# Mitigating Temporal Fragility in the XY Surface Code

Pei-Kai Tsai  Yale University, Department of Applied Physics, New Haven, CT
06511, USA  Yale Quantum Institute, Yale University, New Haven, CT 06511, USA
Yue Wu  Yale Quantum Institute, Yale University, New Haven, CT 06511, USA
Yale University, Department of Computer Science, New Haven, CT 06511, USA
Shruti Puri  Yale University, Department of Applied Physics, New Haven, CT
06511, USA  Yale Quantum Institute, Yale University, New Haven, CT 06511, USA

###### Abstract

An important outstanding challenge that must be overcome in order to fully
utilize the XY surface code for correcting biased Pauli noise is the
phenomenon of fragile temporal boundaries that arises during the standard
logical state preparation and measurement protocols. To address this challenge
we propose a new logical state preparation protocol based on locally
entangling qubits into small Greenberger–Horne–Zeilinger-like states prior to
making the stabilizer measurements that place them in the XY-code state. We
prove that in this new procedure $O(\sqrt{n})$ high-rate errors along a single
lattice boundary can cause a logical failure, leading to an almost quadratic
reduction in the number of fault-configurations compared to the standard
state-preparation approach. Moreover, the code becomes equivalent to a
repetition code for high-rate errors, guaranteeing a $50\%$ code-capacity
threshold during state preparation for infinitely biased noise. With a simple
matching decoder we confirm that our preparation protocol outperforms the
standard one in terms of both threshold and logical error rate in the fault-
tolerant regime where measurements are unreliable and at experimentally
realistic biases. We also discuss how our state-preparation protocol can be
inverted for similar fragile-boundary-mitigated logical-state measurement.

##  I Introduction

Fault-tolerant, scalable quantum computation with noisy physical hardware
relies on encoding quantum information in a large number of physical qubits
making up an error correcting code. Two important performance metrics for an
error correcting code are its threshold, which is the physical error rate
below which error correction becomes successful, and the amount of error
suppression possible for a given number of qubits. These depend on its
distance which sets the minimum weight of an uncorrectable error, the number
of fault-configurations leading to uncorrectable errors, and their likelihood
which heavily depends on the details of underlying noise model [1, 2].

Biased-Pauli noise is a common noise model describing many practical qubit
architectures in which errors that cause bit-flips are far less likely than
those that only lead to phase-flips [3, 4, 5, 6, 7, 8, 9, 10]. When only
phase-flip noise is present, we say that the noise is infinitely biased. The
discovery of native bias-preserving controlled-not gates [11, 9, 10, 12, 13]
has driven research towards tailoring codes to be highly effective at
correcting biased-Pauli noise [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29]. Two leading candidates for such codes are the XY surface code
[19] and XZZX surface code [23]. These are obtained from the standard CSS
surface code by local Clifford-deformation of its stabilizers [30]. Their
favorable properties arise from the underlying symmetry of their stabilizers
due to which these codes reduce to repetition codes when noise is infinitely
biased (see [31] for further discussion on the role of symmetries in error
correction). Thus, these codes have a $50\%$ threshold at infinite bias.
Moreover, compared to the planar XZZX surface code, the XY code can also
tolerate quadratically higher weight phase errors, making it more desirable
for correcting strongly biased noise.

It is natural to ask if the favorable properties of the XY code persist during
logical state preparation, which is the very first step in any quantum
algorithm. In the standard approach, a logical $X$ state
$\ket{+_{\mathrm{L}}}$ is prepared by initializing each physical qubit in the
$\ket{+}$ state, followed by a measurement of all the code stabilizers.
However, in this process half of the stabilizers cannot be used for detecting
phase errors, destroying the symmetry of the XY code, a phenomenon that has
been referred to as temporal fragile boundaries [32]. As a consequence, the
code does not reduce to a repetition code under pure phase noise and, as we
will show, the threshold at infinite bias degrades to $\sim 11\%$. Moreover,
during state preparation the code distance to phase errors scales as
$\sqrt{n}$ where $n$ is the number of physical qubits of the code. In
contrast, if all the stabilizers could be used for error correction then the
code could tolerate more phase-flip errors and its distance to these high-rate
errors would be $n$. Thus, fragile boundaries during state preparation cause
an overall degradation of the XY code. Similar fragile-boundary-induced
degradation also arises during standard approach for logical measurements.

In this work, we propose a new logical state preparation protocol to mitigate
the effect of fragile boundaries. In our protocol, physical qubits are
entangled into local two- and four-body Greenberger–Horne–Zeilinger(GHZ)
states before being entangled into the surface code state by stabilizer
measurements. This initialization pattern allows us to use three-fourths of
the stabilizers for correcting phase noise and, as we prove, the code reduces
to a single repetition code for phase errors with distance $\sqrt{n}$. Thus a
$50\%$ threshold against phase-noise is guaranteed in the state preparation
process, a substantial improvement over the standard preparation scheme.
Moreover, the fact that there is a single repetition code implies that there
are $O(2^{\sqrt{n}})$ fault-configurations. This is almost a quadratic
reduction compared to $O(\alpha^{\sqrt{n}})$ fault-configurations in the
standard preparation protocol where $3.41\leq\alpha\leq 3.67$ [2]. Thus, our
scheme is able to mitigate the degrading effect of fragile-boundaries.

While our analytical results are limited to the case of pure phase noise, we
numerically examine the performance of our scheme under fault-tolerant setting
where some bit-flip errors and measurement errors are present. For decoding we
use a modified minimum-weight matching (MWPM) algorithm as proposed in [22]
and implemented using an open-source library [33]. We find that our code is
successful at reducing the effects of fragile temporal boundaries at
experimentally realistic noise biases. We also discuss how our state-
preparation scheme can be inverted for fragile-boundary-mitigated logical
state measurement. Finally, we present short-depth circuits for Bell-state
preparation that are compatible with the conventional layout of the surface
code and that introduce minimum additional noise into the code.

This paper is organized as follows. Section II starts with a brief outline of
the XY surface code. The standard logical state preparation scheme is
described in section II.1 and our new scheme along with the main theorems are
described in section II.2. Finally, section III presents the results of
numerical simulations and we conclude with discussion on Bell state
preparation and further opportunities in section IV.

##  II XY Code

We focus on the rotated XY code, also referred to as the tailored surface code
[18, 19], which is defined on a $d\times d$ square lattice for odd $d$ with
data qubits on the vertices. The $X$\- and $Y$-type stabilizer generators are
defined on the faces of the lattice in an alternating checkerboard pattern as
shown in Fig. 1. The $X\;(Y)$-type stabilizers are product of Pauli $X\;(Y)$
operators on data qubits around each face. This $d\times d$ rotated XY code
encodes a single logical qubit in $n=d^{2}$ physical qubits. The $X\;(Y)$-type
logical operator is a product of Pauli $X\;(Y)$ acting on qubits on a string
connecting the left (top) and right (bottom) boundary. The distance to $X$ and
$Y$ errors is $d$. The only non-trivial $Z$-type logical operator is a Pauli
$Z$ acting on every qubit [19].

$X\;(Y)$ errors anticommute with the $Y\;(X)$-stabilizers creating pairs of
syndrome defects oriented diagonally as shown by green (blue) stars in Fig. 1.
A $Z$ error anticommutes with both $X$\- and $Y$-type stabilizers leading to
four syndrome defects, with pairs on neighboring rows (or columns) oriented
vertically (or horizontally), as shown by red stars in Fig. 1. The underlying
structure of syndrome defects generated due to pure $Z$ errors leads to
enhanced performance against pure phase noise. More precisely, it has been
shown that a $d\times d$ rotated XY code, reduces to a length-$d^{2}$ or
equivalently a length-$n$ repetition code under pure $Z$ noise [19].
Consequently, its threshold to pure $Z$ noise is $50\%$ and the distance to
$Z$ errors is exactly the number of qubits $n$. Thus, the code also leads to
lower logical error rates under pure $Z$ noise compared to the standard
surface code with $O(\sqrt{n})$ $Z$-distance. However, the high $Z$-distance
in the XY-code is fragile when $X$ or $Y$ are present [32]. At any one of the
four spatial boundaries, $O(\sqrt{n})$ $Z$ errors can combine with a single
$X$ or $Y$ error to cause a undetectable logical error. In addition to the
spatial fragile boundaries, there are temporal fragile boundaries that occur
during state preparation and measurement. While a strategy to mitigate spatial
fragile boundary by modifying the stabilizers has been proposed previously
[32], we present the first approach for mitigating temporal fragile
boundaries.

Figure 1: Layout of a rotated XY code, its stabilizers, and error syndromes
for high-rate($Z$) and low-rate($X,Y$) errors.

###  II.1 Standard Logical State Preparation

We first describe the phenomena of temporal fragile boundaries in the standard
state preparation approach. For concreteness we consider the preparation of
the $\ket{+_{\mathrm{L}}}$ state but the analysis can be extended to
preparation of the
$\ket{-_{\mathrm{L}}},\;\ket{+i_{\mathrm{L}}},\;\ket{-i_{\mathrm{L}}}$ states
as well.

The protocol for the preparation of $\ket{+_{\mathrm{L}}}$ begins by the
initialization of each physical qubit in the $\ket{+}$ state in step 1,
followed by the measurement of all the code stabilizers in step 2 [1]. The
initial unentangled product state $\ket{+}^{\otimes n}$ is the $+1$ eigenstate
of $X_{\mathrm{L}}$ and all the $X$-stabilizers. In the absence of errors, the
outcomes of all the $X$-stabilizer measurements are guaranteed to be $+1$,
while the $Y$-stabilizer measurements result in outcomes randomly chosen from
$(+1,-1)$. Since the stabilizers commute with the logical operators, the
qubits after measurements are projected in the $\ket{+_{\mathrm{L}}}$ state up
to a local gauge determined by the outcomes of the $Y$-stabilizer
measurements. $Z$ or $Y$ errors can be detected as they anticommute with the
$X$-stabilizers and can be corrected. However, because the outcomes of the
$Y$-stabilizer measurements are completely random, they cannot be used to
detect $Z$ errors. As a result, a $Z$ error only produces two syndrome defects
and cannot be differentiated from $Y$ errors. In fact, at this stage the code
appears to be identical to the standard surface code with a $\sim 11\%$
threshold and $d=\sqrt{n}$ distance to $Z$-noise on data qubits [1]. Moreover,
the number of ways to get a minimum-weight $Z$ error scales as
$O(\alpha^{\sqrt{n}})$ where $3.41\leq\alpha\leq 3.67$ [2].

###  II.2 New Protocol

We first describe the new protocol to prepare the $\ket{+_{\mathrm{L}}}$ state
which proceeds in two steps. The protocol begins, in step 1, by initializing
the qubits in Bell states as indicated in Fig. 2. The four physical qubits
around the $Y$-stabilizer plaquettes, highlighted by dark blue squares, are
entangled into the GHZ state
$\ket{\phi_{4}}=\frac{1}{\sqrt{2}}\left(\ket{+}^{\otimes 4}+\ket{-}^{\otimes
4}\right)$. Each pair of qubits involved in the two-body $Y$-stabilizer along
the top $X$-logical boundary, marked by dark blue line, is entangled into the
Bell state $\ket{\phi_{2}}=\frac{1}{\sqrt{2}}\left(\ket{++}-\ket{--}\right)$
and the remaining qubits along a $Y$-logical boundary, marked as blue dots,
are prepared in $\ket{+}$. We will refer to all the qubits except the ones
prepared in single-qubit $\ket{+}$ states as bulk qubits. Subsequently in step
2 all the stabilizers of the code are measured.

Note that $\ket{\phi_{4}}$ and $\ket{\phi_{2}}$ are respectively the $+1$
eigenstates of $X^{\otimes 4}$ and $X^{\otimes 2}$. Thus the initial state is
a $+1$ eigenstate of all the $X$-stabilizers and $X$-type logical operators.
Consequently, in the absence of errors, the post-stabilizer-measurement state
is the $+1$ eigenstate of the $X$-type logical operators and all the
$X$-stabilizer measurement outcomes must be $+1$.

Importantly, $\ket{\phi_{4}}$ and $\ket{\phi_{2}}$ are $+1$ eigenstates of
$Y^{\otimes 4}$ and $Y^{\otimes 2}$, respectively. Consequently, in the
absence of errors, the measurement of outcomes of the marked $Y$-stabilizers
must be $+1$. The measurement outcomes of the unmarked $Y$ stabilizers are
random $(\pm 1)$. Thus, unlike in the standard protocol, half of the
$Y$-stabilizers can be used to detect errors in the new scheme. Note that
although only $\ket{+_{\mathrm{L}}}$ is prepared in Theorem 1,
$\ket{+i_{\mathrm{L}}}$ can be prepared in a similar manner due to the
symmetry of the XY code as shown in Appendix A. We now state the main theorem.

######  Theorem 1.

In the new state-preparation protocol for the square $d\times d$ $XY$ code,
$Z$ errors on all the bulk qubits are correctable (part I) and $Z$ errors on
data qubits at the temporal boundary can be decoded as a single repetition
code of length $d$ (part II).

######  Corollary 1 (Fault-configurations).

There are $O(2^{\sqrt{n}})$ least-weight fault-configurations due to pure $Z$
errors, where $n=d^{2}$ is the total number of qubits. This is nearly
quadratic improvement over the least-weight fault-configurations in the
standard state-preparation approach.

######  Corollary 2 (Threshold).

The threshold to pure $Z$ noise is 50$\%$.

Proof. Consider the square lattice of the $XY$ code in Fig. 2 where the qubits
are placed on the vertices. We will use indices $(i,j)\in\\{1,2,...,d\\}^{2}$
to denote the location of data qubits. $Z$ errors on the data qubits can be
expressed as $Z(\bm{z})=\bigotimes_{i,j}(Z_{i,j})^{z_{i,j}}$ with a
corresponding binary vector
$\bm{z}=(z_{1,1},z_{1,2},...,z_{d,d})\in\\{0,1\\}^{d^{2}}$. $z_{ij}=0\;(1)$
implies no (a $Z$) error on the data qubit located at $(i,j)$. Thus the
probability for $z_{i,j}=1$ is equal to the probability of $Z$ errors and the
problem of decoding $Z$ errors reduces to correctly determining $\bm{z}$.

In the following, we will refer to the $X$ and $Y$ stabilizers with fixed
measurement outcome of $+1$ as fixed stabilizers. Under pure $Z$ errors, the
syndrome measurement of any fixed stabilizer $S$ is
$\prod_{(i,j)\in\mathrm{supp}\,S}(-1)^{z_{i,j}}$, where $\mathrm{supp}\,S$
denotes the set of qubits on which $S$ is supported. Thus, it is possible to
interpret $\bigoplus_{(i,j)\in\mathrm{supp}\,S}z_{i,j}=0$ as the parity checks
of a classical code where $\bigoplus$ denotes summation modulo two. A $-1$
outcome of measuring a fixed stabilizer results in the violation of this
parity check. The parity checks can be decoded to determine $\bm{z}$ and the
location of $Z$ errors. Next we show that these parity checks reduce to a
number of independent classical repetition codes.

First, consider pairs of qubits on the top row for which we have two-bit
parity checks $z_{1,2j-1}\oplus z_{1,2j}$, $j=1,2,...,(d-1)/2$ due to the
fixed $Y$ stabilizers. Each of the two-bit checks forms a classical 2-bit
repetition code $\mathrm{REP}(2)$. By adding the check $z_{1,2j-1}\oplus
z_{1,2j}$ to the four-bit parity check $z_{1,2j-1}\oplus z_{1,2j}\oplus
z_{2,2j-1}\oplus z_{2,2j}$, corresponding to the fixed $X$ stabilizers
directly below the top $Y$ stabilizers, we reduce that four-bit parity check
to a two-bit parity check $z_{2,2j-1}\oplus z_{2,2j}$. Adding this new two-bit
parity check to the next four-bit parity arising due to the fixed $Y$
stabilizer in the next row again reduces the latter to a two-bit parity check.
In this recursive manner, all four-bit parity checks reduce to two-bit parity
checks $z_{i,2j-1}\oplus z_{i,2j}$ for $i=1,2,...,d$ and $j=1,2,...,(d-1)/2$
with support on pairs of adjacent qubits in every row.

We can apply this same procedure but this time starting with pairs of qubits
on the leftmost column for which we have two-bit parity checks $z_{2i,1}\oplus
z_{2i+1,1}$, $i=1,2,...,(d-1)/2$ due to the fixed $X$ stabilizers. By adding
the check $z_{2i,1}\oplus z_{2i+1,1}$ to the four-bit parity check
$z_{2i,1}\oplus z_{2i,2}\oplus z_{2i+1,1}\oplus z_{2i+1,2}$, corresponding to
the fixed $Y$ stabilizers directly to the right of the $X$ stabilizers, we
reduce that four-bit parity check to a two-bit parity check $z_{2i,2}\oplus
z_{2i+1,2}$. Continuing the recursion, all four-bit parity checks this time
reduce to two-bit parity checks $z_{2i,j}\oplus z_{2i+1,j}$ with support on
pairs of adjacent qubits in every column for $i=1,2,...,(d-1)/2$ and
$j=1,2,...,d$. Now consider
$z_{2i,2j-1},\;z_{2i,2j},\;z_{2i+1,2j-1},\;z_{2i+1,2j}$, for
$i,j=1,2,...,(d-1)/2$, which are supported on qubits around the fixed $Y$
stabilizers. These form a classical 4-bit repetition code $\mathrm{REP}(4)$
with parity checks $z_{2i,2j-1}\oplus z_{2i,2j}$, $z_{2i,2j}\oplus
z_{2i+1,2j}$, $z_{2i+1,2j}\oplus z_{2i+1,2j-1}$, and $z_{2i+1,2j-1}\oplus
z_{2i,2j-1}$.

Thus we see that for every fixed $Y$ stabilizer there corresponds a classical
$\mathrm{REP}(2)$ or $\mathrm{REP}(4)$ code. A simple counting shows that
there are $(d-1)/2$ $\mathrm{REP}(2)$ and $(d-1)^{2}/4$ $\mathrm{REP}(4)$
codes.

At the outset it seems that the probability of successful decoding will be
severely limited by the $\mathrm{REP}(2)$ and $\mathrm{REP}(4)$ codes.
However, incorrect decoding of a $\mathrm{REP}(2)$ or $\mathrm{REP}(4)$
results in a flip applied to every bit in its support. Equivalently, this
results in $Z$ error applied to every qubit in the support of the
corresponding fixed $Y$ stabilizer. However, these qubits are prepared in the
entangled states $\ket{\phi_{2}}$ or $\ket{\phi_{4}}$ which are eigenstates of
$Z^{\otimes 2}$ and $Z^{\otimes 4}$ and are thus invariant under these
operators. Hence $Z$ errors on all qubits other than the ones on the last
column are correctable. This proves part I of Theorem 1.

Finally, we consider the last column of qubits. For this column, each fixed
$X$ stabilizer is supported in two qubits. The classical parity checks
$z_{i,d}\oplus z_{i+1,d}$ ($i=1,2,...,d$) form a classical $d$-bit repetition
code $\mathrm{REP}(d)$. Decoding the classical repetition code results in
decoding $Z$ errors on these qubits, which is part II of Theorem 1. Thus,
fewer than $(d-1)/2$-flips on these bits can be corrected. The number of
least-weight fault configurations is
${d\choose{(d+1)/2}}=O(2^{d})=O(2^{\sqrt{n}})$, which is Corollary 1. Since
the threshold for the classical repetition code is $50\%$, the threshold for
$Z$ errors in the quantum code is also $50\%$, from which Corollary 2 follows.

Figure 2: The initialization pattern for $\ket{+_{\mathrm{L}}}$ in our new
state preparation protocol. The fixed $Y$ stabilizers are marked by dark blue
outline. All the $X$ stabilizers are fixed.

##  III Results

###  III.1 Noise model and decoder

The analysis in the previous section demonstrates the advantage of our scheme
over standard state preparation scheme under pure $Z$-noise. In practice there
will be some bit-flip noise affecting the qubits. To compare the performance
of the two approaches in this experimentally relevant situation we resort to
numerical simulation of the state preparation protocol with practical decoding
algorithms. We will use a phenomenological model where (a) $X$, $Y$, and $Z$
errors are applied, with probabilities $p_{x}$, $p_{y}$, and $p_{z}$
respectively, on qubits after they are initialized in the product state
$\ket{+}^{\otimes d^{2}}$ in case of standard scheme or Bell states in case of
our proposed scheme and (b) measurement errors are applied with probability
$p_{\mathrm{m}}$. For fault-tolerance, to measurement errors we perform $d$
rounds of stabilizer measurements after the measurements in step 2. Our aim is
to estimate the logical error rate as a function of the total probability of
errors on the data qubits $p=p_{x}+p_{y}+p_{z}$ for a given bias
$\eta=p_{z}/(p_{x}+p_{y})$. We also assume $p_{x}=p_{y}$ for simplicity.

In the standard state-preparation approach, a $Z$ error on the data qubits
flips two stabilizers in the measurement round in step 2. In contrast, in our
scheme, a $Z$ error on a qubit on the top or right boundary flips two
stabilizers while a $Z$ error on any other qubit flips three stabilizers in
step 2. A $Y$ error in either schemes flips two stabilizer measurement
outcomes. In both the schemes after the first measurement round, a $Z$ error
always flips all neighboring stabilizers since all stabilizers can be used for
error correction. This implies that standard minimum-weight perfect matching
(MWPM) algorithm cannot be used to optimally correct for $Z$-biased noise and
instead we use a modified MWPM algorithm introduced in [22]. We refer to this
as the Tuckett decoder, which we further adapt to account for the fact that
only certain stabilizers can used for error correction in the first
measurement round (see Appendix D for details).

Figure 3: Scaling of logical error rate below threshold for various $p$ with
$p_{\mathrm{m}}=0$. Filled circles show the logical error rates at different
$p$ with the new protocol. For comparison, the subthreshold scaling curve for
$p=0.05$ with standard protocol is also plotted with square markers. The solid
and dotted lines are obtained from linear fits. Filled triangles are the
logical error rates of repetition codes with distance $d$ and bit-flip rate
$p$. The dashed lines through the triangles are just drawn for easy
visualization.

###  III.2 No measurement error, $p_{\mathrm{m}}=0$

We first benchmark the adapted Tuckett decoder under ideal measurements
$p_{\mathrm{m}}=0$ and $\eta=\infty$, so that only pure $Z$ noise is present.
In this limit we find that the adapted Tuckett decoder results in a
$49.1(2)\%$ threshold for our state-preparation scheme and $10.1(1)\%$
threshold for the standard scheme, which are in agreement with our analytical
predictions. Next, we analyze the sub-threshold scaling of logical error rate.
The filled-circles in Fig. 3 show the logical error rate as a function of $d$
for different values of $p$ with $\eta=\infty$ and $p_{\mathrm{m}}=0$ for the
new preparation scheme. For comparison, the filled triangles are the logical
error rates for repetition codes of length $d$ as a lower bound of the
performance of our protocol. We observe that the numerically obtained logical
error rate of our scheme is systematically larger than that for repetition
code, indicating that there is scope to further improve over the decoder.

We fit the data to the ansatz $\log p_{\mathrm{L}}=(\alpha d+\beta)\log
p+(\gamma d+\delta)$. For our preparation scheme we find the fit parameters to
be $(\alpha,\beta,\gamma,\delta)=(0.46(1),0.04(1),0.243(9),-0.7(1))$. Recall
Theorem 1 due to which $(d+1)/2$ phase-flip errors occurring on the last
column of qubits are uncorrectable so that the logical error rate scales as
$p^{(d+1)/2}$ (at low $p$) and $\alpha_{\mathrm{ideal}}=0.5$. The value of the
slope with the adapted Tuckett decoder, $\alpha\sim 0.46$, is close to
$\alpha_{\mathrm{ideal}}$. Despite good agreement with the ideal slope, the
adapted Tuckett decoder does not reduce to an ideal decoder as shown with
examples in the Appendix D.

Nonetheless, at fixed $p$, $p_{\mathrm{L}}$ for our scheme is several orders
of magnitude smaller than with the standard preparation scheme, as seen for
example, by comparing the logical error rate for $p=0.05$ with the standard
preparation scheme (filled red squares) and the new scheme (filled red
circles). For more precise comparison, we examine the error suppression factor
$\Lambda$ which determines how fast the logical error rate decreases as the
distance increases at a given $p$ ($p_{\mathrm{L}}=O(\Lambda^{-d})$) [34]. We
obtain $\Lambda$ for $p=0.05$ from the slopes of the fitted straight lines
through the red circles and squares respectively for the two protocols. For
our protocol we get $\Lambda=2.3(1)$ which is nearly twice of $\Lambda=1.2(1)$
for the standard protocol, indicating a much faster suppression in logical
error rates as larger codes are used with our protocol. The large difference
in the values of $\Lambda$ indicates that the number of fault configurations
with the adapted Tuckett decoder for our scheme is indeed much smaller
compared to that for the standard approach.

Despite the sub-optimality of the Tuckett decoder, it reproduces the
analytically predicted thresholds and overall sub-threshold scaling behavior
for $\eta=\infty$ fairly well. Thus, we also use it for the finite-$\eta$
case. For an experimentally realistic high bias $\eta=10^{4}$ [12, 28], the
threshold for the total error rate with our scheme is $15.1(2)\%$ in
comparison to that of $10.2(1)\%$ with the standard approach. A plot of
threshold as a function of the bias for $p_{\mathrm{m}}=0$ has been shown in
Appendix B.

Figure 4: Noise thresholds as a function of bias $\eta$ for the standard
state-preparation (green) and the new protocol (red) for $p_{\mathrm{m}}=p$.

###  III.3 With measurement error, $p_{\mathrm{m}}=p$

We now consider non-zero measurement errors and assume $p_{\mathrm{m}}=p$
[22]. The plot of threshold as a function of $\eta$ with the two schemes is
shown in Fig. 4. In this case, the threshold difference between the two
schemes is less dramatic. At infinite bias the threshold increases from
$5.66(1)\%$ for the standard scheme to $7.47(5)\%$ for our scheme, while for
$\eta=10^{4}$ the threshold increases from $5.62(2)\%$ to $6.03(4)\%$. Figure
5 compares the logical error rates for the two preparation approaches with
$d=7$. For $\eta=\infty$ we find that the logical error rate with our scheme
is about an order of magnitude smaller compared to the standard approach.
However, the difference between the two approaches becomes smaller as $p$
decreases.

We attribute this effect to the decreasing contribution from temporal
boundaries at smaller $p$ where the gain from our protocol is reduced. For
example, at $\eta=10^{4}$ and $p=4\%$, the state preparation error rate with
our approach almost reaches the floor set by the logical memory error rate
($5.3(1)\times 10^{-3}$) for the same parameters shown as black triangle in
Fig. 5. The standard preparation scheme clearly cannot reach this floor due to
large contribution from temporal boundary errors. On the other hand at
$\eta=10^{4}$ and $p=0.6\%$, the logical memory error rate is $1.5(2)\times
10^{-5}$. Our state preparation reaches this value but the logical error rate
with the standard preparation scheme is slightly higher $2.8(2)\times
10^{-5}$.

These results confirm that our scheme can indeed reduce the amount of
additional state-preparation errors due to temporal fragile boundaries. For
low physical error rates where the improvement looks less significant, the
dominant contribution to logical error rate is mainly from measurement errors
and not due to temporal boundaries.

Figure 5: Logical error rates for the standard (filled-squares) and new
preparation (filled-circles) schemes with $p_{\mathrm{m}}=p$ at $\eta=\infty$
(red) and $\eta=10^{4}$ (green) for $d=7$. The solid and dashed lines are
shown as guides for the eye and are not obtained from any fits. The black
triangles show memory logical error rate at $p=0.6\%,4\%$ for $\eta=10^{4}$.

##  IV Discussion

In this work we proposed a new state-preparation protocol for the XY code
which mitigates the effect of fragile temporal boundaries based on using local
GHZ states. We also studied the performance of our approach with a practical
decoder. In Appendix C we discuss how this protocol can be inverted to realize
fragile-boundary-mitigated logical measurements.

Practically, it is necessary to be able to prepare the Bell states with a
short-depth circuit in a bias-preserving way. One possible circuit is shown in
Fig. 6(a) in which the Bell states are prepared using CX gates between the
data qubits. One drawback of this circuit is that a single high-rate $Z$ error
on the data qubit which is the common target for all the CX gates can spread
to multiple data qubits causing a correlated error. Moreover, this circuit is
also not compatible with the standard connectivity of the surface code where
the data qubits don’t interact with each other directly, but only interact
with an ancilla.

To overcome these shortcomings, we also give an alternative circuit in Fig.
6(b). This circuit can be effectively understood as first creating a four-
(two-)body Bell state on three(one) data qubits and one ancilla and then
swapping the ancilla with the remaining data qubit. Crucially, the ancilla is
left in the $|+\rangle$ state at the end in the absence of noise. A single
$Z$-type error on the ancilla causes it to end up in the $|-\rangle$ state.
Thus a $X$-measurement performed on the ancilla at the very end reveals the
presence of $Z$ errors on it. The Bell state is used in the code only after
the ancilla is measured in $\ket{+}$. Moreover, this heralding also eliminates
error correlations caused by $Z$ errors on the ancilla to first order. The
ancilla-noise robustness only comes at the cost of one extra CX gate compared
to the circuit in Fig. 6(a) and compared to the standard stabilizer
measurement circuit. Ultimately, future work should consider a full circuit-
level simulation of our scheme with additional modifications to mitigate the
fragile spatial boundaries [32].

Moreover, there is considerable room for improving the performance of our
scheme by improving the decoder. One possible path would be to combine the
Tuckett decoder with belief-propagation [35, 36, 37, 32]. Ultimately, in order
to fully understand the advantages and limits of our scheme a hypergraph
decoder will be necessary. While such a decoder may be inefficient,
approximate solutions may be sufficient to reach reasonably low error rates
with reasonable latency [38].

Figure 6: (a) Simple circuits to prepare $\ket{\phi_{4}}$ and
$\ket{\phi_{2}}$. (b) An alternative circuit for $\ket{\phi_{4}}$ which is
compatible of standard surface code qubit layout where data qubits only
connect to ancilla. (c) A similar circuit to prepare $\ket{\phi_{2}}$ on data
qubits.

###### Acknowledgements.

This material is based upon work supported by the National Science Foundation
(CAREER grant no. 2145223) Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the authors and do not
necessarily reflect the views of the National Science Foundation. We also
thank Shilin Huang and Shraddha Singh for useful discussions.

##  Appendix A State preparation of $Y_{\mathrm{L}}$ eigenstate
$\ket{+i_{\mathrm{L}}}$

A similar construction in Fig. 7 shows the local initialization pattern which
is an eigenstate of $Y_{\mathrm{L}}$. Due to the symmetry of $X$ and $Y$ in
the XY code, the same argument as in Theorem 1 applies to prove the
$\mathrm{REP}(d)$ structure and 50% threshold for state preparation.

Figure 7: The initialization pattern for $\ket{+i_{\mathrm{L}}}$ in our new
state preparation protocol where
$\ket{\phi_{4}}_{Y}=\frac{1}{\sqrt{2}}\left(\ket{+i}^{\otimes
4}+\ket{-i}^{\otimes 4}\right)$ and
$\ket{\phi_{2}}_{Y}=\frac{1}{\sqrt{2}}\left(\ket{+i}^{\otimes
2}-\ket{-i}^{\otimes 2}\right)$.

##  Appendix B Threshold when $p_{\mathrm{m}}=0$

Figure 8 shows threshold as a function of $\eta$ when $p_{\mathrm{m}}=0$ and
Pauli errors applied on data qubits.

Figure 8: Noise thresholds as a function of bias $\eta$ for the standard
state-preparation (green) and the new protocol (red) for $p_{\mathrm{m}}=0$.

##  Appendix C Logical Measurement of $X_{\mathrm{L}}$

The standard protocol for $X_{\mathrm{L}}$ measurement proceeds by measuring
each data qubit in the $X$ basis [1]. The measurement outcomes
$x_{i}\in\\{0,1\\}$ can be added to obtain the logical measurement result
$x_{\mathrm{L}}=\bigoplus_{i}x_{i}$.

In the absence of errors, the measurement outcomes of qubits supported by an
$X$-type stabilizer $S$ must sum to zero:
$\bigoplus_{v\in\mathrm{supp}\,S}x_{v}=0$ since all measurements commute with
$S$. The $X$-type stabilizers can thus be effectively used to detect and
correct data qubit $Z$ errors (caused by a physical $Z$ or $Y$ error on the
data qubit or by a measurement errors). The $Y$-type stabilizers do not
provide any information about errors since they don’t commute with the data
qubit $X$ measurements. Thus, there is no way to detect $X$ errors which is
not a problem as these errors don’t affect the $X_{\mathrm{L}}$ measurement
anyway. Nonetheless, only half the stabilizers can be used for correcting
errors and there is no way to distinguish a $Z$ error from $Y$ error. This
results in fragile temporal boundaries similar to the case of state
preparation.

We overcome this challenge by “inverting” the new state preparation protocol.
We measure the local operators that stabilize the Bell states
$|\phi_{4}\rangle$ and $|\phi_{2}\rangle$ which are
$\\{YYYY,XXII,IXXI,IIXX\\}$ and $\\{YY,XX\\}$ respectively. The qubits in the
last column are measured in the $X$ basis. The result of $X_{\mathrm{L}}$
measurement can be inferred from summing over all disjoint $XX$ and $X$
measurements. Moreover, the measurement outcomes obey the set of REP(4),
REP(2), and REP($d$) parity checks as described under Theorem 1.

We know from the discussion under Theorem 1 that incorrect decoding of a
REP(2) or REP(4) results in $Z^{\otimes 2}$ or $Z^{\otimes 4}$ applied to
qubits supporting $\ket{\phi_{2}}$ or $\ket{\phi_{4}}$. However, $Z^{\otimes
2}$ on $\ket{\phi_{2}}$ or $Z^{\otimes 4}$ on $\ket{\phi_{4}}$ commutes with
operators being measured. Thus we conclude that in this new measurement
protocol for the square $d\times d$ $XY$ code, $Z$ errors on data qubits at
the temporal boundary can be decoded as a single repetition code REP($d$) on
the last column of qubits. It also follows that there are $O(2^{\sqrt{n}})$
least-weight fault-configurations, where $n=d^{2}$ is the total number of
qubits and that the threshold to pure $Z$ noise is 50$\%$.

##  Appendix D Adapted Tuckett Decoder

In this work, we apply the XY code decoder exploiting the symmetries of the
code and noise bias [22]. We refer the reader to [22] for details. In this
section we only highlight the modifications made to the original decoder for
state-preparation.

The only difference between the original decoder and the decoder we use is how
the matching graph is weighed in the first time-step to account for the
temporal boundaries. We need to add virtual vertices at the temporal
boundaries. The vertex for an unfixed stabilizer can be either matched to its
virtual temporal vertex with zero weight, or matched to any other vertices
with normal weights corresponding to qubit $X$, $Y$, $Z$ errors. The vertex
for a fixed stabilizer can be matched to its virtual temporal vertex with
weight corresponding to measurement errors $p_{\mathrm{m}}$.

For the standard preparation approach however, the syndromes due to $Z$ errors
are identical to the syndromes due to $Y$ errors in the first time-step. Thus,
we modify the Tuckett decoder in this case so that only diagonal edges
corresponding to $Y$ errors are allowed in the first time-step with weight
corresponding to the probability of $Z$ and $Y$ errors. If we don’t do this
and instead use parallel edges like in the original Tuckett decoder then the
performance of the standard approach degrades substantially.

Recall that in the case of state preparation with the new protocol, an optimal
decoder should correct up to $(d-1)/2$ $Z$ errors on the last column of
qubits, however, with an example shown in Fig. 9 we find that the Tuckett
decoder is unable to achieve this. The unfixed stabilizers or stabilizers
whose measurement outcome is unknown and cannot be used for error correction
are marked with thick black outline in the figure for clarity. Figure 9(a)
shows the syndromes in filled stars due to two $Z$ errors on qubits marked in
red circles. The solid lines shows the possible edges from matching. In this
case the decoder assigns $Z$ errors to qubits correctly. However, there is an
alternate edge-matching of same weight shown in Fig. 9(b). In this case the
decoder assigns $Z$ errors to qubits marked in solid blue, which differs from
the actual $Z$ errors in red by a logical operator.

Figure 9: Decoder confusion during state preparation where the decoder fails
to correct $(d-1)/2$ phase-flip errors. Note that in (b), the syndrome at the
bottom-right corner is connected along time-dimension with zero weight (shown
as dashed line). This is because that stabilizer is unreliable at the first
stage, so it can be seen as a time-like error.

## References

  * Dennis  _et al._ [2002] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological Quantum Memory, J. Math. Phys 43, 4452 (2002).
  * Beverland  _et al._ [2019] M. E. Beverland, B. J. Brown, M. J. Kastoryano, and Q. Marolleau, The Role of Entropy in Topological Quantum Error Correction, J. Stat. Mech.: Theory Exp. 2019 (7), 073404.
  * Shulman  _et al._ [2012] M. D. Shulman, O. E. Dial, S. P. Harvey, H. Bluhm, V. Umansky, and A. Yacoby, Demonstration of Entanglement of Electrostatically Coupled Singlet-Triplet Qubits, Science 336, 202 (2012).
  * Pop  _et al._ [2014] I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf, L. I. Glazman, and M. H. Devoret, Coherent Suppression of Electromagnetic Dissipation due to Superconducting Quasiparticles, Nature 508, 369 (2014).
  * Mirrahimi  _et al._ [2014] M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, Dynamically Protected Cat-Qubits: A New Paradigm for Universal Quantum Computation, New J. Phys. 16, 045014 (2014).
  * Waldherr  _et al._ [2014] G. Waldherr, Y. Wang, S. Zaiser, M. Jamali, T. Schulte-Herbrüggen, H. Abe, T. Ohshima, J. Isoya, J. Du, P. Neumann, _et al._ , Quantum Error Correction in a Solid-State Hybrid Spin Register, Nature 506, 204 (2014).
  * Watson  _et al._ [2018] T. Watson, S. Philips, E. Kawakami, D. Ward, P. Scarlino, M. Veldhorst, D. Savage, M. Lagally, M. Friesen, S. Coppersmith, _et al._ , A Programmable Two-Qubit Quantum Processor in Silicon, Nature 555, 633 (2018).
  * Albert  _et al._ [2019] V. V. Albert, S. O. Mundhada, A. Grimm, S. Touzard, M. H. Devoret, and L. Jiang, Pair-Cat Codes: Autonomous Error-Correction with Low-Order Nonlinearity, Quantum Sci. Technol. 4, 035007 (2019).
  * Puri  _et al._ [2020] S. Puri, L. St-Jean, J. A. Gross, A. Grimm, N. E. Frattini, P. S. Iyer, A. Krishna, S. Touzard, L. Jiang, A. Blais, S. T. Flammia, and S. M. Girvin, Bias-Preserving Gates with Stabilized Cat Qubits, Sci. Adv. 6, eaay5901 (2020).
  * Cong  _et al._ [2022] I. Cong, H. Levine, A. Keesling, D. Bluvstein, S.-T. Wang, and M. D. Lukin, Hardware-Efficient, Fault-Tolerant Quantum Computation with Rydberg Atoms, Phys. Rev. X 12, 021049 (2022).
  * Guillaud and Mirrahimi [2019] J. Guillaud and M. Mirrahimi, Repetition Cat Qubits for Fault-Tolerant Quantum Computation, Phys. Rev. X 9, 041053 (2019).
  * Xu  _et al._ [2022] Q. Xu, J. K. Iverson, F. G. S. L. Brandão, and L. Jiang, Engineering Fast Bias-Preserving Gates on Stabilized Cat Qubits, Phys. Rev. Res. 4, 013082 (2022).
  * Yuan  _et al._ [2022] M. Yuan, Q. Xu, and L. Jiang, Construction of Bias-Preserving Operations for Pair-Cat Codes, Phys. Rev. A 106, 062422 (2022).
  * Aliferis and Preskill [2008] P. Aliferis and J. Preskill, Fault-Tolerant Quantum Computation against Biased Noise, Phys. Rev. A 78, 052331 (2008).
  * Aliferis  _et al._ [2009] P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, Fault-Tolerant Computing with Biased-Noise Superconducting Qubits: A Case Study, New J. Phys. 11, 013061 (2009).
  * Stephens  _et al._ [2013] A. M. Stephens, W. J. Munro, and K. Nemoto, High-Threshold Topological Quantum Error Correction against Biased Noise, Phys. Rev. A 88, 060301 (2013).
  * Robertson  _et al._ [2017] A. Robertson, C. Granade, S. D. Bartlett, and S. T. Flammia, Tailored Codes for Small Quantum Memories, Phys. Rev. Appl. 8, 064004 (2017).
  * Tuckett  _et al._ [2018] D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, Ultrahigh Error Threshold for Surface Codes with Biased Noise, Phys. Rev. Lett. 120, 050505 (2018).
  * Tuckett  _et al._ [2019] D. K. Tuckett, A. S. Darmawan, C. T. Chubb, S. Bravyi, S. D. Bartlett, and S. T. Flammia, Tailoring Surface Codes for Highly Biased Noise, Phys. Rev. X 9, 041031 (2019).
  * Hänggli  _et al._ [2020] L. Hänggli, M. Heinze, and R. König, Enhanced Noise Resilience of the Surface–Gottesman-Kitaev-Preskill Code via Designed Bias, Phys. Rev. A 102, 052408 (2020).
  * Huang and Brown [2020] S. Huang and K. R. Brown, Fault-Tolerant Compass Codes, Phys. Rev. A 101, 042312 (2020).
  * Tuckett  _et al._ [2020] D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, Fault-Tolerant Thresholds for the Surface Code in Excess of 5% under Biased Noise, Phys. Rev. Lett. 124, 130501 (2020).
  * Bonilla Ataides  _et al._ [2021] J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, The XZZX Surface Code, Nat. Commun. 12, 2172 (2021).
  * Darmawan  _et al._ [2021] A. S. Darmawan, B. J. Brown, A. L. Grimsmo, D. K. Tuckett, and S. Puri, Practical Quantum Error Correction with the XZZX Code and Kerr-Cat Qubits, PRX Quantum 2, 030345 (2021).
  * Higgott and Breuckmann [2021] O. Higgott and N. P. Breuckmann, Subsystem Codes with High Thresholds by Gauge Fixing and Reduced Qubit Overhead, Phys. Rev. X 11, 031039 (2021).
  * Guillaud and Mirrahimi [2021] J. Guillaud and M. Mirrahimi, Error Rates and Resource Overheads of Repetition Cat Qubits, Phys. Rev. A 103, 042413 (2021).
  * Huang  _et al._ [2023] E. Huang, A. Pesah, C. T. Chubb, M. Vasmer, and A. Dua, Tailoring Three-Dimensional Topological Codes for Biased Noise, PRX Quantum 4, 030338 (2023).
  * Chamberland  _et al._ [2022] C. Chamberland, K. Noh, P. Arrangoiz-Arriola, E. T. Campbell, C. T. Hann, J. Iverson, H. Putterman, T. C. Bohdanowicz, S. T. Flammia, A. Keller, G. Refael, J. Preskill, L. Jiang, A. H. Safavi-Naeini, O. Painter, and F. G. Brandão, Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes, PRX Quantum 3, 010329 (2022).
  * Xu  _et al._ [2023] Q. Xu, N. Mannucci, A. Seif, A. Kubica, S. T. Flammia, and L. Jiang, Tailored XZZX Codes for Biased Noise, Phys. Rev. Res. 5, 013035 (2023).
  * Dua  _et al._ [2022] A. Dua, A. Kubica, L. Jiang, S. T. Flammia, and M. J. Gullans, Clifford-Deformed Surface Codes (2022), arXiv:2201.07802 [cond-mat, physics:quant-ph].
  * Brown [2023] B. J. Brown, Conservation Laws and Quantum Error Correction: Towards a Generalised Matching Decoder, IEEE BITS the Information Theory Magazine , 1 (2023).
  * Higgott  _et al._ [2023] O. Higgott, T. C. Bohdanowicz, A. Kubica, S. T. Flammia, and E. T. Campbell, Improved Decoding of Circuit Noise and Fragile Boundaries of Tailored Surface Codes, Phys. Rev. X 13, 031007 (2023).
  * [33] Y. Wu and N. Liyanage, QEC-Playground, https://github.com/yuewuo/QEC-Playground.
  * Martinis [2015] J. M. Martinis, Qubit Metrology for Building a Fault-Tolerant Quantum Computer, npj Quantum Inf. 1, 15005 (2015).
  * Criger and Ashraf [2018] B. Criger and I. Ashraf, Multi-Path Summation for Decoding 2d Topological Codes, Quantum 2, 102 (2018).
  * Google Quantum AI [2023] Google Quantum AI, Suppressing Quantum Errors by Scaling a Surface Code Logical Qubit, Nature 614, 676 (2023).
  * Caune  _et al._ [2023] L. Caune, B. Reid, J. Camps, and E. Campbell, Belief Propagation as a Partial Decoder (2023), arXiv:2306.17142 [quant-ph].
  * [38] Y. Wu  _et al._ , (in preparation).


#  $C_{2^{n}}$-equivariant rational stable stems and characteristic classes

Nick Georgakopoulos

###### Abstract.

In this short note, we compute the rational $C_{2^{n}}$-equivariant stable
stems and give minimal presentations for the $RO(C_{2^{n}})$-graded Bredon
cohomology of the equivariant classifying spaces $B_{C_{2^{n}}}S^{1}$ and
$B_{C_{2^{n}}}\Sigma_{2}$ over the rational Burnside functor
$A_{\operatorname{\mathbb{Q}}}$. We also examine for which compact Lie groups
$L$ the maximal torus inclusion $T\to L$ induces an isomorphism from
$H^{*}_{C_{2^{n}}}(B_{C_{2^{n}}}L;A_{\operatorname{\mathbb{Q}}})$ onto the
fixed points of
$H^{*}_{C_{2^{n}}}(B_{C_{2^{n}}}T;A_{\operatorname{\mathbb{Q}}})$ under the
Weyl group action. We prove that this holds for $L=U(m)$ and any $n,m\geq 1$
but does not hold for $L=SU(2)$ and $n>1$.

###### Contents

  1. 1 Introduction
  2. 2 Rational Mackey functors
  3. 3 Euler and orientation classes
  4. 4 Rational stable stems
  5. 5 $C_{2^{n}}$ rational characteristic classes

##  1\. Introduction

This note is the followup to [Geo21c]. We start by computing the
$C_{2^{n}}$-equivariant rational stable stems; this is done in section 4.
While the method employed here is the one used in [Geo21c] and goes back to
[GM95], the result is quite a bit more complicated to state and requires the
notation set up in sections 2 and 3.

We then attempt to generalize the results in [Geo21c] to groups $C_{2^{n}}$.
In [Geo21c], we obtained minimal descriptions of the $C_{2}$-equivariant
Chern, Pontryagin and symplectic characteristic classes associated with
genuine (Bredon) cohomology using coefficients in the rational Burnside Green
functor $A_{\operatorname{\mathbb{Q}}}$. The idea was based on the maximal
torus isomorphism: if $L$ is any one of $U(m),Sp(m),SO(m),SU(m)$, $T$ is a
maximal torus in $L$ and $W$ is the associated Weyl group then the inclusion
$B_{C_{2}}T\to B_{C_{2}}L$ induces an isomorphism
$H^{\bigstar}_{C_{2}}(B_{C_{2}}L;A_{\operatorname{\mathbb{Q}}})\to
H^{\bigstar}_{C_{2}}(B_{C_{2}}T;A_{\operatorname{\mathbb{Q}}})^{W}$. We then
computed $H^{\bigstar}_{C_{2}}(B_{C_{2}}T;A_{\operatorname{\mathbb{Q}}})$ from
$H^{\bigstar}_{C_{2}}(B_{C_{2}}S^{1};A_{\operatorname{\mathbb{Q}}})$ and the
Kunneth formula, which reduced us to the algebraic problem of computing a
minimal presentation of the fixed points
$H^{\bigstar}_{C_{2}}(B_{C_{2}}T;A_{\operatorname{\mathbb{Q}}})^{W}$.

In section 5, we show that the maximal torus isomorphism holds for all groups
$G=C_{2^{n}}$ and $L=U(m)$, but does not hold for $G=C_{2^{n}}$ and $L=SU(2)$
when $n>1$. We also compute the Green functor
$H^{\bigstar}_{G}(B_{G}S^{1};A_{\operatorname{\mathbb{Q}}})$ which turns out
to be algebraically quite a bit more complex compared to the $C_{2}$ case of
[Geo21c]. For that reason, we do not attempt to follow the program in [Geo21c]
and get minimal descriptions of
$H^{\bigstar}_{G}(B_{G}U(m);A_{\operatorname{\mathbb{Q}}})$ from the maximal
torus isomorphism.

### Acknowledgment

We would like to thank Peter May for his numerous editing suggestions,
including the idea to split off this paper from [Geo21c].

##  2\. Rational Mackey functors

The rational Burnside Green functor $A_{\operatorname{\mathbb{Q}}}$ over a
group $G$ is defined on orbits as $G/H\mapsto
A(H)\otimes\operatorname{\mathbb{Q}}$ where $A(H)$ is the Burnside ring of
$H$. A rational $G$-Mackey functor is by definition an
$A_{\operatorname{\mathbb{Q}}}$ module.

We shall use $G$-equivariant _unreduced_ co/homology in
$A_{\operatorname{\mathbb{Q}}}$ coefficients. So if $X$ is an unbased
$G$-space, $H_{\bigstar}^{G}(X)$ is the rational $G$-Mackey functor defined on
orbits as

$H_{\bigstar}^{G}(X)(G/H)=[S^{\bigstar},X_{+}\wedge
HA_{\operatorname{\mathbb{Q}}}]^{H}$ (1)

where $HA_{\operatorname{\mathbb{Q}}}$ is the equivariant Eilenberg-MacLane
spectrum associated to $A_{\operatorname{\mathbb{Q}}}$ and the index
$\bigstar$ ranges over the real representation ring $RO(G)$.

We warn the reader of differing conventions that can be found in the
literature: $H_{\bigstar}(X)$ is sometimes used to denote the _reduced_
homology Mackey functor (the group $G$ being implicit), with
$H_{\bigstar}^{G}(X)$ denoting the value of this Mackey functor on the top
level (i.e. the $G/G$ orbit). In this paper, $H_{\bigstar}^{G}(X)$ always
denotes the Mackey functor and $H_{\bigstar}^{G}(X)(G/G)$ always denotes the
top level. This convention also applies when $\bigstar=*$ ranges over the
integers, in which case $H_{*}(X)$ denotes the nonequivariant rational
homology of $X$.

All these conventions apply equally for cohomology $H^{\bigstar}_{G}(X)$.

The $RO(G)$-graded Mackey functor $H^{G}_{\bigstar}(X)$ is a module over the
homology of a point $H^{G}_{\bigstar}:=H^{G}_{\bigstar}(*)$. This Green
functor agrees with the equivariant rational stable stems:

$\pi_{\bigstar}^{G}S\otimes\operatorname{\mathbb{Q}}=H^{G}_{\bigstar}$ (2)

Two facts about rational Mackey functors that we shall liberally use ([GM95]):

  * •

All rational Mackey functors are projective and injective, so we have the
Kunneth formula:

$H_{\bigstar}^{G}(X\times
Y)=H_{\bigstar}^{G}(X)\boxtimes_{H_{\bigstar}^{G}}H_{\bigstar}^{G}(Y)$ (3)

and duality formula:

$H_{G}^{\bigstar}(X)=\operatorname{Hom}_{H_{\bigstar}^{G}}(H_{\bigstar}^{G}(X),H_{\bigstar}^{G})$
(4)

  * •

For a $G$-Mackey functor $M$ and a subgroup $H$ of $G$ consider the $W_{G}H$
module $M(G/H)/Im(Tr)$ where $W_{G}H=N_{G}H/H$ is the Weyl group and $Im(Tr)$
is the submodule spanned by the images of all transfer maps $Tr_{K}^{H}$ for
$K\subseteq H$. If we let $H$ vary over representatives of conjugacy classes
of subgroups of $G$ then we get a sequence of $W_{G}H$ modules. This functor
from rational $G$-Mackey functors to sequences of
$\operatorname{\mathbb{Q}}[W_{G}H]$-modules is an equivalence of symmetric
monoidal categories.

From now on, we specialize to the case $G=C_{2^{n}}$.

There are two $1$-dimensional $\operatorname{\mathbb{Q}}[G]$ modules up to
isomorphism: $\operatorname{\mathbb{Q}}$ with the trivial action and
$\operatorname{\mathbb{Q}}$ with action $g\cdot 1=-1$ where $g\in G$ is a
generator. We shall denote the two modules by $\operatorname{\mathbb{Q}}$ and
$\operatorname{\mathbb{Q}}_{-}$ respectively; every other module splits into a
sum of these.

The representatives of conjugacy classes of $G=C_{2^{n}}$ are $H=C_{2^{i}}$
for every $0\leq i\leq n$ thus the datum of a rational $G$-Mackey functor is
equivalent to a sequence of rational $W_{G}H=C_{2^{n}}/C_{2^{i}}$ modules.

We let $M_{i}^{+},0\leq i\leq n$, and $M_{i}^{-},0\leq i<n$, be the Mackey
functors corresponding to the sequences
$C_{2^{n}}/C_{2^{j}}\mapsto\delta_{ij}\operatorname{\mathbb{Q}}$ and
$C_{2^{n}}/C_{2^{j}}\mapsto\delta_{ij}\operatorname{\mathbb{Q}}_{-}$
respectively.

For example, $M_{0}^{+},M_{0}^{-}$ are the constant Mackey functors
corresponding to the modules $\operatorname{\mathbb{Q}}$ and
$\operatorname{\mathbb{Q}}_{-}$ respectively.

Observe that:

  * •

The $M_{i}^{\pm}$ are self-dual.

  * •

$M_{i}^{\pm}\boxtimes M_{j}^{\pm}=0$ if $i\neq j$.

  * •

$M_{i}^{\alpha}\boxtimes M_{i}^{\beta}=M_{i}^{\alpha\beta}$ where
$\alpha,\beta\in\\{-1,1\\}$.

Henceforth we shall write $M_{i}$ for $M_{i}^{+}$.

The notation $M_{i}\\{a\\}$ shall mean a copy of $M_{i}$ with a choice of
generator $a\in M_{i}(C_{2^{n}}/C_{2^{i}})=\operatorname{\mathbb{Q}}$. The
element $a$ generates $M_{i}\\{a\\}$ through its transfers:

$M_{i}\\{a\\}(C_{2^{n}}/C_{2^{j}})=\begin{cases}\operatorname{\mathbb{Q}}\\{\operatorname{Tr}_{2^{i}}^{2^{j}}(a)\\}&\textup{if,
}j\geq i\\\ 0&\textup{if, }j<i\end{cases}$ (5)

We analogously define $M_{i}^{-}\\{a\\}$.

The rational Burnside $G$-Green functor is

$A_{\operatorname{\mathbb{Q}}}(C_{2^{n}}/C_{2^{i}})=\frac{\operatorname{\mathbb{Q}}[x_{i,j}]}{x_{i,j}x_{i,k}=2^{i-\max(j,k)}x_{i,\min(j,k)}}$
(6)

where $x_{i,j}=[C_{2^{i}}/C_{2^{j}}]\in A(C_{2^{i}})$ for $0\leq j<i$. To
complete the Mackey functor description, we note that:

$\displaystyle\operatorname{Tr}_{2^{i}}^{2^{i+1}}(x_{i,j})=x_{i+1,j}\text{ ,
}\operatorname{Tr}_{2^{i}}^{2^{i+1}}(1)=x_{i+1,i}$ (7)

Let

$y_{i}=\begin{cases}1-\frac{x_{i,i-1}}{2}&\textup{if, }i\geq 1\\\
1&\textup{if, }i=0\end{cases}$ (8)

living in $A_{\operatorname{\mathbb{Q}}}(C_{2^{n}}/C_{2^{i}})$. We can see
that $y_{i}$ spans a copy of $M_{i}$ in $A_{\operatorname{\mathbb{Q}}}$ and:

$A_{\operatorname{\mathbb{Q}}}=\oplus_{i=0}^{n}M_{i}\\{y_{i}\\}$ (9)

This is an isomorphism of Green functors, where the RHS becomes a Green
functor by setting the product of elements from different summands to be $0$
and furthermore setting the $y_{i}$ to be idempotent ($y_{i}^{2}=y_{i}$).

##  3\. Euler and orientation classes

The real representation ring $RO(C_{2^{n}})$ is spanned by the irreducible
representations $1,\sigma,\lambda_{s,k}$ where $\sigma$ is the $1$-dimensional
sign representation and $\lambda_{s,m}$ is the $2$-dimensional representation
given by rotation by $2\pi s(m/2^{n})$ degrees for $1\leq m$ dividing
$2^{n-2}$ and odd $1\leq s<2^{n}/m$. Note that $2$-locally,
$S^{\lambda_{s,m}}\simeq S^{\lambda_{1,m}}$ as $C_{2^{n}}$-equivariant spaces,
by the $s$-power map. Therefore, to compute $H_{\bigstar}^{C_{2^{n}}}(X)$ it
suffices to only consider $\bigstar$ in the span of
$1,\sigma,\lambda_{k}:=\lambda_{1,2^{k}}$ for $0\leq k\leq n-2$
($\lambda_{n-1}=2\sigma$ and $\lambda_{n}=2$).

We shall now define generating classes for $H_{\bigstar}^{G}$.

We first have Euler classes $a_{\sigma}:S^{0}\hookrightarrow S^{\sigma}$ and
$a_{\lambda_{k}}:S^{0}\hookrightarrow S^{\lambda_{k}}$ given by the inclusion
of the north and south poles; under the Hurewicz map these classes are
$a_{\sigma}\in H_{-\sigma}^{G}(G/G)$ and $a_{\lambda_{k}}\in
H_{-\lambda_{k}}^{G}(G/G)$.

There are also orientation classes $u_{\sigma}\in
H_{1-\sigma}^{G}(C_{2^{n}}/C_{2^{n-1}})$, $u_{2\sigma}\in
H^{G}_{2-2\sigma}(G/G)$ and $u_{\lambda_{k}}\in H^{G}_{2-\lambda_{k}}(G/G)$
but we shall need a small computation in order to define them.

Using the cofiber sequence $C_{2^{n}}/C_{2^{n-1}+}\to
S^{0}\xrightarrow{a_{\sigma}}S^{\sigma}$ we get:

$\displaystyle\tilde{H}_{0}^{G}(S^{\sigma})=M_{n}\\{a_{\sigma}\\}$ (10)
$\displaystyle\tilde{H}_{1}^{G}(S^{\sigma})=\oplus_{i=0}^{n-1}M_{i}^{-}$ (11)

where $\tilde{H}_{*}^{G}(X)$ denotes the reduced homology of a based $G$-space
$X$. We can further see that $\tilde{H}_{1}^{G}(S^{\sigma})$ is generated as a
Green functor module by a class
$u_{\sigma}\in\tilde{H}_{1-\sigma}^{G}(C_{2^{n}}/C_{2^{n-1}})$. So we get

$\tilde{H}_{*}^{G}(S^{\sigma})=M_{n}\\{a_{\sigma}\\}\oplus\oplus_{i=0}^{n-1}M_{i}^{-}\\{y_{i}\operatorname{Res}_{2^{i}}^{2^{n-1}}(u_{\sigma})\\}$
(12)

Using that $S^{2\sigma}=S^{\sigma}\wedge S^{\sigma}$ and the Kunneth formula,
we get a class $u_{2\sigma}$ restricting to $u_{\sigma}^{2}$ and:

$\displaystyle\tilde{H}_{*}^{G}(S^{2\sigma})=M_{n}\\{a_{\sigma}^{2}\\}\oplus\oplus_{i=0}^{n-1}M_{i}\\{y_{i}\operatorname{Res}_{2^{i}}^{2^{n}}(u_{2\sigma})\\}$
(13)

For $0\leq k\leq n-2$ we have a $G$-CW decomposition $S^{0}\subseteq
X\subseteq S^{\lambda_{k}}$ where $X$ consists of the points
$(x_{1},x_{2},x_{3})\in S^{\lambda_{k}}\subseteq\operatorname{\mathbb{R}}^{3}$
with $x_{1}=0$ or $x_{2}=0$. From this decomposition we can see that:

$\displaystyle\tilde{H}_{0}^{G}(S^{\lambda_{k}})=\oplus_{i=k+1}^{n}M_{i}\\{y_{i}\operatorname{Res}_{2^{i}}^{2^{n}}(a_{\lambda_{k}})\\}$
(14)
$\displaystyle\tilde{H}_{2}^{G}(S^{\lambda_{k}})=\oplus_{i=0}^{k}M_{i}\\{y_{i}\operatorname{Res}_{2^{i}}^{2^{n}}(u_{\lambda_{k}})\\}$
(15)

for a class $u_{\lambda_{k}}\in H^{G}_{2-\lambda_{k}}(G/G)$. This also works
for $k=n-1$ and $\lambda_{n-1}=2\sigma$ giving a different way of obtaining
$a_{2\sigma}=a_{\sigma}^{2}$ and $u_{2\sigma}$.

The classes $u_{\sigma},u_{\lambda_{k}}$, $0\leq k\leq n-1$, have not been
canonically defined so far. Once we fix orientations for the spheres
$S^{\lambda_{k}}$, the $u_{\lambda_{k}}$ are uniquely determined by the
following two facts:

  * •

A $G$-self-equivalence of $S^{\lambda_{k}}$ induces the identity map on the
Mackey functor $\tilde{H}^{G}_{2}(S^{\lambda_{k}})$ if it does so on its
bottom level $\tilde{H}^{G}_{2}(S^{\lambda_{k}})(G/e)$.

  * •

An orientation of $S^{\lambda_{k}}$ determines a generator for
$\operatorname{\mathbb{Z}}=\tilde{H}_{2}(S^{2};\operatorname{\mathbb{Z}})$ and
consequently a generator for
$\operatorname{\mathbb{Q}}=\tilde{H}_{2}(S^{2};\operatorname{\mathbb{Q}})=\tilde{H}_{2}^{G}(S^{\lambda_{k}})(G/e)$.

The first fact is proven using that the Mackey functor
$\tilde{H}^{G}_{2}(S^{\lambda_{k}})$ is generated by the transfers of
$y_{i}\operatorname{Res}^{2^{n}}_{2^{i}}(u_{\lambda_{k}})$ where $i\leq k$, so
we only need to check that the induced map is the identity on
$\tilde{H}^{G}_{2}(S^{\lambda_{k}})(G/C_{2^{i}})=\tilde{H}^{C_{2^{i}}}_{2}(S^{\lambda_{k}})(C_{2^{i}}/C_{2^{i}})$
which follows from the fact that $C_{2^{i}}$ acts trivially on
$S^{\lambda_{k}}$ when $i\leq k$.

We can similarly uniquely determine $u_{\sigma}$ upon fixing an orientation of
$S^{\sigma}$ that is compatible with the orientation for
$S^{\lambda_{n-1}}=S^{2\sigma}$, meaning that
$\operatorname{Res}^{2^{n}}_{2^{n-1}}(u_{2\sigma})=u_{\sigma}^{2}$.

The discussion regarding orientation classes can also be performed integrally,
defining $A_{\operatorname{\mathbb{Z}}}$-orientation classes
$u_{\sigma},u_{2\sigma},u_{\lambda_{k}}$ upon fixing orientations for
$S^{\sigma},S^{2\sigma}$, $S^{\lambda_{k}}$ as above. The
$A_{\operatorname{\mathbb{Z}}}$-orientation classes map to the corresponding
$\operatorname{\mathbb{Z}}$-orientation classes of [HHR16] under the map
$HA_{\operatorname{\mathbb{Z}}}\to H\underline{\operatorname{\mathbb{Z}}}$
where $\underline{\operatorname{\mathbb{Z}}}$ is the constant Green functor
corresponding to the trivial $G$-module $\operatorname{\mathbb{Z}}$.

##  4\. Rational stable stems

In this section we shall give a presentation of the Green functor
$H^{G}_{\bigstar}$ with generators and relations.

The generators are elements $r_{k}\in H^{G}_{V_{k}}(C_{2^{n}}/C_{2^{i_{k}}})$
spanning $M_{i_{k}}^{\epsilon_{k}}$, where $\epsilon_{k}=+$ or $-$, such that
every element of $\coprod_{H\subseteq G,\bigstar\in
RO(G)}H_{\bigstar}^{G}(G/H)$ can be obtained from the $r_{k}$ using the
operations of addition, multiplication, restriction, transfer and scalar
multiplication (where the scalars are elements of $\coprod_{H\subseteq
G}A_{\operatorname{\mathbb{Q}}}(G/H)$).

The fact that the $r_{k}$ span $M_{i_{k}}^{\epsilon_{k}}$ gives all the
additive (Mackey functor) relations, but also implies certain multiplicative
relations by means of the Kunneth formula: If $i_{k}<i_{l}$ then
$r_{k}\cdot\operatorname{Res}^{2^{i_{l}}}_{2^{i_{k}}}(r_{l})=0$ and if
$i_{k}=i_{l}$ then $r_{k}r_{l}$ spans $M_{i_{k}}^{\epsilon_{k}\epsilon_{l}}$.

Finally, if $r\in H^{G}_{V_{k}}(C_{2^{n}}/C_{2^{i}})$ and there exists a
unique $r^{\prime}\in H^{G}_{-V_{k}}(C_{2^{n}}/C_{2^{i}})$ with
$rr^{\prime}=y_{i}$, then we shall use the notation $y_{i}/r$ to denote
$r^{\prime}$. If $r,y_{i}/r$ are generators then we have the implicit relation
$r\cdot(y_{i}/r)=y_{i}$.

######  Proposition 4.1.

The Green functor $H_{\bigstar}^{G}$ has a presentation whose generating set
is the union of the following four families:

  * •

$y_{i}\operatorname{Res}_{2^{i}}^{2^{n-1}}(u_{\sigma})$ and
$y_{i}/\operatorname{Res}_{2^{i}}^{2^{n-1}}(u_{\sigma})$ spanning $M_{i}^{-}$,
where $0\leq i<n$.

  * •

$y_{i}\operatorname{Res}^{2^{n}}_{2^{i}}(u_{\lambda_{k}})$ and
$y_{i}/\operatorname{Res}^{2^{n}}_{2^{i}}(u_{\lambda_{k}})$ spanning $M_{i}$,
where $0\leq i\leq k$ and $0\leq k\leq n-2$.

  * •

$y_{i}\operatorname{Res}^{2^{n}}_{2^{i}}(a_{\lambda_{k}})$ and
$y_{i}/\operatorname{Res}^{2^{n}}_{2^{i}}(a_{\lambda_{k}})$ spanning $M_{i}$,
where $k<i\leq n$ and $0\leq k\leq n-2$.

  * •

$a_{\sigma}$ ($=y_{n}a_{\sigma}$) and $y_{n}/a_{\sigma}$ spanning $M_{n}$.

We have implicit relations of the form
$(y_{i}\gamma)\cdot(y_{i}/\gamma)=y_{i}$ in each of the four families. The
remaining multiplicative relations can be obtained using the Kunneth formula.

Two observations:

  * •

For $0\leq i<n$, the square of
$y_{i}\operatorname{Res}_{2^{i}}^{2^{n-1}}(u_{\sigma})$ is
$y_{i}\operatorname{Res}_{2^{i}}^{2^{n}}(u_{2\sigma})$ and spans $M_{i}$.

  * •

The ring $H_{\bigstar}^{G}(G/G)$ has multiplicative relations:
$a_{\sigma}u_{2\sigma}=0$, $a_{\sigma}u_{\lambda_{k}}=0$ and
$a_{\lambda_{k}}u_{\lambda_{s}}=0$ for $s\leq k$.

The Green functor presentation also gives us an additive decomposition of
$H^{G}_{\bigstar}$ into $M_{i},M_{i}^{-}$ but to state it explicitly, we’ll
need some notation: For each integer tuple
$t=(j_{0},...,j_{n-1},j^{\prime}_{0},...,j^{\prime}_{n-1})$ let

$k(t)=\begin{cases}n&\textup{ if $j_{k}=0$ for all $k$}\\\ \min\\{k:j_{k}\neq
0\\}&\textup{ otherwise}\end{cases}$ (16)

and

$k^{\prime}(t)=\begin{cases}-1&\textup{ if $j^{\prime}_{k^{\prime}}=0$ for all
$k^{\prime}$}\\\ \max\\{k^{\prime}:j^{\prime}_{k^{\prime}}\neq 0\\}&\textup{
otherwise}\end{cases}$ (17)

and consider the representation

$\displaystyle
V_{t}^{\pm}=\sum_{k=0}^{n-2}(j_{k}(2-\lambda_{k})-j^{\prime}_{k}\lambda_{k})+j_{n-1}(1-\sigma)-j^{\prime}_{n-1}\sigma$
(18)

where the sign $\pm$ in $V_{t}^{\pm}$ is $+$ if $j_{n-1}$ is even and $-$ if
$j_{n-1}$ is odd.

Let $T$ be the set of all tuples $t$ with $k^{\prime}(t)<k(t)$; as $t$ ranges
over $T$, the $V_{t}^{\pm}$ are pairwise non-isomorphic virtual
representations. We can now state the additive description:

######  Proposition 4.2.

The $C_{2^{n}}$ equivariant rational stable stems are:

$H^{G}_{\bigstar}=\begin{cases}\oplus_{k^{\prime}(t)<i\leq
k(t)}M_{i}&\textup{if }\bigstar=V_{t}^{+}\text{ for }t\in T\\\
\oplus_{k^{\prime}(t)<i\leq k(t)}M_{i}^{-}&\textup{if
}\bigstar=V_{t}^{-}\text{ for }t\in T\\\ 0&\textup{otherwise }\end{cases}$
(19)

###### Proof.

(Of Proposition 4.1). Any representation sphere $S^{V}$ is the smash product
of $S^{\sigma},S^{\lambda_{k}}$ and their duals
$S^{-\sigma},S^{-\lambda_{k}}$. By duality,

$\tilde{H}_{*}^{G}(S^{-\sigma})=\tilde{H}^{-*}_{G}(S^{\sigma})=M_{n}\oplus\oplus_{i=0}^{n-1}M_{i}^{-}\\{y_{i}\operatorname{Res}_{2^{i}}^{2^{n-1}}(u_{\sigma}^{-1})\\}$
(20)

Let $t$ be a generator for this copy of $M_{n}$; then

$\tilde{H}_{0}^{G}(S^{0})=\tilde{H}_{0}^{G}(S^{\sigma})\boxtimes\tilde{H}_{0}^{G}(S^{-\sigma})\oplus\tilde{H}_{1}^{G}(S^{\sigma})\boxtimes\tilde{H}_{-1}^{G}(S^{-\sigma})$
(21)

On the left hand side we have a factor $M_{n}\\{y_{n}\\}$ and on the right
hand side we have $M_{n}\\{a_{\sigma}\\}\boxtimes
M_{n}\\{t\\}=M_{n}\\{a_{\sigma}t\\}$ so $y_{n}=\lambda a_{\sigma}t$ for
$\lambda\in\operatorname{\mathbb{Q}}^{\times}$. Thus we can pick
$t=y_{n}/a_{\sigma}$. The result then follows from the Kunneth formula. ∎

We note that taking geometric fixed points inverts all Euler classes,
annihilating all orientation classes and setting $y_{i}=1$. Therefore:

$\Phi^{C_{2^{n}}}(HA_{\operatorname{\mathbb{Q}}})_{\bigstar}=\operatorname{\mathbb{Q}}[a_{\sigma}^{\pm},a_{\lambda_{k}}^{\pm}]_{0\leq
k\leq n-2}$ (22)

hence
$\Phi^{C_{2^{n}}}HA_{\operatorname{\mathbb{Q}}}=H\operatorname{\mathbb{Q}}$ as
nonequivariant spectra. The homotopy fixed points, homotopy orbits and Tate
fixed points are computed using that $HA_{\operatorname{\mathbb{Q}}}\to
H\underline{\operatorname{\mathbb{Q}}}$ is a nonequivariant equivalence, where
$\underline{\operatorname{\mathbb{Q}}}=M_{0}$ is the constant Green functor.
Thus:

$(HA_{\operatorname{\mathbb{Q}}})_{hC_{2^{n}}\bigstar}=(HA_{\operatorname{\mathbb{Q}}})^{hC_{2^{n}}}_{\bigstar}=\operatorname{\mathbb{Q}}[u_{2\sigma}^{\pm},u_{\lambda_{k}}^{\pm}]_{0\leq
k\leq n-2}$ (23)

and $(HA_{\operatorname{\mathbb{Q}}})^{tC_{2^{n}}}=*$.

##  5\. $C_{2^{n}}$ rational characteristic classes

######  Proposition 5.1.

As a Green functor algebra over the homology of a point:

$H_{G}^{\bigstar}(B_{G}S^{1})=\frac{H_{G}^{\bigstar}[u,\alpha_{m,j}]_{1\leq
m\leq n,1\leq
j<2^{m}}}{\alpha_{m,j}\alpha_{m^{\prime},j^{\prime}}=\delta_{mm^{\prime}}\delta_{jj^{\prime}}\alpha_{m,j}\text{
, }\operatorname{Res}^{2^{n}}_{2^{m-1}}(\alpha_{m,j})=0}$ (24)

for $|u|=2$ and $|\alpha_{m,j}|=0$.

###### Proof.

Note that

$H^{\bigstar}_{G}(X)=H_{G}^{*}(X)\boxtimes_{A_{\operatorname{\mathbb{Q}}}}H^{G}_{\bigstar}$
(25)

so it suffices to describe the integer graded cohomology.

For an explicit model of $B_{G}S^{1}$ we take
$\operatorname{\mathbb{C}}P^{\infty}$ with a $G$ action that can be described
as follows: Let $V_{1},...,V_{2^{n}}$ be an ordering on the irreducible
complex $G$-representations and set $V_{k+2^{n}m}=V_{k}$ for any
$m\in\operatorname{\mathbb{Z}}$, $1\leq k\leq 2^{n}$. The action of $g\in G$
on homogeneous coordinates is $g(z_{1}:z_{2}:\cdots)=(gz_{1}:gz_{2}:\cdots)$
where $g$ acts on $z_{i}$ as it does on $V_{i}$.

Fix a subgroup $H=C_{2^{m}}$ of $G$. The fixed points under the $H$-action
are:

$(B_{G}S^{1})^{H}=\coprod_{j=1}^{2^{m}}\operatorname{\mathbb{C}}P^{\infty}$
(26)

To understand the indexing, let $W_{1},...,W_{2^{m}}$ be an ordering on the
irreducible complex $C_{2^{m}}$-representations; the $j$-th
$\operatorname{\mathbb{C}}P^{\infty}$ in $(B_{G}S^{1})^{H}$ corresponds to the
set of points with homogeneous coordinates $(z_{1}:z_{2}:\cdots)$ such that
$z_{k}=0$ if $\operatorname{Res}_{2^{m}}^{2^{n}}(V_{k})\neq W_{j}$.

By [GM95] we have:

$H^{*}_{G}(B_{G}S^{1})=\oplus_{m=0}^{n}H^{*}((B_{G}S^{1})^{C_{2^{m}}})^{C_{2^{n}}/C_{2^{m}}}$
(27)

where $H^{*}(X)$ is nonequivariant cohomology in $\operatorname{\mathbb{Q}}$
coefficients. The action of $C_{2^{n}}/C_{2^{m}}$ on nonequivariant cohomology
is trivial since it’s determined in degree $*=2$ and thus on the $2$-skeleton,
which itself is the disjoint union of copies of
$S^{2}=\operatorname{\mathbb{C}}P^{1}$ and for each $S^{2}$ the action is a
rotation hence has degree $1$. Thus

$H^{*}_{G}(B_{G}S^{1})=\oplus_{m=0}^{n}\oplus^{2^{m}}_{j=1}H^{*}(\operatorname{\mathbb{C}}P^{\infty})=\oplus_{m=0}^{n}\oplus^{2^{m}}_{j=1}\operatorname{\mathbb{Q}}[e_{m,j}]$
(28)

where each $e_{m,j}$ spans $M_{m}$. Set $\alpha_{m,j}=e_{m,j}^{0}$ and
$u=\sum_{m,j}e_{m,j}$; then

$\sum_{j=1}^{2^{m}}\alpha_{m,j}=\frac{\operatorname{Tr}_{2^{m}}^{2^{n}}(y_{m})}{2^{m}}$
(29)

so the $\alpha_{m,2^{m}}$ are superfluous. Thus we can take $1\leq m\leq n$
and $1\leq j<2^{m}$ in the indexing for $\alpha_{m,j}$. ∎

We can similarly prove that:

######  Proposition 5.2.

We have an isomorphism of Green functor algebras over $H^{\bigstar}_{G}$:

$H_{G}^{\bigstar}(B_{G}\Sigma_{2})=\frac{H^{\bigstar}_{G}(B_{G}S^{1})}{u}$
(30)

where the quotient map $H^{\bigstar}_{G}(B_{G}S^{1})\to
H_{G}^{\bigstar}(B_{G}\Sigma_{2})$ is induced by complexification:
$B_{G}\Sigma_{2}=B_{G}O(1)\to B_{G}U(1)=B_{G}S^{1}$.

The set of generators $\\{u,\alpha_{m,j}\\}$ for $H^{*}_{G}(B_{G}S^{1})$ is
not minimal. Indeed, whenever we have generators $e_{1},...,e_{s}$ with
$e_{i}e_{j}=\delta_{ij}e_{i}$, we can replace them by a single generator
defined by $e=e_{1}+2e_{2}+\cdots+se_{s}$:

$\displaystyle\frac{\operatorname{\mathbb{Q}}[e_{1},...,e_{s}]}{e_{i}e_{j}=\delta_{ij}e_{i}}=\frac{\operatorname{\mathbb{Q}}[e]}{e(e-1)\cdots(e-s)}$
(31)

This isomorphism follows from the fact that any polynomial $f$ on
$e_{1},...,e_{n}$ satisfies:

$f(e)=f(0)+(f(1)-f(0))e_{1}+\cdots+(f(s)-f(0))e_{s}$ (32)

and thus

$e_{i}=\frac{f_{i}(e)}{f_{i}(i)}\text{ where
}f_{i}(x)=\frac{x(x-1)\cdots(x-s)}{x-i}$ (33)

In this way, $H^{*}_{G}(B_{G}S^{1})$ is generated as an
$A_{\operatorname{\mathbb{Q}}}$ algebra by two elements $u,\alpha$ but now
with $\alpha$ satisfying some rather complicated relations. If $n=1$ i.e.
$G=C_{2}$, we only have one $\alpha_{m,j}$ element, namely
$\alpha=\alpha_{1,1}$ satisfying $\alpha^{2}=\alpha$.

######  Proposition 5.3.

The inclusion $B_{G}U(1)^{m}\to B_{G}U(m)$ induces an isomorphism of Green
functor algebras over $H^{\bigstar}_{G}$:

$H^{\bigstar}_{G}(B_{G}U(m))=(\otimes^{m}H^{\bigstar}_{G}(B_{G}U(1)))^{\Sigma_{m}}$
(34)

###### Proof.

Let $V_{i}$ be the complex $G$-representation corresponding to the root of
unity $e^{2\pi i/n}$. The Grassmannian model for $B_{G}U(m)$ uses complex
$m$-dimensional subspaces of $\operatorname{\mathbb{C}}^{\infty\rho_{G}}$; a
$G$-fixed point $W$ of $B_{G}U(m)$ is then a $G$-representation and thus as
splits as $W=\oplus_{i=1}^{2^{n}}k_{i}V_{i}$ for $k_{i}=0,1,...$ with
$\sum_{i}k_{i}=m$. An automorphism of $W$ is made out automorphisms for each
$k_{i}V_{i}$ hence

$B_{G}U(m)^{G}=\coprod_{\sum k_{i}=m}\prod_{i=1}^{2^{n}}BU(k_{i})$ (35)

Following [Geo21c] and inducting on the $n$ in $G=C_{2^{n}}$, it suffices to
show that

$H^{*}((B_{G}U(m))^{G})\to H^{*}(\prod^{m}(B_{G}S^{1})^{G})$ (36)

is an isomorphism after taking $\Sigma_{m}$ fixed points on the RHS. Spelling
this out, we have:

$\prod_{\sum
k_{i}=m}\otimes_{i=1}^{2^{n}}H^{*}(BU(k_{i}))\to\prod^{2^{nm}}\otimes^{m}H^{*}(BS^{1})$
(37)

where the product on the right is indexed on configurations
$(V_{r_{1}},...,V_{r_{m}})$. If we fix $k_{i}$ with $\sum_{i}k_{i}=m$ then we
get

$\otimes_{i=1}^{2^{n}}H^{*}(BU(k_{i}))\to\prod\otimes^{m}H^{*}(BS^{1})$ (38)

where the product on the right is indexed on configurations
$(V_{r_{1}},...,V_{r_{m}})$ where $k_{i}$ many of the $r_{j}$’s are equal to
$i$. Taking $\Sigma_{m}$ fixed points is equivalent to fixing a configuration
and then taking $\Sigma_{k_{1}}\times\cdots\times\Sigma_{k_{2^{n}}}$ fixed
points, where each $\Sigma_{k_{i}}$ permutes the $k_{i}$ many coordinates that
are $V_{i}$ in the configuration. Thus we are reduced to the nonequivariant
isomorphism:

$H^{*}(BU(k_{i}))=(\otimes^{k_{i}}H^{*}(BS^{1}))^{\Sigma_{k_{i}}}$ (39)

∎

For $n=1$, $G=C_{2}$ and $H^{*}_{G}(B_{G}S^{1})$ has a simple enough
description to allow the computation of an explicit minimal presentation of
$H^{*}_{G}(B_{G}U(m))=H^{*}_{G}(B_{G}U(1)^{m})^{\Sigma_{m}}$. Due to the
greater algebraic complexity of $H^{*}_{G}(B_{G}S^{1})$ for $n\geq 2$
($G=C_{2^{n}}$), we do not attempt to generalize this and the rest of [Geo21c]
to groups $G=C_{2^{n}}$ for $n\geq 2$.

We note that the maximal torus isomorphism does not work $C_{2^{n}}$
equivariantly for the Lie group $L=SU(2)=Sp(1)$ and $n\geq 2$. The reason is
that a $C_{2^{n}}$ representation in $SU(2)$ is $2,2\sigma$ or $V_{i}\oplus
V_{-i}$, $1\leq i<2^{n-1}$, using the notation of the proof above. Thus:

$B_{G}SU(2)^{G}=BSU(2)\coprod BSU(2)\coprod\coprod_{i=1}^{2^{n-1}-1}BS^{1}$
(40)

so $H^{0}(B_{G}SU(2)^{G})$ has dimension $2^{n-1}+1$. On the other hand, the
maximal torus is $U(1)\subseteq SU(2)$ with Weyl group $C_{2}$ and

$B_{G}U(1)^{G}=\coprod_{i=1}^{2^{n}}BS^{1}$ (41)

The $C_{2}$ action does not affect $H^{0}(B_{G}U(1)^{G})$ which has dimension
$2^{n}$. Finally, $2^{n-1}+1=2^{n}$ only when $n=1$.

## References

  * [Geo21c] N. Georgakopoulos, _$C_{2}$ equivariant characteristic classes over the rational Burnside ring_, available here
  * [GM95] J.P.C. Greenlees, J.P. May, _Generalized Tate Cohomology_ , Memoirs of the American Mathematical Society 543 (1995).
  * [HHR16] M. A. Hill, M. J. Hopkins, D. C. Ravenel, _On the non-existence of elements of Kervaire invariant one_ , Annals of Mathematics, Volume 184 (2016), Issue 1

1

Department of Mathematics, University of Chicago
E-mail<EMAIL_ADDRESS>
Website: math.uchicago.edu/$\sim$nickg