manifold $\textrm{Span}\\{|0\rangle,|1\rangle\\}$, so that the initial density
matrix can be written as

$\hat{\rho}_{0}=\begin{bmatrix}0&0&0\\\ 0&\rho_{11}&\rho_{10}\\\
0&\rho_{01}&\rho_{00}\end{bmatrix}$ (24)

(we order the matrix columns and rows as
$\\{|r\rangle,|1\rangle,|0\rangle\\}$). Upon solving the resulting coupled
first-order differential equations, we find that the final state after the
$2\pi$ pulse with decay is, to leading order in $\gamma/\Omega$,

$\hat{\rho}_{f}=\begin{bmatrix}3\gamma t_{\pi}\rho_{11}/4&0&0\\\ 0&(1-3\gamma
t_{\pi}/4)\rho_{11}&-e^{-\gamma t_{\pi}/2}\rho_{10}\\\ 0&-e^{-\gamma
t_{\pi}/2}\rho_{01}&\rho_{00}\end{bmatrix}.$ (25)

Here, $t_{\pi}=\frac{\pi}{2\Omega}$ is the duration of a $\pi$ pulse. Indeed,
Eq. (25) confirms our intuition from Section III.2 that the coherences
$\rho_{r1}$, $\rho_{1r}$ vanish upon “averaging” over all possible transition
times during the $2\pi$ pulse.

One can then verify that, to leading order in $\gamma/\Omega$, the Kraus
operators

$\displaystyle M_{0}=$ $\displaystyle|r\rangle\langle
r|+\sqrt{1-p_{2}}|0\rangle\langle 0|$ (26)
$\displaystyle+\sqrt{(1-p_{1})(1-p_{2})}|1\rangle\langle 1|$
$M_{r}=\sqrt{p_{1}(1-p_{2})}|r\rangle\langle 1|$ (27)
$M_{1}=\sqrt{p_{2}}|1\rangle\langle 1|$ (28)
$M_{2}=\sqrt{p_{2}}|0\rangle\langle 0|$ (29)

give rise to the desired evolution from $\rho_{0}$ to $\rho_{f}$ provided we
take $p_{1}=3\gamma t_{\pi}/4$ and $p_{2}=\gamma t_{\pi}/8$.

##  Appendix C Atom Loss Errors

Figure 13: Circuit for detecting atom loss.

As mentioned in Section III.4 of the main text, neutral atom setups can also
suffer from atom loss errors if the trapping is imperfect, or if the trapping
lasers need to be turned off during Rydberg excitation (e.g. as done for 87Rb
in Ref. Bernien  _et al._ (2017)). Fortunately, such errors can also be
detected and corrected within our FTQC framework at the cost of one ancilla
qubit and some extra gates for each operation. In particular, an atom loss
event can be detected by applying the circuit of Figure 13 for each data qubit
after using the optical pumping technique to correct for leakage out of the
computational subspace. The ancilla measurement will then produce $+1$ in the
presence of atom loss, and $-1$ if such an error did not occur. Once detected,
an atom loss error can be converted to a single-qubit Pauli-$Z$ or $X$ type
error if a reservoir of atoms is available, for instance by replacing the lost
atom with a new atom initialized to the $\ket{0}$ state.

We now discuss the steps needed for establishing robustness against errors
occurring during this circuit. As in the case of fault-tolerant Rydberg
leakage detection discussed in Appendix D, to protect against ancilla errors
in Figure 13, we again adopt a multi-step ancilla measurement protocol and
require two positive ancilla measurements to confirm an atom loss error. On
the other hand, any phase-flip error on the data qubit cannot propagate to
more than a single physical qubit error per logical qubit in the universal
gate set implementations for Ryd-7 or Ryd-3. Leakage errors (Rydberg or non-
Rydberg) can be addressed by repeating the respective re-pumping procedures
after applying the atom loss detection circuit. Thus, by incorporating this
circuit into the implementation of fault-tolerant stabilizer measurements and
logical operations in Sections IV and V, we can also address atom loss errors
in our FTQC protocols.

Notice that this circuit can be used for atom loss after correcting for
leakage into atomic states outside the computational subspace by using the
blockade effect and optical pumping techniques. In addition, this approach
does not distinguish between atom loss and leakage into other hyperfine
states, so it can also be used to suppress any residual hyperfine leakage
errors.

##  Appendix D Converting Rydberg Leakage to Pauli Errors

Once a Rydberg leakage error is detected, it can be converted to an atom loss
error by ejecting the Rydberg atom, which is naturally done by the anti-
trapping potential from the tweezer Bluvstein  _et al._ (2021a) and can be
expedited by pulsing a weak, ionizing electric field Beguin (2013); Cohen
(2003); Wu  _et al._ (2022). The exact location of the ejected atom can be
determined by following the atom loss protocol outlined in Appendix C and
Figure 13 444In this case, the atom loss protocol does not need to be applied
in a robust fashion, since an error has already occurred.; subsequently, the
ejected atom can be replaced with a fresh atom prepared in the $|1\rangle$
state. Although this process simply replaces the Rydberg atom by an atom in
the $\ket{1}$ state, by using the operator identity $|1\rangle\langle
1|=\frac{1}{2}\left(\mathbf{1}-Z\right)$, it is straightforward to see that
the resulting state is now a superposition of the original state without
error, and the same state with a $Z$ error on this physical qubit. Such
$Z$-type errors can be detected and corrected for using stabilizer
measurements in both the seven-qubit and three-qubit codes. This procedure can
also be modified to convert the Rydberg leakage error to a Pauli $X$-type
error by applying Hadamard gates at the beginning and end; this is used to in
the logical CCZ gate for Ryd-7 (see Algorithm 3).

To reduce the need for applying the atom loss correction circuit, one could
add a preventative step after every entangling gate which incoherently re-
pumps any remnant population in several most probable Rydberg states into the
qubit $|1\rangle$ state. This re-pumping can be implemented via the following
three-step procedure:

  1. 1.

Swap the population in $|1\rangle$ and the stretched ground state
$|F=I+1/2,m_{F}=I+1/2\rangle$.

  2. 2.

For the most probable final states $\ket{r^{\prime}}$ of a BBR transition (or
the Rydberg state $\ket{r}$ in the case of RD), perform a Rydberg laser pulse
that sends $\ket{r^{\prime}}$ (or $\ket{r}$) to a short-lived $P$ state. In
particular, we choose the $P$ state with the smallest possible $n$, largest
possible $F$, and largest possible $m_{F}$. This state will quickly decay to
the stretched state $|F=I+1/2,m_{F}=I+1/2\rangle$, and cannot decay to any
other ground state.

  3. 3.

Repeat Step (1).

By applying this procedure preventatively, one can convert a large fraction of
Rydberg leakage errors to $Z$-type errors without the need for the atom loss
correction circuit of Figure 13.

##  Appendix E Fault-Tolerant Detection of Rydberg Leakage Errors

As mentioned in the main text, for fault-tolerant error detection and
correction, it is important to address any errors that may occur on an ancilla
used to probe for Rydberg population. This can be done by using a multi-step
measurement procedure to detect leakage for the ancilla qubit:

  1. 1.

Perform a Hadamard gate on the ancilla.

  2. 2.

Check whether the ancilla is in the $\ket{1}$ state (e.g. by coupling
$\ket{1}$ to a cycling transition and detecting fluorescence).

  3. 3.

Perform an $X$ gate on the ancilla.

  4. 4.

Check for $\ket{1}$ population again.

If neither the second nor the last step yields $\ket{1}$, the ancilla atom
must have undergone a leakage error. In that case, we convert any possible
ancilla atom Rydberg error to a possible $Z$-type error as described in
Appendix D. Similarly, because the Rydberg pulses can potentially cause a
phase-flip error on the ancilla qubit, if a Rydberg leakage error is detected
by the ancilla, the detection protocol must be repeated once more to ensure
that the outcome did not result from such an error.

##  Appendix F Error Syndromes with Postponed Measurements

In Section IV.1, we discussed how Rydberg leakage detection can be postponed
in the Ryd-7 stabilizer measurement and controlled-phase gate protocols to
facilitate experimental implementation. This relied on the ability to use
stabilizer measurements to distinguish between the possible correlated errors
that can result from postponed detection of a Rydberg leakage error. Here, we
present details on how to use error syndromes to identify the corresponding
correlated error in each case. As in the main text, we assume the stabilizers
for the Steane code are ordered as

$\displaystyle g_{1}=IIIXXXX$ $\displaystyle g_{2}$ $\displaystyle=IXXIIXX$
$\displaystyle g_{3}$ $\displaystyle=XIXIXIX$ $\displaystyle g_{4}=IIIZZZZ$
$\displaystyle g_{5}$ $\displaystyle=IZZIIZZ$ $\displaystyle g_{6}$
$\displaystyle=ZIZIZIZ.$ (30)

Table 5: Using error syndromes to distinguish between correlated errors resulting from postponed detection of Rydberg leakage during measurement of the $X_{4}X_{5}X_{6}X_{7}$ stabilizer in the Ryd-7 FTQC protocol. Because the possible correlated errors are all products of Pauli-$X$ errors, we show the corresponding values of $Z^{\otimes 4}$ stabilizer measurements. Error | $g_{4}$ | $g_{5}$ | $g_{6}$
---|---|---|---
$X_{5}X_{6}X_{7}$ | $-1$ | $+1$ | $+1$
$X_{6}X_{7}$ | $+1$ | $+1$ | $-1$
$X_{7}$ | $-1$ | $-1$ | $-1$

For the stabilizer measurement, we will consider (without loss of generality)
the measurement of $g_{1}$ on qubits $4,5,6,7$ using a circuit of the form
shown in Figure 1b. If a Rydberg leakage error occurs on the ancilla atom at
any point, the data atoms do not suffer any correlated errors. On the other
hand, if a data atom suffers a Rydberg leakage error during the circuit, the
possible correlated errors that can result are $X_{5}X_{6}X_{7}$,
$X_{6}X_{7}$, or $X_{7}$. These errors can be distinguished by measuring the
$Z^{\otimes 4}$ stabilizers of the seven-qubit code; the corresponding error
syndromes are shown in Table 5.

For the case of the logical CCZ gate, we grouped the 27 physical Rydberg gates
into groups $\mathcal{G}_{i}$ of three, and performed Rydberg leakage
detection after each group. Without loss of generality, we will consider the
group $\mathcal{G}_{1}$ in Figure 3. There are two possible correlated errors
that could result from the delayed detection of Rydberg leakage in this case
(up to a single-qubit error within each logical qubit):
$R(2_{A},2_{B};2_{C})R(3_{A},3_{B};3_{C})$ and $R(3_{A},3_{B};3_{C})$. By
writing the Rydberg gate as

$R(j,k;l)=(\mathbf{1}+Z_{j})(\mathbf{1}+Z_{k})(\mathbf{1}+Z_{l})-\mathbf{1},$
(31)

we find that the two cases can be distinguished by measuring the stabilizers
$g_{2}$ and $g_{3}$ for each of the logical qubits. In the former case, at
least one of the the logical qubits would have either a $Z_{2}$ or
$Z_{2}Z_{3}$ error, giving rise to stabilizer eigenvalues
$(g_{2},g_{3})=(-1,+1)$ or $(+1,-1)$, while in the latter scenario, all three
sets of stabilizer measurements would yield $(-1,-1)$ or $(+1,+1)$.

##  Appendix G Implementation of a Bias-Preserving Toffoli Gate

Figure 14: Using two ancilla qubits and multiple Rydberg states to implement a
bias-preserving Toffoli gate between control atoms $C_{1}$, $C_{2}$, and
target atom $T$. The ancilla atoms ($A_{1}$ and $A_{2}$) are chosen to lie on
either side of the target atom. The dotted boxes indicate the most natural
bias-preserving three-qubit gate for Rydberg systems, where $\pi$ pulses
$|1\rangle\leftrightarrow|r_{+}\rangle$ are applied to each of the first two
(i.e., the upper two) involved atoms, the bias-preserving pulse sequence of
Figure 7 is applied to the third (lower) atom, and $-\pi$ pulses
$|1\rangle\leftrightarrow|r_{+}\rangle$ are applied to the first two qubits;
the Rydberg states $|r_{\pm}\rangle$ are chosen to be either
$|r_{1,\pm}\rangle$ or $|r_{2,\pm}\rangle$ for each such gate. In this
circuit, we set $|r_{\pm}\rangle=|r_{1,\pm}\rangle$ in the first, second,
fourth, and fifth cases, while choosing $|r_{\pm}\rangle=|r_{2,\pm}\rangle$
for the third one. With this choice of Rydberg levels, the two ancillas will
not interact with each other during the third Rydberg gate. However, we note
that the two control atoms may interact with each other during the first,
second, fourth, and fifth entangling gates if the distance between them is
less than one blockade radius; this is not problematic because Rydberg errors
can occur during at most one of these gates, so at least one ancilla atom will
generate the correct interaction with the target atom during the third gate.

In Figure 8, we showed how an ancilla atom can be used to eliminate $X$-type
errors resulting from control atom decay in the implementation of a bias-
preserving CNOT gate. Analogously, a bias-preserving Toffoli gate can be
implemented by making use of two ancilla atoms which lie on either side of the
target atom. This protocol is illustrated in Figure 14.

As with the case of the bias-preserving CNOT, the choice of Rydberg states
differs throughout the procedure. By coupling the atoms to $|r_{2,\pm}\rangle$
during the third gate of Figure 14 and using ancilla atoms on opposite sides
of the target atom, we ensure that the ancilla atoms do not interact with each
other via Rydberg blockade during this gate; this is important in case one of
the ancilla atoms undergoes a radiative decay transition during this gate. On
the other hand, the other entangling gates in Figure 14 all use the Rydberg
states $|r_{1,\pm}\rangle$, due to larger distances between the atoms during
these gates. We note that the two control atoms may interact with each other
during these four gates if the distance between them is less than one blockade
radius, which is different from the case of the third gate. This is acceptable
because Rydberg errors can occur during at most one of these four gates, so at
least one ancilla atom will generate the correct interaction with the target
atom during the third gate.

##  Appendix H Computing Resource Costs for Rydberg FTQC Protocols

We now provide details on how to obtain the resource costs for Ryd-7 and Ryd-3
presented in Tables 1 and 2.

For the Ryd-7 protocol, each stabilizer measurement requires four two-qubit
Rydberg gates in the absence of errors (see Algorithm 1); thus, 24 two-qubit
gates are required to measure all stabilizers. If an error occurs, the worst
case scenario for the stabilizer measurement is when the first five
stabilizers all have $+1$ eigenvalues, while the very last stabilizer is
measured to be $-1$. In this case, $g_{4}$, $g_{5}$, and $g_{6}$ need to be
re-measured, which requires 12 additional two-qubit gates. The logical CCZ
gate for Ryd-7 is implemented using 27 physical three-qubit gates in the
absence of error, as described in Algorithm 3. The worst case error in this
case is a Rydberg leakage error that occurred during the first entangling gate
in the final group $\mathcal{G}_{9}$ of Figure 3. In this scenario,
identifying the location of the Rydberg leakage error requires up to 18
additional two-qubit gates, while measuring the stabilizers
$g_{2},g_{3},...,g_{6}$ for all three logical qubits would amount to 60
additional two-qubit gates; the correction circuit could require up to two
additional three-qubit gates.

In the Ryd-3 protocol, each of the two stabilizer measurements requires two
bias-preserving CNOT gates (Figure 1e), and each bias-preserving CNOT gate is
broken down to two two-atom gates and one three-atom entangling gate (see
Section V.1). Thus, in the absence of error, the stabilizer measurements would
require eight two-qubit gates and four three-qubit gates. If an error occurs,
the worst case scenario is if the second stabilizer is measured to be $-1$; in
this case, both stabilizers need to be re-measured, and the gate cost is
doubled. The Ryd-3 CCZ gate can be implemented in a round-robin fashion in the
same way as the Ryd-7 CCZ, which is bias-preserving and uses 27 physical
three-qubit gates.

Finally, the Ryd-3 Hadamard gate consists of a fault-tolerant, bias-preserving
Toffoli gate followed by single-qubit measurements and rotations (Figure 10).
The pieceable fault-tolerant Toffoli gate in the Ryd-3 code consists of nine
physical bias-preserving Toffoli gates and two rounds of error correction. As
discussed above, each round of error correction involves eight two-atom
Rydberg gates and four three-atom Rydberg gates. When the data atoms within
each logical qubit are indexed as in Figure 15b and we are implementing a
logical Toffoli gate $\textrm{CCX}_{ABC}$ between the three qubits $A$, $B$,
$C$ highlighted in bold, the number of Rydberg gates required to implement
each physical Toffoli gate depends on the blockade radius $R_{B,1}$. If the
blockade radius $R_{B,1}$ is larger than $3.61d$, each physical Toffoli gate
can be implemented using two ancilla atoms (one on either side of the target
atom) and five three-atom Rydberg gates, as described in Appendix G; this is
because the distance between any physical control atom $C_{i}$ and any ancilla
$A_{j}$ in Figure 14 will always be less than the blockade radius $R_{B,1}$,
so the entangling gates can be implemented directly. In this case, each
physical Toffoli gate involves five three-atom Rydberg gates, so the total
gate count (upon including the QEC steps) is 16 two-atom gates and 53 three-
atom gates in the absence of errors. On the other hand, if we wish to reduce
the blockade radius requirement to $R_{B,1}>3d$, there are two physical
Toffoli gates (corresponding to the choices $j_{A}=l_{C}=1$, $k_{B}=2$ and
$j_{A}=l_{C}=3$, $k_{B}=2$), where the distance between one of the physical
control atoms and one of the ancilla atoms ($2_{B}$ and $A_{3}$ in Figure 15b)
would be too large to directly implement a Rydberg entangling gate required
for the physical Toffoli gate. Instead, in place of the first (respectively,
second) three-atom Rydberg gate involving $A_{3}$, we would implement a
Rydberg gate with the same two control atoms and one of the ancilla atoms
$A_{1}$ or $A_{2}$, whichever is not involved in the rest of the Figure 14
circuit, followed (respectively, preceded) by a bias-preserving CNOT gate
between that ancilla and $A_{3}$. These gates can be implemented directly
because both $A_{1}$ and $A_{2}$ are within the blockade radius of $2_{B}$,
$1_{A}$, $2_{A}$, $3_{A}$, and $A_{3}$. In this way, four extra two-atom gates
are required for the logical Toffoli (two for the physical Toffoli with
$j_{A}=l_{C}=1$, $k_{B}=2$ and two for the physical Toffoli with
$j_{A}=l_{C}=3$, $k_{B}=2$), which increases the total gate count to 20 two-
atom gates and 53 three-atom gates in the absence of error, as shown in Table
2. With errors, the worst case scenario is if the final stabilizer measurement
in the second round of QEC yields $-1$, in which case the stabilizers need to
be measured again; this adds another eight two-atom gates and four three-atom
gates to the total resource cost.

Figure 15: Example labeling of atoms for the Ryd-7 and Ryd-3 FTQC protocols
used to derive the gate counts and blockade radius requirements. As in Figure
1, data atoms are shown in blue, while ancilla atoms are shown in pink. (a) In
the Ryd-7 protocol, each logical qubit consists of seven data atoms (blue
dotted hexagons). For each data atom, a number is used to indicate which
physical qubit of the seven-qubit logical state the atom encodes. With this
labeling, the blockade radius $R_{B}$ is defined by the interaction range
needed to perform a logical CCZ gate between three neighboring logical qubits
such as $A$, $B$, and $C$. Using the specific CCZ protocol given in Algorithm
3, the blockade radius requirement is then $R_{B}>3.61d$, where $d$ is the
spacing between nearest neighbors on the lattice; this is determined by the
distance between physical atoms $3_{A}$ and $1_{C}$ (thinner, light grey
dotted line). However, by using a different set of physical CCZ gates to
implement the logical CCZ, this requirement can be reduced to $R_{B}>3d$
(thicker, dark grey dotted line). (b) In the Ryd-3 protocol, each logical
qubit consists of three data atoms (blue dotted triangles). For each data
atom, a number is used to indicate which physical qubit of the three-qubit
logical state the atom encodes. With this labeling, the larger blockade radius
$R_{B,1}$ is determined by the interaction range required for performing a
logical Toffoli gate between three neighboring logical qubits such as $A$,
$B$, and $C$. In this case, there are two possibilities for $R_{B,1}$—either
$R_{B,1}>3.61d$ (thinner, light grey dotted line) or $R_{B,1}>3d$ (thicker,
dark grey dotted line). When the larger blockade radius of $3.61d$ can be
realized, the resource cost for the logical Toffoli and Hadamard gates can be
reduced by four two-qubit entangling gates compared to the numbers presented
in Table 2 (see also Appendix I).

##  Appendix I Computing Rydberg Blockade Radius Requirements for Rydberg FTQC
Protocols

To obtain the blockade radius requirement for the Rydberg FTQC protocols, we
must identify each physical qubit with an atom on the lattice, and then
determine the maximum distance between two atoms which must interact with each
other during a Rydberg gate. When the underlying atoms are placed in a
triangular lattice, Figure 15 depicts convenient identifications for both the
Ryd-7 and Ryd-3 codes. In this figure, numbers are used indicate the indices
of data atoms within each logical qubit. (The index of a physical qubit within
each logical qubit is the position, counting from the left, of that qubit in
the definition of the logical states; see Equations (6) and (7) for the seven-
qubit code, or Equation (12) for the three-qubit code.)

In the Ryd-7 protocol, the blockade radius is defined by the interaction range
needed to perform a logical CCZ gate between three neighboring logical qubits
such as $A$, $B$, and $C$. Using the specific protocol given in Algorithm 3,
which involves 27 physical CCZ gates between atoms
$j_{A},k_{B},l_{C}\in\\{1,2,3\\}$, we find the largest interaction range is
required to perform the physical CCZ gate between farthest-separated triples
such as $(j_{A},k_{B},l_{C})=(3,3,1)$. For this specific case, the distances
between atom pairs are $\textrm{dist}(j_{A},k_{B})=3d$,
$\textrm{dist}(j_{A},l_{C})=\sqrt{(7/2)^{2}+3/4}d\approx 3.61d$, and
$\textrm{dist}(k_{B},l_{C})=4d$. To apply the three-qubit Rydberg gate
$R(j_{A},k_{B};l_{C})$ as defined in Section II, this would require a blockade
radius of $R_{B}>4d$. However, this is not entirely necessary for our
purposes: instead, it is sufficient that two out of the three distances
$\textrm{dist}(j_{A},k_{B})$, $\textrm{dist}(j_{A},l_{C})$, and
$\textrm{dist}(k_{B},l_{C})$ be less than the blockade radius. To see this,
let us suppose, for example, that the distance between the two control atoms
$j_{A}$ and $k_{B}$ is greater than $R_{B}$. In this case, applying the same
pulse sequence as illustrated in Figure 2b would result in a three-qubit gate
$R=\textrm{diag}(1,-1,-1,-1,-1,-1,1,1)$, which can also be obtained from the
CCZ gate by single-qubit unitaries ($R\propto
Y_{1}Y_{2}(\textrm{CCZ})X_{1}X_{2}$).

The argument above allows the blockade radius requirement for Ryd-7 to be
reduced to $R_{B}>3.61d$ (thinner, light grey dotted line in Figure 15). In
fact, by modifying the implementation of the logical CCZ gate, it is possible
to further reduce this requirement to $R_{B}>3d$ (thicker, dark grey dotted
line in Figure 15); this is shown in Appendix J.

In the Ryd-3 protocol, the blockade radius $R_{B,1}$ is determined by the
interaction range required to implement the logical Toffoli gate between
neighboring logical qubits (e.g., $A$, $B$, and $C$ in Figure 15). As
discussed in Appendix H, there are two possibilities in this case. To directly
implement every physical bias-preserving Toffoli gate using the circuit of
Figure 14, the distance between $2_{B}$ and $A_{3}$ must be less than
$R_{B,1}$; this requires $R_{B,1}>\sqrt{(7/2)^{2}+3/4}d\approx 3.61d$
(thinner, light grey dotted line in Figure 15). However, this requirement can
be reduced to $R_{B,1}>3d$ (thicker, dark grey dotted line in Figure 15) at
the expense of four additional two-atom entangling gates per logical Toffoli
or Hadamard operation.

##  Appendix J Blockade Radius Reduction for Ryd-7

To reduce the blockade radius requirement from $R_{B}=3.61d$ to $R_{B}=3d$ in
the Ryd-7 protocol, we must modify the implementation of the logical CCZ
operation. Recall that Algorithm 3 implements a logical CCZ gate using 27
physical CCZ gates between the first three physical qubits of every logical
qubit. This round-robin decomposition makes use of Eq. (11), which we now
derive:

$\textrm{CCZ}_{ABC}=\prod_{j_{A},k_{B},l_{C}\in\\{1,2,3\\}}\textrm{CCZ}(j_{A},k_{B},l_{C}),$
(32)

To begin the derivation, we first recall that the logical states (6) and (7)
of the seven-qubit code have well-defined parity: the number of physical
qubits in the $|1\rangle$ state is always even for $|0\rangle_{L}$ and odd for
$|1\rangle_{L}$. It then follows that the logical CCZ gate can be implemented
in a fully round-robin fashion involving all physical qubits

$\textrm{CCZ}_{ABC}=\prod_{j_{A},k_{B},l_{C}\in\\{1,2,...,7\\}}\textrm{CCZ}(j_{A},k_{B},l_{C}).$
(33)

This is because the round-robin implementation results in a $-1$ phase
accumulation for each triple $(j_{A},k_{B},l_{C})$ of physical qubits in the
$|1\rangle$ state, and the number of such triples is odd if all logical qubits
are in the $|1\rangle_{L}$ logical state, while it is even if at least one
logical qubit is in the $|0\rangle_{L}$ state. To reduce this to Eq. (32), we
notice that for each choice of $j_{A}$ and $k_{B}$, the product

$\prod_{l_{C}\in\\{4,5,6,7\\}}\textrm{CCZ}(j_{A},k_{B},l_{C})$ (34)

acts as an identity operation on the logical qubits, because
$g_{4}=Z_{4}Z_{5}Z_{6}Z_{7}$ is a stabilizer of the seven-qubit code. We then
multiply both sides of Eq. (33) by this operator, and use the fact that all
the CCZ gates commute with each other and square to the identity operator. In
this way, the product over $l_{C}$ in the logical CCZ gate can be reduced from
$l_{C}\in\\{1,2,...,7\\}$ to $l_{C}\in\\{1,2,3\\}$. Because the CCZ gate is
symmetric in the three involved qubits, this same argument can be applied to
reduce the products over $j_{A}$ and $k_{B}$ to obtain Eq. (32).

To reduce the blockade radius requirement from $R_{B}=3.61d$ to $R_{B}=3d$, we
can replace the product (34) by

$\prod_{l_{C}\in\\{1,2,4,7\\}}\textrm{CCZ}(j_{A},k_{B},l_{C})$ (35)

in our derivation for one of the logical qubits, say qubit $C$. This is
because the single-qubit operator $Z_{1}Z_{2}Z_{4}Z_{7}=g_{2}g_{3}$ is the
product of two stabilizers, so the operator (35) also acts trivially on the
logical subspace. It follows that

$\textrm{CCZ}_{ABC}=\prod_{\begin{subarray}{c}j_{A},k_{B}\in\\{1,2,3\\}\\\
l_{C}\in\\{3,5,6\\}\end{subarray}}\textrm{CCZ}(j_{A},k_{B},l_{C}).$ (36)

Thus, the 27 physical CCZ gates in Algorithm 3 may be replaced by the 27 CCZ
gates used in the right hand side of Eq. (36).

Given the geometrical layout of individual atoms within each logical qubit
shown in Figure 15a, we see that the required interaction range for
implementing the logical CCZ operation using these 27 gates is smaller than
the interaction range required to perform the 27 gates of Algorithm 3.
Furthermore, following the observation made in Appendix I, we notice that
these $9$ physical qubits need not all be within the blockade radius of each
other, so long as every physical qubit $j_{A}\in\\{1,2,3\\}$ is within
distance $R_{B}$ of every $l_{C}\in\\{3,5,6\\}$, and every
$k_{B}\in\\{1,2,3\\}$ is within distance $R_{B}$ of every
$l_{C}\in\\{3,5,6\\}$. This requirement is satisfied for any $R_{B}>3d$, as
shown in Figure 15a.

##  Appendix K Square Lattice Geometry for Ryd-3

Figure 16: Square lattice geometry for the Ryd-3 FTQC protocol. Data (blue)
and ancilla (pink) atoms are placed on the vertices of a square lattice in an
alternating fashion, with three data atoms comprising a logical qubit (blue
dotted boxes). The numbers on each data atom indicate the index of that atom
within each logical qubit; this is relevant for the implementation of
stabilizer measurements and logical operations. Two Rydberg states with
different blockade radii are required to implement the bias-preserving CNOT
and Toffoli gates. The larger blockade radius $R_{B,1}$ must be larger than
$\sqrt{10}d$ (dark grey), where $d$ is the nearest-neighbor spacing on the
lattice, while the smaller blockade radius must satisfy $d<R_{B,2}<2d$ (light
grey). The interaction range $R_{B,1}$ is needed to perform a logical CCZ gate
between the three logical qubits indicated in bold.

As mentioned in Section V.3 of the main text, the Ryd-3 protocol can also be
implemented when the underlying physical atoms are placed on a square lattice.
In this case, the data and ancilla atoms are placed on the vertices of the
lattice in an alternating fashion as shown in Figure 16. The stabilizer
measurements can be implemented as discussed in Section V.2 if the smaller
blockade radius $R_{B,2}$ satisfies $d<R_{B,2}<2d$. The logical operation
requiring the largest interaction range is the logical CCZ gate

$\textrm{CCZ}_{ABC}=\prod_{j_{A},k_{B},l_{C}\in\\{1,2,3\\}}\textrm{CCZ}(j_{A},k_{B},l_{C}),$
(37)

which is implemented from 27 physical CCZ gates. To implement each physical
gate, the distance between every pair $(j_{A},l_{C})$ and $(k_{B},l_{C})$ must
be less than the larger blockade radius $R_{B,1}$, as discussed in Appendix I.
The longest such distance is $\sqrt{10}d$ as shown in the dark grey dotted
line of Figure 16, so the corresponding blockade radius requirement for this
geometry is $R_{B,1}>\sqrt{10}d$.

With these blockade radii, the protocols of Section V.2 can be directly
applied to perform all logical operations. We note that the higher density of
ancilla atoms in this arrangement allows us to implement every physical
Toffoli gate in the logical Toffoli operation directly using the circuit of
Figure 14, without the need for additional ancilla atoms or CNOT gates (as was
the case for two physical Toffoli operations under the triangular lattice
geometry). In this way, for the square lattice geometry, the number of two-
qubit entangling operations required for the logical Hadamard or Toffoli
operations may be reduced by 4 compared to the numbers shown in Table 2.

##  Appendix L Optical Pumping Procedure for the Bias-Preserving CNOT

To implement the bias-preserving CNOT pulse sequence shown in Figure 7 of the
main text, it is important that the optical pumping procedure in the final
step pumps only the $m_{F}>0$ states to the $|1\rangle$ state, and only the
$m_{F}<0$ states to the $|0\rangle$ state. This requirement is essential to
ensuring that the CNOT does not generate any $X$\- or $Y$-type errors. For
magnetic field regimes typically used in alkali atom Rydberg experiments, this
state selectivity may not be straightforward to implement, as the level
separation between different $m_{F}$ states within a single hyperfine manifold
may be much smaller than the linewidth of the lasers used for optical pumping.
To address this challenge, we can utilize a Rydberg state as a shelving state
(due to its long lifetime) to avoid unwanted pumping of $m_{F}<0$
(respectively, $m_{F}>0$) states to $|1\rangle$ ($|0\rangle$). Thus, in Step 6
of Figure 7, the optical pumping of $m_{F}>0$ states into the $|1\rangle$
state can be implemented for 85Rb as follows:

  1. 1.

Swap the population between the $|1\rangle$ state and the stretched ground
state $|F=I+1/2,m_{F}=I+1/2\rangle$.

  2. 2.

Swap the population between the $|0\rangle$ state and the ground state
$|F=3,m_{F}=0\rangle$.

  3. 3.

Apply a resonant $\pi$ pulse to shelve any population in the
$|F=2,m_{F}=-2\rangle$ state into the Rydberg state
$|nS_{1/2},m_{J}=-1/2,m_{I}=-5/2\rangle$.

  4. 4.

Use $\sigma^{+}$ light to excite states in the $F=2$ ground state manifold to
the $5P_{3/2}$ $F=3$ manifold; these excited states decay quickly back to the
ground state.

  5. 5.

Apply resonant $\pi$ pulses
$|F=3,m_{F}=1\rangle\leftrightarrow|F=2,m_{F}=1\rangle$ and
$|F=3,m_{F}=2\rangle\leftrightarrow|F=2,m_{F}=2\rangle$.

  6. 6.

Repeat Steps 4 and 5 as necessary; after several iterations, all population
that started with $m_{F}>0$ will be in the $|F=3,m_{F}=3\rangle$ state.

  7. 7.

Repeat Steps 1, 2, and 3.

Because the $|F=2,m_{F}=-2\rangle$ state can only be populated if a Rydberg
error occurred in one of the earlier steps of the bias-preserving CNOT, to
leading order in the total error probability, we may assume that the Rydberg
state $|nS_{1/2},m_{J}=-1/2,m_{I}=-5/2\rangle$ will not decay if it is
populated in the above procedure. In this way, the only $F=2$ states that can
be populated at the beginning of Step 4 above will be the $m_{F}>0$ states, so
the optical pumping will work in the same way as the protocol described in
Section IV.2 (Figure 5). An analogous procedure can then be applied to pump
the $m_{F}<0$ states into $|0\rangle$. In this latter case, it will not be
necessary to shelve population in the Rydberg state, as all $m_{F}>0$
population will already have been transferred to the $|1\rangle$ state.

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