\prime}{\rm d}u_{3}^{\prime},$

where $P_{\alpha}^{2}(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})$ is the success probability for a given point $(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})$ when $M_{1}\in\text{PZ}$ and $M_{2}\in\text{PZ}$,

	$\displaystyle P_{\alpha}^{2}(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})=\left(\sum_{n_{1}}\int_{\sqrt{\pi}/2+2n_{1}\sqrt{\pi}-u_{1}^{\prime}-u_{2}^{\prime}}^{3\sqrt{\pi}/2+2n_{1}\sqrt{\pi}-u_{1}^{\prime}-u_{2}^{\prime}}f_{q_{1}^{\prime}}(\alpha_{1}){\rm d}\alpha_{1}\right)\left(\sum_{n_{2}}\int_{\sqrt{\pi}/2+2n_{2}\sqrt{\pi}-u_{1}^{\prime}-u_{3}^{\prime}}^{3\sqrt{\pi}/2+2n_{2}\sqrt{\pi}-u_{1}^{\prime}-u_{3}^{\prime}}f_{q_{2}^{\prime}}(\alpha_{2}){\rm d}\alpha_{2}\right)$
	$\displaystyle\approx\frac{1}{4}\left[{\rm erf}\left(\frac{\frac{3\sqrt{\pi}}{2}-u_{1}^{\prime}-u_{2}^{\prime}}{\tilde{\Delta}}\right)-{\rm erf}\left(\frac{\frac{\sqrt{\pi}}{2}-u_{1}^{\prime}-u_{2}^{\prime}}{\tilde{\Delta}}\right)\right]\left[{\rm erf}\left(\frac{\frac{3\sqrt{\pi}}{2}-u_{1}^{\prime}-u_{3}^{\prime}}{\tilde{\Delta}}\right)-{\rm erf}\left(\frac{\frac{\sqrt{\pi}}{2}-u_{1}^{\prime}-u_{3}^{\prime}}{\tilde{\Delta}}\right)\right].$		(49)

$P_{\alpha}^{3}(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})$ and $P_{\alpha}^{4}(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})$ can also be calculated by the similar way, and it can be shown that the failure probabilities of cases 3 and 4 are the same as that of the case 2, namely,

$$P_{f,\text{3-rep}}^{2}=P_{f,\text{3-rep}}^{3}=P_{f,\text{3-rep}}^{4}.$$

(50)

The failure probability of case 5 is

$$P_{f,\text{3-rep}}^{5}=3P_{F}^{2}(1-P_{F})+P_{F}^{3}.$$

(51)

Finally, the total failure probability of the 3-qubit GKP repetition code is

	$\displaystyle P_{f,\text{3-rep}}(\Delta,\tilde{\Delta})$	$\displaystyle=$	$\displaystyle P_{f,\text{3-rep}}^{1}+P_{f,\text{3-rep}}^{2}+P_{f,\text{3-rep}}^{3}+P_{f,\text{3-rep}}^{4}+P_{f,\text{3-rep}}^{5}$		(52)
		$\displaystyle\approx$	$\displaystyle\int_{u_{1}^{\prime}=-\sqrt{\pi}/2}^{\sqrt{\pi}/2}\int_{u_{2}^{\prime}=-\sqrt{\pi}/2}^{\sqrt{\pi}/2}\int_{u_{3}^{\prime}=-\sqrt{\pi}/2}^{\sqrt{\pi}/2}F(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})\left[1-P_{\alpha}^{1}(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})\right]{\rm d}u_{1}^{\prime}{\rm d}u_{2}^{\prime}{\rm d}u_{3}^{\prime}$
			$\displaystyle+6\int_{u_{1}^{\prime}=\sqrt{\pi}/2}^{3\sqrt{\pi}/2}\int_{u_{2}^{\prime}=-\sqrt{\pi}/2}^{\sqrt{\pi}/2}\int_{u_{3}^{\prime}=-\sqrt{\pi}/2}^{\sqrt{\pi}/2}F(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})\left[1-P_{\alpha}^{2}(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})\right]{\rm d}u_{1}^{\prime}{\rm d}u_{2}^{\prime}{\rm d}u_{3}^{\prime}$
			$\displaystyle+3P_{F}^{2}(1-P_{F})+P_{F}^{3}.$

The last two terms correspond to the failure probability of the classical 3-qubit repetition code, and the first two terms correspond to the failure probability when error occurs on no more than one data qubit, which is what makes GKP repetition code different from the classical repetition code.