Encoding an oscillator into many oscillators
Kyungjoo Noh, S. M. Girvin, Liang Jiang
TL;DR
This paper introduces non-Gaussian oscillator-into-oscillators quantum error correction codes using GKP states, enabling correction of Gaussian errors and noise in bosonic systems, surpassing Gaussian-only limitations.
Contribution
It proposes the first non-Gaussian oscillator-into-oscillators codes, including the GKP-repetition and GKP-stabilizer codes, to correct Gaussian errors in bosonic quantum systems.
Findings
The two-mode GKP-repetition code corrects additive Gaussian noise.
The GKP-two-mode-squeezing code quadratically suppresses Gaussian noise.
Logical Gaussian operations are implementable with physical Gaussian operations.
Abstract
An outstanding challenge for quantum information processing using bosonic systems is Gaussian errors such as excitation loss and added thermal noise errors. Thus, bosonic quantum error correction (QEC) is essential. Most bosonic QEC schemes encode a finite-dimensional logical qubit or qudit into noisy bosonic oscillator modes. In this case, however, the infinite-dimensional bosonic nature of the physical system is lost at the error-corrected logical level. On the other hand, there are several proposals for encoding an oscillator mode into many noisy oscillator modes. However, these oscillator-into-oscillators encoding schemes are in the class of Gaussian quantum error correction. Therefore, these codes cannot correct practically relevant Gaussian errors due to the established no-go theorems which state that Gaussian errors cannot be corrected by using only Gaussian resources. Here, we…
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Encoding an oscillator into many oscillators
Kyungjoo Noh
Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
Yale Quantum Institute, Yale University, New Haven, Connecticut 06520, USA
S. M. Girvin
Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
Yale Quantum Institute, Yale University, New Haven, Connecticut 06520, USA
Liang Jiang
Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
Yale Quantum Institute, Yale University, New Haven, Connecticut 06520, USA
Pritzker School of Molecular Engineering, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA
Abstract
An outstanding challenge for quantum information processing using bosonic systems is Gaussian errors such as excitation loss and added thermal noise errors. Thus, bosonic quantum error correction (QEC) is essential. Most bosonic QEC schemes encode a finite-dimensional logical qubit or qudit into noisy bosonic oscillator modes. In this case, however, the infinite-dimensional bosonic nature of the physical system is lost at the error-corrected logical level. On the other hand, there are several proposals for encoding an oscillator mode into many noisy oscillator modes. However, these oscillator-into-oscillators encoding schemes are in the class of Gaussian quantum error correction. Therefore, these codes cannot correct practically relevant Gaussian errors due to the established no-go theorems which state that Gaussian errors cannot be corrected by using only Gaussian resources. Here, we circumvent these no-go results and show that it is possible to correct Gaussian errors by using Gottesman-Kitaev-Preskill (GKP) states as non-Gaussian resources. In particular, we propose a non-Gaussian oscillator-into-oscillators code, the two-mode GKP-repetition code, and demonstrate that it can correct additive Gaussian noise errors. In addition, we generalize the two-mode GKP-repetition code to an even broader class of non-Gaussian oscillator codes, namely, GKP-stabilizer codes. Specifically, we show that there exists a highly hardware-efficient GKP-stabilizer code, the GKP-two-mode-squeezing code, that can quadratically suppress additive Gaussian noise errors in both the position and momentum quadratures up to a small logarithmic correction. Moreover, for any GKP-stabilizer code, we show that logical Gaussian operations can be readily implemented by using only physical Gaussian operations. Furthermore, we show that our non-Gaussian oscillator encoding scheme can also be used to correct excitation loss and thermal noise errors, which are dominant error sources in many realistic bosonic systems.
I Introduction
Continuous-variable (CV) bosonic quantum systems are ubiquitous in various quantum computing and communication architectures Braunstein and van Loock (2005); Weedbrook et al. (2012) and provide unique advantages to, e.g., quantum simulation of bosonic systems such as boson sampling Aaronson and Arkhipov (2011); Hamilton et al. (2017); García-Patrón et al. (2019); Wang et al. (2019) and simulation of vibrational quantum dynamics of molecules Huh et al. (2015); Sparrow et al. (2018); Clements et al. (2018); Wang et al. (2019). Since CV quantum information processing tasks are typically implemented by using harmonic oscillator modes in photonic or phononic systems, realistic imperfections such as excitation loss and added thermal noise errors are major challenges for realizing large-scale and fault-tolerant CV quantum information processing.
Quantum error correction (QEC) is essential for scalable and fault-tolerant quantum information processing Gottesman (2009). Recently, there have been significant advances in using bosonic systems to realize QEC in a more hardware-efficient manner Albert et al. (2018). In many bosonic QEC schemes proposed so far, a finite-dimensional quantum system (e.g., a qubit) is encoded into an oscillator Cochrane et al. (1999); Gottesman et al. (2001); Leghtas et al. (2013); Mirrahimi et al. (2014); Michael et al. (2016); Li et al. (2017) or into many oscillators Chuang et al. (1997); Harrington and Preskill (2001); Bergmann and van Loock (2016); Fukui et al. (2017); Niu et al. (2018); Fukui et al. (2018a, b); Vuillot et al. (2019); Fukui (2019); Noh and Chamberland (2019). For example, the four-component cat code Leghtas et al. (2013) encodes a logical qubit into an oscillator using cat states with even number of excitations, i.e., and , which can correct single-excitation loss errors by monitoring the parity of the excitation number. Thanks to the inherent hardware efficiency, various qubit-into-an-oscillator bosonic QEC schemes have been realized experimentally Leghtas et al. (2015); Ofek et al. (2016); Touzard et al. (2018); Hu et al. (2019); Flühmann et al. (2018, 2019); Flühmann and Home (2019); Grimm et al. (2019); Campagne-Ibarcq et al. (2019). However, in such qubit-into-an-oscillator encoding schemes, the infinite-dimensional bosonic nature of the physical oscillator modes is lost at the logical level because the error-corrected logical system is described by discrete variables (DV) such as Pauli operators. Therefore, the error-corrected logical DV system is not itself tailored to CV quantum information processing tasks.
On the other hand, if an infinite-dimensional oscillator mode is encoded into many noisy oscillator modes (i.e., oscillator-into-oscillators encoding), such an error-corrected oscillator mode will still be tailored to various continuous-variable quantum information processing tasks. So far, there have been several proposals for encoding an oscillator into many oscillators Lloyd and Slotine (1998); Braunstein (1998a, b); Aoki et al. (2009); Hayden et al. (2016, 2017); Faist et al. (2019); Woods and Alhambra (2019). For example, in the case of the three-mode Gaussian-repetition code Lloyd and Slotine (1998); Braunstein (1998a), an infinite-dimensional oscillator mode is encoded into three oscillators by repeatedly appending the position eigenstates: . Note that in this case, the logical Hilbert space is infinite-dimensional because can be any real number.
In general, oscillator-into-oscillators bosonic QEC is more challenging than qubit-into-oscillators bosonic QEC because in the former we aim to protect an infinite-dimensional bosonic Hilbert space against relevant errors, whereas in the latter we only aim to protect a finite-dimensional Hilbert space embedded in infinite-dimensional bosonic modes. Indeed, while there exist many qubit-into-oscillators codes that can correct experimentally relevant Gaussian errors, none of the previously proposed oscillator-into-oscillators codes can correct Gaussian errors. This is because they are Gaussian quantum error correction schemes and the established no-go theorems state that Gaussian errors cannot be corrected by using only Gaussian resources Eisert et al. (2002); Niset et al. (2009); Vuillot et al. (2019). Since Gaussian errors include excitation losses, thermal noise and additive Gaussian noise errors which are ubiquitous in many realistic CV quantum systems, these no-go results set a hard limit on the practical utility of the proposed Gaussian QEC schemes.
We circumvent the established no-go results and provide non-Gaussian oscillator-into-oscillators codes that can correct Gaussian errors using Gottesman-Kitaev-Preskill (GKP) states Gottesman et al. (2001) as non-Gaussian resources. Our paper is organized as follows: In Section II, we briefly review common non-Gaussian resources and summarize known properties of the GKP state that will be used in later sections. In Section III, we introduce a non-Gaussian oscillator-into-oscillators code, namely, the two-mode GKP-repetition code and demonstrate that it is indeed possible to correct Gaussian errors using the two-mode GKP-repetition code. In Section IV, we generalize the two-mode GKP-repetition code and propose an even broader class of non-Gaussian oscillator codes, called GKP-stabilizer codes. In particular, in Subsection IV.1, we show that there exists a highly hardware-efficient GKP-stabilizer code, the GKP-two-mode-squeezing code, that can quadratically suppress additive Gaussian noise errors in both the position and momentum quadrature. In Subsection IV.2, we show that, for any GKP-stabilizer codes, logical Gaussian operations can be readily implemented by using only physical Gaussian operations which are available in many experimental systems. In Section V, we discuss experimental realization of our schemes and analyze the adverse effects of realistic imperfections. We also discuss potential applications of our schemes.
II GKP states as non-Gaussian resources
The established no-go theorems on Gaussian QEC schemes Eisert et al. (2002); Niset et al. (2009); Vuillot et al. (2019) make it clear that non-Gaussian resources Zhuang et al. (2018); Takagi and Zhuang (2018) are necessary for correcting Gaussian errors while preserving the bosonic nature at the error-corrected logical level. Examples of non-Gaussian resources include the single-photon Fock state and photon-number-resolving measurements Knill et al. (2001); Aaronson and Arkhipov (2011), Kerr nonlinearities Lloyd and Braunstein (1999), cubic phase state and gate Gottesman et al. (2001), SNAP gate Krastanov et al. (2015), Schrödinger cat states Cochrane et al. (1999), and GKP states Gottesman et al. (2001); Baragiola et al. (2019). Among these non-Gaussian resources, we demonstrate that GKP states are particularly useful for encoding an oscillator into many oscillators in a robust way. Specifically, we cast the GKP state as a tool to work around the Heisenberg uncertainty principle.
The Heisenberg uncertainty principle states that the position and momentum operators \hat{q}\equiv(\hat{a}^{\dagger}+\hat{a})/\mathchoice{{\hbox{\displaystyle\sqrt{2,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}} and \hat{p}\equiv i(\hat{a}^{\dagger}-\hat{a})/\mathchoice{{\hbox{\displaystyle\sqrt{2,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}} cannot be measured simultaneously because they do not commute with each other (i.e., ). Despite the Heisenberg uncertainty principle, the following displacement operators
[TABLE]
do commute with each other and therefore can be measured simultaneously Gottesman et al. (2001). Note that measuring \hat{S}_{q}=\exp[i\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\hat{q}] and \hat{S}_{p}\equiv\exp[-i\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\hat{p}] is equivalent to measuring their exponents (or phase angles) i\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\hat{q} and -i\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\hat{p} modulo . Thus, the commutativity of and implies that the position and momentum operators can indeed be measured simultaneously if they are measured modulo . The canonical GKP state (or the grid state) Gottesman et al. (2001); Duivenvoorden et al. (2017) is defined as the unique (up to an overall phase) simultaneous eigenstate of the two commuting displacement operators and with unit eigenvalues. Explicitly, the canonical GKP state is given by
[TABLE]
and thus clearly has definite values for both the position and momentum operators modulo , i.e., mod .
Ideally, the canonical GKP state has an infinite average photon number because it is superpositions of infinitely many () infinitely squeezed states (|\hat{q}=\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}n\rangle or |\hat{p}=\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}n\rangle). However, one can define an approximate GKP state with a finite average photon number by applying a non-unitary operator to the canonical GKP state and then normalizing the output state: Gottesman et al. (2001). In Fig. 1 (a), we plot the Wigner function of the canonical GKP state with an average photon number . Negative peaks in the Wigner function clearly indicate that the canonical GKP state is a non-Gaussian state.
There have been many proposals for preparing an approximate GKP state in various experimental platforms Gottesman et al. (2001); Travaglione and Milburn (2002); Pirandola et al. (2004, 2006); Vasconcelos et al. (2010); Terhal and Weigand (2016); Motes et al. (2017); Weigand and Terhal (2018); Arrazola et al. (2019); Su et al. (2019); Eaton et al. (2019); Shi et al. (2019); Weigand and Terhal (2019); Hastrup et al. (2019). Notably, the proposal in Ref. Travaglione and Milburn (2002) has recently been realized in a trapped ion system Flühmann et al. (2018, 2019); Flühmann and Home (2019) and a variation of the scheme in Ref. Terhal and Weigand (2016) has recently been realized in a circuit QED system Campagne-Ibarcq et al. (2019). In the interest of clarity, we only consider the ideal canonical GKP state when we present our main results. Issues related to the use of the use of more realistic approximate GKP states will be addressed in Section V.
Clearly, the ability to measure the position and momentum operators modulo allows us to prepare the canonical GKP state. Remarkably, the converse is also true. That is, we can measure the quadrature operators modulo given GKP states and Gaussian operations as resources: As shown in Fig. 1 (b), one can measure the position (momentum) operator modulo by using a canonical GKP state, the SUM (inverse-SUM) gate and the homodyne measurement of the position (momentum) operator. The SUM gate is a Gaussian operation and is defined as , which maps to . The inverse-SUM gate is defined as the inverse of the SUM gate. The canonical GKP state and the modulo simultaneous quadrature measurement are the key non-Gaussian resources of our oscillator-into-oscillators encoding schemes which we introduce below.
In the following section, we construct a non-Gaussian oscillator-into-oscillators code, namely, the two-mode GKP-repetition code and demonstrate that it can correct additive Gaussian noise errors. That is, we circumvent the established no-go results on Gaussian QEC schemes Eisert et al. (2002); Niset et al. (2009); Vuillot et al. (2019) by using the canonical GKP state as a non-Gaussian resource.
III The two-mode GKP-repetition code
Quantum error-correcting codes work by hiding the quantum information from the environment by storing the logical state in a non-local entangled state of the physical components. In the case of the scheme we present below, a data oscillator mode is entangled via a Gaussian operation with an ancillary oscillator mode, which is initially in the canonical GKP state, in a manner that prevents the environment from learning about the logically encoded state. Like qubit-into-an-oscillator GKP codes Gottesman et al. (2001), our oscillator-into-oscillators codes are specifically designed to protect against random displacement errors. Also, it succeeds because we assume access to non-Gaussian resources (GKP states and modular quadrature measurements) that are unavailable to the environment.
More explicitly, we propose the following encoding of an arbitrary bosonic state into two oscillator modes,
[TABLE]
where is the canonical GKP state in the second mode. We call this encoding the two-mode GKP-repetition code (see Fig. 2 (a) for the encoding circuit). Also, we refer to the first mode as the data mode and the second mode as the ancilla mode because the information was stored only in the first mode before the application of the encoding circuit. The SUM gate () is a CV analog of the CNOT gate which, in the Heisenberg picture, transforms and into and and leaves all the other quadrature operators unchanged. The quadrature operators are transformed by the encoding circuit into
[TABLE]
Note that the SUM gate in the encoding circuit is analogous to the CNOT gate in the encoding circuit of the two-bit repetition code for qubit bit-flip errors: This is why we refer to the encoding in Eq. (3) as the two-mode GKP-repetition code.
Before moving on to the detailed analysis of the two-mode GKP-repetition code, we point out that our two-mode GKP-repetition code can be regarded as a non-Gaussian modification of the two-mode Gaussian-repetition code (see Refs. Lloyd and Slotine (1998); Braunstein (1998a) for the three-mode Gaussian-repetition code and Appendix A for its generalization to modes). We also remark that the two-mode GKP-repetition code is not very efficient as it is significantly outperformed by the GKP-two-mode-squeezing code introduced in Subsection IV.1. Nevertheless, we introduce the two-mode GKP-repetition code to explain the key elements of our non-Gaussian oscillator encoding schemes and to contrast our schemes with the previous Gaussian encoding schemes.
Let us now analyze the performance of the two-mode GKP-repetition code. We assume that the oscillator modes undergo independent and identically distributed (iid) additive Gaussian noise errors (or Gaussian random displacement errors) . Here, is an additive Gaussian noise error acting on the mode which, in the Heisenberg picture, adds Gaussian random noise and to the position and momentum operators of the mode, i.e.,
[TABLE]
where . Also, and are independent Gaussian random variables with zero mean and variance . That is, .
We emphasize that additive Gaussian noise errors are generic in the sense that any excitation loss and thermal noise errors can be converted into an additive Gaussian noise error by applying a suitable quantum-limited amplification channel Albert et al. (2018); Noh et al. (2019). For example, a pure excitation loss error with loss probability can be converted into an additive Gaussian noise error with \sigma=\mathchoice{{\hbox{\displaystyle\sqrt{\gamma,}}\lower 0.4pt\hbox{\vrule height=4.30554pt,depth=-3.44446pt}}}{{\hbox{\textstyle\sqrt{\gamma,}}\lower 0.4pt\hbox{\vrule height=4.30554pt,depth=-3.44446pt}}}{{\hbox{\scriptstyle\sqrt{\gamma,}}\lower 0.4pt\hbox{\vrule height=3.01389pt,depth=-2.41113pt}}}{{\hbox{\scriptscriptstyle\sqrt{\gamma,}}\lower 0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}} (see Lemma 6 and Table 1 in Ref. Noh et al. (2019)).
The decoding procedure (shown in Fig. 2 (b)) begins with the inverse of the encoding circuit, i.e., with . Upon the inverse of the encoding circuit, the transformed quadrature operators in Eq. (4) are transformed back to the original quadrature operators but the added quadrature noise and in Eq. (5) are reshaped, i.e.,
[TABLE]
for where the reshaped quadrature noise and are given by
[TABLE]
Note that the position quadrature noise of the data mode is transferred to the position quadrature of the ancilla mode (see in ), whereas the momentum quadrature noise of the ancilla mode is transferred to the momentum quadrature of the data mode (see in ).
In the remainder of the decoding procedure, both the position and momentum quadrature noise of the ancilla mode are measured simultaneously modulo (using the measurement circuits shown in Fig. 1 (b)). By doing so, we measure both and modulo . Note that such measurements of and modulo are equivalent to measurements of only the reshaped ancilla quadrature noise and modulo . This is because the ancilla modes were initially in the canonical GKP state and thus mod holds. The extracted information about and will then be used to estimate the data position quadrature noise and the ancilla momentum quadrature noise such that the uncertainty of the data position quadrature noise is reduced while the ancilla momentum quadrature noise (transferred to the data momentum quadrature) does not degrade the momentum quadrature of the data mode. Below, we provide a detailed description of this estimation procedure.
From the outcomes of the measurements of and modulo , we assume that the true values of and are the ones with the smallest length among the candidates that are compatible with the modular measurement outcomes. That is,
[TABLE]
where and . More concretely, equals a displaced sawtooth function with an amplitude and period and is given by if . Then, based on these estimates, we further estimate that the position quadrature noise of the data mode and the momentum quadrature noise of the ancilla mode are
[TABLE]
The latter estimate was chosen simply because and the former choice is based on a maximum likelihood estimation explained in detail in Appendix A.
Finally, based on the estimates and , we apply counter displacement operations and to the data mode and end up with the following logical position and momentum quadrature noise
[TABLE]
In Appendix B, we provide explicit expressions for the probability density functions of and (which are used to obtain Fig. 3) in the most general case. Here, we instead focus on a simple (but important) case where is much smaller than . In this case, the reshaped ancilla quadrature noise and lie in the unambiguously distinguishable range [-\mathchoice{{\hbox{\displaystyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\textstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\scriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=5.25pt,depth=-4.20003pt}}}{{\hbox{\scriptscriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}},\mathchoice{{\hbox{\displaystyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\textstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\scriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=5.25pt,depth=-4.20003pt}}}{{\hbox{\scriptscriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}}] with a very high probability and thus we have R_{\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.15778pt,depth=-2.52625pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=2.25555pt,depth=-1.80446pt}}}}(\xi_{q}^{(2)}-\xi_{q}^{(1)})=\xi_{q}^{(2)}-\xi_{q}^{(1)} and R_{\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.15778pt,depth=-2.52625pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=2.25555pt,depth=-1.80446pt}}}}(\xi_{p}^{(2)})=\xi_{p}^{(2)}. Then, the logical position and momentum quadrature noise are given by
[TABLE]
That is, the variance of the logical position quadrature noise is reduced by a factor of . This is due to the syndrome measurement of the reshaped ancilla position quadrature noise modulo which is then used to reduce the uncertainty of the data position quadrature noise . Moreover, the variance of the logical momentum quadrature noise remains unchanged despite the temporary increase () during the decoding procedure. Again, this is due to the syndrome measurement of the reshaped ancilla momentum quadrature noise modulo which fully captures the transferred ancilla momentum quadrature noise if \sigma\ll\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}.
In Fig. 3, we plot the standard deviations of the output logical quadrature noise for the two-mode GKP-repetition code. The standard deviation of the output logical position quadrature noise is indeed reduced by a factor of (i.e., \sigma_{q}=\sigma/\mathchoice{{\hbox{\displaystyle\sqrt{2,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}), while the standard deviation of the logical momentum quadrature noise remains unchanged (i.e., ) for . Note that the condition is translated to in the case of pure excitation losses, where is the pure-loss probability Albert et al. (2018); Noh et al. (2019). Thus, if the standard deviation of an additive Gaussian noise error is sufficiently small, our GKP-repetition coding scheme can successfully reduce the noise of the position quadrature, while keeping the momentum quadrature noise unchanged. That is, our (non-Gaussian) GKP-repetition codes can correct additive Gaussian noise errors.
We remark that in an analogous two-mode Gaussian-repetition coding scheme (presented in detail in Appendix A), the variance of the position quadrature noise is reduced by a factor of (i.e., ) similarly as in Eq. (11) but the variance of the momentum quadrature noise is increased by the same factor (i.e., ). This implies that, in the case of Gaussian-repetition codes, the position and momentum quadrature noise are only squeezed () instead of being corrected (): This reaffirms the previous no-go results on Gaussian QEC schemes Eisert et al. (2002); Niset et al. (2009); Vuillot et al. (2019).
The key difference between our GKP-repetition code and the previous Gaussian-repetition code is that in the latter case the ancilla momentum quadrature noise that is transferred to the data mode (see in ) is left completely undetected, whereas in our case it is captured by measuring the ancilla momentum quadrature operator modulo . Such a limitation of the Gaussian-repetition code is in fact a general feature of any Gaussian QEC scheme which relies on homodyne measurements of ancilla quadrature operators. With homodyne measurements, one can only monitor the noise in one quadrature of a bosonic mode, while the noise in the other conjugate quadrature is left completely undetected.
On the other hand, we have worked around this barrier by using the canonical GKP state as a non-Gaussian resource which allows simultaneous measurements of both the position and momentum quadrature operators modulo (see Fig. 1 (b)). In this regard, we emphasize that such non-Gaussian modular simultaneous measurements of both position and momentum quadrature operators (using canonical GKP states) are fundamentally different from Gaussian heterodyne measurements Shapiro et al. (1979) where both quadrature operators are measured simultaneously but in a necessarily noisy manner.
Finally, we compare our two-mode GKP-repetition code with the conventional three-bit repetition code (i.e., and ) which can correct single bit-flip errors Nielsen and Chuang (2000). First, we observe that only two bosonic modes are sufficient to reduce the variance of the position quadrature noise in the case of the GKP-repetition code, whereas at least three qubits are needed to suppress qubit bit-flip errors in the case of mutli-qubit repetition code. However, while the GKP-repetition code can be implemented in a more hardware-efficient way, it does not reduce the variance of the position quadrature noise quadratically but instead only reduce the variance by a constant factor, i.e., if \sigma\ll\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}. On the other hand, the three-qubit repetition code can reduce the bit-flip error probability quadratically from to if .
Note that such a quadratic (or even higher order) suppression of Pauli errors is a key step towards qubit-based fault-tolerant universal quantum computation Gottesman (2009). Thus, it is also highly desirable in the oscillator encoding case to have such a quadratic suppression of additive Gaussian noise errors, going way beyond the reduction by a constant factor which was shown above.
Below, we introduce the GKP-two-mode-squeezing code and show that it can suppress both the position and momentum quadrature noise quadratically up to a small logarithmic correction. In particular, the proposed scheme can achieve such a quadratic noise suppression by using only two bosonic modes (one data mode and one ancilla mode) and therefore is hardware-efficient. In the case of qubit-based QEC, on the contrary, at least five qubits and high-weight multi-qubit gate operations are needed to suppress both the bit-flip and phase-flip errors quadratically Laflamme et al. (1996); Bennett et al. (1996).
IV GKP-stabilizer formalism
In this section, we generalize our GKP-repetition code. Specifically, in Subsection IV.1, we introduce the GKP-two-mode-squeezing code and show that it can suppress both the position and momentum quadrature noise quadratically up to a small logarithmic correction. In Subsection IV.2, we further generalize our non-Gaussian QEC schemes and provide an even broader class of non-Gaussian oscillator(s)-into-oscillators codes, namely, GKP-stabilizer codes. We also show that logical Gaussian operations can be readily implemented by using only physical Gaussian operations for any GKP-stabilizer code.
IV.1 The GKP-two-mode-squeezing code
We define the GKP-two-mode-squeezing code as follows:
[TABLE]
where is an arbitrary bosonic state of the first mode, and is the canonical GKP state of the second mode. Also, is the two-mode squeezing operation acting on the modes and with a gain (hence the name of the code; see Fig. 4 (a)). In the Heisenberg picture, the two-mode squeezing operation transforms the quadrature operators into , where the symplectic matrix associated with is given by
[TABLE]
Here, is the identity matrix and is the Pauli matrix.
The two-mode squeezing operation can be decomposed into a sequence of beam splitter operations and single-mode squeezing operations:
[TABLE]
where \lambda=\mathchoice{{\hbox{\displaystyle\sqrt{G,}}\lower 0.4pt\hbox{\vrule height=6.83331pt,depth=-5.46667pt}}}{{\hbox{\textstyle\sqrt{G,}}\lower 0.4pt\hbox{\vrule height=6.83331pt,depth=-5.46667pt}}}{{\hbox{\scriptstyle\sqrt{G,}}\lower 0.4pt\hbox{\vrule height=4.78333pt,depth=-3.82668pt}}}{{\hbox{\scriptscriptstyle\sqrt{G,}}\lower 0.4pt\hbox{\vrule height=3.41666pt,depth=-2.73334pt}}}+\mathchoice{{\hbox{\displaystyle\sqrt{G-1,}}\lower 0.4pt\hbox{\vrule height=6.83331pt,depth=-5.46667pt}}}{{\hbox{\textstyle\sqrt{G-1,}}\lower 0.4pt\hbox{\vrule height=6.83331pt,depth=-5.46667pt}}}{{\hbox{\scriptstyle\sqrt{G-1,}}\lower 0.4pt\hbox{\vrule height=4.78333pt,depth=-3.82668pt}}}{{\hbox{\scriptscriptstyle\sqrt{G-1,}}\lower 0.4pt\hbox{\vrule height=3.41666pt,depth=-2.73334pt}}}. Here, is the single-mode squeezing operation that transforms and into and , respectively. Also, is the beam splitter interaction between the modes and with a transmissivity and is associated with a symplectic matrix
[TABLE]
Note that the squeezing parameter (or the gain ) can be chosen at will to optimize the performance of the error correction scheme.
Upon the addition of independent and identically distributed Gaussian noise errors, the quadrature operator is further transformed into , where is the quadrature noise vector obeying . Then, the inverse of the encoding circuit in the decoding procedure (shown in Fig. 4 (b)) transforms the quadrature operator into , where is the reshaped quadrature noise vector which is given by
[TABLE]
As in the case of GKP-repetition code, information about the reshaped ancilla quadrature noise and is extracted by simultaneously measuring the position and momentum quadrature operators of the ancilla mode modulo . These modular measurement outcomes are then used to estimate the reshaped data quadrature noise and .
Before elaborating the detailed estimation strategy from the obtained syndrome measurement outcomes, let us explain the key idea behind it. Note that the bare quadrature noise is uncorrelated and thus its covariance matrix is proportional to the identity matrix: . On the other hand, the covariance matrix of the reshaped quadrature noise is given by
[TABLE]
Thus, the reshaped data and ancilla quadrature noise are correlated whenever . In other words, the inverse of the encoding circuit converts the uncorrelated quadrature noise into a correlated noise . In particular, in the limit, the covariance matrix is asymptotically given by
[TABLE]
and therefore the data and ancilla position quadrature noise are perfectly anti-correlated (i.e., ) and the data and ancilla momentum quadrature noise are perfectly correlated (i.e., ). These strong correlations in the limit allow us to reliably estimate the data quadrature noise and based solely on the knowledge of the ancilla quadrature noise and .
We may be tempted to choose as large gain as possible since the data and ancilla quadrature noise are perfectly correlated (or anti-correlated) in the limit. However, we cannot increase the gain indefinitely because then the reshaped ancilla quadrature noise and (whose variances are given by ) are not contained within the unambiguously distinguishable range [-\mathchoice{{\hbox{\displaystyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\textstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\scriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=5.25pt,depth=-4.20003pt}}}{{\hbox{\scriptscriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}},\mathchoice{{\hbox{\displaystyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\textstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\scriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=5.25pt,depth=-4.20003pt}}}{{\hbox{\scriptscriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}}]. Therefore, we have to choose the gain such that the reshaped ancilla quadrature noise lie mostly in the unambiguously distinguishable range. Below, we describe the decoding strategy and the parameter optimization in more detail.
Based on the outcomes of the measurements of the ancilla quadrature operators modulo , we estimate that the reshaped ancilla noise and are the smallest ones that are compatible with the modular measurement outcomes, i.e.,
[TABLE]
similarly as in Eq. (8). Then, we further estimate that the reshaped data position and momentum quadrature noise and are
[TABLE]
The underlying reasons behind our choice of these estimates are explained in detail in Appendix C. In the limit, the estimates and in Eq. (20) are respectively reduced to and , which are reasonable because then the data and ancilla position quadrature noise are perfectly anti-correlated whereas the data and ancilla momentum quadrature noise are perfectly correlated.
Finally, applying the counter displacement operations and , we end up with the following output logical quadrature noise
[TABLE]
As shown in detail in Appendix C, the probability density functions of the output logical quadrature noise and are identical to each other and are given by a mixture of Gaussian distributions:
[TABLE]
Here, p[\sigma](z)\equiv\frac{1}{\mathchoice{{\hbox{\displaystyle\sqrt{2\pi\sigma^{2},}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\textstyle\sqrt{2\pi\sigma^{2},}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptstyle\sqrt{2\pi\sigma^{2},}}\lower 0.4pt\hbox{\vrule height=3.15778pt,depth=-2.52625pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi\sigma^{2},}}\lower 0.4pt\hbox{\vrule height=2.40916pt,depth=-1.92735pt}}}}\exp[-z^{2}/2\sigma^{2}] is the probability density function of the Gaussian distribution and and are given by
[TABLE]
Thus, the mixture probabilities sum up to unity, i.e., . One can understand the probability density function in Eq. (22) in the following way: First of all, is the probability that the ancilla position quadrature noise (whose standard deviation is given by \mathchoice{{\hbox{\displaystyle\sqrt{2G-1,}}\lower 0.4pt\hbox{\vrule height=6.83331pt,depth=-5.46667pt}}}{{\hbox{\textstyle\sqrt{2G-1,}}\lower 0.4pt\hbox{\vrule height=6.83331pt,depth=-5.46667pt}}}{{\hbox{\scriptstyle\sqrt{2G-1,}}\lower 0.4pt\hbox{\vrule height=4.78333pt,depth=-3.82668pt}}}{{\hbox{\scriptscriptstyle\sqrt{2G-1,}}\lower 0.4pt\hbox{\vrule height=3.41666pt,depth=-2.73334pt}}}\sigma) lies in the range [(n-1/2)\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}},(n+1/2)\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}]. In this case, we have R_{\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.15778pt,depth=-2.52625pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=2.25555pt,depth=-1.80446pt}}}}(z_{q}^{(2)})=z_{q}^{(2)}-\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}n and therefore misidentify by a finite shift \mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}n. Secondly, such a misidentification results in an undesired shift to the data mode through a miscalibrated counter displacement (see Eq. (20)).
Now, we consider the variance of the output logical quadrature noise. By using the probability density function in Eq. (22), we find that the variance is given by
[TABLE]
Recall that we can freely choose the gain to optimize the performance of the GKP-two-mode-squeezing code. Here, we choose such that the standard deviation of the output logical quadrature noise is minimized. In Fig. 5, we plot the minimum standard deviation of the output logical quadrature noise (see Fig. 5 (a)) and the optimal gain that achieves the minimum standard deviation (see Fig. 5 (b)). These optimal values are obtained via a brute-force numerical optimization. Note that in Fig. 5 (b), we show the strength of the required single-mode squeezing operations to achieve the optimal gain in the unit of decibel (i.e., where \lambda^{\star}=\mathchoice{{\hbox{\displaystyle\sqrt{G^{\star},}}\lower 0.4pt\hbox{\vrule height=8.1361pt,depth=-6.50891pt}}}{{\hbox{\textstyle\sqrt{G^{\star},}}\lower 0.4pt\hbox{\vrule height=8.1361pt,depth=-6.50891pt}}}{{\hbox{\scriptstyle\sqrt{G^{\star},}}\lower 0.4pt\hbox{\vrule height=5.71388pt,depth=-4.57114pt}}}{{\hbox{\scriptscriptstyle\sqrt{G^{\star},}}\lower 0.4pt\hbox{\vrule height=4.34721pt,depth=-3.4778pt}}}+\mathchoice{{\hbox{\displaystyle\sqrt{G^{\star}-1,}}\lower 0.4pt\hbox{\vrule height=8.1361pt,depth=-6.50891pt}}}{{\hbox{\textstyle\sqrt{G^{\star}-1,}}\lower 0.4pt\hbox{\vrule height=8.1361pt,depth=-6.50891pt}}}{{\hbox{\scriptstyle\sqrt{G^{\star}-1,}}\lower 0.4pt\hbox{\vrule height=5.71388pt,depth=-4.57114pt}}}{{\hbox{\scriptscriptstyle\sqrt{G^{\star}-1,}}\lower 0.4pt\hbox{\vrule height=4.34721pt,depth=-3.4778pt}}}; see Eq. (14)).
We find that for , the optimal gain is trivially given by and thus the GKP-two-mode-squeezing code cannot reduce the noise: . On the other hand, if the standard deviation of the input noise is small enough, i.e., , the optimal gain is strictly larger than and the minimum standard deviation of the output logical quadrature noise can be made smaller than the standard deviation of the input quadrature noise : . Note that the condition corresponds to for the pure excitation loss error where is the loss probability (see Lemma 6 and Table 1 in Ref. Noh et al. (2019)). Below, we provide asymptotic expressions for and in the limit and show that the GKP-two-mode-squeezing code can quadratically suppress additive Gaussian noise errors up to a small logarithmic correction.
We observe that the optimal solutions are found in the regime where \mathchoice{{\hbox{\displaystyle\sqrt{2G-1,}}\lower 0.4pt\hbox{\vrule height=6.83331pt,depth=-5.46667pt}}}{{\hbox{\textstyle\sqrt{2G-1,}}\lower 0.4pt\hbox{\vrule height=6.83331pt,depth=-5.46667pt}}}{{\hbox{\scriptstyle\sqrt{2G-1,}}\lower 0.4pt\hbox{\vrule height=4.78333pt,depth=-3.82668pt}}}{{\hbox{\scriptscriptstyle\sqrt{2G-1,}}\lower 0.4pt\hbox{\vrule height=3.41666pt,depth=-2.73334pt}}}\sigma\ll\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}} holds and thus the ancilla quadrature noise and are contained within the range [-\mathchoice{{\hbox{\displaystyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\textstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\scriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=5.25pt,depth=-4.20003pt}}}{{\hbox{\scriptscriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}},\mathchoice{{\hbox{\displaystyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\textstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\scriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=5.25pt,depth=-4.20003pt}}}{{\hbox{\scriptscriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}}] with a very high probability. In this regime, the variance is approximately given by
[TABLE]
where \textrm{erfc}(x)\equiv\frac{2}{\mathchoice{{\hbox{\displaystyle\sqrt{\pi,}}\lower 0.4pt\hbox{\vrule height=3.01389pt,depth=-2.41113pt}}}{{\hbox{\textstyle\sqrt{\pi,}}\lower 0.4pt\hbox{\vrule height=3.01389pt,depth=-2.41113pt}}}{{\hbox{\scriptstyle\sqrt{\pi,}}\lower 0.4pt\hbox{\vrule height=2.10971pt,depth=-1.68779pt}}}{{\hbox{\scriptscriptstyle\sqrt{\pi,}}\lower 0.4pt\hbox{\vrule height=1.50694pt,depth=-1.20557pt}}}}\int_{x}^{\infty}dze^{-z^{2}} is the complementary error function. Further assuming , we find that the variance in Eq. (25) is minimized when the gain is given by
[TABLE]
(See Appendix C for the derivation.) Then, the optimal standard deviation of the output logical quadrature noise is given by
[TABLE]
As can be seen from Fig. 5, these asymptotic expressions agree well with the exact numerical results. In particular, the minimum standard deviation of the logical quadrature noise decreases quadratically as decreases (i.e., ) up to a small logarithmic correction. For example, when (which corresponds to for the pure excitation loss error), the optimal gain is given by which requires single-mode squeezing operations. In this case, the resulting standard deviation of the output noise is given by which corresponds to the loss probability . This corresponds to a QEC “gain” for the protocol of in terms of the loss probability and in terms of displacement errors.
We emphasize again that our scheme is hardware-efficient because only two bosonic modes (one data mode and one ancilla mode) are needed to achieve the quadratic suppression of both quadrature noise up to a small logarithmic correction. In contrast, in the case of multi-qubit QEC, at least five qubits and high-weight multi-qubit operations are needed to suppress both the bit-flip and phase-flip errors quadratically Laflamme et al. (1996); Bennett et al. (1996).
On the other hand, while the GKP-two-mode-squeezing code is relatively simple and achieves a quadratic noise suppression, one might at some point want to have a more complicated but even more powerful codes that can achieve a higher (than second) order noise suppression. Below, having this in mind, we generalize our non-Gaussian quantum error-correcting codes to an even broader class of non-Gaussian quantum error-correcting codes.
IV.2 GKP-stabilizer codes
In this subsection, we further generalize the GKP-repetition code (Section III) and the GKP-two-mode-squeezing code (Subsection IV.1) to an even broader class of non-Gaussian oscillator codes, namely, GKP-stabilizer codes: We define logical code states of a general GKP-stabilizer code (encoding oscillator modes into oscillator modes) as
[TABLE]
where is an arbitrary -mode bosonic state in the first modes (data modes) and is the copies of the canonical GKP states in the last modes (ancilla modes) and is an encoding Gaussian unitary operation (see Fig. 6 for the encoding circuit). For example, for the two-mode GKP-repetition code we have , and . For the GKP-two-mode-squeezing code, we have , and .
Since one can choose any Gaussian operation as an encoding circuit , our GKP-stabilizer formalism is at least as flexible as the stabilizer formalism for qubit-based QEC Gottesman (1997) which encompasses almost all conventional multi-qubit QEC schemes. Therefore, our GKP-stabilizer formalism has a great deal of potential. As an illustration, in Appendix D, we introduce a family of GKP-stabilizer codes, GKP-squeezed-repetition codes, and show that the -mode GKP-squeezed-repetition code can suppress additive Gaussian noise errors to the order, as opposed to the second order for the GKP-two-mode-squeezing code. In addition to the GKP-squeezed-repetition codes, there will be many other interesting families of GKP-stabilizer codes. For example, it will be an interesting future research direction to consider an encoding circuit which is analogous to the encoding circuit of a multi-qubit surface code Kitaev (1997); Fowler et al. (2012) and define an oscillator-into-oscillators GKP-surface code which can be implemented locally on a -dimensional plane (see Refs. Vuillot et al. (2019); Noh and Chamberland (2019) for the qubit-into-oscillators toric-GKP code and the surface-GKP code). In the following subsection, we will discuss implementation of logical Gaussian operations for a general GKP-stabilizer code, while leaving the problem of finding useful GKP-stabilizer codes as a future research direction.
IV.3 Logical Gaussian operations
Here, we discuss implementation of logical Gaussian operations. In general, for any GKP-stabilizer code with a Gaussian encoding circuit , a logical Gaussian operation can be readily implemented by using only physical Gaussian operations: , where is the Gaussian operation that we want to apply to the data modes and is an auxiliary Gaussian operation to the ancilla modes (see Fig. 6). Physical Gaussian operations are sufficient for the implementation of logical Gaussian operations because the input ancilla GKP state is the only non-Gaussian resource in the encoding scheme and the remaining circuit is Gaussian. We remark that auxiliary Gaussian operations can be used to simplify the form of the required physical Gaussian operation (see below for an illustration).
Many interesting CV quantum information processing tasks such as the KLM protocol Knill et al. (2001), boson sampling Aaronson and Arkhipov (2011) and simulation of vibrational quantum dynamics of molecules Huh et al. (2015); Sparrow et al. (2018) require Gaussian operations and, as non-Gaussian resources, input single- or multi-photon Fock states and photon number measurements of the output modes. Since the non-Gaussian resources are consumed locally only in the beginning and the end of the computation, any imperfections in these non-Gaussian resources can be addressed independently to the imperfections in the Gaussian operations. Below, we focus on improving the quality of Gaussian operations, beam splitter interactions specifically, having the KLM protocol and boson sampling in mind.
Let us recall the GKP-two-mode-squeezing code introduced in Subsection IV.1. For simplicity, we consider two logical oscillator modes encoded in four physical oscillator modes via the encoding Gaussian circuit . Here, the modes are the data modes and the modes are the ancilla modes which are used for error syndrome detection. Note that the encoding Gaussian circuit of the GKP-two-mode-squeezing code has the following nice property:
[TABLE]
where . That is, by choosing , we can can implement the logical beam splitter interaction transversally, simply by applying a pair of beam splitter interactions and to the data modes and the ancilla modes (see Fig. 7 (a)).
We now discuss how the transversality in Eq. (29) can be used to correct additive Gaussian noise errors that occur during the implementation of a beam splitter interaction. Note that any passive beam splitter interactions commute with iid additive Gaussian noise errors (see Appendix E for more details). Therefore, the beam splitter interaction continuously corrupted by iid additive Gaussian noise errors can be understood as the ideal beam splitter interaction followed by an iid additive Gaussian noise channel where the variance of the additive noise is proportional to the time needed to complete the beam splitter interaction.
As shown in Fig. 7 (b), the uncorrelated additive Gaussian noise error that has been accumulated during the logical beam splitter interaction is converted into a correlated additive Gaussian noise error through the inverse of the encoding circuit in the decoding procedure: That is, after the noise reshaping, the quadrature noise in the data modes and are correlated with the ones in the ancilla modes and , respectively. Thanks to the correlation between the data modes and the ancilla modes, we can learn about the reshaped data quadrature noise and solely by extracting information about the reshaped ancilla quadrature noise and . This information can then be used to suppress the additive Gaussian noise accumulated in the data mode quadratically up to a small logarithmic correction, similarly as in Subsection IV.1.
Note that we have so far assumed ideal GKP states to clearly demonstrate the error-correcting capability of GKP-stabilizer codes. In the following section, we discuss experimental realization of GKP-stabilizer codes. Especially, we address issues related to the use of realistic noisy GKP states and discuss a key factor that should be taken into account in the design of practical GKP-stabilizer codes.
V Discussion
Here, we discuss experimental realization of our GKP-stabilizer codes and the effects of realistic imperfections. Recall that the only required non-Gaussian resource for implementing GKP-stabilizer codes is the preparation of a canonical GKP state. While Gaussian resources are readily available in many realistic bosonic systems, preparing a canonical GKP state is not strictly possible because it would require infinite squeezing. Recently, however, finitely-squeezed approximate GKP states have been realized in a trapped ion system Flühmann et al. (2018, 2019); Flühmann and Home (2019) by using a heralded preparation scheme with post-selection Travaglione and Milburn (2002) and in a circuit QED system by using a deterministic scheme Terhal and Weigand (2016); Campagne-Ibarcq et al. (2019). Thus, our GKP-two-mode-squeezing code can in principle be implemented in the state-of-the-art quantum computing platforms.
Imperfections in realistic GKP states such as finite squeezing will add additional quadrature noise to the system. Therefore in near-term experiments, the performance of the GKP-two-mode-squeezing code will be mainly limited by the finite squeezing of the approximate GKP states. Indeed, we show below that a non-trivial QEC gain is achievable with the GKP-two-mode-squeezing code only when the supplied GKP states have a squeezing larger than dB. On the other hand, the squeezing of the experimentally realized GKP states ranges from dB to dB Flühmann et al. (2019); Campagne-Ibarcq et al. (2019). In this regard, we emphasize that our oscillator encoding scheme is compatible with non-deterministic GKP state preparation schemes. This is because the required GKP states can be prepared offline and then supplied to the error correction circuit in the middle of the decoding procedure (similar to the magic state injection for the qubit-based universal quantum computation Bravyi and Kitaev (2005)). Thus in near-term experiments, it will be more advantageous to sacrifice the success probability of the GKP state preparation schemes and aim to prepare a GKP state of higher quality (with a squeezing larger than the critical value dB) by using post-selection.
In general, the imperfections in GKP states may be especially detrimental to a GKP-stabilizer code involving a large squeezing parameter. This is because such imperfections may be significantly amplified by the large squeezing operations. With this concern in mind, let us revisit the GKP-two-mode-squeezing code in Subsection IV.1 and recall that the optimal gain is asymptotically given by in the limit. Therefore, if the standard deviation of the input noise is very small, we indeed have a huge gain parameter (or \lambda^{\star}=\mathchoice{{\hbox{\displaystyle\sqrt{G^{\star},}}\lower 0.4pt\hbox{\vrule height=8.1361pt,depth=-6.50891pt}}}{{\hbox{\textstyle\sqrt{G^{\star},}}\lower 0.4pt\hbox{\vrule height=8.1361pt,depth=-6.50891pt}}}{{\hbox{\scriptstyle\sqrt{G^{\star},}}\lower 0.4pt\hbox{\vrule height=5.71388pt,depth=-4.57114pt}}}{{\hbox{\scriptscriptstyle\sqrt{G^{\star},}}\lower 0.4pt\hbox{\vrule height=4.34721pt,depth=-3.4778pt}}}+\mathchoice{{\hbox{\displaystyle\sqrt{G^{\star}-1,}}\lower 0.4pt\hbox{\vrule height=8.1361pt,depth=-6.50891pt}}}{{\hbox{\textstyle\sqrt{G^{\star}-1,}}\lower 0.4pt\hbox{\vrule height=8.1361pt,depth=-6.50891pt}}}{{\hbox{\scriptstyle\sqrt{G^{\star}-1,}}\lower 0.4pt\hbox{\vrule height=5.71388pt,depth=-4.57114pt}}}{{\hbox{\scriptscriptstyle\sqrt{G^{\star}-1,}}\lower 0.4pt\hbox{\vrule height=4.34721pt,depth=-3.4778pt}}}\gg 1). However, we explain in detail below that we have designed the GKP-two-mode-squeezing code very carefully so that any imperfections in GKP states are not amplified by the large squeezing operations.
With these potential issues in mind, let us now analyze the adverse effects of the finite squeezing in a rigorous and quantitative manner. Recall that an approximate GKP state with a finite squeezing can be modeled by . As shown in Ref. Noh and Chamberland (2019), one can convert the finitely-squeezed GKP state via a noise twirling into
[TABLE]
i.e., an ideal canonical GKP state corrupted by an incoherent random shift error . Here, is the variance of the additive noise associated with the finite GKP squeezing. The noise standard deviation characterizes the width of each peak in the Wigner function of an approximate GKP state. The GKP squeezing is then defined as . The GKP squeezing quantifies how much an approximate GKP state is squeezed in both the position and the momentum quadrature in comparison to the vacuum noise variance .
In Fig. 8, we present the full circuit for the implementation of the GKP-two-mode-squeezing code. Note that the third mode (or the measurement mode) in the decoding scheme is introduced to simultaneously measure the position and momentum operators of the ancilla mode modulo . That is, we consume one GKP state to perform the simultaneous and modular position and momentum measurements.
We remark that we would have consumed two GKP states for the simultaneous and modular quadrature measurements if we were to use the measurement circuits in Fig. 1 (b) (i.e., one for the modular position measurement and the other for the modular momentum measurement). While this scheme certainly works, it is not the most efficient strategy. This is because the measurement circuits in Fig. 1 (b) are for non-destructive measurements. While the first measurement (e.g., the modular position measurement) has to be performed in a non-destructive way, the following measurement (e.g., the modular momentum measurement) can be done in a destructive way since we no longer need the quantum state in the ancilla mode and instead only need the classical measurement outcomes and . This is the reason why we simply measure the momentum quadrature of the second mode (i.e., the ancilla mode) in a destructive way after the modular position measurement (see Fig. 8 (b)). Such a non-Gaussian resource saving is especially crucial in the regime where the finite squeezing of an approximate GKP state is the limiting factor.
Thanks to the resource saving described above, we only need to supply two GKP states to implement the GKP-two-mode-squeezing code (one in the input of the ancilla mode and the other for the simultaneous and modular ancilla quadrature measurements). We assume that these two GKP states are corrupted by an additive Gaussian noise channel , i.e., (see Eq. (30) and Fig. 8). Due to this additional noise associated with the finite squeezing of the GKP states, the estimated reshaped ancilla quadrature noise in Eq. (19) is corrupted as follows:
[TABLE]
Here, and are the additional noise due to the finite GKP squeezing and follow . Such additional noise will then be propagated to the data mode through the miscalibrated counter displacement operations based on noisy estimates. In the presence of additional GKP noise, the sizes of the optimal counter displacements and are given by
[TABLE]
and do not explicitly depend on in the limit (see Eq. (20)). Therefore, the additional GKP noise and will simply be added to the data quadrature operators without being amplified by the large gain parameter . This absence of the noise amplification is a critically important feature of our scheme and is generally not the case for a generic GKP-stabilizer code involving large squeezing operations (see Appendix D for an illustration).
In Appendix C, we provide a through analysis of the adverse effects of the finitely-squeezed GKP states. In particular, we derive the variance of the output logical quadrature noise as a function of the input noise standard deviation , the GKP noise standard deviation , and the gain of the two-mode squeezing (see Eq. (65)). Similarly as in Subsection IV.1, we optimize the gain of the two-mode squeezing to minimize the output logical noise standard deviation for given and .
In Fig. 9 (a), we plot the maximum achievable QEC gain as a function of the input noise standard deviation for various values of the GKP squeezing ranging from dB to dB. In Fig. 9 (b), we plot the optimal gain of the two-mode squeezing, translated to the required single-mode squeezing in the unit of decibel. We observe that the non-trivial QEC gain can be achieved only when the supplied GKP states have a squeezing larger than the critical value dB. Also, when the supplied GKP states have a squeezing of dB, the maximum QEC gain is given by , which is achieved when . For comparison, the QEC gain at the same input noise standard deviation is given by when the ideal canonical GKP states are used to implement the GKP-two-mode-squeezing code (see Subsection IV.1). The fact that these two values ( verses ) are close to each other is an indicative of the fact that the additional GKP noise is not catastrophically amplified by the large (dB) single-mode squeezing operations needed in this regime.
Lastly, we outline several open questions. First, note that our scheme works with imperfect GKP states but is not truly fault tolerant, in the sense that the added noise due to the finitely-squeezed GKP states is not suppressed to an arbitrarily small value. An important open question is whether a fault-tolerant scheme can be found in which the effects of added noise due to non-ideal GKP ancilla states can be made arbitrarily small. Moreover, it will be an interesting research avenue to search for a family of efficient GKP-stabilizer codes by exploring various encoding circuits . For example, one can look for GKP-stabilizer codes that can be implemented locally in a low dimensional space, or for ones with low resource overheads, or ones with high fault-tolerant threshold, if any. In addition, while logical Gaussian operations can be readily implemented by using only Gaussian operations for any GKP-stabilizer codes, implementation of logical non-Gaussian operations will require some non-Gaussian resources. Thus, it will also be interesting to explore whether logical non-Gaussian operations can be implemented efficiently by using, e.g., GKP states or cubic phase states as non-Gaussian resources.
VI Conclusion
We have worked around the previous no-go theorems on Gaussian QEC schemes Eisert et al. (2002); Niset et al. (2009); Vuillot et al. (2019) and proposed several non-Gaussian oscillator-into-oscillators codes, i.e., the two-mode GKP-repetition code and the GKP-two-mode-squeezing code, that can correct additive Gaussian noise errors. We generalized them to an even broader class of non-Gaussian oscillator codes, namely, GKP-stabilizer codes. In particular, we explicitly constructed a family of -mode GKP-stabilizer codes that can efficiently suppress additive Gaussian noise errors to the order. We also showed that our proposed QEC schemes can also correct excitation loss and thermal noise errors as well as additive Gaussian noise errors by a suitable noise conversion through a quantum-limited amplification channel. The only required non-Gaussian resource for our GKP-stabilizer QEC schemes is the preparation of the canonical GKP state. We showed that, for any GKP-stabilizer codes, logical Gaussian operations can be readily implemented by using only physical Gaussian operations. Therefore, our GKP-stabilizer QEC schemes will be useful for realizing error-corrected boson sampling and simulation of bosonic systems. In addition, our GKP-stabilizer QEC schemes may also be able to suppress errors for quantum metrology with bosonic sensors. Indeed a recent follow-up work Zhuang et al. (2019) has theoretically demonstrated that the GKP-two-mode-squeezing code can be used to enhance the robustness of CV distributed sensing protocols. Moreover, our oscillator codes may remove the need for CV-DV-CV conversion for quantum communication over bosonic channels. Lastly, our oscillator encoding schemes can be useful for overcoming loss errors in transduction protocols Zhang et al. (2018); Lau and Clerk (2019) where a microwave-frequency bosonic mode should be transferred to an optical-frequency bosonic mode.
Acknowledgments
We would like to thank Rob Schoelkopf for helpful discussions. We acknowledge support from the ARL-CDQI (W911NF-15-2-0067, W911NF-18-2-0237), ARO (W911NF-18-1-0020, W911NF-18-1-0212), ARO MURI (W911NF-16-1-0349 ), AFOSR MURI (FA9550-14-1-0052, FA9550-15-1-0015), DOE (DE-SC0019406), NSF (EFMA-1640959, DMR-1609326), and the Packard Foundation (2013-39273). K.N. acknowledges support through the Korea Foundation for Advanced Studies.
Appendix A Gaussian-repetition codes
Here, we introduce and analyze the -mode Gaussian-repetition code, which is a straightforward generalization of the three-mode Gaussian-repetition code introduced in Refs. Lloyd and Slotine (1998); Braunstein (1998a). In particular, we introduce the maximum likelihood estimation of the data position quadrature noise for the -mode Gaussian-repetition code (Eq. (38)), which is a key motivation behind our choice of the estimate in Eq. (9).
Consider an arbitrary oscillator state . The state can be embedded in oscillator modes via the -mode Gaussian-repetition code as follows:
[TABLE]
where the first mode is the data mode and the rest are the ancilla modes. Note that the data position eigenstate is mapped into through the encoding procedure. As shown in Fig. 10, this encoding can be realized by applying a sequence of Gaussian SUM gates where . Upon the encoding circuit, the quadrature operators are transformed into
[TABLE]
as in Eq. (4) where . We then assume that the oscillator modes undergo independent and identically distributed additive Gaussian noise errors , i.e.,
[TABLE]
as in Eq. (5). The added noise follow an independent and identically distributed Gaussian random distribution .
The goal of the decoding procedure (shown in Fig. 10) is to extract some information about the added noise and () through a set of syndrome measurements. The decoding procedure begins with the inverse of the encoding circuit, i.e., by a sequence of inverse-SUM gates for . Upon the inverse of the encoding circuit, the quadrature operators are transformed into and , where the reshaped quadrature noise is given by
[TABLE]
as in Eq. (7) where . Then, by performing homodyne measurements of the ancilla position quadrature operators, we can exactly extract the values of
[TABLE]
for all . This is because the ancilla modes are initially in the position eigenstates and thus measuring is equivalent to measuring for all . Note, however, that we cannot extract any information about the reshaped momentum quadrature noise . This will later turn out to be the key limitation of Gaussian-repetition codes.
From the extracted values of , we can infer that position quadrature noise is given by . Then, the undetermined data position quadrature noise can be estimated by a maximum likelihood estimation: Since noise with smaller are more likely, we estimate that is
[TABLE]
from the syndrome measurement outcomes . Note that for , Eq. (38) reduces to : This is the main reason why we chose in Eq. (9) as the estimate of . Finally, the decoding procedure ends with an application of the counter displacement operation to the data oscillator mode (see Fig. 10).
As a result of the encoding and decoding procedures, we end up with a logical additive noise error and of the data oscillator mode, where the output logical position and momentum quadrature noise are given by
[TABLE]
Since , we have
[TABLE]
Thus, the variance of the position quadrature noise is reduced by a factor of , but the variance of the momentum quadrature noise is increased by the same factor. The latter increase is due to the fact that the ancilla momentum quadrature noise which are transferred to the data momentum quadrature (see in ) are left completely undetected by the position homodyne measurements during the syndrome extraction stage. As a result, the product of the noise standard deviations remains unchanged at the end of the error correction procedure (i.e., ). This implies that Gaussian-repetition codes can only squeeze the Gaussian quadrature noise but cannot actually correct them. This reaffirms the previous no-go results Eisert et al. (2002); Niset et al. (2009); Vuillot et al. (2019).
In Section III in the main text, we modify the two-mode Gaussian-repetition code (i.e., ) and introduce the two-mode GKP-repetition code that can indeed correct additive Gaussian noise errors. Specifically, we replace several Gaussian elements in the Gaussian-repetition code by non-Gaussian ones involving the canonical GKP state, such that we can only benefit from the decreased position noise variance by a factor of , while preventing the momentum noise variance from increasing by the same factor.
Appendix B Supplementary material for the two-mode GKP-repetition code
Here, we provide explicit expressions for the probability density functions of the logical quadrature noise and for the two-mode GKP-repetition code. Recall Eq. (10) and note that the logical quadrature noise and for the two-mode GKP-repetition are given by
[TABLE]
where and and . Let denote the probability density function of a Gaussian distribution with zero mean and variance , i.e., p_{\sigma}(z)\equiv(1/\mathchoice{{\hbox{\displaystyle\sqrt{2\pi\sigma^{2},}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi\sigma^{2},}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi\sigma^{2},}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi\sigma^{2},}}\lower 0.4pt\hbox{\vrule height=3.44165pt,depth=-2.75334pt}}})\exp[-z^{2}/(2\sigma^{2})]. Then, the probability density functions and of the logical quadrature noise and are given by
[TABLE]
and
[TABLE]
where is the Dirac delta function. Note that can be expressed as
[TABLE]
where is an indicator function, i.e., if is true if is false. Then, using Eq. (44), we can make and in Eqs. (42),(43) more explicit as follows:
[TABLE]
and
[TABLE]
Fig. 3 in the main text is obtained by numerically computing these probability density functions and then evaluating the standard deviations of the obtained probability density functions.
Appendix C Supplementary material for the GKP-two-mode-squeezing code
Here, we first explain the underlying reasons behind our choice of the estimates and in Eq. (20) for the GKP-two-mode-squeezing code. Recall that the covariance matrix of the reshaped noise is given by
[TABLE]
For now, let us ignore the fact that we can measure and only modulo and instead assume that we know their exact values. Note that is only correlated with , whereas is only correlated with . Consider the estimates of the form and , where and are constants. We choose and such that the variances of and are minimized: Since and are given by
[TABLE]
they are minimized when
[TABLE]
Therefore, if both and are precisely known, the optimal estimates of and are given by
[TABLE]
Since, however, we can only measure and modulo , we replace and by \bar{z}_{q}^{(2)}=R_{\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.15778pt,depth=-2.52625pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=2.25555pt,depth=-1.80446pt}}}}(z_{q}^{(2)}) and \bar{z}_{p}^{(2)}=R_{\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.15778pt,depth=-2.52625pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=2.25555pt,depth=-1.80446pt}}}}(z_{p}^{(2)}) and get the estimates and in Eq. (20).
Now we provide explicit expression for the probability density functions of the logical quadrature noise and for the GKP-two-mode-squeezing code. Recall Eq. (21):
[TABLE]
where follows a Gaussian distribution with zero means and the covariance matrix . By using Eq. (44), we find that the probability density functions of the quadrature noise are given by
[TABLE]
and similarly
[TABLE]
where and are as defined in Eq. (23).
Finally, we derive the asymptotic expressions for the optimal gain and the minimum standard deviation given in Eqs. (26), (27). Recall that assuming \mathchoice{{\hbox{\displaystyle\sqrt{2G-1,}}\lower 0.4pt\hbox{\vrule height=6.83331pt,depth=-5.46667pt}}}{{\hbox{\textstyle\sqrt{2G-1,}}\lower 0.4pt\hbox{\vrule height=6.83331pt,depth=-5.46667pt}}}{{\hbox{\scriptstyle\sqrt{2G-1,}}\lower 0.4pt\hbox{\vrule height=4.78333pt,depth=-3.82668pt}}}{{\hbox{\scriptscriptstyle\sqrt{2G-1,}}\lower 0.4pt\hbox{\vrule height=3.41666pt,depth=-2.73334pt}}}\sigma\ll 1, we have Eq. (25):
[TABLE]
Assuming further that (which is relevant when ) and using the asymptotic formula for the complementary error function, i.e.,
[TABLE]
we can simplify Eq. (54) as
[TABLE]
where . The optimum can be found by solving . Note that is given by
[TABLE]
Thus, should satisfy
[TABLE]
where we iteratively plugged in the first equation into itself to get the second equation. Since , we can disregard the second term in the second line. By further neglecting the logarithmic factor , we get
[TABLE]
Since , Eq. (26) follows:
[TABLE]
The optimal value is then approximately given by
[TABLE]
Since , we can disregard the second term and obtain Eq. (27):
[TABLE]
Let us now consider the case with noisy GKP states (see Fig. 8 and Eq. (31)). Then, we have
[TABLE]
where the GKP noise is independent of and and follows . Then, the probability density function is given by
[TABLE]
Thus, the variance of the output logical quadrature noise is given by
[TABLE]
Appendix D GKP-squeezed-repetition codes
Here, we introduce a family of GKP-stabilizer codes, GKP-squeezed-repetition codes, and briefly explain that the -mode GKP-squeezed-repetition code can suppress additive Gaussian noise errors to the order, i.e., , where is the standard deviation of the input additive Gaussian noise and and are the standard deviations of the output logical position and momentum quadrature noise, respectively.
Let be an arbitrary one-mode bosonic state. We define the encoded logical state of the -mode GKP-squeezed-repetition code as
[TABLE]
where the encoding Gaussian circuit is recursively defined as
[TABLE]
for (see Fig. 11 (a)). In the base case (), is given by (see Fig. 11 (b)) and the symplectic matrix associated with the encoding Gaussian circuit is given by
[TABLE]
To illustrate how this family of codes works, let us focus on the case: The symplectic matrix associated with the encoding Gaussian circuit (see Fig. 11 (c)) is explicitly given by
[TABLE]
Thus, the reshaped quadrature noise vector is given by
[TABLE]
where is the original quadrature noise vector whose elements follow independent and identically distributed Gaussian random distributions with variance .
Through the noise reshaping, the data position quadrature noise is amplified by a factor of (i.e., ). Note that, by choosing \lambda=\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}c/\sigma with a small constant , we can make sure that the ancilla position quadrature noise and are contained within the unambiguously distinguishable range [-\mathchoice{{\hbox{\displaystyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\textstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\scriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=5.25pt,depth=-4.20003pt}}}{{\hbox{\scriptscriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}},\mathchoice{{\hbox{\displaystyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\textstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\scriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=5.25pt,depth=-4.20003pt}}}{{\hbox{\scriptscriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}}] with a very high probability. Then, by measuring the reshaped ancilla position quadrature noise of the second mode , (modulo ) we can learn about the amplified position quadrature noise up to an error . Then, by further measuring the ancilla position quadrature noise of the third mode, , (modulo ) we can learn about the residual error up to an even smaller error . As a result, we can reduce the position quadrature noise of the data mode by a factor of despite the temporary increase by the same factor before the correction. Since , this implies that the standard deviation of the position quadrature noise is suppressed cubically, i.e., .
Through the noise reshaping, the data momentum quadrature noise is immediately reduced by a factor of (see in ) but there are transferred ancilla momentum quadrature noise (see and in ). Note that by measuring the ancilla momentum quadrature noise of the third mode, (which is contained within the range [-\mathchoice{{\hbox{\displaystyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\textstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\scriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=5.25pt,depth=-4.20003pt}}}{{\hbox{\scriptscriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}},\mathchoice{{\hbox{\displaystyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\textstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{\scriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=5.25pt,depth=-4.20003pt}}}{{\hbox{\scriptscriptstyle\sqrt{\pi/2,}}\lower 0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}}] with a very high probability), we can precisely extract the value of and then eliminate the transferred ancilla momentum quadrature noise in and also in . Then, by further measuring the reshaped ancilla momentum quadrature noise of the second mode (after eliminating ), we can precisely extract the value of and then eliminate the transferred ancilla momentum quadrature noise in . As a result, we end up with the reduced date momentum quadrature noise without any transferred ancilla momentum quadrature noise. Similarly as above, since , we can suppress the momentum quadrature noise cubically, i.e., .
Generalizing these arguments, one can inductively show that the -mode GKP-squeezed-repetition code can suppress additive quadrature noise errors to the order, i.e., for any . We emphasize, however, these codes are susceptible to the realistic noise in GKP states (except for the case), because the GKP noise can be amplified by large squeezing operations. For example, in the case of the three-mode GKP-squeezed-repetition code discussed above, one has to multiply a factor of into the obtained measurement outcome of to eliminate the transferred noise in the reshaped data momentum quadrature noise . Therefore, the GKP noise that corrupt the measurement outcome of are amplified by a large squeezing parameter \lambda=\mathchoice{{\hbox{\displaystyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\textstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{\scriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{\scriptscriptstyle\sqrt{2\pi,}}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}c/\sigma\xrightarrow{\sigma\rightarrow 0}\infty when they are propagated to the data mode via miscalibrated counter displacement operations.
Thus, although the -mode GKP-squeezed-repetition code can correct additive Gaussian noise errors to the order, it is sensitive to the realistic noise in GKP states. It will therefore be interesting to look for a family of GKP-stabilizer codes that can achieve higher order error suppression () while not amplifying the GKP noise.
Appendix E Commutativity of beam splitter interactions and iid additive Gaussian noise errors
Here, we show that any passive beam splitter interactions commute with iid additive Gaussian noise errors. Consider bosonic modes described by bosonic annihilation operators . A general passive beam splitter interaction among these modes is generated by a Hamiltonian of the following form:
[TABLE]
where for all . Note that the total excitation number is conserved under a general beam splitter interaction: . Also, iid additive Gaussian noise errors are generated by the Lindbladian
[TABLE]
where . Specifically, the Lindbladian generator is related to the additive Gaussian noise error by the relation
[TABLE]
where is the diffusion rate and is the time elapsed.
Having introduced the generators of the beam splitter interactions and iid additive Gaussian noise errors, we now prove the following commutation relation:
[TABLE]
where is the Lindbladian superoperator associated with a beam splitter Hamiltonian . Note that for a general and we have
[TABLE]
and similarly
[TABLE]
Therefore, we have
[TABLE]
Using this general relation, we find
[TABLE]
Similarly, we also have
[TABLE]
Then, since , Eq. (74) follows:
[TABLE]
Since the generators of a beam splitter interaction and an iid additive Gaussian noise error commute with each other at the superoperator level, we have
[TABLE]
where and and is the desired -mode beam splitter unitary operation. This implies that the noisy beam splitter interaction continuously corrupted by iid additive noise errors can be understood as the desired noiseless beam splitter interaction followed by an iid additive Gaussian noise channel with variance , where is the diffusion rate and is the time needed to complete the beam splitter interaction.
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