Hardware-Efficient Autonomous Quantum Memory Protection

We propose to encode a quantum bit of information in a superposition of coherent states of an oscillator, with four different phases. Our encoding in a single cavity mode, together with a protection protocol, significantly reduces the error rate due to photon loss. This protection is ensured by an efficient quantum error correction scheme employing the nonlinearity provided by a single physical qubit coupled to the cavity. We describe in detail how to implement these operations in a circuit quantum electrodynamics system. This proposal directly addresses the task of building a hardware-efficient quantum memory and can lead to important shortcuts in quantum computing architectures.

Figure 1

Our AQEC scheme is composed of three operations: encoding, correcting, and decoding. The joint cavity-qubit state is represented by a generalized Fresnel diagram. Fresnel plane positions carry the description of the cavity mode, while colors carry the description of the qubit. Our protocol only requires that we represent superpositions of coherent states entangled with the qubit degrees of freedom. A circle whose center is positioned at $\alpha$ in the diagram corresponds to a coherent state component of amplitude $\alpha$. For example, the diagram in frame 2 represents the state $c_g|C_{\alpha }^{+}⟩+c_e|C_{i\alpha }^{+}⟩=\mathcal{N}(c_g|g,\alpha ⟩+c_g|g,-\alpha ⟩+c_e|g,i\alpha ⟩+c_e|g,-i\alpha ⟩)$, where $\mathcal{N}\approx 1/\sqrt{2}$ is a normalization factor. Each component of this state corresponds to a circle whose color refers to whether the qubit is in $|g⟩$ (blue areas) or $|e⟩$ (red areas). The rim of each circle indicates whether the prefactor is $c_g$ (single line) or $c_e$ (double line). Finally, the fraction of the colored disk represents the total weight $|\mathcal{N}{c}_{g,e}{|}^2$ of each coherent component. Here, quarter filled circles correspond to $|\mathcal{N}{c}_{g,e}{|}^2=1/4$. Initially (frame 1), the qubit is in $c_g|g⟩+c_e|e⟩$ and the cavity is in vacuum. The plus (respectively, minus) sign in the 2 and 3 diagrams indicates whether the logical qubit is encoded in an even (respectively, odd) cavity state. A jump from a plus to a minus sign is induced by a photon loss error, which we aim to correct.

Figure 2

Sequence of operations which generate ${\mathcal{U}}_{\mathrm{encode}}$. See the caption of Fig. 1 for an explanation of the diagram notation. The symbol given in the $k$th frame corresponds to the operation performed to go from frame $k-1$ to $k$. The curved arrow corresponds to the rotation of the excited state component of the state. We denote $\beta =\alpha (-1+i)$ and $\overline{n}=|\alpha {|}^2$. The frames are ordered from left to right and top to bottom.

Figure 3

(a)–(c) Full correcting sequence obtained by concatenating the three sequences of pulses. (a) The entropy is transferred from the cavity to the qubit. (b) First, the qubit is reset to its ground state and then energy is repumped into the coherent component to compensate the deterministic decay due to damping during the waiting time $T_w$ between two correction sequences. (c) The cavity state is mapped back onto the initial cavity logical 0 and logical 1. See the caption of Fig. 1 for an explanation of the diagram notation. Here, we denote ${\alpha }^{\prime }=e^{-\kappa T_w/2}\alpha$ the damped amplitude after the waiting time $T_w$, ${\overline{n}}^{\prime }=|{\alpha }^{\prime }{|}^2$, and ${\beta }^{\prime }={\alpha }^{\prime }(i-1)$. In order to compensate the damping during $T_w$, during the repumping step, we take ${\beta }_d^{\prime }=({\beta }^{\prime }-\beta )/2$. In the last frame of (a), the error is encoded in the phase of the qubit superposition and is not represented in this diagram. After qubit reset [first frame of (b)], this phase information is erased.

Figure 4

Diagram of our proposal to store a quantum bit of information in a single cavity mode. The information is encoded in a superposition of cat states, represented by the $I$-$Q$ diagram. The cross symbolizes a superconducting qubit. The wavy line represents the readout resonator which measures and resets the ancilla qubit state. This hardware-efficient quantum memory could significantly reduce the complexity of quantum computing architectures.