Quantum Computing with Rotation-Symmetric Bosonic Codes

Bosonic rotation codes, introduced here, are a broad class of bosonic error-correcting codes based on phase-space rotation symmetry. We present a universal quantum computing scheme applicable to a subset of this class—number-phase codes—which includes the well-known cat and binomial codes, among many others. The entangling gate in our scheme is code agnostic and can be used to interface different rotation-symmetric encodings. In addition to a universal set of operations, we propose a teleportation-based error-correction scheme that allows recoveries to be tracked entirely in software. Focusing on cat and binomial codes as examples, we compute average gate fidelities for error correction under simultaneous loss and dephasing noise and show numerically that the error-correction scheme is close to optimal for error-free ancillae and ideal measurements. Finally, we present a scheme for fault-tolerant, universal quantum computing based on the concatenation of number-phase codes and Bacon-Shor subsystem codes.

Popular Summary

The fundamental challenge to realizing robust quantum technology is protecting quantum systems from noise. Bosonic error-correcting codes have recently emerged as a solution to this problem. In a bosonic error-correcting code, quantum information is encoded redundantly in collections of indistinguishable bosons. Large numbers of bosons can be trapped in a single quantum system, such as vibrations in a mechanical oscillator or photons trapped in a high-quality cavity, making this approach both practical and hardware efficient. In our work, we unify many previously studied bosonic codes under a common framework that allows for easy discovery and construction of new codes.

Our common framework is based on rotation symmetry, which can be harnessed for quantum computation. For any code that obeys rotation symmetry, we construct an explicit toolbox of quantum gates. To mitigate noise, we propose a scheme that detects and corrects errors during the computation, and we show numerically that this scheme performs nearly as well as the fundamental laws of quantum mechanics allow. Finally, we provide a detailed blueprint for combining bosonic codes with other error-correcting codes to construct a fault-tolerant quantum computer.

Our work allows for a large class of experimentally relevant bosonic codes to be studied in a fully fault-tolerant context for the first time. In future work, we will study how large the benefit is when using a bosonic code at the ground level in a fault-tolerant quantum computer and how different bosonic codes stack up against each other.

Figure 1

Graphical summary of several $N=4$ rotation codes: cat and squeezed cat (Appendix pp2-s1), binomial (Appendix pp2-s2), and Pegg-Barnett (Appendix pp2-s3). Logical code words are $+1$ eigenstates of the discrete rotation operator [Eq. (1)] and exhibit $N$-fold rotation symmetry. Top row: Ball-and-stick diagrams illustrating $|{+}_N⟩$ (orange) and $|{-}_N⟩$ (blue). Indicated on each is the primitive $|\mathrm{\Theta }⟩$ for the code. Bottom row: Wigner functions for the $|{+}_N⟩$ state $W_{|{+}_N⟩}(\alpha )$. Red (blue) is positive (negative), the color scale on each plot is different, and $Q=\frac{1}{2}(\alpha +{\alpha }^{*})$ and $I=(1/2i)(\alpha -{\alpha }^{*})$ are the real and imaginary parts, respectively, of $\alpha$.

Figure 2

Graphical summary of the Fock-space and phase-space structure of code words for an $N=2$ rotation code. (a) The computational-basis code words $|0_N⟩$ and $|1_N⟩$ have support on every $2kN$ and $(2k+1)N$ Fock state for $k=0,1,2,\dots$, respectively. Up to $N-1$, loss or gain errors can, in principle, be detected. (b) The dual code words $|{\pm }_N⟩$ are related by a rotation in phase space by $\pi /2$. Rotation errors small compared to $d_{\theta }=\pi /2$ are detectable with a code-dependent uncertainty.

Figure 3

Distinguishing the $|{\pm }_N⟩$ code words. By design, $|{-}_N⟩=e^{i(\pi /N)\widehat{n}}|{+}_N⟩$, such that distinguishing the code words can be viewed as a phase-estimation problem. The phase precision ${\mathrm{\Delta }}_N(\theta )$ depends on the measurement scheme, with a canonical scheme given by the POVM in Eq. (14). (a) Measuring $\theta$ to distinguish the code words for an $N=2$ code ($d_{\theta }=\pi /2$). If the measurement result falls in a white wedge, we declare “$+$,” and in a gray wedge, we declare “$-$.” (b) The same procedure for a code word that has undergone a rotation error $e^{i\theta \widehat{n}}|{+}_N⟩$. For large rotations $\theta \sim d_{\theta }/2$, the logical state is misidentified. If the noise model is biased in such a way that clockwise rotations are more likely than counterclockwise (or vice versa), the decoder should be biased accordingly by rotating the white and gray wedges.

Figure 4

(a) Magnitude of the Fock-state amplitudes for a $|{+}_N⟩$ state for $N=3$ cat, binomial, and Pegg-Barnett codes with parameters that give ${\overline{n}}_{\mathtt{code}}=19.5$ for each. The amplitudes are nonzero only on Fock states $|3k⟩$ for $k=0,1,2,\dots$. (b) The embedded phase uncertainty [Eq. (15)] for each family of $N=3$ codes as ${\overline{n}}_{\mathtt{code}}$ increases. For each code, a proper asymptotic limit, given in Table 1, yields no embedded phase uncertainty, ${\mathrm{\Delta }}_N(\theta )\to 0$. In the opposite regime, ${\overline{n}}_{\mathtt{code}}\to \frac{3}{2}$, all three codes limit to the 03 encoding. Note that $N=3$ codes with ${\overline{n}}_{\mathtt{code}}<\frac{3}{2}$ do not exist and that binomial and Pegg-Barnett codes are defined only at discrete excitation numbers; connecting lines are guides for the eye.

Figure 5

Measurement of $\widehat{n}\mathrm{mod}4$ using an ancilla prepared in (a) a coherent state and (b) a two-lobe cat state. Shown are the initial ancilla states (solid circle), and each circle is a potential rotation of the ancilla that depends on the data-rail state being measured. They are labeled by outcomes $\ell$ for a measurement of $\widehat{n}\mathrm{mod}4$. In both (a) and (b), the specific ancilla rotation shown, ${\theta }_{\mathrm{anc}}$, corresponds to an $\ell =1$ outcome indicating that the data rail is off the order-$N$ Fock grid.

Figure 6

Propagation of errors through gates. An error ${\widehat{E}}_k(\theta )$ at the input of a gate (left column) is propagated to new errors at the output (right column). The circuit identities hold up to an overall phase factor (see the text for the exact relations). The new error ${\widehat{E}}_0({\theta }_k)$ for ${\widehat{S}}_N$ and crot is a pure rotation error with ${\theta }_k=(\pi k/NM)$, with $N=M$ for ${\widehat{S}}_N$.

Figure 7

(a) Schematic illustration of telecorrection using number-phase codes, where ${\mathcal{M}}_X$ are phase measurements with outcomes $x_i$. The scheme is based on how errors spread through the crot gate. An arbitrary error ${\widehat{E}}_k({\theta }_a)$ on the order-$N$ data rail (b) induces a rotation of the order-$M$ upper ancilla rail by ${\theta }_b=(\pi k/NM)$ (c). Phase measurements on the data rail and the upper ancilla rail extract information about ${\theta }_a$ and ${\theta }_b\sim k$. By the end of the circuit, the logical information is teleported to the bottom order-$L$ ancilla rail (typically, $L=N$).

Figure 8

Average gate infidelity as a function of the average excitation number in the code, ${\overline{n}}_{\mathtt{code}}$ [Eq. (11)], for an $N=3$ cat code (top row) and binomial code (bottom row). We compare the theoretically optimal error-correction scheme found numerically (optimal) to our telecorrection scheme using pretty good measurements (pretty good) and canonical phase measurements (phase). The encoded data rail is subject to noise through evolution under the master equation in Eq. (49) with equal loss and dephasing strength, ${\kappa }_{\varphi }t=\kappa t$. Each column shows results for a different amount of total noise $\kappa t$ before error correction is performed. A code performs better than break even (the uncorrected trivial encoding) whenever the gate infidelity is below the dashed line (falls outside the shaded region).

Figure 9

Average gate infidelity as a function of noise strength $\kappa t$ for cat codes (top row) and binomial codes (bottom row). For each $N$ and each ${\kappa }_{\varphi }t=\kappa t$, the optimal average excitation number ${\overline{n}}_{\mathtt{code}}$ is used for each code. A code performs better than break even whenever the gate infidelity is below the dashed line (outside the shaded region). We show only results for $1-F\ge 2.5\times {10}^{-8}$, $\kappa t\ge 0.5\times {10}^{-3}$, and $N\le 4$ due to the prohibitively large Fock-space truncation needed for numerical simulations and numerical accuracy issues for very small infidelity.

Figure 10

Building blocks of the error-correction scheme from Ref. [36]. (a) The wires represent number-phase codes, and the ${\widehat{C}}_Z$ gates are crot gates. A ${\widehat{C}}_Z^{\otimes n}$ gate enacted between a repetition code block and a single ancilla acts as a logical ${\widehat{C}}_Z$. (b) The same circuit as in (a), where the thick wires represent length-$n$ repetition code blocks. (c),(d) By repeating a nondestructive measurement of ${\widehat{Z}}_{\mathcal{L}}\otimes {\widehat{Z}}_{\mathcal{L}}$ $r$ times, we get a robust ${\mathcal{M}}_{Z_{\mathcal{L}}{Z}_{\mathcal{L}}}$ gadget. (e) The error-correction gadget from Ref. [36]. Here, $|{+}_{\mathcal{L}}⟩=|{+}_N{⟩}^{\otimes n}$, and the ${\widehat{X}}_{\mathcal{L}}$ measurement is independent, destructive phase measurement of each mode in the repetition code block.

Figure 11

An extension of the error-correction scheme from Ref. [36] where each ancilla is encoded in the state $|{{+}_{\mathcal{L}}}^{\prime }⟩$ [Eq. (52)]. On the left, the rails represent number-phase codes, the ${\widehat{C}}_Z$ gates are crot gates, and the measurement is an independent destructive phase measurement of each mode in a block. The parity of the $n$ digitized measurement outcomes is computed to perform a measurement in the $|{{\pm }_{\mathcal{L}}}^{\prime }⟩$ basis. On the right, the thick lines represent repetition code blocks, and the gates are logical ${\widehat{C}}_Z$ between the code blocks. Error correction is otherwise performed exactly as in Fig. 10: The circuit is repeated $r$ times, and, finally, the upper data block is measured ${{\mathcal{M}}_X}^{\otimes n}$, with the lower data block prepared in $|{+}_N{⟩}^{\otimes n}$.

Figure 12

Comparison between the number-phase codes introduced in this work and GKP codes [3]. For simplicity, we consider the square-lattice GKP code. The cartoon shows code words for an $N=4$ number-phase code, with $d_n=4$ and $d_{\theta }=\pi /4$, and a GKP code, with $d_q=d_p=\sqrt{\pi }$. For each code type, there are unitary gates that are “natural” in the sense that they are generated by physically natural Hamiltonians and that they map small errors to small errors. To achieve universality, the unitary gates are supplemented with state preparation and Pauli measurements for both code types. Note that it was recently shown that the GKP magic state $|T_{\mathtt{gkp}}⟩$ can be distilled using only Gaussian operations [114]. The number-phase-code analogs to the envelopes in the approximate GKP code words are number-amplitude envelopes such as those in Fig. 4.

Figure 13

Code lattices ${\mathcal{L}}_0$ (filled circles) and ${\mathcal{L}}_1$ (open circles) in the complex plane for (a) square-lattice GKP and (b) hexagonal-lattice GKP. The area of the lattice parallelogram is $\pi /2$. The code space lattice $\mathcal{L}$ is also apparent in the Wigner function $W_{\mathtt{gkp}}(\alpha )$ of the completely mixed state $\frac{1}{2}{\widehat{\mathrm{\Pi }}}_{\mathtt{gkp}}$. Using quadrature axes $Q$ and $I$, as defined in Fig. 1, $W_{\mathtt{gkp}}(\alpha )$ is identical to the above plots with positive $\delta$ functions at each point in $\mathcal{L}$. For the Wigner functions, arrows indicate the direction of displacement for ${\widehat{X}}_{\mathtt{gkp}}$ and ${\widehat{Z}}_{\mathtt{gkp}}$.

Figure 14

A hybrid Steane-Knill error-correction scheme for rotation codes.