New Class of Quantum Error-Correcting Codes for a Bosonic Mode

We construct a new class of quantum error-correcting codes for a bosonic mode, which are advantageous for applications in quantum memories, communication, and scalable computation. These “binomial quantum codes” are formed from a finite superposition of Fock states weighted with binomial coefficients. The binomial codes can exactly correct errors that are polynomial up to a specific degree in bosonic creation and annihilation operators, including amplitude damping and displacement noise as well as boson addition and dephasing errors. For realistic continuous-time dissipative evolution, the codes can perform approximate quantum error correction to any given order in the time step between error detection measurements. We present an explicit approximate quantum error recovery operation based on projective measurements and unitary operations. The binomial codes are tailored for detecting boson loss and gain errors by means of measurements of the generalized number parity. We discuss optimization of the binomial codes and demonstrate that by relaxing the parity structure, codes with even lower unrecoverable error rates can be achieved. The binomial codes are related to existing two-mode bosonic codes, but offer the advantage of requiring only a single bosonic mode to correct amplitude damping as well as the ability to correct other errors. Our codes are similar in spirit to “cat codes” based on superpositions of the coherent states but offer several advantages such as smaller mean boson number, exact rather than approximate orthonormality of the code words, and an explicit unitary operation for repumping energy into the bosonic mode. The binomial quantum codes are realizable with current superconducting circuit technology, and they should prove useful in other quantum technologies, including bosonic quantum memories, photonic quantum communication, and optical-to-microwave up- and down-conversion.

Popular Summary

A functioning quantum computer must be capable of maintaining coherent quantum superpositions of the 0 and 1 states of its quantum bits until an algorithm has completed. However, interactions of the computer with its environment introduce errors that destroy these superpositions. Quantum error-correction codes exist to protect against these errors, and a typical approach distributes a single “logical” quantum bit nonlocally across a large set of physical quantum bits. This approach is powerful, but its realization is a daunting engineering challenge. Here, we explore an alternative approach that relies on embedding an error-resistant logical quantum bit in photon states of an electromagnetic cavity to correct photon loss, gain, and dephasing errors.

Current technology has made remarkable progress in developing superconducting cavities that are controllable on the quantum level, with lifetimes longer than the best corresponding physical quantum bits. As a result, cavities have become very appealing as long-lived quantum memories. The remaining limiting factor is the loss of energy due to emitted radiation and errors introduced by imperfect quantum control. In this work, we address these issues using a novel quantum error-correction process for a logical quantum bit stored in a superposition of different numbers of photons. Importantly, our codes have a lower mean number of photons and, consequently, a significantly lower rate of uncorrectable errors than earlier codes. Our codes can protect against a wide range of errors, but we tailor them to be experimentally advantageous for the detection of, and recovery from, the dominant errors.

We expect that our codes may have immediate applications in improving the fidelity of quantum memories and communication, which represents a fundamental step toward achieving scalable quantum computation.

Figure 1

Two-parameter $(N,S)$ space of the binomial codes (7). The largest blue circle denotes the code (2) protected against a photon loss error $L=1$, the blue square is the code (3) protected against ${\overline{\mathcal{E}}}_2=\{\widehat{I},\widehat{a},{\widehat{a}}^2,\widehat{n}\}$ or ${\overline{\mathcal{E}}}_2^{\prime }=\{\widehat{I},\widehat{a},{\widehat{a}}^{†},\widehat{n}\}$, and the green diamond denotes the code (5) protected against ${\overline{\mathcal{E}}}_1^{\prime }=\{\widehat{I},\widehat{a},\widehat{n}\}$. The parameter $S=L+G$ sets the total number of detectable photon loss $L$ and gain $G$ errors. The parameter $N$ sets the maximum order the code is protected against photon loss, gain, and dephasing errors, $N=\max \{L,G,2D\}$. The codes denoted with blue have protection against photon loss and gain errors set by $S=L+G$, and in addition, they are protected against dephasing up to ${\widehat{n}}^{\lfloor \max \{L,G\}/2\rfloor }$. The codes denoted with red allow, in addition, heralding of $S-2N$ uncorrectable photon loss or gain errors. The codes denoted with green are protected against a total of $S$ photon loss and gain errors, as well as against up to ${\widehat{n}}^{N/2}$ dephasing errors. In the limit $N\to \infty$, the binomial codes asymptotically approach the $2(L+1)$-legged cat codes [17, 18, 19] protected against $L$ photon loss errors.

Figure 2

The rate of entanglement infidelity ${\tilde{F}}_{\mathrm{e}}/\delta t$ for a fully mixed logical qubit state plotted as a function of the time step $\delta t$ (notice the logarithmic scale of both axes) for the binomial codes (7) with $S=N=L=1$ (black), 2 (gray), 3 (brown), 4 (red), and 5 (orange). Here, we have assumed a perfectly faithful recovery process. The dashed line shows the performance of the naive encoding $L=0$, $|W_{\uparrow /\downarrow }^{\mathrm{i}}⟩=|0/1⟩$, whose rate of entanglement infidelity at small $\delta t$ approaches $\kappa /2$ corresponding to the rate of a photon loss with $\overline{n}=1/2$. Entanglement infidelity [92] is calculated as ${\tilde{F}}_{\mathrm{e}}=1-F_{\mathrm{e}}=1-\sum _{k=0}^L\sum _{\ell =0}^{\infty }|\mathrm{Tr}({\widehat{R}}_k{\widehat{E}}_{\ell }{\widehat{\rho }}_{\mathrm{c}}){|}^2$, where ${\widehat{\rho }}_{\mathrm{c}}=\frac{1}{2}\sum _{\sigma }|W_{\sigma }⟩⟨W_{\sigma }|$ is the fully mixed state of the logical code words. The entanglement infidelity for a fully mixed state is equal to the process infidelity $1-{\chi }_{II}$ of a quantum memory. For an ideal quantum memory, ${\chi }_{II}=1$ and the full quantum process is just an identity operation [64]. At a small time step $\delta t$, the slopes of ${\tilde{F}}_{\mathrm{e}}/\delta t$ agree well with the slopes for the rate of the leading uncorrectable error $P_{L+1}/\delta t$. The binomial code with $L=1$ outperforms the naive encoding for time steps $\delta t\lesssim 0.4{\kappa }^{-1}$, and the codes with $L>1$ become favorable when $\delta t\lesssim 0.2{\kappa }^{-1}$.

Figure 3

(a) Sketch of a circuit QED hardware proposal and (b) schematic of a quantum-state transfer scenario utilizing quantum error correction for the binomial quantum code words. First, the state of qubit A is encoded in the send cavity S using code words $|W_{\uparrow ,\downarrow }^{\mathrm{S}}⟩$. The state in S is allowed to leak into the flying oscillator mode F (pitch). By controlling the cavity decay rate, one can precisely tune a time-reversal symmetric temporal mode such that the quantum information is fully absorbed in the receiving cavity R (catch). Before decoding the received cavity state to the qubit B, the fidelity of the transfer process can be improved by a round of QEC in the received cavity state.

Figure 4

Sketch of a two-cavity configuration with a dispersively coupled common transmon qubit [43], which is sufficient for realizing the two-mode codes. Each of the elements has a separate drive, denoted by ${ϵ}_j$ and $\overrightarrow{n}$, respectively. Alternatively, one could use two distinct modes of the same cavity.