Transversality and lattice surgery: Exploring realistic routes toward coupled logical qubits with trapped-ion quantum processors

M. Gutiérrez, M. Müller, and A. Bermúdez

Phys. Rev. A 99, 022330 – Published 25 February 2019

Abstract

Active quantum error correction has been identified as a crucial ingredient of future quantum computers, motivating the recent experimental efforts to encode logical quantum bits using small topological codes. In addition to the demonstration of the beneficial role of the encoding, a break-even point in the progress towards large-scale quantum computers will be the implementation of a universal set of gates. This midterm challenge will soon be faced by various quantum technologies, which urges the need of realistic assessments of their prospects. In this work, we pursue this goal by assessing the capability of current trapped-ion architectures in facing one of the most demanding parts of this quest: the implementation of an entangling controlled-NOT (CNOT) gate between encoded logical qubits. We present a detailed comparative study of two alternative strategies for trapped-ion topological color codes, either a transversal or a lattice-surgery approach, characterized by a detailed microscopic modeling of both current technological capabilities and experimental sources of noise afflicting the different operations. Our careful fault-tolerant design, together with a low-resource optimization, allows us to determine via exhaustive numerical simulations the experimental regimes where each of the approaches becomes favorable. We hope that our study thereby contributes to guiding the future development of trapped-ion quantum computers.

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Received 11 February 2018

DOI:https://doi.org/10.1103/PhysRevA.99.022330

Physics Subject Headings (PhySH)

Cooling & trapping Quantum error correction Quantum gates Quantum information processing

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

M. Gutiérrez^1,2, M. Müller¹, and A. Bermúdez^1,3,4

¹Department of Physics, College of Science, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom
²Escuela de Química, Universidad de Costa Rica, San José, 2060 Costa Rica
³Instituto de Física Fundamental, IFF-CSIC, Madrid E-28006, Spain
⁴Departamento de Física Teórica, Universidad Complutense de Madrid, 28240 Madrid, Spain

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Vol. 99, Iss. 2 — February 2019

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Figure 1
Schematics of high-optical access (HOA) segmented ion traps for topological QEC: Microfabricated segmented traps connected through $Y$ junctions forming a honeycomb lattice structure to manipulate mixed-species ion register for quantum information processing. We note that the particular honeycomb tiling is a visual simplification of a larger-scale design, which will certainly require modifications to exploit the high-optical access. (Inset) Each arm of the trap corresponds to an HOA-2 Sandia trap [66], which consists of a slotted linear section with individually controllable electrodes used for crystal reconfiguration operations among three storage (S1,S2,S3) and two manipulation (M1,M2) regions. These operations are schematically shown by black arrows with the following letters: (r) rotation of an ion crystal, (sh) shuttling of an ion(s), (s) splitting an ion crystal, and (m) merging of sets of ions into a single crystal. The $Y$ junctions contain additional electrodes that can be used to (j) shuttle ion(s) between neighboring central regions. Additionally, in the manipulation zones, laser beams can propagate across the surface, or normal to it via the slotted region, and can be focused onto the ions to manipulate their electronic and motional degrees of freedom via (1) single-qubit gates, (2) two-qubit (multiqubit) entangling gates, and (c) sympathetic laser cooling of the ion crystal. These operations form the toolbox for scalable QEC with trapped atomic ions that is considered in this paper.
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Figure 2
Triangular color codes on the 4.8.8 lattice: One logical qubit is encoded into several data qubits forming a 2D triangular code structure. The code space is defined in the usual way via $S_{x}^{(p)}$ and $S_{z}^{(p)}$ stabilizer operators (1), each acting on a plaquette $p$ that involves either 4 or 8 data qubits. We represent instances of distance $d = 3, 5, 7$ triangular color codes on the 4.8.8. lattice, which allow one to encode a single logical qubit with increasing levels of redundancy and protection. Logical operators such as $Z_{L} = ⨂_{i} Z_{i}$ , and similarly the other logical single-qubit Clifford gate generators $X_{L} : = ⨂_{i} X_{i}, H_{L} : = ⨂_{i} H_{i} = ⨂_{i} \frac{1}{\sqrt{2}} (X_{i} + Z_{i})$ , and $S_{L} : = ⨂_{i} S_{i}^{†} = ⨂_{i} e^{- i \frac{π}{4} (1 - Z_{i})}$ , can be realized transversally, i.e., in a bitwise manner. In the lower panel we show two examples of how the logical operators can be deformed into strings of Pauli operators along the boundaries of the triangular lattice, as expressed in Eq. (2).
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Figure 3
Flag-based FT readout of weight-4 stabilizers: (Upper panel) Scheme for the parity-check measurement of $S_{x}^{(p)} = X_{i_{1}} X_{i_{2}} X_{i_{3}} X_{i_{4}}$ by the sequential application of CNOT gates, represented by a solid line with a filled and a crossed circle on the corresponding qubits. This circuit maps the stabilizer information into the syndrome qubit, which is measured in the $X$ basis. The data-qubit indexes ${i_{1}, i_{2}, i_{3}, i_{4}}$ belonging to $p th$ plaquette are sorted in ascending order according to our choice in Fig. 2, which will be important for the decoding described in the lookup Table 1. A dangerous single bit-flip error is depicted by a red $X$ star at the middle of the circuit. This error would propagate into a pair of bit-flip errors on two data qubits (semitransparent red $X$ stars), as can be seen by using the CNOT gate conjugation identity $U_{CNOT}^{c, t} X_{c} = X_{c} X_{t} U_{CNOT}^{c, t}$ between control $c$ and target $t$ qubits. To identify this dangerous error, one introduces the (f-1) CNOT gate between the syndrome and the extra flag qubit, such that a bit-flip error also cascades into the flag (semitransparent red $X$ star), and can be captured by a $- 1$ measurement in the $Z$ basis, signaling that this correlated error has indeed propagated into the code. The (f-2) CNOT gate is required to map correctly the stabilizer information into the syndrome qubit. (Lower panel) Analogous scheme for the parity-check measurement corresponding to stabilizer $S_{z}^{(p)} = Z_{i_{1}} Z_{i_{2}} Z_{i_{3}} Z_{i_{4}}$ , where we use the CNOT identity $U_{CNOT}^{c, t} Z_{t} = Z_{c} Z_{t} U_{CNOT}^{c, t}$ . Notice that an error after the first CNOT can propagate three errors to the data qubits without triggering the flag. This situation is not fatal since those three errors are equivalent to a single error (up to the stabilizer being measured), which can be corrected at a later stage.
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Figure 4
Transversal CNOT gate between two seven-qubit color codes: (Left panel) Two logical qubits are each embedded in seven data qubits from a couple of triangular planar codes. The transverse CNOT operator can be visualized by stacking the two logical qubits forming a bilayer, such that each pair of equivalent data qubits is coupled by a bare CNOT gate (yellow arrows). (Right panel) Transversal CNOT circuit based on fully entangling two-qubit MS gates $X_{i, j}^{2}$ defined below Eq. (3), and represented by a solid line with two filled circles (not to be confused with a controlled-phase gate). Additionally, single-qubit rotations along the $x$ and $y$ axis, $X (- π / 2)$ and $Y (\pm π / 2)$ , defined below Eq. (5), are required to obtain the transversal CNOT. The shaded gray region, labeled by (b), includes a set of operations that will be used repeatedly in the modular microscopic trapped-ion schedules of the following sections.
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Figure 5
Teleportation-based circuit for implementing a CNOT gate: $M_{X X}$ ( $M_{Z Z}$ ) corresponds to the measurement of the joint operator $X^{t} X^{a}$ ( $Z^{c} Z^{a}$ ). Intuitively, the $M_{X X}$ operation pushes a $Z$ ( $| 0 〉$ ) state from the target to the control while the $M_{Z Z}$ operation pushes a $X$ ( $| + 〉$ ) state from the control to the target. The stepwise evolution of the canonical operators is presented explicitly on Table 2. The final measurement in the $X$ basis, $M_{X}$ , is necessary to decouple the ancillary qubit from the other two. Since the outcome of each measurement operation is random, the final state must be corrected conditionally upon these outcomes: a $Z$ operation must be performed on the control if the total parity of $M_{X X}$ and $M_{X}$ is odd. Likewise, an $X$ operation must be performed on the target if the parity of $M_{Z Z}$ is odd.
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Figure 6
Lattice-surgery-based CNOT gate between two $d = 3$ color-code logical qubits: scheme for the 2D arrangement of the three logical qubits required for the realization of a logical CNOT between the control and target qubit, via intermediate coupling to an ancillary logical qubit. The gray-shaded two- and four-qubit operators with dashed borders are measured in order to merge the ancilla qubit with the target and control qubit, respectively. This yields the desired measurements of the joint logical operators $X_{L}^{a} X_{L}^{t}, Z_{L}^{c} Z_{L}^{a}$ .
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Figure 7
Full FT circuit for the lattice-surgery-based CNOT: The ancillary logical qubit is initialized in the product state ${| 0 〉}^{\otimes 7}$ . A round of $X$ -type stabilizer measurement ${QEC}_{X}^{1}$ is first performed to project the ancilla state to ${| 0 〉}_{L}$ . It is safe to do this by only a single round of $X$ stabilizer measurements since single-qubit $X$ errors can be caught at a later stage. The fault-tolerant implementation of the merging process $M_{X X}$ ( $M_{Z Z}$ ) is shown in detail in Fig. 8. Notice that during the splitting process the error syndromes for each logical qubit involved in the gate must be shared via classical communication to ensure that the procedure is fault tolerant. Also, since the stabilizers during the splitting process are measured using flags, it might be necessary to measure the conjugate stabilizers in the event of a flag being triggered (hook-error detection substep). The red dashed regions denote that the operations inside them are only conditionally applied. The superscripts denote the number of times the stabilizers need to be measured. During the hook-error detection stage, the stabilizers can be measured only once and the circuit construction does not require the previous FT constraints. In the trapped-ion realization, as detailed in sections below, we use a single ancillary qubit and two five-qubit entangling gates in order to minimize the complexity of the microscopic schedules. This still ensures that the overall protocol is FT at level $t = 1$ since these operations will be performed only if an error has already occurred [68].
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Figure 8
FT circuit to measure the joint logical operator $X_{L}^{t} X_{L}^{a}$ : $M_{x (w - 2)}$ and $M_{x (w - 4)}$ refer to the measurement of the merging operators of weight-2 and weight-4 depicted in Fig. 6. ${QEC}_{X}^{2}$ denotes two rounds of $X$ -type stabilizers, which are necessary to distinguish between $Z$ errors that occurred before and after the measurement of the merging operators, as explained in the main text and illustrated in Fig. 10. The merging operators have to be measured at least twice to account for detection errors. For each operator, if the two first measurement outcomes coincide, we trust them. Otherwise, if the two outcomes disagree, the operator is measured a third time (the red dashed regions denote that the operations inside them are only conditionally applied). This scheme is tolerant to a single detection fault. Notice that measuring $X$ stabilizers does not affect the merging action of the $M_{x (w - 2)}$ and $M_{x (w - 4)}$ operators since they all commute. Splitting is only caused by measuring the $Z$ stabilizers. For the $Z$ -type joint logical operator $Z_{L}^{a} Z_{L}^{c}$ , one uses an analogous FT circuit with the roles of the $X$ and $Z$ basis interchanged. Measuring the $X$ -type stabilizers must be done in a FT fashion, which requires at least two rounds. In the first round, the stabilizers are measured with the flag-based readout depicted in Fig. 3. If no error is detected during the first round, we stop and trust the result. Otherwise, we measure the stabilizers a second time to account for detection errors and trust the outcomes of the second round. Following our low-resource philosophy, since an error has already occurred, during the second round the stabilizers are measured without the previous FT constraints. If a flag is triggered during the first round of stabilizer measurements, implying that an $X$ error might have propagated from the ancilla, this information is passed on to the splitting substep, where the $Z$ stabilizers are measured, in order to correctly interpret the syndrome.
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Figure 9
Circuit for the FT readout of the weight-4 operator: (Upper panel) By sequencing the CNOT gates for the measurement of $X_{a_{1}} X_{a_{2}} X_{t_{1}} X_{t_{2}}$ , a single $X$ error on the readout qubit after the second CNOT will propagate into a single bit-flip error on the logical ancillary qubit and a single bit-flip error on the logical target qubit, both of which are correctable. All other single-qubit errors propagate to form single-qubit errors, up to the operator $X_{a_{1}} X_{a_{2}} X_{t_{1}} X_{t_{2}}$ itself. (Lower panel) For the measurement of weight-4 of $Z_{a_{1}} Z_{a_{2}} Z_{c_{1}} Z_{c_{2}}$ operators, one would use instead a circuit that inverts the sense of the CNOT gates, and a readout qubit that is initialized in $| 0 〉$ and measured in the $Z$ basis.
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Figure 10
Dangerous boundary error in the merging substep: Example of a single error that might result in a logical $Z$ error if not handled properly. If the $Z$ error occurs after the second measurement of the weight-4 boundary operator, applying $Z_{5}^{t}$ results in the right correction. On the other hand, if the error occurs before the first measurement of the weight-4 boundary operator, applying a $Z_{5}^{t}$ will result in a logical $Z$ error. Instead, one should apply $Z_{6}^{t} Z_{7}^{t}$ (see the main text for a more detailed explanation). If the $Z$ error occurs on any of the nonboundary qubits ( $d 1 - d 4$ ), the traditional correction works because the error commutes with the joint logical operator on the boundary.
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Figure 11
Trapped-ion flag-based FT readout of weight-4 stabilizers: (Upper panel) MS-based circuit for the parity-check measurements based on the flag-readout schemes of Fig. 3. A dangerous phase-flip error is depicted as a blue $Z$ star at the middle of the circuit, which would propagate into a pair of bit-flip error (red stars) on the data qubits. This can be seen using the MS-gate propagation identities $Z_{i} X_{i, j}^{2} (π / 2) \to X_{i, j}^{2} (π / 2) Y_{i} X_{j}$ , and $Y_{i} X_{i, j}^{2} (π / 2) \to X_{i, j}^{2} (π / 2) Z_{i} X_{j}$ . To identify this dangerous error, the (f-1) MS gate between the syndrome and an extra flag qubit is introduced, such that a $+ 1$ measurement in the $Z$ basis signals that this correlated error may have occurred. The (f-2) MS gate is required to reverse the effect of the (f-1) MS gate, such that the stabilizer is correctly mapped into the syndrome qubit. Notice that it is possible to switch off the (f-1) and (f-2) MS gates to recover the traditional single-ancilla non-FT stabilizer measurement circuit. (Lower panel) The stabilizer measurement with unflagged qubits can be implemented using a pair of five-ion fully entangling MS gates $U_{MS, 0} (π / 2) [U_{MS, 0} (- π / 2)]$ , hereby represented by a solid line with five filled (empty) circles. By sandwiching a single-qubit $Z$ rotation with the two MS gates [80], the stabilizer information can be mapped onto the syndrome qubit directly. Since this unflagged readout is only performed when an error has already occurred, one preserves the FT nature of the multiqubit readout to level $t = 1$ .
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Figure 12
Schedule for the flag-based measurement of $S_{x}^{(1)}$ : Data qubits $d 1, d 2, d 3, d 4, d 5, d 6, d 7$ (blue circles) are distributed among the storage zones $S 2$ and $S 3$ , whereas the ancillary flag $f$ and syndrome $s$ qubits (red circles) occupy the storage $S 1$ and manipulation $M 2$ zones, respectively. A cooling ion of a different species is cotrapped in region $M 2$ . The different lines represent the scheduling of operations required to measure $S_{x}^{(1)} = X_{1} X_{2} X_{3} X_{4}$ using the flag-based circuit of Fig. 11. Black straight arrows joining two different steps represent the splitting, shuttling, and merging of a set of ions that are being transported between two different regions. Blue curved arrows represent a rotation of the ion crystal, which is listed as $R o t$ in the rightmost column. Fully entangling MS gates between the $i th$ and $j th$ ions $X_{i j}^{2} (π / 2)$ are represented by a red oval, prior to which a sympathetic recooling of the ion crystal must be applied (blue oval), and are listed as $MS (i, j)$ , and $c o o l$ , in the rightmost column. Finally, single-qubit gates listed as $X_{\pm} (j) = X_{j} (\pm π / 2)$ in the rightmost column act on an specific $j th$ ion. The fluorescence measurements, listed as $M e a s$ , of the flag and syndrome ions are depicted by black detectors and are followed by a reset operation via optical pumping.
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Figure 13
Ion layout for the logical CNOT in HOA-2 traps: (Upper panel) The seven-qubit color codes for the control and target logical qubits of a transversal CNOT are stored in the central regions of the two lower arms using the 10-ion distribution of Fig. 1. The storage and manipulation zones $M 1, S 1$ of the upper arm are vacated, accommodating the ions of the corresponding logical qubit in the remaining zones, which are mere spectators during the transversal CNOT. Accordingly, the $M 1, S 1$ ones can be used to simplify the required crystal reconfiguration operations following the schedule described in the main text. (Lower panel) For a lattice-surgery CNOT, the logical qubit stored in the upper arm is no longer a mere spectator, but acts as the ancillary qubit in the teleportation-based circuit of Fig. 5.
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Figure 14
Schedule for the transversal CNOT gate between two seven-qubit color codes: A pair of logical qubits, each encoded into seven ions, are distributed in two neighboring arms of the trap together with their corresponding ancillary and cooling ions. We use the same notation as in Fig. 12, but eliminate the circles denoting the ions, and simply use their labels (i.e., data $d1 - d7$ , ancillary flag f, and syndrome s, and cooling c) with a different font and coloring to denote the two logical blocks. We list the operations for the FT transverse CNOT routine in the leftmost column which, in addition to those of Fig. 12, include Sp (split), Sh (shuttle), M (merge), $J cross$ (junction crossing), and $Y_{\pm} = Y (\pm π / 2)$ (single-qubit rotations applied to the specific ion in the red oval). The remaining columns contain the real-space scheme of the ion-crystal configurations in the different zones of the neighboring arms across a $Y$ junction. On the rightmost column, we group operations into four modules (a)–(d).
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Figure 15
Schedule for the lattice-surgery measurement of low-weight operators for $M_{X X}$ : Three logical qubits (i.e., control ancilla and target), each encoded into seven ions, are distributed in three neighboring arms of the trap together with their corresponding ancillary and cooling ions. We use the same notation as in Figs. 12 and 14, with a different font and coloring to denote the three logical blocks. For the measurement of the $X_{a} X_{t}$ logical operator, we need to sequentially measure weight-2 $M_{x (w - 2)} and$ weight-4 $M_{x (w - 4)}$ operators.
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Figure 16
Schedule for the lattice-surgery measurement of low-weight operators for $M_{Z Z}$ : Three logical qubits (i.e., control ancilla and target), each encoded into seven ions, are distributed in three neighboring arms of the trap together with their corresponding ancillary and cooling ions. We use the same notation as in Figs. 12 and 14 with a different font and coloring to denote the three logical blocks. For the measurement of the $Z_{c} Z_{a}$ logical operator, we need to sequentially measure weight-2 $M_{z (w - 2)} and$ weight-4 $M_{z (w - 4)}$ operators. The core module to implement such operations is described by the subsets of operations $(Z Z)$ and $(Z Z Z Z)$ in the rightmost column, whereas the remaining operations correspond to the reconfiguration steps needed to bring the required ions close to each other, such that these modules can be applied.
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Figure 17
Schematics of all subsets of microscopic error configurations: (Left panel) The individual boxes labeled as $w - n$ correspond to the set of error configurations of weight- $n$ errors that may occur during the execution of one complete circuit. The area $A_{n}$ of the subsets is meant to indicate (not to scale) the relative probability of occurrence of each subset. In the limit of small physical error rates, the $w - 0$ subset dominates and all other subsets become vanishingly small. In this limit, the traditional sampler becomes inefficient, as it predominantly samples from the zero-error subset. In contrast, the subset-based sampling strategy explores in a targeted way the specific subsets chosen by the user, typically corresponding to the lowest-weight, noncorrectable error subsets. (Right panel) Multiparameter generalization of the error subsets. Every noise parameter adds an extra dimension to the subset structure, which can be depicted as a hypercube (e.g., cube for a three-parameter error model as shown in the picture). As in the single-parameter case, the hypervolume of each subset indicates (not to scale) its relative probability of occurrence, which depends on the error rates vector $\vec{p}$ .
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Figure 18
Comparison of the logical CNOT strategies as a function of the junction-crossing errors: Probability of a logical $Z$ (left) and logical $X$ (right) error after the preparation of a logical Bell pair using two alternative CNOT strategies as a function of the error strength associated with a trap-junction crossing. All the other error strengths and reordering durations are set to the anticipated experimental values. The horizontal black curve corresponds to the probability of causing a physical $Z$ or $X$ error during the preparation of a physical Bell pair assuming the same error model employed in the simulations. This error rate is $8 p_{2 q} / 15$ because of the possible 15 Pauli errors after a two-qubit gate, 8 cause a $Z$ error on a Bell pair ( $I Y, I Z, X Y, X Z, Y I, Y X, Z I, Z X$ ), and a different set of 8 causes an $X$ error ( $I X, I Y, X I, X Z, Y I, Y Z, Z X, Z Y$ ). For each logical error rate, the subset sampler returns a lower and an upper bound, which essentially coincide tightly at low physical error rates and start to diverge at increasingly larger physical error rates where the importance of the error subsets, which have not been included in the sampling, becomes important.
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Figure 19
Comparison of the logical CNOT strategies as a function of the entangling MS gate errors: Probability of a logical $Z$ (left) and a logical $X$ (right) error after the preparation of a logical Bell pair using two alternative CNOT strategies as a function of the two-qubit MS gate error strength. All the other error strengths and reordering durations are set to the anticipated experimental values. The error strength of the five-qubit MS gate is set to be $p_{5 q} = 5 p_{2 q}$ . As before, the black curve corresponds to the probability of causing a physical $Z$ error (left) or physical $X$ error (right) during the preparation of a physical Bell pair, which is given by $8 p_{2 q} / 15$ , as explained in Fig. 18.
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Figure 20
Gate and junction-crossing resources in the lattice-surgery CNOT: Average number of one-qubit gates, two- and five-qubit MS gates, and trap-junction crossings performed during the lattice-surgery CNOT as a function of two different error strengths, the one associated with the trap-junction crossing $p_{cross}$ (left) and the error rate of the two-qubit MS gate (right). All the other error strengths and durations are set to the anticipated experimental values. For the figures on the right, the error strength of the five-qubit MS gate is set to be $p_{5 q} = 5 p_{2 q}$ . In this case, for high-error crossing we set $p_{cross} = 10^{- 3}$ , while for the low-error crossing we set $p_{cross} = 10^{- 5}$ . The black curves correspond to an error-free regime where every step in the protocol is FT because we never detect an error. In the realistic (faulty) setting, we instead switch to non-FT circuits as soon as an error is detected. This decreases the total number of one-qubit rotations and two-qubit MS gates that we have to perform on average. In contrast, it increases the number of non-FT five-qubit MS gates, which we use only after an error has been detected. The total number of trap-junction crossings also increases, but only slightly.
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Figure 21
Comparison of the durations of the transversal and lattice-surgery CNOT gates: Average duration of the two alternative CNOT strategies as a function of two different error strengths. the one associated with the trap-junction crossing $p_{cross}$ (left) and the error rate of the 2-q MS gate (right). All the other error strengths and durations are set to the anticipated experimental values. For the figures on the right, the error strength of the five-qubit MS gate is set to be $p_{5 q} = 5 p_{2 q}$ . In this case, for high-error crossing we set $p_{cross} = 10^{- 3}$ , while for the low-error crossing we set $p_{cross} = 10^{- 5}$ . The durations of every gate and reordering operation are taken from Table 4. The duration of the trap-junction crossing is set to $t_{cross} = - T_{2} ln (1 - 2 p_{cross})$ , and therefore increases with $p_{cross}$ .
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