Assessing the Progress of Trapped-Ion Processors Towards Fault-Tolerant Quantum Computation

A quantitative assessment of the progress of small prototype quantum processors towards fault-tolerant quantum computation is a problem of current interest in experimental and theoretical quantum information science. We introduce a necessary and fair criterion for quantum error correction (QEC), which must be achieved in the development of these quantum processors before their sizes are sufficiently big to consider the well-known QEC threshold. We apply this criterion to benchmark the ongoing effort in implementing QEC with topological color codes using trapped-ion quantum processors and, more importantly, to guide the future hardware developments that will be required in order to demonstrate beneficial QEC with small topological quantum codes. In doing so, we present a thorough description of a realistic trapped-ion toolbox for QEC and a physically motivated error model that goes beyond standard simplifications in the QEC literature. We focus on laser-based quantum gates realized in two-species trapped-ion crystals in high-optical aperture segmented traps. Our large-scale numerical analysis shows that, with the foreseen technological improvements described here, this platform is a very promising candidate for fault-tolerant quantum computation.

Popular Summary

The advantages provided by exploiting the quantum-mechanical representation of information have motivated major investments in quantum computation, including those from many of the world’s largest companies. To date, various small-scale prototype quantum computers have been built, boosting expectations that systems capable of performing useful computations will be possible to construct. As prototype systems scale up to larger sizes, one must carefully consider and counteract the detrimental effects of noise and hardware errors. The framework of fault-tolerant quantum error correction (QEC) provides the foundational analytic tool to project current capabilities to large systems, with a central goal of achieving experimental noise and error levels below critical thresholds. We introduce new criteria to assess the progress of quantum-computing architectures and use these to study the near-term prospects of trapped-ion quantum-computing technology.

We present a versatile and realistic trapped-ion toolbox for QEC based on current and near-term projections of experimental capability, including high fidelity, dynamically protected gate operations, techniques for reconfiguring ion crystals, and fast classical control and electronic feed-forward. We derive detailed microscopic error models for the various elements of this toolbox, which feed into comprehensive numerical simulations of various QEC protocols.

Our investigations demonstrate that today’s state of the art—two-species ion crystals operated in segmented traps with high optical access—constitutes a viable platform for large-scale fault-tolerant quantum computation.

Figure 1

The Sandia HOA2 trap as a QEC platform: In our envisioned scheme, ${}^{40}{\mathrm{Ca}}^{+}$ ions (blue and red dots) are cotrapped with ${}^{88}{\mathrm{Sr}}^{+}$ ions (green dots) in a quantum zone divided into three storage regions $S_1$, $S_2$, $S_3$ and two manipulations zones $M_1$, $M_2$. Some of the ${}^{40}{\mathrm{Ca}}^{+}$ ions can be used as data qubits to encode quantum information according to a QEC code (blue dots), while others (red dots) can be used as ancilla qubits for syndrome extraction. The ${}^{88}{\mathrm{Sr}}^{+}$ ions (green dots) are used as sympathetic coolants to reduce the number of phonons prior to the entangling gates. Possible crystal-reconfiguration operations are shown in the panel in the lower-right corner: (a) splitting of an ion crystal, (b) shuttling of an ion and subsequent merging with another ion to form a crystal, and (c) rotation (swapping) of a mixed species crystal. Schematics of the trap are adapted from a micrograph in Ref. [21].

Figure 2

Cartoon illustration of the protocol for assessing the efficacy of our QEC cycle. Strictly speaking, Alice and Bob have ideal experimental equipment capable of encoding and decoding a quantum state perfectly, whereas only Igor has the imperfections of our real laboratory setting. The stages of the protocol are detailed in Table 1.

Figure 3

Scheme of the seven-qubit color code: One logical qubit is embedded in seven data qubits forming a 2D triangular color-code structure of three plaquettes. The code space is defined via $S_i^x$ and $S_i^z$ stabilizer operators (generators), each acting on a plaquette that involves four physical qubits. Logical operators such as $Z_L=\prod _i{Z}_i$, and similarly the other logical single-qubit Clifford-gate generators $X_L≔\prod _i{X}_i$, $H_L≔\prod _i{H}_i=\prod _i(1/\sqrt{2})(X_i+Z_i)$, and $S_L≔\prod _i{S}_i^{†}=\prod _i{e}^{-i(\pi /4)(1-Z_i)}$, can be realized in a transversal, i.e., bit-wise, way.

Figure 4

Error propagation through cnot and ms gates. (Left panel) An incoming Pauli error of the $X$ ($Z$) type at the control (target) qubit propagates onto a target (control) error of the same $X$ ($Z$) type after a cnot gate. Conversely, an incoming Pauli error of the $Z$ ($X$) type at the control (target) qubit does not propagate into the target (control) qubit. (Right panel) Error propagation through an ms gate up to phase factors irrelevant for fault-tolerant arguments. The left column corresponds to $XX$-type ms gates $X_{i,j}^2$, whereas the right column describes $YY$-type ms gates $Y_{i,j}^2$. Pauli errors that anticommute with the ms-gate basis are rotated and propagated to the other qubit. Pauli errors in the same ms-gate basis do not propagate.

Figure 5

Stabilizer readout based on multiqubit ms gates: The yellow box contains two five-ion ms gates, interspersed by a single-qubit rotation of the ancillary qubit around the $Z$ axis. This sequence realizes a coherent mapping of the $+1/-1$ eigenvalue information of $X$-type stabilizers [Eq. (23)] onto two orthogonal states of the ancilla qubit (red dot), initially prepared in a superposition state on the equator of the Bloch sphere. Note that the basis of the initial rotation $U_{R,\varphi }(\pi /2)$ required to prepare the ancilla qubit in this way, specified by the angle $\varphi$, can be arbitrary as long as the final rotation $U_{R,\varphi +\pi /2}(\pi /2)$ of the ancilla is shifted by $\pi /2$. In other words, the relative phase between these two single-qubit pulses on the ancilla needs to be well defined, but there does not need to be a fixed phase reference between the addressed laser, driving resonant single-qubit rotations, and the lasers driving global rotations and the entangling ms gate. For the readout of $Z$-type stabilizers [Eq. (23)], the data qubits must be rotated via $Y(\pm \pi /2)$ at the beginning and at the end of the mapping, to effectively switch between $X$- and $Z$-type stabilizers.

Figure 6

Stabilizer readout based on sequential two-qubit ms gates: $X$-type ($\alpha =x$) and $Z$-type ($\alpha =z$) stabilizer readout circuits, using sequential two-ion ms $X_{0,j}^2$ gates [Eq. (8)], depicted by solid lines with black dots between the single ancilla qubit and each of the data qubits involved in a particular stabilizer, and with an $XX$ label that defines the basis of the entangling gate [Eq. (4)]. The ms gates have to be combined with single-qubit rotations $X_j(\pi /2),Y_j(\pm \pi /2)$ on the data qubits [Eq, (25)] to achieve the stabilizer mapping [Eq. (26)]. The measurement of the ancillary qubit in the computational basis is denoted as $M_0^z$.

Figure 7

Non-fault tolerance of single-ancilla MS-based stabilizer readout: Illustration of an error event, in which a single phase-flip error $Z$ on the ancilla qubit occurs between the second and the third ms gates (a). This error propagates into the data qubit layer (b) after the third ms gate, where it results in two bit single-qubit flip errors on the data qubits (c). These two errors correspond ultimately to a logical bit-flip error (d) that cannot be corrected by the code.

Figure 8

MS-based circuit for preparation and verification of a GHZ state in the DiVincenzo-Shor scheme: The first three MS gates create a four-qubit GHZ state, which is, up to single-qubit $Z$ rotations, equivalent to the desired GHZ state $(1/\sqrt{2})(|x_{+},x_{+},x_{+},x_{+}⟩+|x_{-},x_{-},x_{-},x_{-}⟩)$. The state is verified by coupling it via two additional MS gates involving the first and the fourth ancilla qubit to a verification qubit. Note that the $Z$-type rotation on the ancilla verification qubit is incorporated to remove a relative phase in the GHZ state. Some parts of the circuit can be executed in parallel, such as part of the verification circuit, while the GHZ state is still being built up.

Figure 9

MS-based circuit for transversal coupling in the DiVincenzo-Shor scheme: The stabilizer $S_{\alpha }^{(n)}$ information is encoded in the relative phase of the ancilla GHZ state $(1/\sqrt{2})(|x_{+}{x}_{+}{x}_{+}{x}_{+}⟩\pm |x_{-}{x}_{-}{x}_{-}{x}_{-}⟩)$ by a sequence of two-ion ms gates and local rotations (25). This relative phase is revealed by the even or odd parity of the $Z$-basis measurements of the four ancilla qubits.

Figure 10

cnot-based circuit for DiVincenzo-Aliferis-type QEC: This circuit involves the preparation of an ancilla four-qubit GHZ state by cnot gates, its transversal coupling to the data qubit layer, the ancilla state decoding, and the measurements of the ancillas.

Figure 11

MS-based circuit for DiVincenzo-Aliferis-type QEC: Initially, a sequence of three two-ion ms gates and local $Z$ rotations is used to prepare the ancilla $|\mathrm{GHZ}⟩$ [see Eq. (28)]. This state is subsequently coupled transversally by four $X$-type ms gates to the data qubits, thereby mapping the $S_x$ stabilizer eigenvalue information onto the relative phase of the GHZ state. The decoding circuit, formed by local $Z$ rotations and two two-ion ms gates, is constructed in such a way that a harmful two-qubit error, which has propagated to the code layer, is unambiguously signaled by the $M_3^z=+1,M_4^z=-1$ measurement outcome of the third and fourth ancilla qubits. The measurement of $Z$-type stabilizers ($\alpha =z$) is almost identical, also introducing $Y$ rotations of the four data qubits before and after applying the four $X$-type ms gates of the coupling step.

Figure 12

Dangerous error propagation during the encoding step of the DiVincenzo-Aliferis-type scheme: Subfigures (a)–(d) illustrate how a single phase-flip error $Z_{a_3}$, occurring after the $YY$ ms gate of the encoding step, results in two phase-flip errors $Z_{a_3}{Z}_{a_4}$ that feed into the data qubits. Note that in the last step [from (c) to (d)], we have made use of the fact that a two-qubit $Y_{a_3}{Y}_{a_4}$ error is equivalent to $Z_{a_3}{Z}_{a_4}$. This can be seen, e.g., by recognizing that the $Y_{a_3}{Y}_{a_4}$ error corresponds to simultaneous bit- and phase-flip errors on both qubits, $Y_{a_3}{Y}_{a_4}\equiv Z_{a_3}{Z}_{a_4}{X}_{a_3}{X}_{a_4}$, and that $X_{a_3}{X}_{a_4}$ leaves the ancilla GHZ state $(1/\sqrt{2})(|x_{+}{x}_{+}{x}_{+}{x}_{+}⟩+|x_{-}{x}_{-}{x}_{-}{x}_{-}⟩)$ invariant.

Figure 13

Dangerous error propagation during the coupling step of the DiVincenzo-Aliferis-type scheme: The circuits show how an incoming $Z_{a_3}{Z}_{a_4}$ error, resulting, e.g., from a single phase-flip error in the encoding step in Fig. 12, is converted into a pair of $Y$-type errors on the ancilla qubits and furthermore propagate into the data qubit layer, where they result in two bit-flip $X_{j_3}{X}_{j_4}$ errors.

Figure 14

Detection of the dangerous two-qubit errors in the DiVincenzo-Aliferis-type scheme: The circuits show how an incoming $Y_{a_3}{Y}_{a_4}$ error, resulting from the transversal-coupling step in Fig. 13, results in bit flips ($Y_{a_3}$ and $X_{a_4}$) on the third and fourth ancilla qubits. However, in this case, the third and fourth qubits end up in $|{0⟩}_{a_3}$ and $|{1⟩}_{a_4}$, respectively, which results in an $M_3^z=+1,M_4^z=-1$ measurement outcome.

Figure 15

Real-space representation of shuttling-based one-species QEC cycle with multiqubit ms gates: Data and ancillary ions are represented by blue and red dots, respectively, and are distributed within the storage and manipulation regions of the first row. The black arrows pointing to the storage zone represent splitting operations followed by a shuttling of a subset of physical ions onto the processing zone, after which they all merge with the static ancillary ion. The black arrows towards the storage zones represent splitting operations, followed by a shuttling of the physical ions back to the storage zone, where they are merged with any physical ions that might already be present there. The blue arrows within the storage zone represent crystal rotations that reorder the physical ions. On the right column, we specify the times where the stabilizer mappings Map $S_{\alpha }^{(m)}$, implemented by multiqubit entangling gates and represented by red ellipses, and ancillary measurements or reset (Meas or reset), represented by a black detector, are applied.

Figure 16

Circuit representation of shuttling-based one-species QEC cycle with multiqubit ms gates: The data $\{d_1,\dots ,d_7\}$ and ancillary $a_0$ ions are arranged vertically, and a set of boxes represents the elementary operations taking place at a particular time step: “Sh.” stands for shuttling of the ions within the green boxes, “M.” stands for the merging of the two sets of ions separated by a dotted line within the blue boxes, “Sp.” stands for splitting of the crystal into two sets of ions separated by a dotted line within the yellow boxes, “Meas.” stands for the ancillary ion measurement in the orange boxes, and “Rot.” stands for the rotation of the ion crystal within the purple boxes, with arrows representing the corresponding crystal reordering. Note that some crystal-reconfiguration operations, such as merging or splitting and shuttling, can be operated simultaneously in different trap zones. Finally, Map $S_{\alpha }^{(m)}$ stands for the mapping of a particular stabilizer $S_{\alpha }^{(m)}$ involving the data and ancillary ions within the red boxes. These red boxes contain the sequence of quantum gates in Fig. 5.

Figure 17

Quantum-channel representation of shuttling-based one-species QEC cycle with multiqubit ms gates: The ion reconfiguration steps of Fig. 16 lead to an increase of the phonon populations and a dephasing of the idle qubits during the time required for these reconfigurations to take place. Additionally, idle ions also dephase during the time lapse of the stabilizer mapping. These time intervals are $T_{\mathrm{sh}}$ for shuttling, $T_M$ for merging or splitting, $T_{\text{Map}}$ for stabilizer mapping, $T_{\mathrm{meas}}$ for measuring, and $T_{\mathrm{rot}}$ for rotations. The dephasing channel applied during a certain period is depicted by light grey boxes, with the channel ${ϵ}_{\mathrm{d}}(n_{\mathrm{sh}}{T}_{\mathrm{sh}}+n_M{T}_M+n_{\mathrm{Map}}{T}_{\mathrm{Map}}+n_{\mathrm{meas}}{T}_{\mathrm{meas}}+n_{\mathrm{rot}}{T}_{\mathrm{rot}})$ acting on a particular qubit, where $n_o$ is the number of operations of the type $o$ that occur within that period. The actual channel corresponds to Eq. (13) with a probability $p_d=\sum _o{n}_o{T}_o/2T_2$, with $T_2$ being the coherence time of the qubits. The ancilla readout and reset measure is modeled by a bit-flip channel (22) acting on the ancillary qubit ${ϵ}_b$ with a probability given by the sum of the measurement and reset errors $p_b={ϵ}_{\mathrm{meas}}+{ϵ}_{\mathrm{res}}$, as reported in Table 2, and represented by dark grey boxes after each measurement. Finally, map is modeled by an ideal stabilizer mapping acting on the particular set of qubits, followed by a depolarizing channel ${ϵ}_{\mathrm{MS}}$ (i.e., grey box) given by one of the three channels of Eqs. (14)–(16) with an error probability $p_d={ϵ}_m+{ϵ}_d+{ϵ}_I$ that depends on the current phonon number through Eq. (17), and the gate time via Eqs. (19) and (20).

Figure 18

Real-space representation of shuttling-based, two-species QEC cycle with multi-qubit ms gates: We use the same conventions as in Fig. 15. As announced in the main text, the number of crystal reconfigurations is reduced with respect to the shuttling-based one-species protocol. In the rightmost column, we specify the times when crystal rotations, stabilizer mappings, ancillary measurements, and the new sympathetic (Recooling), represented by blue rectangles, are applied.

Figure 19

Real-space representation of shuttling-based two-species QEC cycle with two-qubit ms gates: We represent half of a cycle of QEC for the sequential measurement of $\{S_{\alpha }^{(1)},S_{\alpha }^{(2)},S_{\alpha }^{(3)}\}$, either for $\alpha =x$ or for $z$, and use the same conventions as in Figs. 15 and 18, placing the extra cooling ion (green circle) in the manipulation zone $M_2$. In the rightmost column, we specify the time steps when crystal-reconfiguration operations, sympathetic recooling, and ancilla measurement/reset take place. The new mapping functions, represented by red ellipses, consist of the data-ancilla conditional gate $\mathrm{Map}(a0,dj)=U_x^{(a_0,d_j)}$ for a $X$-type stabilizer or its combination with local rotations $\mathrm{Map}(a0,dj)=Y_{d_j}(+\pi /2)U_x^{(a_0,d_j)}{Y}_{d_j}(-\pi /2)$ for a $Z$-type stabilizer. We recall that these local rotations can be obtained from the available set of gates in Eqs. (5) and (6) by simple refocusing sequences $Y_{d_j}(\theta )=U_{\mathrm{R},\pi /2}(\theta /2)U_{{\mathrm{R}}_{d_j},z}(\pi )U_{\mathrm{R},\pi /2}(-\theta /2)U_{{\mathrm{R}}_{d_j},z}(-\pi )$.

Figure 20

Circuit representation of the hiding-based, two-species QEC cycle with multiqubit ms gates: We use the same conventions as in Fig. 16. The additional hiding $h$ and unhiding operations $h^{-1}$, represented by green boxes, are realized by composite pulse sequences to coherently map the state of physical qubits from $4S_{1/2}(m_f=-1/2)$ and $3D_{5/2}(m_f=-1/2)$ to a set of storage $D$ states (see Sec. 3c).

Figure 21

Real-space representation of the shuttling-based, two-species QEC cycle with a DiVincenzo-Shor scheme: We represent the sequence of operations for the fault-tolerant DiVincenzo-Shor readout of a single stabilizer operator $S_{\alpha }^{(1)}$ for $\alpha =x$ or $z$ in two panels, and we use the same conventions as in Figs. 15, 18, and 19. The extra cooling ions c1, c2 are placed in the manipulation zones $M_1$, $M_2$, and are again depicted by a green circle. The rightmost columns of both panels contain the sequence of operations that take place. In addition to the ones described in previous figures, we include $X$- and $Y$-type ms gates $XX(i,j)$ and $YY(i,j)$ depicted by red ellipses, as well as single-qubit rotations $X(j)$, $X\_\mathrm{inv}(j)$ corresponding to $X_j(\pm \pi /2)$ (7), and analogously for $Y(j),Y\_\mathrm{inv}(j)$, and $Z(j),Z\_\mathrm{inv}(j)$, all of which are depicted by yellow ellipses. We also introduce $Z2(j)$, which corresponds to $Z_j(\pi )$. Finally, some $Y(j),Y\_\mathrm{inv}(j)$ rotations on the rightmost columns of each panel are shown inside a yellow rectangle, which implies that they are only applied for a $Z$-type stabilizer readout ($\alpha =z$). We have also included a classical relabeling operation, Relabel, conditional on the measurement outcome $M_0^z$.

Figure 22

Real-space representation of the shuttling-based, two-species QEC cycle with a DiVincenzo-Aliferis scheme: We represent the operations for the fault-tolerant DiVincenzo-Aliferis readout of a single stabilizer operator $S_{\alpha }^{(1)}$, either for $\alpha =x$ or $z$, in two panels, and we use the same conventions as in Figs. 15, 18, 19, and 21.

Figure 23

Success probability ${\mathcal{P}}_{\mathrm{B}}$ under the hiding-based, two-species protocol QEC cycle with multiqubit ms gates (cf. Fig. 20): The parameters underlying the simulation correspond to the current values from Tables 2, 3, 4, 5. Here and in the following figures, the $x$ axis gives time in units of the environmental dephasing time $T_2$. The noise model for the imperfect five-ion ms gate operations corresponds to the quantum channel (i) of independent depolarizing noise [Eq. (14)]. Results show that even for this optimistic noise model, there is no time window in which the application of an imperfect QEC cycle is beneficial as compared to not applying it [i.e., the regime in Eq. (2) is never attained]. The underlying reason for this is that the current five-ion ms gate fidelity is insufficient to reach the crossover point. Although not shown in the figure, we note that similar results are found for the shuttling-based protocol 2, for which reaching the crossover point with current parameters is not possible. Here and elsewhere, each random qubit selected for Alice to encode is $U|0⟩$, where unitary $U$ is formed by selecting three angles ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$ uniformly from 0 to $2\pi$ and setting $U=\cos {\varphi }_1(\cos {\varphi }_{2}I+i\sin {\varphi }_{2}Z)+i\sin {\varphi }_1(\cos {\varphi }_{3}Y+\sin {\varphi }_{3}X)$, where $I$ is the identity and $X$, $Y$, $Z$ are the Pauli operators.

Figure 24

Success probability ${\mathcal{P}}_{\mathrm{B}}$ under the single-species, shuttling-based QEC cycle with multiqubit ms gates (cf. Fig. 17): The parameters underlying the simulation correspond to the anticipated improved values from Tables 2, 3, 4, 5. We use two noise models for the imperfect five-ion ms gate operations involved in the stabilizer mappings: the optimistic model (i) of independent depolarizing channels [Eq. (14)] and the pessimistic model (iii) with a multiqubit depolarizing channel [Eq. (16)]. When we adopt the optimistic noise model and we employ Igor (purple curve), then for $t/T_2>0.3$ there exists a small advantage as compared to not using Igor to correct the logical qubit (red curve). Thus, Eq. (2) is fulfilled. However, when multiple qubit errors are fully enabled by the noise model, and we use the multiqubit depolarizing channel (“Igor,” blue data points), the advantage disappears. This highlights the importance of modeling correlated errors appropriately, thus going beyond simplified error models that use the same single-qubit channel after each elementary operation in the quantum circuit. For reference, the behavior of an unencoded, bare physical qubit under the same environmental (dephasing) noise is also shown (grey data points). The inset shows a zoom of the parameter interval in which QEC becomes advantageous: For a total waiting time $\tau$ larger than about 300 ms, it becomes advantageous to apply an imperfect Igor QEC cycle but only for independent depolarizing noise. In this figure and those following, the inset shares the same axes labels as the main plot.

Figure 25

Success probability ${\mathcal{P}}_{\mathrm{B}}$ under the two-species, shuttling-based QEC cycle with multiqubit ms gates (cf. Fig. 18): The parameters underlying the simulation correspond to the anticipated improved values from Tables 2, 3, 4, 5. The noise model for the imperfect five-ion ms gate operations corresponds to the worst-case noise model (iii) of multiqubit depolarizing noise [Eq. (16)]. Results show that there exists an ample parameter region (at times larger than about 100 ms) in which the application of an imperfect Igor QEC cycle becomes advantageous [Eq. (2)] as compared to not applying it. Note that this takes place at ${\mathcal{P}}_{\mathrm{B}}$ values of 0.981, much higher than in the shuttling-based scenario 1, with a marginal gain at ${\mathcal{P}}_{\mathrm{B}}$ values of about 0.92. Note that in the present scheme, for not-too-long total times $\tau$, below about 200 ms, applying an imperfect QEC cycle is advantageous even as compared to a single, nonencoded physical qubit undergoing dephasing noise of the same strength, such that the more-stringent regime [Eq. (3)] for beneficial QEC can also be achieved.

Figure 26

Success probability ${\mathcal{P}}_{\mathrm{B}}$ under repetitive, two-species, shuttling-based QEC cycles with multiqubit ms gates (cf. Fig. 18): We consider the same scenario as in Fig. 25 with the anticipated improved values of Tables 2, 3, 4, 5, but the model for the imperfect five-ion ms gate operations corresponds to the physically motivated noise model (ii) of one- and two-qubit depolarizing quantum channels [Eq. (15)]. For a single QEC cycle, direct comparison to the results of Fig. 25 shows no appreciable difference. Hence, we can conclude that using the more-pessimistic noise model with a multiqubit depolarizing channel [Eq. (16)] or using the one with equally likely one- and two-qubit errors [Eq. (15)] does not make any difference. We also depict the results for two rounds of QEC (green dots), where one sees an increase of the region [Eq. (3)] where QEC is advantageous compared to a single nonencoded physical qubit with respect to the case with a single QEC round. The inset plots $\mathrm{\Delta }$, which is ${\mathcal{P}}_{\mathrm{B}}$ relative to that for the single-qubit memory. We see that multiple rounds of QEC allow us to sustain the logical qubit for a longer period of time. In the inset, we display the relative performance $\mathrm{\Delta }$ of the encoded (red), single-cycle QEC (blue), and two-cycle QEC (green) with respect to the unprotected physical qubit, which is obtained by subtracting the bare qubit ${\mathcal{P}}_{\mathrm{B}}$ of the main panel (grey line), from ${\mathcal{P}}_{\mathrm{B}}$ for the different schemes, also in the main panel. As emphasized before, we observe a wider region of advantage for the two cycles of QEC.

Figure 27

Success probability ${\mathcal{P}}_{\mathrm{B}}$ under the hiding-based, two-species QEC cycle with multiqubit ms gates (cf. Fig. 20): The parameters underlying the simulation correspond to the future improved values from Tables 2, 3, 4, 5. Experimental capabilities (two-species, cooling, etc.) are the same as in Fig. 23, and we use the more challenging (worst-case) noise model (iii) of multiqubit depolarizing noise [Eq. (16)]. Results show that there exists a clear parameter window for which the application of an imperfect Igor QEC cycle becomes advantageous both as compared to not applying it [Eq. (2)] and compared to an unprotected single physical qubit [Eq. (3)]. Note that the ${\mathcal{P}}_{\mathrm{B}}$ value where the QEC cycle crossover towards a beneficial Igor takes place is around 0.982, which is very similar to the behavior found for the shuttling-based protocol 2 with future parameters (cf. Fig. 25).

Figure 28

Success probability ${\mathcal{P}}_{\mathrm{B}}$ under two-species, shuttling-based QEC cycles with only QEC errors: This graph shows how ${\mathcal{P}}_{\mathrm{B}}({\rho }_L,{\rho }_{\mathrm{QEC}})$ changes with parameter $\lambda$, which is defined in the main text and adjusts error rates within Igor’s cycle. The three plots show a non-fault-tolerant, two-species, shuttling-based scheme based on two-qubit ms gates (cf. Fig. 19), and the fault-tolerant DV-S scheme (cf. Fig. 21), and the fault-tolerant DV-A scheme (cf. Fig. 22). As described in the text, the inverted quadratic curves, as compared to the linear behavior, are the signature of a correct fault-tolerant circuit. The simulation parameters correspond to the future improved values from Tables 2, 3, 4, 5. The noise applied for the imperfect two-ion ms gate operations follows the standard depolarizing model.

Figure 29

Today’s hardware: Success probability ${\mathcal{P}}_{\mathrm{B}}$ with QEC according to the shuttling-based, two-species protocol. The standard Alice-Igor-Bob protocol with the parameters underlying the simulation corresponding to the current values from Tables 2, 3, 4, 5. The red curve shown here is Bob’s performance in the absence of Igor, such that the occurring errors only come form the environment (see Table 1). For reference, the equivalent plot for a single physical qubit with the same environmental noise is also shown (grey curve). We see that using a logical qubit and the non-FT QEC cycle (yellow) can produce a small positive effect; the shaded area indicates this beneficial region. However, the fully FT protocol (purple line), when it is possible (see main text), is always inferior.

Figure 30

Future hardware: Success probability ${\mathcal{P}}_{\mathrm{B}}$ with QEC according to the shuttling-based, two-species protocol. The scenario simulated in this case is the same as in Fig. 29, except that the parameters applied correspond to the future improved values from Tables 2, 3, 4, 5. Performance is obviously profoundly improved, as discussed in the main text.

Figure 31

Inferior future hardware: Success probability ${\mathcal{P}}_{\mathrm{B}}$ with QEC according to the shuttling-based, two-species protocol. The error model and scenarios simulated here are the same as in Fig. 30. The operational numbers are scaled to yield a performance that is 3 times worse than those of the future improved values: Each operation takes 3 times longer, and the gate fidelities are 3 times worse (the environmental dephasing rate remains unchanged). As discussed in the main text, while the performance here is degraded vs the preceding figure, it is nevertheless sufficient to largely demonstrate the goal of beneficial QEC.

Figure 32

Multiple cycles of future hardware: Success probability ${\mathcal{P}}_{\mathrm{B}}$ with QEC according to the shuttling-based, two-species protocol. The parameters underlying the simulation correspond to the future improved values from Tables 2, 3, 4, 5. The QEC method used here is the non-fault-tolerant one (cf. Fig. 19), as it takes the shortest possible time and thus has more potential if more error-correction cycles are to be applied within one round. Results show that the application of more QEC cycles sustains the logical qubit for a longer time, as depicted by the grey dashed line, which is drawn based on the envelope of the different curves. As discussed in the main text, the dashed line allows us to infer the effective rate of decay of integrity of the logical qubit.