Repetition Cat Qubits for Fault-Tolerant Quantum Computation

We present a 1D repetition code based on the so-called cat qubits as a viable approach toward hardware-efficient universal and fault-tolerant quantum computation. The cat qubits that are stabilized by a two-photon driven-dissipative process exhibit a tunable noise bias where the effective bit-flip errors are exponentially suppressed with the average number of photons. We propose a realization of a set of gates on the cat qubits that preserve such a noise bias. Combining these base qubit operations, we build, at the level of the repetition cat qubit, a universal set of fully protected logical gates. This set includes single-qubit preparations and measurements, not, controlled-not, and controlled-controlled-not (Toffoli) gates. Remarkably, this construction avoids the costly magic state preparation, distillation, and injection. Finally, all required operations on the cat qubits could be performed with slight modifications of existing experimental setups.

Popular Summary

Quantum computers are expected to solve problems that classical computers cannot, but their implementation is hampered by a need to protect their fragile quantum states from the noisy environment. Error-correcting codes can help preserve the quantum-encoded information but come at a cost of enormous physical resources. In recent years, a type of error-correcting code known as a “cat code” (named for Schrödinger’s famous feline) has shown success in limited situations. Here, we propose a new implementation of cat codes that can be extended to a fully fault-tolerant and universal quantum computer with efficient use of hardware.

Our design makes use of cat qubits, which are stabilized by an engineered quantum “friction” that suppresses one of the two components of noise. Then, we demonstrate that by combining these qubits in a simple 1D error-correcting code, it is possible to enable a universal set of fully protected logical operations. Remarkably, this construction has the potential of avoiding the significant overhead that plagues other error-correcting codes.

While colleagues develop an experimental demonstration of these concepts, we are working to quantitatively assess the expected performance of our proposal while also finding the best architecture for scaling.

Figure 1

(a) Bloch sphere representation of a cat qubit. (b) Vector field associated to the semiclassical dynamics behind the master equation (1) represented in the phase space of the harmonic oscillator. This vector field governs the dynamics of coherent states. It admits two stable equilibria $|\pm \alpha ⟩$ and one saddle point at zero. The exponential suppression of the bit-flip errors can be understood by the fact that any local perturbation of the state $|0{⟩}_c\approx |\alpha ⟩$ (respectively, $|1{⟩}_c\approx |-\alpha ⟩$) keeps the state in the domain of attraction of $|\alpha ⟩$ (respectively, $|-\alpha ⟩$).

Figure 2

Layout of a repetition cat qubit using high-$Q$ 3D cylindrical postcavities [40]. Each data cat qubit (in blue cavities) is connected to a pair of ancilla cat qubits (in green cavities) for the joint-parity measurement. The results of the parity measurement are read out using the low-$Q$ strip-line resonators (in red) coupled to green cavities. Each cat qubit is continuously driven via the two-photon driven-dissipative scheme (arrows). The couplings between cavity modes are mediated by a Josephson circuit and extra microwave drives, required for bias-preserving cnot operations as detailed in Sec. 4 and Fig. 11. The choice of cylindrical postcavities is to ensure high-quality factors, but a similar layout could be thought of in a 2D architecture.

Figure 3

Overall scheme for achieving fault-tolerant universal quantum computation using cat qubits. The fundamental operations (left-hand side) are performed on the cat qubits, in a bias-preserving manner. Fault-tolerant logical operations acting on the repetition cat qubits (right-hand side) are built out of these operations, as depicted by the arrows. This construction is detailed in Sec. 5.

Figure 4

Wigner function of the state of a cat qubit during the execution of an $X$ operation. The green dots are the Wigner functions of the instantaneous steady states of the dynamics $\dot{\rho }={\kappa }_2\mathcal{D}[{\widehat{a}}^2-\alpha (t{)}^2]$. These attractive points are slowly rotated from $\pm \alpha$ to $\mp \alpha$ on the dashed circle, as shown by the green arrows. When this rotation is performed slowly, the cat follows the attractors (red arrows).

Figure 5

Joint-parity measurement between two neighboring cat qubits $j$ and $j+1$ of a repetition cat qubit, using one ancilla cat qubit. Note that the error propagation from ancilla cat qubit to data ones is exponentially suppressed by the cat size.

Figure 6

Quantum error correction circuit for a repetition cat qubit. One repetition cat qubit, composed of data cat qubits and ancilla cat qubits (in green), is protected from errors by performing $r$ rounds of error detection.

Figure 7

Transversal implementation of the logical cnot. For clarity, the number of cat qubits per repetition cat qubit is $n=3$. Here, the three upper lines represent the control repetition cat qubit, and the three bottom ones represent the target one.

Figure 8

Pieceable fault-tolerant Toffoli circuit. Each block of $n=3$ cat qubits represents one repetition cat qubit.

Figure 9

Logical Hadamard circuit (a). The circuits (b)–(d) are equivalent circuits that help to understand why the circuit (a) implements a logical Hadamard gate. In circuit (b), the cnot gate between the repetition cat qubits 2 and 3 is replaced by the equivalent circuit where the control and target roles are switched with the addition of Hadamard gates before and after the gate. It becomes clear in circuit (b) that the second repetition cat qubit plays no role: It is initialized in $|-{⟩}_L$, transformed to $|1{⟩}_L$ after the first Hadamard, and, thus, always triggers the corresponding part of the control of the Toffoli, before being converted to $|-{⟩}_L$ by the second Hadamard. Actually, the second repetition cat qubit is needed only because we do not readily have a controlled-phase gate available at the repetition cat qubit level (but here we show that it can be done with the Toffoli gate plus one ancilla repetition cat qubit). The idle role of the second repetition cat qubit is represented more simply in circuit (c), where again the cnot between the repetition cat qubits 1 and 3 is replaced by its equivalent circuit where the control and target roles are exchanged with the addition of Hadamard gates. Finally, in circuit (d) (where we omit the second repetition cat qubit for clarity), the first Hadamard of the first line and the preparation of $|+{⟩}_L$ are replaced by the equivalent preparation of $|0{⟩}_L$ and the second Hadamard is commuted through the $X_L$, producing a $Z_L$ gate. The remaining circuit in the dashed box is just a teleportation of the state $|\psi {⟩}_L$ of the second repetition cat qubit to the first one. After the state is teleported, the remaining Hadamard gate $H_L$ is applied, thus establishing the equivalence between circuit (a) and a logical Hadamard.

Figure 10

Process tomography of the cnot gate in the presence of noise. The cnot process presented in Sec. 4 is numerically simulated for $\overline{n}=7$ photon cat qubits using two different error models. First, we consider photon loss on both modes ${\kappa }_{1\mathrm{ph}}\mathcal{D}[\widehat{a}]+{\kappa }_{1\mathrm{ph}}\mathcal{D}[\widehat{b}]$ (a)–(c). Then, we consider a more elaborate error model including photon loss ${\kappa }_{1\mathrm{ph}}(1+n_{\mathrm{th}})\mathcal{D}[\widehat{a}]+{\kappa }_{1\mathrm{ph}}(1+n_{\mathrm{th}})\mathcal{D}[\widehat{b}]$, thermal excitations ${\kappa }_{1\mathrm{ph}}{n}_{\mathrm{th}}\mathcal{D}[{\widehat{a}}^{†}]+{\kappa }_{1\mathrm{ph}}{n}_{\mathrm{th}}\mathcal{D}[{\widehat{b}}^{†}]$ ($n_{\mathrm{th}}=10\%$) and dephasing on both modes ${\kappa }_{\varphi }\mathcal{D}[{\widehat{a}}^{†}\widehat{a}]+{\kappa }_{\varphi }\mathcal{D}[{\widehat{b}}^{†}\widehat{b}]$ (d)–(f). In both cases, we set ${\kappa }_{1\mathrm{ph}}/{\kappa }_{2\mathrm{ph}}={10}^{-3}$, and the gate time is chosen optimal $T^{*}=[2\overline{n}\sqrt{\pi {\kappa }_{1\mathrm{ph}}{\kappa }_{2\mathrm{ph}}}{]}^{-1}\approx 1.27{\kappa }_{2\mathrm{ph}}^{-1}$ (see the main text). We plot the real part of the process matrix $\chi$ (a),(d) and the real (b),(e) and imaginary (c),(f) parts of the error matrix ${\chi }^{\mathrm{err}}$. In the lower row (g)–(i), we check the validity of our theoretical error model for photon loss for various gate times and cat sizes. While the dots illustrate the simulation results where the full master equation in the presence of loss are considered, the plain lines correspond to the analytical formula provided in the main text. The blue dots correspond to the diagonal process matrix element corresponding to the $Z_1$, and the red dots correspond to the coinciding diagonal matrix elements corresponding to $Z_2$ and $Z_1{Z}_2$. The green dots correspond to the off-diagonal elements corresponding to the coherence between $Z_2$ and $Z_1{Z}_2$ errors. The pale magenta dots correspond to the off-diagonal elements corresponding to coherence between $Z_1$ and $I$; this coherence is due to high-order nonadiabatic effects and is not included in our theory. The black dots correspond to all remaining errors, including bit-flip-type ones. It is clear that such errors remain exponentially suppressed.

Figure 11

Proposal for an experimental implementation of bias-preserving cnot and Toffoli gates for cat qubits. (a) Setup for implementing a bias-preserving cnot gate. The cat qubits are encoded in high-$Q$ cylindrical postcavities (in blue, resonance frequencies ${\omega }_a$ and ${\omega }_b$). The two cavities are coupled via a $\mathsf{Y}$-shape transmon as in Ref. [58] to a low-$Q$ strip-line resonator (in red, resonance frequency ${\omega }_d$) playing the role of the dump mode. The system is driven with three microwave pumps at frequencies ${\omega }_1=2{\omega }_b-{\omega }_d$, ${\omega }_2=({\omega }_a-{\omega }_d)/2$, and ${\omega }_3={\omega }_d$. (b) A similar setup for implementing a bias-preserving Toffoli gate with three cat qubits encoded in high-$Q$ postcavities (frequencies ${\omega }_a$, ${\omega }_b$, and ${\omega }_c$) all coupled to a single strip-line resonator (frequency ${\omega }_d$). The system is driven with five microwave pumps at frequencies ${\omega }_1=2{\omega }_c-{\omega }_d$, ${\omega }_2={\omega }_a+{\omega }_b-{\omega }_d$, ${\omega }_3=({\omega }_a-{\omega }_d)/2$, ${\omega }_4=({\omega }_b-{\omega }_d)/2$, and ${\omega }_5={\omega }_d$.