Quantum Error Correction with the Gottesman-Kitaev-Preskill Code

The Gottesman-Kitaev-Preskill (GKP) code was proposed in 2001 by Daniel Gottesman, Alexei Kitaev, and John Preskill as a way to encode a qubit in an oscillator. The GKP codewords are coherent superpositions of periodically displaced squeezed vacuum states. Because of the challenge of merely preparing the codewords, the GKP code was for a long time considered to be impractical. However, the remarkable developments in quantum hardware and control technology in the last two decades has made the GKP code a frontrunner in the race to build practical, fault-tolerant bosonic quantum technology. In this Perspective, we provide an overview of the GKP code with emphasis on its implementation in the circuit-QED architecture and present our outlook on the challenges and opportunities for scaling it up for hardware-efficient, fault-tolerant quantum error correction.

Popular Summary

The introduction of quantum-error-correcting (QEC) codes in the early–mid 1990s transformed the field of quantum information science. The principle behind QEC is simple yet remarkably powerful: quantum information is stored redundantly in nonlocal observables of a many-body quantum system, such that common errors, which damage parts of the system locally do not destroy the information globally.

Although the early quantum-error-correcting codes were designed with information stored in discrete variables, very soon the idea was applied to information stored in quantum states characterized by continuous variables. An example is the state of a quantum harmonic oscillator, which is characterized by its position and momentum. One of the earliest such continuous-variable error-correcting codes was introduced by Gottesman, Kitaev, and Preskill in 2001, now known as the GKP code after the name of its inventors. There are strong reasons to believe that if implemented successfully, the GKP code could lead to a hardware-efficient solution for scalable quantum error correction. Like many revolutionary ideas, at the time of its invention it was met with some reservation due to the complexity of realizing the quantum states defining the code. However, remarkable advances in quantum systems’ design and control over the last two decades has led to the realization of the GKP code in two very different experimental platforms: trapped ions and superconducting circuits. These developments have laid a fertile ground for future research on scalable quantum computing with the GKP code.

In this Perspective, we provide an overview of the GKP code, focusing on its realization in the superconducting circuit platform. We present our perspective on the challenges that lay ahead and discuss possible opportunities to meet them in order to harness the power of the GKP code, and build a large-scale fault-tolerant quantum computer.

Figure 1

Square (a) and hexagonal (b) GKP lattices (adapted from Ref. [29]). In Eq. (6a) coherent states are placed on the filled (empty) circles for the $|0_L⟩$ ($|1_L⟩$) state. The area of the lattice parallelogram is $\pi /2$, which ensures the correct anticommutation relation for the logical operators defined in Eq. (3). Wigner functions $W(\zeta )$ for approximate $|{\tilde{0}}_L⟩$ states with $\mathrm{\Delta }=0.3$ are shown for square (c) and hexagonal (d) GKP codes. Position [$\hat{q}=(\hat{a}{}^{†}+\hat{a})/\sqrt{2}$] and momentum [$\hat{p}=i(\hat{a}{}^{†}-\hat{a})/\sqrt{2}$] wave functions are shown in blue.

Figure 2

Average gate fidelity for an approximate square GKP code as a function of photon number $n_{\mathrm{code}}=(⟨0_L|\hat{n}|0_L⟩+⟨1_L|\hat{n}|1_L⟩)/2$, for simultaneous loss and dephasing quantified by $\kappa t$ and ${\kappa }_{\varphi }t$, followed by optimal error correction. The GKP code outperforms the “trivial encoding” in Fock states $|0⟩$ and $|1⟩$ whenever the fidelity dips below the gray shaded region.

Figure 3

(a) Position wave functions for logical zero (blue) and one (red) for a square GKP code with $\mathrm{\Delta }=0.3$. A logical ${\mathcal{M}}_Z$ measurement can be executed by first measuring position (homodyne measurement) and binning the result. The white (gray) regions correspond to a logical zero (one) outcome. (b) Probability of mistaking logical zero for one under a homodyne measurement with measurement efficiency $\eta$ for $\mathrm{\Delta }=0.3$ ($\mathcal{S}=10.1$ dB, blue solid) and $\mathrm{\Delta }=0.2$ ($\mathcal{S}=13.8$ dB, orange solid). Note that the bin boundaries are scaled by $\sqrt{\eta }$ to account for measurement inefficiency. For comparison, the dotted lines show the corresponding measurement error with one round of noiseless phase estimation using the scheme in Fig. 4. (c) Same as (b) but for a majority vote over $n$ rounds of noiseless phase estimation using the schemes in Figs. 4 (solid) and 4 (dotted).

Figure 4

(a) One round of phase estimation. The controlled displacement is defined as $C\hat{D}(\zeta )=\hat{D}(\zeta /2)\otimes |0_a⟩⟨0_a⟩+\hat{D}(-\zeta /2)\otimes |1_a⟩⟨1_a⟩$, with $|0_a/1_a⟩$ the state of the ancilla. For $\overline{X}$ measurement $\zeta =\alpha$ and for $\overline{Z}$ measurement $\zeta =\beta$. (b) Improved phase-estimation scheme. Here, $ϵ$ is a small displacement orthogonal to $\zeta$, i.e., $\arg ϵ=\arg \zeta +\pi /2$. For $\overline{X}$ measurement we set $\hat{D}(ϵ/2)=e^{-i\lambda \hat{Q}}$ and for $\overline{Z}$ measurement $\hat{D}(ϵ/2)=e^{i\lambda \hat{P}}$, with $\lambda \in \mathbb{R}$. The ${\hat{R}}_x$ gate is defined as ${\hat{R}}_x=e^{-i\pi {\hat{\sigma }}_x/4}$. Note that with this definition of the controlled displacement gate, the state is displaced by $\zeta /2$ from the codespace after the measurement. This can be corrected for with another displacement if necessary.

Figure 5

Example computation in the (single-qubit) Clifford frame. A general quantum circuit (a) can be rewritten using only (adaptive) Clifford gates and magic states $|A⟩=T|+⟩$, with $T=\mathrm{diag}(1,e^{i\pi /4})$ (b). The circuit in (b) can moreover be rewritten as either (c) or (d), depending on the measurement outcome $Z_1=\pm 1$ of the first qubit. The generalized control gates are defined in Eq. (22). For GKP qubits, updating the Clifford frame amounts to changing the phase of local oscillators (c.f. Fig. 6) so that single-qubit Clifford gates are done in software.

Figure 6

Illustration of how the Hamiltonian ${\hat{H}}_{\theta ,\varphi }\propto e^{i\theta }\hat{a}\hat{b}{}^{†}+e^{i\varphi }\hat{a}\hat{b}+\mathrm{H.c.}$, required for realization of the two-qubit Clifford gates ${\overline{C}}_{{\sigma }_i{\sigma }_j}$ in Eq. (23), may be realized in a cQED architecture. The two GKP resonators, shown in orange and blue, have frequencies ${\omega }_{c,1}$ and ${\omega }_{c,2}$, respectively. In (a), a superconducting nonlinear asymmetric inductive element (SNAIL) is used for the coupling [51]. At an appropriate flux bias, the SNAIL exhibits a strong third-order nonlinearity. Two microwave drives are applied at frequencies ${\omega }_1$ and ${\omega }_2$ so that ${\omega }_1=|{\omega }_{c,1}-{\omega }_{c,2}|$ and ${\omega }_2={\omega }_{c,1}+{\omega }_{c,2}$. Because of the third-order nonlinearity, a single photon at ${\omega }_1$ is consumed to convert a photon at ${\omega }_{c,1}$ to that at ${\omega }_{c,2}$. This leads to an interaction of the form ${\hat{a}}^{†}\hat{b}{e}^{i\theta }+\mathrm{H.c.}$, where the strength of the interaction and $\theta$ depend on the strength of the microwave drive at ${\omega }_1$ and its phase, respectively. Similarly, a single photon at ${\omega }_2$ is consumed to create two photons, one at ${\omega }_{c,1}$ and the other at ${\omega }_{c,2}$. This leads to an interaction of the form ${\hat{a}}^{†}{\hat{b}}^{†}{e}^{i\varphi }+\mathrm{H.c.}$, where, again, the strength of the interaction and $\varphi$ is set by the drive strength and phase, respectively. Alternatively, it is possible to engineer ${\hat{H}}_{\theta ,\varphi }$ using a transmon as shown in (b). In this case, four pump tones can be used to generate the interaction. Due to the fourth-order nonlinearity of the transmon, a photon each from the pumps at ${\omega }_3$ and ${\omega }_4$ such that ${\omega }_3+{\omega }_4={\omega }_{c,1}+{\omega }_{c,2}$, are consumed to generate two photons at ${\omega }_{c,1}$ and ${\omega }_{c,2}$. On the other hand, photons from the pumps at ${\omega }_1$ and ${\omega }_2$ such that $|{\omega }_1-{\omega }_2|=|{\omega }_{c,1}-{\omega }_{c,2}|$, convert a photon at ${\omega }_{c,1}$ to ${\omega }_{c,2}$ via four-wave mixing.

Figure 7

“Sharpen” and “Trim” protocols used to prepare approximate GKP codestates, the controlled displacement gate is defined as in Fig. 4. The parameter $ϵ$ is small with $\arg ϵ=\arg \zeta +\pi /2$, such that $\hat{D}(ϵ)$ is a small displacement orthogonal to $\hat{D}(\zeta )$. The $\hat{S}=\mathrm{diag}(1,i)$ gate is the usual phase gate on the ancilla.

Figure 8

Preparation of a logical state can be done by alternating “Sharpen” (“$S$”) and “Trim” (“$T$”) cycles for the two stabilizers ${\hat{S}}_X,{\hat{S}}_Z$ to project onto the codespace, and finally perform a nondestructive $\overline{Z}$ measurement (“$M$”) using, e.g., one of the two circuits in Fig. 4.

Figure 9

Implementation of controlled-displacement gate, $C\hat{D}(\zeta )$, between a GKP resonator and a transmon (a) and Kerr-cat realized in a SNAIL (b). The frequencies of the resonator, transmon, and SNAIL are ${\omega }_c$, ${\omega }_t$, and ${\omega }_s$, respectively. The desired effective interaction for $C\hat{D}$ gate is $\propto ({\hat{a}}^{†}+\hat{a})\hat{Z}$, where $\hat{Z}$ is the Pauli operator of the transmon or Kerr-cat qubit. The scheme shown in (a) is from Ref. [3] and uses the cross-Kerr coupling between the transmon and the resonator $=\chi {\hat{a}}^{†}\hat{a}\hat{Z}$. The underlying principle of the scheme is as follows. In the presence of the microwave pulse ${\mathcal{E}}_c(t)$, the Hamiltonian of the system (in a frame rotating at ${\omega }_c$) is $\hat{H}=\chi {\hat{a}}^{†}\hat{a}\hat{Z}+{\mathcal{E}}_c(t)({\hat{a}}^{†}+\hat{a})$. In a displaced frame the effective Hamiltonian becomes $\hat{D}(\alpha )\hat{H}{\hat{D}}^{†}(\alpha )=\chi {\hat{a}}^{†}\hat{a}\hat{Z}+\chi (\hat{a}{\alpha }^{\ast }+{\hat{a}}^{†}\alpha )\hat{Z}+\chi |\alpha {|}^2\hat{Z},\mathrm{where}\alpha =\int {\mathcal{E}}_c(t)dt$. Clearly, we obtain the important interaction for the $C\hat{D}(\zeta )$ gate given by the second term in this equation. In order to cancel the effect of the other terms, the qubit can be flipped in the middle using the resonant $\pi$ pulse ${\mathcal{E}}_t$. For more details see Ref. [3]. In (b), a Kerr-cat qubit is realized in the SNAIL flux biased at a point where both third- and fourth-order nonlinearities are large. To realize the qubit Hamiltonian, a microwave drive ${\mathcal{E}}_s$ of frequency $2{\omega }_s$ is applied to the SNAIL. Another microwave drive at frequency $|{\omega }_c-{\omega }_s|$ is applied to generate Eq. (25). Due to the three-wave mixing, a photon from the latter drive is consumed to convert a photon at the SNAIL to that in the resonator. This effectively creates the desired coupling $\propto ({\hat{a}}^{†}+\hat{a})\hat{Z}$ between the resonator and the Kerr-cat qubit [53].

Figure 10

Illustration of “Steane” (a) and “Knill” (b) error-correction circuits. The labels $\{u,v\}$ next to a rail indicates a general displacement error $e^{-iu\hat{P}}{e}^{iv\hat{Q}}$, and the diagram indicates how the incoming errors propagate through the circuit. The two measurements are of the $\hat{P}$ and $\hat{Q}$ quadratures of the GKP code, respectively, and for the correction shifts one uses the measurement outcomes modulo the GKP lattice spacing $\sqrt{\pi }$ [1].

Figure 11

Building blocks of ${\mathcal{C}}_{\mathrm{GKP}}▹{\mathcal{C}}_{\mathrm{surface}}$ with All-GKP surface code and Hybrid-GKP surface code. Of the required operations, only the controlled-displacement gate between a discrete qubit and a GKP-encoded qubit have been demonstrated experimentally [2, 3]. Not shown in the figure above are single-qubit gates and measurement of the regular ancilla qubits that may be required. We assume these are readily available.

Figure 12

(a) Illustration of possible realization of ${\mathcal{C}}_{\mathrm{GKP}}▹{\mathcal{C}}_{\mathrm{surface}}$ in cQED. The yellow and white regions indicate $X$ and $Z$ stabilizer checks of the surface code. Scaling up an architecture based on high-$Q$ 3D microwave resonators is still a very active area of research. It is also possible to have high-$Q$ resonators on chip as an alternative to the 3D approach. The purpose of this figure is to indicate the resources that would be required to implement the widely studied All-GKP surface code. Resonators for GKP states used as data qubits in the surface code are indicated in orange. The ancilla GKP states required for realizing Steane-based ${\mathcal{C}}_{\mathrm{GKP}}$ and ${\mathcal{C}}_{\mathrm{surface}}$ are shown in green and blue, respectively. State preparation, gates, and measurements are mediated by nonlinear couplers, such as a transmon or a SNAILmon. (b) Stabilizer measurement circuits and gate ordering for surface-code error correction. The controlled-$\ominus$ gate represents the ${\overline{C}}_X^{†}$ gate between the ancilla and data GKP states.

Figure 13

One-bit teleportation circuit to teleport an arbitrary state from a two-level ancilla (bottom) to an encoded GKP state (top). The ${\hat{C}}_X$ gate is implemented by a controlled displacement $C\hat{D}(\alpha )$. This can be used for preparation of arbitrary states, including magic states, assuming we have universal control over the ancilla.