# Subsystem surface and compass code sensitivities to non-identical infidelity
distributions on heavy-hex lattice

M. Carroll  J. R. Wootton  A. W. Cross

###### Abstract

Logical qubits encoded into a quantum code exhibit improved error rates when
the physical error rates are sufficiently low, below the pseudothreshold.
Logical error rates and pseudothresholds can be estimated for specific
circuits and noise models, and these estimates provide approximate goals for
qubit performance. However, estimates often assume uniform error rates, while
real devices have static and/or dynamic distributions of non-identical error
rates and may exhibit outliers. These distributions make it more challenging
to evaluate, compare, and rank the expected performance of quantum processors.
We numerically investigate how the logical error rate depends on parameters of
the noise distribution for the subsystem surface code and the compass code on
a subdivided hexagonal lattice. Three notable observations are found: (1) the
average logical error rate depends on the average of the physical qubit
infidelity distribution without sensitivity to higher moments (e.g., variance
or outliers) for a wide parameter range; (2) the logical error rate saturates
as errors increase at one or a few ’bad’ locations; and (3) a decoder that is
aware of location specific error rates modestly improves the logical error
rate. We discuss the implications of these results in the context of several
different practical sources of outliers and non-uniform qubit error rates.

##  I Introduction

Quantum computing promises computational speed up in numerous special purpose
applications Reiher  _et al._ ; Childs  _et al._ ; Bravyi  _et al._ . An
outstanding challenge is that noisy physical operations limit the depth of
reliable quantum circuits. Quantum error correction (QEC) provides a fault-
tolerant approach to construct deep, reliable circuits and is believed to be
an essential tool for scaling Dennis  _et al._ ; Fowler  _et al._ ; Bravyi
_et al._ .

A wide range of qubit gate infidelity distributions are observed in practical
devices iOlius  _et al._ ; Hertzberg  _et al._ . A general concern is how a
distribution affects logical qubit performance. More specifically, predicting
relative performance of sets of qubits with different infidelity distributions
is of practical importance for steps like screening selection and acceptance
criteria (e.g., quantifying yield and predicting whether outlier infidelities
will be tolerable). In this context the qubit set would be an appropriately
connected set to implement a quantum error correction code and for which the
infidelity distribution was obtained directly or estimated by indirect means
Zhang  _et al._ (2022). Recent work has begun to address independent and non-
independent, non-identically distributed noise Hanks  _et al._ ; iOlius  _et
al._ ; Clader  _et al._ ; Berke  _et al._ , as well as the existence of
thresholds in the presence of inoperable qubits and gates Strikis  _et al._
(2023); Nagayama  _et al._ ; Fowler and Gidney ; Auger .

To provide insight about how logical error rates depend on changing parameters
of different distributions of physical qubit error rates, we numerically
simulate the sensitivity of logical error rates to changes in parameters of
several important types of non-identical independent distributed noise. Here
we define sensitivity as the change in logical error to any change in a
parameter of a noise distribution. The numerics show trends that provide
qualitative guidance about how to rank sets of qubits intended for QEC codes
such as surface code. Ranking of sets of qubits with non-uniform gate
infidelity distributions is in the context of predicting a relative ordering
of logical error rates for the different sets of qubits (e.g., predicting
which part of a chip will perform a small code better). In the context of
providing practical guidance, we choose to examine circuit-level simulations
of surface codes and compass codes Li  _et al._ (2019) mapped to the heavy hex
latticeChamberland  _et al._ (2020). The heavy hex lattice is chosen, in part,
because of its immediate utility in presently available devices
sundaresan_matching_2022. To provide additional experimental context, we
discuss the relationship of these non-identical error distributions to forms
of decoherence such as energy relaxation in superconducting qubits, $T_{1}$,
which are known to introduce both spatial and temporal infidelity
distributions, Fig. 1 (a) Klimov  _et al._ ; Carroll  _et al._ .

(a)  Sample of measured $T_{1}$s (b)  Heavy hex layout

Figure 1: (a) Violin plots of an illustrative distribution of $T_{1}$
measurements of 20 qubits for which other related statistics of individual
qubits in the device were previously published Carroll  _et al._ . (b)
Schematic of the rotated subsystem surface code (RSSC) for a distance 5
layout. The qubit types are distinguished by color and shape. Data qubits are
black circles and measurement qubits are purple squares.

We begin with a discussion of the error correction codes used in this work.
This includes the first presentation of a rotated subsystem code mapping to
the heavy hex lattice and an improved schedule for the heavy hex code
Chamberland  _et al._ (2020), section II. We then introduce the different
error rate distribution cases used in this work, section III. Key results are
highlighted for each of the cases in section IV, while supporting details are
placed in supporting appendices. We provide some perspective on the meaning of
these results for areas such as pseudothresholds, screening and modularity in
the discussion, section V followed by a brief conclusion.

##  II Quantum error correction on a heavy hexagonal layout

Control and fabrication constraints can impact the yield of large quantum
computing devices based on fixed-frequency transmon qubits Hertzberg  _et al._
. This has led to devices with reduced qubit connectivity, using so-called
“heavy” (or subdivided) lattices where qubits are placed on vertices and edges
of a low-degree planar graph (see Fig. 1b). The lattice’s reduced degree of
connectivity eases physical implementation Hertzberg  _et al._ ; Zhang  _et
al._ (2022).

There is interest to design fault-tolerant operations adapted to these
constraints. For example, Chamberland et al. proposed flag error correction
circuits for surface codes and compass codes on heavy lattices Chamberland
_et al._ (2020). Here we consider two codes adapted specifically to the heavy
hexagonal lattice, a compass code called the heavy hexagon code (HHC)
Chamberland  _et al._ (2020) and the subsystem surface code (SSC) Bravyi  _et
al._ (2013).

(a) HHC gauge operators

(b) HHC stabilizers

(c) RSSC gauge operators

(d) RSSC stabilizers

Figure 2: Heavy-hexagon code (HHC) and rotated subsystem surface code (RSSC).
Data qubits are placed on vertices of each graph. Pauli Z (X) operators are
shown in blue (red), and logical operators are indicated by lines. (a,b) HHC
has weight-2 and 4 gauge operators, weight-$2d$ Bacon-Shor stabilizers, and
weight-4 surface code bulk stabilizers. (c,d) RSSC has weight-3 (triangular)
bulk gauge operators and weight-6 (hexagonal) bulk stabilizers.

###  II.1 Heavy hexagon code (HHC)

The heavy hexagon code is a subsystem stabilizer code Poulin (2005) defined by
the gauge group

$\displaystyle G=\langle$ $\displaystyle
X_{i,j}X_{i,j+1},Z_{i,j}Z_{i,j+1}Z_{i+1,j}Z_{i+1,j+1},$ (1) $\displaystyle
Z_{2m-1,1}Z_{2m,1},Z_{2m,d}Z_{2m+1,d}\rangle$

where $i,j=1,2,\dots,d-1$, $m=1,2,\dots,\frac{d-1}{2}$, and $i+j$ is odd for
the second term, as shown in Fig. 2a. We choose odd distances $d$ throughout
this paper. The corresponding stabilizer group, illustrated in Fig. 2b, is

$\displaystyle S=\langle$ $\displaystyle
X_{i,j}X_{i,j+1}X_{i+1,j}X_{i+1,j+1},X_{1,2m}X_{1,2m+1},$ (2) $\displaystyle
X_{d,2m-1}X_{d,2m},\prod_{j}Z_{i,j}Z_{i+1,j}\rangle$

with $i+j$ even for the first term. The X stabilizers are the same as the
surface code whereas the Z stabilizers are those of the Bacon-Shor code. This
code has a threshold of $p_{th}\sim 0.0045$ on the heavy hexagonal lattice for
errors detected by the surface code stabilizers, and although there is no
corresponding threshold for the Bacon-Shor code stabilizers, low logical error
rates can still be achieved Chamberland  _et al._ (2020). Quantum error
correction has been experimentally demonstrated on the $d=3$ HHC Sundaresan
_et al._ (2023).

In this work, we apply the idea of schedule-induced gauge-fixing Higgott and
Breuckmann (2021) to optimize the total logical error probability of the HHC.
This is a general idea that can be applied to CSS subsystem codes wherein
gauge operators with deterministic eigenvalues are treated as stabilizers
during syndrome processing. For the HHC, the circuits for each gauge operator
measurement round are identical to Sundaresan  _et al._ (2023), but the
schedule may now repeat the same set of gauge operator measurements multiple
times (see Appendix C for details). The decoding algorithm is also identical
to the matching decoder in Sundaresan  _et al._ (2023), but detection events
are defined with respect to the instantaneous stabilizer groups as described
in Higgott and Breuckmann (2021).

###  II.2 Rotated subsystem surface code (RSSC)

The subsystem surface code (SSC) Bravyi  _et al._ (2013) and the standard
surface code are related to each other by a local, constant depth stabilizer
circuit. However, the gauge operators of the SSC have weight-3 or less and
therefore can be measured by circuits that are naturally fault-tolerant.
Higgott and Breuckmann have shown that subsystem surface codes can have high
thresholds of $0.85\%$ under circuit model depolarizing noise Higgott and
Breuckmann (2021), which exceeds the $0.67\%$ threshold of the surface code.

We focus on a rotated form of the SSC Brown  _et al._ (2019) that is defined
by the gauge group

$\displaystyle G=\langle$ $\displaystyle
Z_{i,j}Z_{i,j+1}Z_{t(i,j)},Z_{i+1,j}Z_{i+1,j+1}Z_{t(i,j)},$ (3) $\displaystyle
Z_{2m-1,1}Z_{2m,1},Z_{2m,d}Z_{2m+1,d},$ (4) $\displaystyle
X_{i,j}X_{i+1,j}X_{t(i,j)},X_{i,j+1}X_{i+1,j+1}X_{t(i,j)},$ (5) $\displaystyle
X_{1,2m-1}X_{1,2m},X_{d,2m}X_{d,2m+1}\rangle,$ (6)

where $i,j=1,2,\dots,d-1$, $m=1,2,\dots,\frac{d-1}{2}$, and $i+j$ is odd for
the weight-3 operators, as shown in Fig. 2c. The index $t(i,j)$ refers to the
qubit in the center of the face with corners $(i,j)$, $(i+1,j)$, $(i,j+1)$,
$(i+1,j+1)$, where $i+j$ is odd. A distance $d$ code has $(d-1)$ weight-2 X
(Z) gauge operators and $(d-1)^{2}$ weight-3 X (Z) gauge operators. We use odd
distances $d$ throughout the paper. The corresponding stabilizer group is

$\displaystyle S=\langle$ $\displaystyle
Z_{t(i,j-1)}Z_{i,j}Z_{i+1,j}Z_{i,j+1}Z_{i+1,j+1}Z_{t(i,j+1)},$ (7)
$\displaystyle
X_{t(i-1,j)}X_{i,j}X_{i,j+1}X_{i+1,j}X_{i+1,j+1}X_{t(i+1,j)}\rangle,$ (8)

where $i,j=1,2,\dots,d$ with $i+j$ is even. The operators $X_{i,j}$,
$Z_{i,j}$, $X_{t(i,j)}$, and $Z_{t(i,j)}$ are defined to be identity whenever
$i,j\notin\\{1,2,\dots,d\\}$. Example stabilizers, which are hexagonal in the
bulk, are drawn in Fig. 2d. Since each pair of X and Z stabilizers that
overlaps on four qubits has an associated pair of independent, anticommuting
gauge operators, an RSSC with odd minimum distance $d$ encodes one logical
qubit and $(d-1)^{2}/2$ gauge qubits into $d^{2}+(d-1)^{2}/2$ data qubits.

The RSSC is a promising option for implementing fault-tolerant quantum
computing on a heavy hexagonal lattice for several reasons. First, the gauge
operators can be measured by simple circuits. Second, the RSSC has a
threshold, so it is expected to asymptotically out-perform the compass code.
Finally, techniques for fault-tolerant computing with the surface code carry
over to the subsystem surface code and differ only in implementation details.

The RSSC naturally overlays the heavy hexagonal lattice as shown in Fig. 1b.
We refer to extra qubits for making or mediating measurements as measurement
qubits. Each X gauge operator uses a single measurement qubit on the interior,
and the weight-2 Z gauge operators on the boundary use three measurement
qubits, assuming translation symmetry. Counting these additional $d^{2}+2d-3$
ancillary qubits, we operate a distance-$d$ code using a subset of the lattice
containing $5d^{2}/2+d-5/2$ qubits.

(a) X-type gauge operators

(b) Z-type gauge operators

Figure 3: Circuits for measuring gauge operators of the subsystem surface code
on the heavy-hexagon lattice

Due to the natural mapping to the heavy hexagon lattice, the X gauge operators
can be measured without extra gates using the circuit in Fig. 3a. Bit-flip
errors on the measurement qubit propagate to at most one bit-flip error on the
data modulo the gauge group, so there are no space-like hook errors. However
the circuits can produce space-time hook errors from a single fault event,
just like the surface code.

The Z gauge operators are more difficult to measure but each can be measured
using the circuit in Fig. 3b. By inspecting all single-fault events, we
conclude that this circuit has the property that any single fault gives rise
to at most one X and one Z error on the data modulo the gauge group. Since
these are correctable errors for CSS codes, the syndrome measurement circuit
is fault-tolerant in the following sense. If $w$ faults occur then the output
state deviates by at most $w$ X errors and $w$ Z errors modulo the gauge
group. These X and Z errors can be corrected independently as long as there is
no mechanism to convert them into Y errors.

On the heavy hexagon lattice, we find that the RSSC has a threshold of nearly
$0.3\%$ for circuit-level noise. Additional details and a comparison with the
HHC can be found in Appendix C.

###  II.3 Other relevant codes

It is worth mentioning other codes that can be directly implemented on the
heavy hex layout, which are not included in this study. In particular, Floquet
codes Hastings and Haah (2021), Floquet color codes Kesselring  _et al._
(2022), matching codes Wootton (2022) and repetition codes Liepelt  _et al._
(2023). In all these cases, syndrome measurements are made via two-qubit
parity operators, for which the qubits on the edges of the heavy hex lattice
can be used as auxiliaries.

Repetition codes are very much an outlier in this group since they protect
only a logical bit rather than a logical qubit, and have a distance equal to
the number of data qubits. We therefore do not include them in our study,
since the results are likely be unrepresentative of truly quantum codes.

For Floquet codes, Floquet color codes and matching codes, though a direct
implementation of these is possible on the heavy hex lattice, they are more
ideally suited to systems in which direct parity measurements are possible
Paetznick  _et al._ (2023). Compared to this, implementation with the heavy
hex would result in a significant decrease in threshold and performance, and
significantly less resilience when there is a bias towards different noise
sources Hetényi and Wootton (2023). It is for this reason that we do not
include such codes in this study.

##  III Noise models and methods

In this section we introduce general noise models used in this work. For all
models, the circuits are decomposed into a set of operation types,
$\mathcal{C}$ = {cx,h,s,id,x,y,z,measure,initialize,reset}, the first seven
abbreviated operation types being the cnot, hadamard, phase, identity, $x,y,z$
rotations, respectively. For iid, each of the operation types experiences a
fault with a depolarizing noise error rate specific to the operation type.
Non-identical, independent, distribution (niid) assignments used in this work
will be further detailed below. Stabilizer simulations of the noisy circuits
are done using the Qiskit software stack.

Decoding is done using minimum weight perfect matching (MWPM) Dennis  _et al._
. The decoder is provided with either all location specific error rate
information, called aware condition, or limited to using the lattice averages
for each operation type, called naive. Briefly the MWPM method forms a
decoding hypergraph of nodes representing Z or X stabilizer measurements.
Edges of the graph are weighted according to the available error rate
information. A perfect matching algorithm returns a minimum weight set of
connecting edges for each syndrome of stabilizer measurements at the nodes
that highlight the presence of an error. Aware decoding is used unless
otherwise noted. Further details of the simulation method can be found in
Sundaresan et al. Sundaresan  _et al._ (2023).

We simulate logical error rates, $\langle p_{out}\rangle$, for cases composed
of different conditions defined from: (i) type of lattice distribution of
error rates (i.e., noise model), (ii) code, (iii) decoder type (aware or
unaware), and (iv) which distribution parameter is varied, Table 1.

We label the distributions categories as: uniform, normal, non-normal and
location specific. The uniform signals that error rates for the faulty circuit
operation, $cx,h$ and $id$ (unless otherwise noted), are set to the same input
error rate, $p_{in}$, at all lattice locations. The errors are modeled as
depolarizing noise. A background error rate, typically negligibly small
compared to the $p_{in}$ is used for the other faulty circuit operations for
these test cases unless otherwise noted.

The normal distribution signals that each faulty $cx,h$ and $id$ location is
assigned an error rate drawn from the absolute value of a normal distribution
(unless otherwise noted). The normal distribution is defined with an average
error rate, $\langle p_{in}\rangle$ and a standard deviation
$\sigma=\alpha\langle p_{in}\rangle$. The errors are again modeled as
depolarizing noise. Two steps of averages are done to obtain an output logical
error rate. In the first step, the results are averaged for a single device
instance of unique fixed error rates at each faulty location. Each shot of a
circuit includes $d$ rounds of zzxx(xxzz) stabilizer measurements for X(Z)
intialization and measurement, respectively. For the second step, an average
is formed over a number of device instances that we define as a new
distribution of error rates from the error distribution category.

A non-normal, reciprocal-normal distribution is considered in this work that
is a proxy, for example, for coherence limited errors and more generally
distributions with heavy tails at higher error rates, see appendix A and B. We
also examine cases where one to four locations have error rates that are
either fixed at $p_{in}=0.5$ or varied between $\langle
p_{in}\rangle<p_{in}<0.5$, labeled location (e.g., ’bad’ sites). An errant
location assigns increased error rates for the operations $h,id$ and the two
qubit operations, $cx$, connected to the qubit site. The rest of the lattice
error rates are constant, $\langle p_{in}\rangle$. Further details for each
case are indicated in the results sections.

Variable |  $p_{in}$ |  $\sigma$ |  N loc. |  Code |  Decode
---|---|---|---|---|---
\Distribution |  |  |  |  | comp.
Uniform | S.II,A.C | - | - | R,H | -
Normal | - | S.IV.3, A.D | - | R,H | S.IV.3
Non-normal | S.IV.1 | S.IV.1 | - | R | -
Location | S.IV.2, A.F | - | S.IV.2 | R,H | A.F
Table 1: Table of section numbers and codes examined cross referenced to the
category of error distribution and parameters varied. The letters S or A
indicate section or appendix, respectively, and R or H indicate, RSSC,
respectively. - indicates not applicable or unavailable.

##  IV Results

###  IV.1 Impact of normally distributed coherence times on logical error
rates

In this section we describe results from a parameterized error distribution
for which the error rate is reciprocally related to a normally distributed
random variable, $\mathbf{p}\propto 1/|\mathbf{X}|$. A physical motivation for
this distribution is randomly distributed coherence times, like energy
relaxation with a coherence time $T_{1}$. We assign an error rate
$\mathbf{p}=(1-e^{\tau/|\mathbf{T_{1}}|})\sim\tau/|\mathbf{T_{1}}|$, where
$\mathbf{T_{1}}$ is a random variable pulled from a normal distribution with
mean, $\langle T_{1}\rangle$, standard deviation, $\sigma$ and $\tau$ is a
constant representative of an effective gate time of the qubit operation. We
simulate output error dependence on the standard deviation of the normal
distribution, $\sigma$, for the RSSC code.

The logical output error rate dependence on uniform $p_{in}$ is shown in Fig.
4 (a) (see appendix C for more details) and the dependence on varying $\sigma$
is shown in Fig. 4 (b). We show the dependence normalized to the average and
therefore vary a parameter $\alpha=\sigma/\langle T_{1}\rangle$. The average
error rate in Fig. 4 (b) is $\mu=\tau/\langle T_{1}\rangle=10^{-3}$, an
average cx error rate foreseeable in the near future Stehlik  _et al._ (2021).
We show the logical error for both X or Z initialization and measurement
cases. The output error rate shows an onset of increasing average logical
error rate when $\sigma$ approaches $\alpha\sim 0.2-0.5$.

We now discuss the numerics in the context of a simple model that illustrates
that the onset of increasing $\langle p_{out}\rangle$ may be understood as the
onset of the mean of the input error rate, $\langle p_{in}\rangle$ increasing
(i.e., $\mu\neq\langle p_{in}\rangle$). We first observe that numerical
simulations of logical error rates for normally distributed error rates show
no dependence on the standard deviation of the distribution, see appendix D,
in contrast with the $\sigma$ dependence of the reciprocal normal
distribution. To understand this observation, we note that a phenomenological
model of the repetition code with normally distributed error rates results in
an $\langle p_{out}^{rep}\rangle=3\bar{\epsilon}^{2}-2\bar{\epsilon}^{3}$
irrespective of the standard deviation of the error rates, see appendix E.
Here we define an average output error of the phenomenological model, $\langle
p_{out}^{rep}\rangle$, with contributing data sites at indices $i$. Each data
site has normally distributed error rates, $\epsilon_{i}$ and an average error
rate of all the data, $\bar{\epsilon}$. The phenomenological model provides
the insight that random distributed error rates of a faulty type of component
will result in an averaged contribution to the output error rate consistent
with the numeric result. We conjecture that for a wide range of uncorrelated
error rates the average logical error is a function of the average physical
qubit error rate, $\langle p_{out}\rangle=F(\langle p_{in}\rangle)$, in
contrast to dependence on higher moments of the distribution. On the other
hand, we also note that although $\langle p_{out}^{rep}\rangle$ is not
sensitive to changes in $\sigma_{\epsilon}$, the ’device-to-device’ variance
of $\langle p_{out}^{rep}\rangle$ is predicted to be dependent on
$\sigma_{\epsilon}$, see appendix E.

We return to the observation that the logical error rate increases for
$\alpha\sim 0.2-0.3$ when the error rate distribution of the faulty locations
is proportional to $\tau/|\mathbf{T_{1}}|$. The mean of such a distribution
depends on the harmonic mean of the $\mathbf{T_{1}}$ distribution. The average
is done for a normally distributed $T_{1}$ distribution,
$P(\mathbf{T_{1}}=T_{1})=\frac{1}{\sqrt{2\pi\sigma}}e^{\frac{-(T_{1}-\langle
T_{1}\rangle)^{2}}{2\sigma^{2}}}$, see appendix B for more details. The
$T_{1}$ values sampled above versus below $\langle T_{1}\rangle$ are unevenly
weighted and this leads to a $\sigma$ dependence of $\langle p_{in}\rangle$
despite $\langle T_{1}\rangle$ being independent of $\sigma$ consequently
leading to an increase in $\langle p_{out}\rangle$.

This behavior is of general interest for any error process that depends
reciprocally on a random process such as electronics noise impacting dephasing
(e.g., $T_{2}$), although relative weights in the tails could lead to a
deviation from the simple picture that the output error rate can be predicted
by the average input error rates as if they were uniform error rates. Effects
of outliers will be discussed in the following section.

(a)  RSSC $p_{out}$ for uniform $p_{in}$ (b)  RSSC $p_{out}$ response to
varying $\sigma$

Figure 4: (a) Logical output error rates for the RSSC for numerical
simulations of distance 3, 5, and 7. The (red) X-measurement and (blue)
Z-measurement cases are both shown. (b) Logical output error dependence on an
infidelity distribution generated as $p_{in}=\frac{\tau}{T_{1}}$ for which
$T_{1}$ is a normally distributed variable. The error rate is sampled uniquely
for each operation and held constant throughout the simulation for distance 3
and 7. X initialization, measurement (red) an Z initialization, measurement
(blue) bases. The simulations were run for 4, 64 lattice instances, 250k, 10k
shots each and $\langle p_{in}\rangle=2\times 10^{-4},1\times 10^{-3}$,
respectively. A single instance for the X and Z basis are compared to the
average cases $\langle X\rangle$ and $\langle Z\rangle$. The standard
deviation of the distribution is normalized to the mean input error rate
(i.e., $\langle p_{in}\rangle=\frac{\tau}{\langle T_{1}\rangle}$,
$\sigma=\alpha\langle\epsilon\rangle$).

###  IV.2 Dependence of output error on high infidelity outliers at specific
qubit sites

In this section we examine the effect of changing error rate at a limited
number of sites while the rest of the error rates for faulty locations are
held constant. We ask, for example, how sensitive the logical error rate is to
deviations of error rate at a single faulty location from the average?

We first examine the effect of increasing the error rates, $p_{in,0}$, of h,
id and all cx operations that include the data qubit located at site 0, Fig. 1
(a). All other error rates are set to $5\times 10^{-4}$. A S-curve behavior is
observed as $p_{in,0}$ is increased. While $p_{in,0}$ is similar to $\langle
p_{in}\rangle$, little effect is observed on $p_{out}$, consistent with
$\langle p_{in}\rangle$ not being significantly changed, see discussion above.
At higher $p_{in,0}$, $p_{out}$ begins to rise and then saturates. Rapid
increase in output error rate begins at an input error rate of $\sim 10^{-2}$
approximately 40 times larger than $\langle p_{in}\rangle$ for $d=5$. A single
data qubit location can be relatively faulty compared to the devices average
(i.e., $\langle p_{in}\rangle\sim 5\times 10^{-4}$) before the output error
rate begins to be substantially degraded. The maximum increase in error is
bounded further by the commensurate loss in distance of the code.

To further illustrate the effect of ’knocking out’ data qubits, we simulate
output error as a function of uniform input error for instances where 0 to 4
data qubits are set to $p_{in}=0.5$, Fig. 5. We note that the sites 0 to 4 are
along the logical Z operator in the RSSC. The measure X output error jumps
after the addition of a single ’bad’ qubit at site 0 and ouput error increases
quasi-linearly with the device $\langle p_{in}\rangle$, Fig. 5 (b). The end
points of the sensitivity analysis of output error to varying the $p_{in,i}$
of a single ’bad’ qubit at site $i$, Fig. 5 (a), are identifiable in Fig. 5
(b) as indicated. We infer that there are similar S-curves between the other
points in Fig. 5 (b).

Increasing the number of ’bad’ data qubits along the Z logical operator leads
to monotonic jumps in output error rate. Qualitatively this could be
interpreted as the sequential reduction of the effective distance of the code
through ’knocking out’ code qubits along a logical operator. The 0 site is
along both the logical X and Z operators. ’Knocking out’ the 0 site with a
’bad’ qubit reduces the correcting power of the logical qubit for both X and Z
cases. The output error rate of the Z-measurement does not increase very
rapidly, however. This behavior qualitatively may be understood as the code
maintaining a $\sim 4$ distance correction for X-like errors combined with a
lack of sensitivity of the logical Z-measure to logical-Z-like errors. Further
discussion of site specific sensitivity can be found in appendix F.

###  IV.3 Improved decoding with non-uniform edge weights

Minimum weight perfect matching (MWPM) is used for decoding in the
simulations. Weights for each edge in the decoder graph are assigned according
to available information about error rates. We now investigate the decoder
performance for the case that error rates for each gate operation are
available, aware decoding (i.e., Dijkstra algorithm applied to unique edge
weights in the decoding graph), in contrast to assuming an average error rate
for the gate operations, naive (i.e., edge weights using the average error
rate for i, h, cx operations).

An important way that decoding can improve is through resolving ambiguous
syndromes for error chains greater than $(d+1)/2$. Random error rate
distributions can offer ’tie breaking’ information (see for example appendix
G). Improved error rates using aware decoding for random distributions for a
phenomenological analysis have already been reported Tiurev  _et al._ (2023).
The prominence of ambiguous cases that can be resolved using this information
is likely code dependent. To further probe the question of relative decoder
effectiveness of aware decoding, we investigate the response of a heavy hex
code (HHC) to these two different decoding approaches. The HHC is a hybrid of
a surface code and compass code with flag qubits Chamberland  _et al._ (2020);
Sundaresan  _et al._ (2023). This combination provides a useful relative view
of the effectiveness of aware decoding for these two different codes.

Simulations were carried out for aware and naive and were done for the the
$|0\rangle$ (Z) and $|+\rangle$ (X), four different cases, for several
distances, Fig. 6. The Z basis corresponds to stronger sensitivity to the
surface code like checks and the X basis corresponds to compass code like
checks with flag qubits. In both X and Z bases, the aware decoder reduces
$p_{out}$ compared to the naive case and the benefit increases with both
decreasing $p_{in}$ and increasing $d$, consistent with predictions from
phenomenological modeling Tiurev  _et al._ (2023). The reduction in $p_{out}$
is, however, relatively weak for x basis cases and is in general relatively
small for these small distance codes. Qualitatively weak improvements from
decoder improvements for the X basis were reported in experiments on the $d=3$
HHC even when applying maximum likelihood methods. One contribution noted in
that work was that the Z check circuit is susceptible to introducing a high
rate of Z errors on the data, particularly because of how deflagging is done,
which is difficult to improve with better decoding Sundaresan  _et al._
(2023).

Figure 5: (a) Output error rate dependence on a ’bad’ qubit at site 0 (see
Fig. 1(a) for distance 5 using $\langle p_{in}\rangle=5\times 10^{-4}$. (b)
Output error for distance 5 with $p_{in}=0.5$, ’bad’, data qubits at positions
0, 1, 2, 3 and 4 (Fig. 1 (a)) and variable $\langle p_{in}\rangle$ for gate
operations associated with qubits sites that do not interact with the ’bad’
sites. Two sets of 4 lines are shown. From lowest to increasing output error,
the 0 to 4 sites are populated with ’bad’ qubits. Red indicates X-measurement
and blue indicates Z-measurement.

##  V Discussion

###  V.1 Screening

During the fabrication process of QPUs, checkpoints can be established where
infidelities of qubit operations can be estimated Zhang  _et al._ (2022).
Imperfect fabrication processes lead to sets of QPUs each with their own
estimated infidelity distributions. Ranking of the QPUs within these sets is
useful for selection of which QPUs to ultimately deploy. A predictive pre-
selection is desirable, all other considerations assumed to be the same.In the
context of these findings, a paradigm of ranking by average input error rate
of a dominant faulty circuit operations (e.g., two qubit gate operations) may
therefore be an effective starting strategy in the absence of other supporting
indicators like logical qubit error simulation.

This strategy might be further refined in the presence of a model for the
logical error rate dependence on average input error rate, for example,
circuit simulations of a particular code and device layout. In such
circumstances, more quantitative evaluation of the sensitivity to differences
in average input error, QPU to QPU, come into consideration including a better
understanding of shot to shot standard deviation in the logical error rate.

###  V.2 Sensitivity to ’bad’ location

The impact of one or a limited number of ’bad’ locations is a common question
due to many factors ranging from fabrication imperfections to instability in
device performance (e.g., spectral diffusion of two level systems Carroll  _et
al._ ). Here we loosely define a ’bad’ location as a faulty location for which
the infidelity of the gate operation appears to be an outlier relative to the
average error rate.

We have observed, in section IV.2, that the sensitivity of the logical error
can be relatively weak to an outlier location until the error rate is
appreciably larger than the contribution of the average error distribution.
The ’bad’ location contribution to the logical error rate is, furthermore,
limited as the error rate saturates at roughly the equivalent of the reduction
of the code distance by one.

For a simplified order of magnitude estimation example, we might consider
identification of ’bad’ two qubit gate locations by assuming a case where the
two qubit gate locations are the dominant error rate for the logical error.
Then a two qubit location is identifiable as a ’bad’ location when its error
rate is: $\epsilon_{bad}\sim N_{q}\bar{\epsilon}$, where $N_{q}$ is the number
of qubits in the encoding. In general the average error distribution has many
contributions from the different circuit operations. Each of the operation
types have their own relative contributions to the logical error rate.

###  V.3 Time dynamic noise

Error rates are time dependent in realistic devices. There are a number of
sources of time dependence Witzel  _et al._ ; Müller  _et al._ ; de Graaf  _et
al._ ; Krantz  _et al._ (2019). A natural question is what logical error rates
to expect in the presence of time dynamic noise. In many cases the time
dynamics are correlated, which is out of the scope of this analysis. This
work, however, does provide insight about the limit of uncorrelated time
fluctuations. In this context, the observation that the average logical error
rate is not dependent on higher moments of the distribution provides an
indication that the logical error rate will converge to an average based on
the average of the input error rates when uncorrelated noise processes are
also stationary, while the variance of the logical error rates, in contrast,
will depend on higher moments. It is left to future work to establish how
close this limit approximates experimental cases that can have time and space
correlations.

###  V.4 Pseudothresholds and guidance for design

A device’s qubits performance relative to the pseudothreshold of a code is a
predominant concern for design, fab and operation of quantum error correction
on a QPU. Pseudothresholds are often estimated using simulations with forms of
uniform error rates for types of circuit operations (e.g., two qubit
operations). How to assess an actual device’s non-uniform infidelity
distibution relative to a code’s pseudothreshold, particularly time varying
distributions, without measuring or simulating the specific case (when
measurement is not readily available) is a practical problem. This work
provides the observation that for uncorrelated noise and surface-like codes,
$\langle p_{out}\rangle=F(\bar{\epsilon})$ over a relative wide error rate
range. The concept of a pseudothreshold therefore is also applicable when
framed as an outcome of an average error rate of faulty locations in the error
correction circuit. This observation may also be of utility in the context of
system design, for which variability in system components are their own source
of qubit operation infidelity distributions. System designs therefore will
need to assess impact of outliers and variances of their component
performances relative to their targets. Analysis based on the average error
rates would greatly simplify the challenge of multi-distribution problems, in
contrast to analysis of the contributions of multiple distribution each with
their own multi-moment parameterizations.

###  V.5 Modularity

Interest has increased recently in noisy connections between devices to form
extended modular quantum error corrected patches Ramette  _et al._ ; Auger ;
Nickerson  _et al._ . The impact of a sparse number of high error rate
locations (i.e., connection points) on output logical error rates is of
central interest in order to provide guidance about design and fabrication
tolerances. The logical $\langle p_{out}\rangle$ dependence on ’bad’ qubit
sites placed along a logical operator, in this work, is of tangential
relevance as it highlights that if the intersecting ’seam’ between modules is
placed along a logical operator, it may unnecessarily exaggerate the
deleterious impact of the ’seam’ compared to staggering the intersections in a
less spatially correlated mode (i.e., sparse random).

(a)  aware vs. naive (b)  Standard deviation (c) Ratio =
$\frac{p_{\textit{naive}}}{p_{\textit{aware}}}$

Figure 6: (a) Comparison of aware, blue, and naive, green, decoding of
lattices with i,h and cx operations selected randomly from a normal
distribution for multiple distances of the heavy hex code. All other operation
error rates are set to a negligible contribution. Results for initialization
and measurement in the $|0\rangle$ and $|+\rangle$ basis are labeled as z and
x, respectively. The standard deviation of the noise is swept and is set as
$\sigma=\alpha\langle p_{in}\rangle$. The cx, i and h operations are set to a
constant mean error rate, $\langle p_{in}\rangle=10^{-3}$. Each point
represents 512 iterations of 50k shots. Standard error bars are approximately
the size of the symbols. (b) The standard deviation of $\langle
p_{out}\rangle$ from the 4 iterations. (c) The ratio between aware and naive
decoders.

##  VI Conclusion

We have studied independent non-identical noise distributions at circuit level
for proxy cases of the surface and compass code families. The circuits are
mapped to a heavy hex layout and for this work it is shown how to map a
rotated subsystem surface code (RSSC) onto the heavy hex lattice. These cases
are of general interest to understand surface and compass code trends, while
the particular cases are of direct interest to the application of
superconducting qubit quantum error correction Sundaresan  _et al._ (2023).

A central result is that the $\langle p_{out}\rangle\approx F(\langle
p_{in}\rangle)$ for the distributions studied in this work, where $\langle
p_{in}\rangle$ is the average error rate of one or a combination of dominant
faulty circuit operations. Notably, $\langle p_{out}\rangle$ is not a strong
function of higher moments of the error distribution over the range of
parameters studied. The standard deviation of $p_{out}$, $\sigma_{p-out}$,
however, does depend on higher moments of the distribution. The independence
of $\sigma_{p-out}$ is consistent with a phenomenological model of the
repetition code which, for a three qubit example is: $\langle
p_{out}^{rep}\rangle=3\bar{\epsilon}^{2}-2\bar{\epsilon}^{3}$ (see appendix
E).

The effect of outliers is examined with simulations of the sensitivity to
varying error on a single or few ’bad’ sites. The simulations highlight that
$\epsilon_{bad}\sim\mathcal{O}(N_{q}\bar{\epsilon})$ begin to appreciably
change the logical error rate. The logical error increase, furthermore, is
limited and saturates. The saturation qualitatively behaves as if the code
distance is reduced by the introduction of the ’bad’ qubit,
$\epsilon_{bad}\gg\mathcal{O}(N_{q}\bar{\epsilon})$ .

For reference, we discuss a proxy physical example of non-identical error
rate, decoherence from energy relaxation which, for example, produces a time
dynamic distribution of error rates in superconducting qubits. A normal
distribution loosely fits the histogram frequency of $T_{1}$ measured across
devices. We use this example to highlight the importance of the harmonic mean
of coherence times as being a relevant measure for estimating $\langle
p_{out}\rangle$. A number of studies in the literature emphasize normally
distributed infidelity distributions, which may overlook the importance of the
role of higher moments in $p_{in}$ (e.g., $\sigma$). Error rates are often
reciprocally dependent on coherence times, for example, which are determined
by common intrinsic (e.g., fab defects) and extrinsic (e.g., electronics
noise) distributions of noise.

Tantalizingly, the non-identical distributions of error rates offer an
opportunity in decoding approaches like minimum weight perfect matching.
Decoding of syndromes can be hampered by ambiguities for error chains of
$(d+1)/2$ or greater with the same syndrome. Non-uniform error rates introduce
additional information, aware, that might improve decoding relative to naive.
As with previous investigators, we observe an improvement in decoding with
aware information relative to naive. For normally distributed error rates
across the probe operations (i.e., $i,h$ and $cx$ operations), the improvement
increases with increasing $\sigma$ of the error rate distribution. It is a
possibly interesting future direction to assess how much of a decoding utility
the non-uniformity can be.

In summary, we have numerically studied non-identical, independent noise
distributions for surface and compass-like codes at a circuit level. We
observe several trends that provide heuristic guidance regarding the role of
the average in contrast with higher moments of error rate distributions play
on the logical error. We also probe the sensitivity of logical error rates to
one or a few outlier error rate device locations, which provides additional
insight about the role of ’bad’ locations and identification of error rates at
these locations that cross-over to being non-negligible relative to the weight
of the rest of the error contributions in the circuit. These observation
provide practical insights into areas ranging from screening, modularity and
system design.

##  VII Acknowledgements

We thank Ted Yoder for improvements to the circuit shown in Fig. 3b. We thank
Kenny Tran for access to computing resources for the numerical simulations. We
acknowledge insightful discussions with A. Corcoles, B. Brown, L. Govia, M.
Takita, D. McKay, J. Tersoff and T. Yoder. Parts of this research was
sponsored by the Army Research Office accomplished under Grant Number
W911NF-21-1-0002 and by IARPA under LogiQ (contract W911NF-16-1-0114). The
views and conclusions contained in this document are those of the authors and
should not be interpreted as representing the official policies, either
expressed or implied, of the Army Research Office or the U.S. Government. The
U.S. Government is authorized to reproduce and distribute reprints for
Government purposes notwithstanding any copyright notation herein.

##  VIII References

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##  Appendix A $T_{1}$ coherence example

###  A.1 Averaged $T_{1}$ distribution

Energy relaxation is a leading source of decoherence in superconducting
qubits. A spread of relaxation times, $T_{1}$, are observed across many qubit
devices at present. The single and two qubit gate infidelities depend on these
coherence times leading to a source of non-uniform infidelity distribution
across a device. The energy relaxation rates, furthermore, fluctuate in time
over a wide range of time scales Klimov  _et al._ ; Müller  _et al._ ;
Paladino  _et al._ .

An example distribution for a 20 qubit device is shown in figure 1 (a) Carroll
_et al._ . The distribution is representative of several common features of
$T_{1}$ distributions observed in devices. We note that the histogram and
’normal’-like distribution is, however, not necessarily an absolutely complete
representation of the actual $T_{1}$ distribution. That is the distribution
(1) may include some cases that are non-exponential decays due to strong
couplings to defects; (2) may miss some fluctuations due to limited
measurement time resolution of a non-white power spectral distribution; and
(3) may miss some cases in the low coherence time tail of the distribution
because the $T_{1}$ was immeasurably short.

In this work we primarily focus on the normal-like part of the distribution as
this is sufficient to highlight some of the key implications of independent
but non-identically distributions (i.n.d.) for quantum error correction. We
note that there are a variety of sources of non-uniform infidelity
distribution in devices Krantz  _et al._ (2019); Paladino  _et al._ . The
energy relaxation distribution is a proxy for the more general problem of non-
uniform, fluctuating infidelity distributions.

###  A.2 How normal are the measurable $T_{1}$ distributions?

The $T_{1}$ times for all qubits were collected for a period over 9 months, a
set of 6140 values, shown in the main text, Fig. 1. A representative histogram
of a $T_{1}$ measured daily for just one qubit is shown in Fig. 7. A
Kolmogorov-Smirnov null hypothesis test was done for each qubit, for which the
null hypothesis is that the $T_{1}$ distribution is indistinguishable from a
normal distribution. The test statistic was 0 and the p-value was greater than
0.999 indicating the normal distribution was indistinguishable from the
measured $T_{1}$ distribution to the limits of accuracy of this statistical
measure. We caution, as noted above, that the $T_{1}$ distribution has some
bias through neglect of outlier instances when the qubit’s $T_{1}$ is so short
that it is immeasurable.

The standard deviation of the $T_{1}$ distribution for each qubit is shown in
Fig. 7 (b). Most qubits show standard deviations of $\sim 0.25-0.3\langle
T_{1}\rangle$. The details of the $T_{1}$ measurement are described in a
previous publication Carroll  _et al._ .

Figure 7: (a) histogram of $T_{1}$s measured on qubit labeled 0 of the 20
qubits described in Carroll et al. Carroll  _et al._ and in Fig. 1 in the main
text. (b) The ratio $\sigma_{T1}/\langle T_{1}\rangle$ plotted according to
its corresponding the qubit’s $\langle T_{1}\rangle$ for the device.

##  Appendix B Mean of distribution dependent on reciprocal of normally
distributed random variable

Many error rates depend on the reciprocal of a parameter, for example when the
error rate is dominated by coherence time it can be approximated as
$\epsilon\propto 1/\tau$, where $\epsilon$ is the error rate and $\tau$ is a
coherence time.

In this appendix we are concerned with the error rate mean when the
’reciprocal parameter’ is a normally distributed random variable. We will show
that for a normally distributed ’reciprocal parameter’ (e.g., $T_{1}$) that
the new random variable’s mean is no longer constant rather it depends on
$\sigma$, the standard deviation of the ’reciprocal parameter’.

We start by defining a ’reciprocal parameter’ random variable $T_{1}$ that is
normally distributed:

$\displaystyle
P({\bf{T_{1}}}=T_{1})=\frac{1}{\sqrt{2\pi\sigma}}\exp{\frac{-(T_{1}-\mu)^{2}}{2\sigma^{2}}}$
(9)

where $\mu$ is $\langle T_{1}\rangle$ and $\sigma$ is the standard deviation
of the distribution.

We now consider a new random variable, the error rate, formed with the
’reciprocal parameter’, ${\bf{e}}=t_{gate}/{\bf{|T_{1}|}}$. The probability of
a qubit site having an error rate of $P({\bf{e}}=e(T_{1}))$ is therefore
$P({\bf{T_{1}}}=T_{1})$.

The mean of the error rate distribution is:

$\displaystyle\langle
e\rangle=\int_{-\infty}^{\infty}\frac{\tau}{T_{1}}P(T_{1})\,dP(T_{1})$ (10)
$\displaystyle=\int_{-\infty}^{\infty}\frac{\tau}{T_{1}}\frac{1}{\sqrt{2\pi\sigma}}e^{\frac{-(T_{1}-\mu)^{2}}{2\sigma^{2}}}d(T_{1})$

To see analytically that the mean will depend on $\sigma$ we evaluate the
integral in the approximation of very small $\sigma$ for which we may Taylor
expand $\frac{\tau}{T_{1}}$ around $\mu=\langle T_{1}\rangle$. This
simplification allows integration over all values but is implicitly inaccurate
for $T_{1}$ approaching 0, however, it qualitatively shows the trend for the
largest weight of the distribution. The average error in this simplistic
approximation is:

$\displaystyle\langle
e\rangle\approx\frac{\tau}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\left[\frac{1}{\mu}-\frac{(T_{1}-\mu)}{\mu^{2}}+\frac{(T_{1}-\mu)^{2}}{\mu^{3}}+\dots\right]$
(11) $\displaystyle\times
e^{\frac{-(T_{1}-\mu)^{2}}{2\sigma^{2}}}\frac{dT_{1}}{\sigma}$

Using well established identities for the integration of the product of the
Gaussian function and polynomials, for example:

$\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}x^{2}e^{-(ax)^{2}}dx=\frac{1}{2a}\left(\frac{\pi}{a}\right)^{\frac{1}{2}}$
(12)

where $a=\frac{1}{2}$, $x=\frac{(T_{1}-\mu)}{\sigma}$ and
$dx=\frac{dT_{1}}{\sigma}$. To 2nd order, dropping higher terms, the average
of the error becomes:

$\displaystyle\langle
e\rangle\approx\tau\left(\frac{1}{\mu}+\frac{\sigma^{2}}{8\mu^{3}}\right)$
(13)

for which we see that $\langle e\rangle$ increases as a quadratic function of
$\sigma$. Odd polynomials do not contribute to the integration over the even
Gaussian interval and better approximation would include higher even order
terms. Overall the 2nd order approximation grossly underestimates the
quantitative dependence of $\langle e\rangle$ on $\sigma$ because the
approximation neglects contributions from the $T_{1}$ values closer to 0 that
can be very large but qualitatively confirms that an anticipated increase in
average error as $\sigma_{T1}$ increases. For reference we numerically
evaluate $e=\frac{\tau}{T_{1}}$ truncated for $T_{1}>1~{}\mu$s and
$\tau=100~{}ns$ in Fig. 8 and show its dependence on $\sigma$.

(a) Comparison of distributions (b)  Mean shift by $\alpha$

Figure 8: (a) Histogram of error rates produced with either a normal
distribution ($\mu_{0}=10^{-3}$ and $\sigma=0.3\mu$) or from a scaled
reciprocal normal distribution with the same $\sigma$. That is, $f=$
normal$(\mu_{1},\sigma)$ and the error is $1/f$, where $\mu_{1}=1/\mu_{0}$.
(b) The resulting mean error of the distribution as a function of the standard
deviation, $\sigma=\alpha\mu$.

##  Appendix C Numerical threshold estimates for the RSSC on the heavy-hexagon
lattice

The RSSC syndrome measurement circuit is a sequence of X and Z gauge operator
measurement cycles. The X measurements are scheduled in 3 parallel CNOT layers
using time steps shown in Fig. 10. The Z measurements occur in two stages
where we measure all of the left-pointing triangles followed by all of the
right-pointing triangles. In each stage, the circuits act on disjoint sets of
qubits, so we can schedule them independently according to Fig. 9.

(a) Right-pointing gauge operators

(b) Left-pointing gauge operators

(c) Boundary gauge operators

Figure 9: Scheduled circuits for measuring Z-type gauge operators. (a) The
second auxiliary qubit is reset in the prior circuit, so we need not account
for idle time during that reset operation. (b) Similarly, the first auxiliary
qubit is reset within the prior. (c) On the boundary there are three auxiliary
qubits and two data qubits. Figure 10: Measuring RSSC checks on the heavy
hexagon lattice. Each gauge operator type (X, Z) has two distinct orientations
(up/down, left/right). X gauge operators (red) can be measured in parallel
using the circuit in Fig. 3a. The numbers label the time step for each CNOT
gate. Z gauge operators (blue) with the same orientation can be measured in
parallel with the circuit in Fig. 3b since those operators involve disjoint
data and auxiliary qubits.

To evaluate the logical error probability, we construct quantum memory
circuits that prepare, store, and measure logical $|0\rangle$ or $|+\rangle$
states. These circuits prepare all of the data qubits in either
$|0^{n}\rangle$ or $|+^{n}\rangle$, respectively, measure X and Z gauge
operators in some sequence, and measure all of the data qubits in the $Z$ or
$X$ basis, respectively. We consider sequences of X and Z gauge measurements
represented by the string $(Z^{s}X^{s})^{t}$ for positive integers $s$ and
$t$. This string means we apply the Z measurements $s$ times followed by the X
measurements $s$ times, and the whole sequence is repeated $t$ times. We set
the total number of ZX measurements to $st=12$ and iterate over $s=1,2,3,4$.
As in Higgott and Breuckmann (2021), we find that $s=2$ optimizes the logical
error rate. In this case, roughly half the syndrome bits are computed from the
product of eigenvalues of a pair of gauge operators, while the other half is
given directly by the eigenvalues of the gauge operators. The matching
subroutine in the decoder is implemented using PyMatching Higgott (2021).

We simulate these circuits using a standard Monte-Carlo simulation wherein we
sample from a collection of faulty circuits. Faulty gates are modeled as ideal
gates followed by a Pauli channel that applies with probability $p$ a
uniformly random non-identity Pauli error on all output qubits. Each type of
operation and idle qubit can fail. Faulty preparations and measurements flip
their outputs with probability $p$. The simulation is considered a failure if
the logical measurement outcome is incorrect. The logical qubit we store in
the memory is exposed to a logical Pauli channel with parameters $p_{x}$,
$p_{y}$, and $p_{z}$. When we prepare and measure in the $Z$ basis, the
simulation provides an estimate of the logical Pauli error probability
$p_{x}+p_{y}$, and when we measure in the $X$ basis we estimate $p_{z}+p_{y}$.

The logical $X$ and $Z$ operators have minimum weight $d$ and are related by a
transversal Hadamard gate followed by a reflection about the diagonal, so they
we expect them to have comparable logical error probabilities $p_{X}=O(p_{Z})$
that are suppressed to the same order in the code distance. The logical $Y$
operators of the code also have minimum weight $d$, but there is only one
coset representative with this weight, whereas there are $O(2^{d})$ coset
representatives of logical $X$ weight $d$. Therefore, we expect logical Y
errors to be suppressed. For this reason, we choose to (over)estimate the
total logical error rate as the total $p_{x}+p_{z}+2p_{y}$ of the estimates
from the $Z$ and $X$ basis circuits.

Simulation results are shown in Fig. 11 and Fig. 12. These results suggest a
threshold of nearly $0.3\%$ for the RSSC on the heavy-hexagon lattice using a
circuit-level noise model. A distance-$11$ code is close to break-even at
$p=10^{-3}$ and its logical error rate rapidly decreases to below $10^{-4}$ as
$p$ approaches $6\times 10^{-4}$.

Figure 11: Threshold estimate for RSSC on heavy-hexagon lattice. The total
logical error rate $p_{x}+p_{z}+2p_{y}$ is plotted versus the physical error
rate for codes with distances $d=3,5,7,9,11$ (circle, cross, square, plus,
star). Data points with the same code distance are connected by line segments.
The syndrome measurement schedule $(ZZXX)^{6}$ corresponds to $2$ gauge
measurement repetitions and $12$ total rounds of syndrome measurements. This
schedule produces the highest threshold (approximately $0.3\%$) of all
schedules we considered. We take 10,000 samples per point for physical error
rates $0.0005\leq p\leq 0.0045$ and 100,000 samples per point for physical
error rates $0.0001\leq p<0.0005$. Figure 12: Logical error rates for RSSC on
heavy-hexagon lattice. The bit-flip (dotted line) and phase-flip (dashed line)
logical error rates are plotted versus the physical error rate for codes with
distances $d=3,5,7,9,11$ (circle, cross, square, plus, star). Data points with
the same code distance are connected by line segments. The syndrome
measurement schedule $(ZZXX)^{6}$ corresponds to $2$ gauge measurement
repetitions and $12$ total rounds of syndrome measurements. We take 10,000
samples per point for physical error rates $0.0005\leq p\leq 0.0045$ and
100,000 samples per point for physical error rates $0.0001\leq p<0.0005$.

For the purpose of comparison, we carry out simulations of the HHC using the
same parameters and syndrome measurement schedules as the RSSC. When we decode
the HHC, we choose to use a simple deflagging procedure Sundaresan  _et al._
(2023) rather than dynamically modifying the edge weights in the decoding
graph as in Chamberland  _et al._ (2020). As before, the logical error rate is
optimized by choosing $s=2$, so we conclude that both codes’ logical error
rates are improved by schedule-induced gauge fixing. Figure 13 compares the
estimated total logical error rate. We find that the HHC and RSSC have nearly
identical total logical error rates for distances $3$, $5$, and $7$, but the
RSSC exhibits superior logical error rates at higher distances. This is
expected behavior because the RSSC has an asymptotic threshold whereas HHC
does not.

Figure 13: Logical error probabilities for RSSC and HHC on heavy-hexagon
lattice. The total logical error rate $p_{x}+p_{z}+2p_{y}$ is plotted versus
the physical error rate for codes with distances $d=3,5,7,9,11$. Data points
with the same code distance are connected by line segments. The syndrome
measurement schedule $(ZZXX)^{6}$ corresponds to $2$ gauge measurement
repetitions and $12$ total rounds of syndrome measurements. This schedule
produces the lowest logical error of all schedules we considered. Figure 14:
(a) Average logical output dependence on normally distributed error rate,
depolarized noise, for the RSSC. The standard deviation of the error rates is
parameterized as $\sigma=\alpha\times\langle p_{in}\rangle$. The average is
for 64 device instances. The randomly chosen error rate is held constant for
each simulation instance. $p_{in}=10^{-3}$. For reference, one of the
instances for each $\alpha$ is overlaid. (b) Standard deviation of the logical
output error rates, instance to instance.

##  Appendix D Normal distributed error rate simulations

We numerically examine the dependence of the logical output error rate on
error rates selected randomly for h, i and cx at each site for the RSSC. The
error rate is drawn from a normal distribution with a standard deviation
parameterized as $\sigma=\alpha\langle p_{in}\rangle$ and the error rate is
held constant through out the simulated circuit schedule for each instance.
The average output error rate converges to the uniform case, Fig. 14 (a).
Increasing standard deviation, $\alpha$, in the input error distribution does
show a dependence in standard deviation of $\sigma_{out}$, instance to
instance, Fig. 14 (b). Qualitatively this is also consistent with what is
expected from a phenomenological repetition code, appendix E.

##  Appendix E Noise model cases of the three qubit code

###  E.1 Case 0: Uniform error for 3 qubit repetition code

We consider a few simple noise models for a three qubit majority vote code to
provide insight about the effect of bias on logical qubit error rate when
there is a normally distributed persistent bias selected for each qubit
operation.

We ask what is the error rate, $E$, defined as the probability that the code
reports an incorrect output value from the majority vote after a single round
of measurements of the three qubits. For a simple case, case 0, the qubits
have the same probability of error, $p$ (e.g., a Bernoulli-like binary trial).
The probability of an error can be expressed as $E=3p^{2}(1-p)+p^{3}$.

###  E.2 Case 1: Persistent biased single qubit noise

We now turn to a second case, case 1, where we examine how $E$ is effected by
adding a random error bias on each qubit site. We substitute unique and
persistent error probabilities at each of the $i$ sites, $\epsilon_{i}$ (e.g.,
biases on the single qubit operations). The probability of failure for a three
qubit example with $\vec{\epsilon}=[\epsilon_{1},\epsilon_{2},\epsilon_{3}]$
can then be expressed as,

$E(\vec{\epsilon})=\epsilon_{1}\epsilon_{2}(1-\epsilon_{0})+\epsilon_{0}\epsilon_{1}(1-\epsilon_{2})+\epsilon_{0}\epsilon_{2}(1-\epsilon_{1})+\epsilon_{0}\epsilon_{1}\epsilon_{2}$
(14)

simplifying to:

$E(\vec{\epsilon})=\epsilon_{0}\epsilon_{1}+\epsilon_{1}\epsilon_{2}+\epsilon_{0}\epsilon_{2}-2\epsilon_{0}\epsilon_{1}\epsilon_{2}.$
(15)

We now define the error rate at the $i^{th}$ qubit as
$\epsilon_{i}=\bar{\epsilon}+\delta_{i}$ for which $\delta_{i}$ is the
explicit bias error rate on the $i^{th}$ qubit. We consider the case where the
$\delta_{i}$ is persistent shot to shot for a device instance $j$. The unique
draw for all $i$ qubits, $\vec{\delta}$, describes an error rate, $E_{j}$.
That is, a particular $j^{th}$ device instance is described by
$\vec{\delta}_{j}$ with an error rate $E_{j}$.

We are interested in how $E_{j}$, the error rate for the $j^{th}$ device
instance and $\sigma_{E_{j}}$ depend on the biased error. More explicitly we
have in mind that the probability of drawing $\delta_{i}$ for a particular
site $i$ is described by the function,
$f(\delta_{i})=\frac{1}{2\pi\sqrt{\alpha\bar{\epsilon}}}e^{-\frac{1}{2}(\frac{\delta_{i}}{\alpha\bar{\epsilon}})^{2}}$
(i.e., $\sigma_{\epsilon}=\alpha\bar{\epsilon}$). We now explicitly express
the error rate, $E_{j}$, as a function of the bias error rates for a
particular device instance $\vec{\delta}_{j}$ and in the context of the three
qubit example:

$\displaystyle
E_{j}(\vec{\delta}_{j})=(\bar{\epsilon}+\delta_{0})(\bar{\epsilon}+\delta_{1})$
(16) $\displaystyle+(\bar{\epsilon}+\delta_{0})(\bar{\epsilon}+\delta_{2})$
$\displaystyle+(\bar{\epsilon}+\delta_{1})(\bar{\epsilon}+\delta_{2})$
$\displaystyle-2(\bar{\epsilon}+\delta_{0})(\bar{\epsilon}+\delta_{1})(\bar{\epsilon}+\delta_{2})$

and after reorganizing

$\displaystyle
E_{j}(\vec{\delta}_{j})=3\bar{\epsilon}^{2}+2(\delta_{0}+\delta_{1}+\delta_{2})\bar{\epsilon}+(\delta_{0}\delta_{1}+\delta_{0}\delta_{2}+\delta_{1}\delta_{2})-$
(17) $\displaystyle
2(\bar{\epsilon}^{3}+(\delta_{0}+\delta_{1}+\delta_{2})\bar{\epsilon}^{2}+(\delta_{0}\delta_{1}+\delta_{0}\delta_{2}+\delta_{1}\delta_{2})\bar{\epsilon}+\delta_{0}\delta_{1}\delta_{2}).$

The probability of a particular error rate $E_{j}$ is now a random variable
defined by the sum and product of independent normal distributions of
$\epsilon_{i}$. Different devices will have different biases leading to a set
of device instances with error rates of $\\{...,E_{j-1},E_{j},E_{j+1},\\}$. A
particular $j^{th}$ instance will have an error rate set by $\bar{\epsilon}$
and $\vec{\delta}_{j}$ leading to a sequence of Bernouli ’trials’ with average
error $E_{j}$ and $\sigma_{E_{j}}=\sqrt{E_{j}(1-E_{j})}$.

Through averaging $\langle E_{j}\rangle=\sum_{j}^{N_{j}}{E_{j}}/N_{j}$ over
the span of the set of $N_{j}$ device instances
$\\{...,E_{j-1},E_{j},E_{j+1},\\}$ and using the property $\langle
XY\rangle=\langle X\rangle\langle Y\rangle$, for independent random variables
and $\langle\delta_{i}\rangle_{j}\rightarrow 0$ for $N_{j}\rightarrow\infty$,
the error rate averaged over many device instances simplifies eqn. 17 to:

$\langle E_{j}\rangle=3\bar{\epsilon}^{2}-2\bar{\epsilon}^{3}$ (18)

We also ask how the average variance may depend on $\sigma_{\epsilon}$. The
variance is $\langle\sigma_{E_{j}}^{2}\rangle=\langle(E_{j}-\langle
E_{j}\rangle)^{2}\rangle_{j}$ averaged over all $j$ instances. Explicitly this
becomes:

$\displaystyle\langle\sigma_{Ej}^{2}\rangle\sim\langle(4\delta_{0}^{2}\bar{\epsilon}^{2}+4\delta_{1}^{2}\bar{\epsilon}^{2}+4\delta_{2}^{2}\bar{\epsilon}^{2})$
(19)
$\displaystyle+(\delta_{0}^{2}\delta_{1}^{2}+\delta_{0}^{2}\delta_{2}^{2}+\delta_{1}^{2}\delta_{2}^{2})$
$\displaystyle-(8\delta_{0}^{2}\bar{\epsilon}^{3}+8\delta_{1}^{2}\bar{\epsilon}^{3}+8\delta_{2}^{2}\bar{\epsilon}^{3})$
$\displaystyle-(4\delta_{0}^{2}\delta_{1}^{2}\bar{\epsilon}+4\delta_{0}^{2}\delta_{2}^{2}\bar{\epsilon}+4\delta_{1}^{2}\delta_{2}^{2}\bar{\epsilon})$
$\displaystyle+(4\delta_{0}^{2}\bar{\epsilon}^{4}+4\delta_{1}^{2}\bar{\epsilon}^{4}+4\delta_{2}^{2}\bar{\epsilon}^{4})$
$\displaystyle+(4\delta_{0}^{2}\delta_{1}^{2}\bar{\epsilon}^{2}+4\delta_{0}^{2}\delta_{2}^{2}\bar{\epsilon}^{2}+4\delta_{1}^{2}\delta_{2}^{2}\bar{\epsilon}^{2})$
$\displaystyle+4\delta_{0}^{2}\delta_{1}^{2}\delta_{2}^{2}\rangle$

Observing that
$\sigma_{\epsilon_{i}}^{2}=\langle(\epsilon_{i}+\delta_{i})^{2}\rangle-\langle(\epsilon_{i}+\delta_{i})\rangle^{2}=\langle\delta_{i}^{2}\rangle$
and keeping only the $4^{th}$ order terms eqn. 19 simplifies to:

$\displaystyle\langle\sigma_{Ej}^{2}\rangle\sim
4(\sigma_{\epsilon_{0}}^{2}\bar{\epsilon}^{2}+\sigma_{\epsilon_{1}}^{2}\bar{\epsilon}^{2}+\sigma_{\epsilon_{2}}^{2}\bar{\epsilon}^{2})+$
(20)
$\displaystyle(\sigma_{\epsilon_{0}}^{2}\sigma_{\epsilon_{1}}^{2}+\sigma_{\epsilon_{0}}^{2}\sigma_{\epsilon_{2}}^{2}+\sigma_{\epsilon_{1}}^{2}\sigma_{\epsilon_{2}}^{2})\propto\langle(\alpha
E)E\rangle$

where for illustration we parameterize the standard deviation as
$\sigma=\alpha\langle E\rangle$ as in the body of the paper. The variance
averaged over all instances depends on the standard deviation,
$\sigma_{\epsilon}$ and higher moments of $\delta_{i}$ in contrast to the
$\langle E_{j}\rangle$ that depends only on the input average error rate
$\bar{\epsilon}$.

##  Appendix F Output error sensitivity to different functional sites

In this appendix we illustrate the general trends and relative sensitivity of
’bad’ functional sites in a distance 3 HHC, Fig. 15. The S-shape behavior is
generally repeated and the onset of increased output error is also
qualitatively similar as described in the main text. That is, there is
relatively small increase in $p_{out}$ until $p_{in}$ is large enough to begin
to shift $\langle p_{in}\rangle$. We see, however, that there are differences
in magnitude of sensitivity and some sites are more or less sensitive to X or
Z initialization and measurement, a consequence of their specific roles in the
measurement of the stabilizers. We also show the difference in error rates for
different decoding choices aware (red or blue) and naive (green).

Figure 15: (a) The layout of the distance 3 heavy hex code with labelled
sites, following the same notation as Sundaresan et al. Sundaresan  _et al._
(2023). Z (blue) and X (red) gauge operators. (b) The logical output error
rate dependence on $P_{in}$ of the cx involved in operations with the qubit at
the site labeled b, a data qubit. (c) Logical output error rate dependence on
a flag qubit at the location marked c and the ancilla measure qubit for the Z
check at the side marked d. The $\langle p_{in}\rangle=5\times 10^{-4}$ for
all three figures. Error rates shown in green are for naive decoding.

##  Appendix G Decoder notes

A minimum weight perfect matching decoder was used for all results in this
work. As described in Sundaresan et al. Sundaresan  _et al._ (2023) the
perfect matching algorithm considers X and Z errors separate. The algorithm
finds the minimum weight perfect matching in a graph to associate the syndrome
with a particular error. A graph is formed of vertices representing error-
sensitive events, V, and hyperedges representing the correlations between the
events caused. The probabilistic circuit errors of each operation combine to
form the correlation for each edge. There is a graph for each the X and Z
errors. Edge weights are set as $w_{e}=\log((1-p_{e})/p_{e})$, where $p_{e}$
is the edge probability estimated from the leading order of the polynomial
error rate associated with the parameterized gate operation error rates. The
naive decoder assumes that the error rates are the same for gates of the same
kind of operation, while the aware decoder uses the error rate that is
actually at each site (e.g., selected from the random distribution). The
decoder used in this work is the same as that used in Sundaresan et al.
Sundaresan  _et al._ (2023) in which the details of the decoder are more
extensively discussed.

##  Appendix H Complementary simulations for bad qubits with bias noise and
distance dependences

In this appendix we illustrate the $p_{out}$ dependence on different aspects
of high infidelity outlier error, complementing the example discussed in the
main text. In figure 16 (a) we first show the distance dependence of $p_{out}$
when there is a single ’bad’ qubit in the corner site 0. The $p_{in}$ of the
’bad’ qubit is increased while the other errors are held constant as in the
main text. One observation is that the error saturates in a qualitatively
similar way for each distance.

In figure 16 (b) we examine the bias dependence by setting the error rate for
ZI, ZZ and IZ to $p_{in}=0.5$ when linked to the ’bad’ qubit site(s). The bad
qubits are sequentially chosen along the Z logical operator. Depolarizing
noise is still applied to all other locations. When the logical state is
prepared and measured in the Z basis, it is insensitive to the additional
’bad’ Z errors. Qualitatively, when preparing and measuring in the X basis, a
similar weak response to XI, XX and IX error is correspondingly observed (not
shown). The relative sensitivity is an example of the Z vs. X related
$p_{out}$ response to biased ’bad’ qubit noise.

(a)  (b)

Figure 16: (a) Output error rate dependence on a ’bad’ qubit at site 0 (see
Fig. 1(a)) for distances of 3, 5 and 7 using $\langle p_{in}\rangle=5\times
10^{-4}$. The number of shots were 5k, 5k and 10k, respectively. (b) Output
error for distance 5 with IZ, ZI and ZZ set to $p_{in}=0.5$, ’bad’, data
qubits at positions 0, 1, 2, 3 and 4 (Fig. 1 (a)). Red indicates X-measurement
and blue indicates Z-measurement.