# Flexible layout of surface code computations using AutoCCZ states

Craig Gidney<EMAIL_ADDRESS>Google Inc., Santa Barbara, California
93117, USA  Austin G. Fowler  Google Inc., Santa Barbara, California 93117,
USA

###### Abstract

We construct a self-correcting CCZ state (the “AutoCCZ”) with embedded delayed
choice CZs for completing gate teleportations. Using the AutoCCZ state we
create efficient surface code spacetime layouts for both a depth-limited
circuit (a ripply-carry addition) and a Clifford-limited circuit (a QROM
read). Our layouts account for distillation and routing, are based on
plausible physical assumptions for a large-scale superconducting qubit
platform, and suggest that circuit-level Toffoli parallelism (e.g. using a
carry-lookahead adder instead of a ripple-carry adder) will not reduce the
execution time of computations involving fewer than five million physical
qubits. We reduce the spacetime volume of delayed choice CZs by a factor of 4
compared to techniques from previous work (Fowler 2012), and make several
improvements to the CCZ magic state factory from (Gidney 2019).

##  1 Introduction

An interesting consequence of the topological nature of quantum computation in
the surface code [7] is that, in spacetime diagrams describing the
computation, there is very little distinction between spacelike and timelike
directions. It is valid to redirect a qubit’s worldline “backwards in time” so
that its next operation actually happened earlier. There is of course a
forward-in-time description of the computation occurring, where a magic state
representing an operation is prepared, applied to the qubit via gate
teleportation, and then completed via classically controlled fixup operations.
But conceptually it is helpful to treat the time direction as just another
space direction.

Computations that only involve Clifford operations are particularly easy to
treat purely topologically, with time equivalent to space, because when
Clifford operations are applied via gate teleportation the resulting fixup
operations are always classically controlled Paulis (which can be applied
entirely within the classical control system). In order to treat time like
space when performing more general operations, such as T gates and Toffoli
gates, it is necessary to use techniques such as the selective routing
construction described by Fowler in [8].

Fowler’s technique allows general operations to be laid out arbitrarily
through spacetime, but produces a series of “routing qubits” that must be
stored until an adaptive measurement process determines whether to measure
them in the X or Z basis. The measurement depth (e.g. T depth) of the circuit
determines how many times the classical control system will have to perform a
set of measurements, decide which basis to use for the next set of
measurements, and start those measurements. The speed at which the control
system can run this loop, and work through the measurements, determines the
speed of computation.

The characteristic time it takes the control system to react to a measurement,
and perform the following dependent measurement, is the control system’s
“reaction time”. We refer to the general paradigm of performing a quantum
computation whose speed is limited only by the measurement depth of the
circuit and the reaction time of the control system as “reaction limited
computation”.

Figure 1:  Equivalent concepts expressed in quantum circuit diagrams, ZX
calculus graphs, and 3d topological diagrams. In circuit diagrams, time goes
from left to right. In ZX calculus graphs, there is no preferred time
direction. In 3d topological diagrams, times goes from bottom to top. We never
show Pauli operations in lattice surgery diagrams or in ZX calculus graphs,
because they are performed by the classical control system instead of by
operating on qubits. Our usage of the ZX calculus is somewhat non-standard in
that we consider ZX graphs to be equivalent if they are equal modulo Pauli
operations, we use a non-standard node coloring that matches the coloring of
our topological diagrams, and we introduce a “delayed choice node” to
represent adaptive effects coming from the classical control system. We
exaggerate the spacing of our 3d topological diagrams, as in [11], so that it
is possible to see how the components are interconnected.

Our contributions in this paper relate to decreasing the space overhead of
reaction limited computation, and making it easier to route. The paper is
organized as follows. In Section 1 we describe the context in which we are
operating and note some notational conventions. In Section 2 we present our
optimized version of a reaction limited selective CZ, which we refer to as a
delayed choice CZ. In Section 3 we show how to produce and consume AutoCCZ
states, which make routing easier because they decouple the consumption of the
CCZ state from the fixup operations needed to complete a gate teleportation.
Section 4 presents several improvements to the CCZ distillation factory from
[11]. In Section 5 we lay out a reaction limited ripple-carry adder that
relies heavily on AutoCCZ states. In Section 6 we lay out a QROM read that
uses multiple access hallways in order to _nearly_ transition from being
Clifford limited to being reaction limited. Finally, in Section 7 we summarize
our contributions and discuss their implications.

In this paper we will represent quantum computations using quantum circuit
diagrams, ZX calculus graphs [5] and 3d topological diagrams. Figure 1 shows
how to translate between the three representations.

When making estimates, we will be using assumptions that are plausible for a
future large-scale superconducting qubit platform: a reaction time of 10
microseconds, a surface code cycle time of 1 microsecond, and a characteristic
gate error rate of $10^{-3}$. We show these quantities in Table 1, which also
notes the effect of changing each value.

Quantity | Value | Effect of 10x decrease | Effect of 10x increase
---|---|---|---
Physical gate error rate | $10^{-3}$ | 4x less space, same time, fewer factories | Too close to threshold for tractable computation
|  | Reaction limited at 7 factories |
Surface code cycle time | 1 $\mu$s | Same time, fewer factories | 10x more spacetime volume
|  | Reaction limited at 2 factories | Reaction limited at 135 factories
Reaction time | 10 $\mu$s | Easy to trade space for time | Hard to trade space for time
|  | Reaction limited at 135 factories | Reaction limited at 2 factories
Physical connectivity | planar | N/A | N/A

Table 1:  Physical assumptions we make in this paper, and the effect of
varying them.

##  2 A compact delayed choice CZ

Fowler’s selective routing technique from [8] is based on controllable
multiplexers and demultiplexers (see Figure 2). Each (de)multiplexer produces
two “routing qubits” which, depending on whether they are measured in the X or
Z basis at a later time, can route a data qubit through one of multiple
precomputed worldlines. By chaining these selective computations together
through space instead of through time, the computation becomes reaction
limited.

For example, consider a series of Toffoli gates where the output of each gate
affects the control of a following Toffoli gate, such as the Toffolis in a
ripple-carry adder. Normally, the current Toffoli would need to be completely
finished (including fixup operations necessitated due to using gate
teleportation) before it was possible to start the next Toffoli gate. But, by
using multiplexers routing through multiple possible precomputed fixups, it is
possible to apply all of the Toffoli operations simultaneously while delayed-
choice routing through all the various possible fixups.

Although the (de)multiplexer construction from [8] is very general, it is
often not optimal. For example, teleporting a CCZ gate produces up to three
possible CZ fixup operations. Using the multiplexer construction to delay the
choice of whether or not the various CZ fixups should be applied would produce
eight routing qubits per potential CZ (because there are two qubits involved
in a CZ and each must go through a multiplexer/demultiplexer pair). In Figure
3, we present a more efficient construction for performing delayed choice CZs
that only uses two routing qubits.

Figure 2:  Delayed choice multiplex/demultiplex construction from [8]. Top
left is a circuit diagram directly from [8]. The bottom left shows the process
as a ZX calculus graph which, unlike the circuit diagram, is identical for
multiplexing and demultiplexing. Known rewrite rules are used to show
equivalence with the claimed “choose which route is connected” functionality.
On the right side is a 3d topological diagram of a lattice surgery
implementation of the construction. The vertical poles coming out of the
branches of the fork are the routing qubits used to control which of the two
branches connects to the rear trunk. They can be extended arbitrarily. The red
squares atop the routing qubit columns are placeholders for an eventual X or Z
basis measurement, to be determined by classical control software.

Figure 3:  Our optimized delayed choice CZ as a circuit and as a lattice
surgery construction. The two forms are shown to be equivalent via ZX graph
identities. During execution, the choice of whether or not to apply the CZ is
delayed by extending the red-topped columns (the “routing qubits”) in the 3d
topological diagram. Once the choice is known, the columns are terminated with
the red square replaced either by a white square (activates the CZ) or a black
square (skips the CZ). The circuit can be opened in the online simulator Quirk
by following this link.

##  3 The AutoCCZ state

We can embed three instances of the delayed choice CZ construction directly
into a CCZ state, so that there is one delayed choice CZ for each CZ fixup
that may be needed when performing gate teleportation. This augments the CCZ
state into an “AutoCCZ” state, so named because it automatically cleans up its
own CZ fixup garbage. This makes consuming the state much simpler, because no
corrections are needed at the consumption site. (Note that any magic state can
be augmented into an auto magic state. For example, Figure 17 of [15] defines
an auto-corrected T gate.)

In Figure 4, we show a circuit diagram producing and consuming an AutoCCZ
state. The figure also shows how to represent this concept as a ZX calculus
graph. Then, in Figure 5, we construct a compact “CCZ fixup box” with contents
equivalent to the CZ fixup part of the ZX graph. AutoCCZ states are produced
by linking the CCZ state output from a CCZ factory to the ports of the fixup
box as the CCZ state is routed to its destination.

In Figure 6, we show how to use a CCZ state to perform a Toffoli operation. In
Figure 7, we show the resulting reaction limited control system cycle.

We believe that it is possible to reduce the amount of spacetime volume we use
to transform CCZ states into AutoCCZ states. For example, our current
construction uses six routing qubits (two for each possible CZ fixup) but
intuition would suggest that only three qubits should be needed in order to
distinguish between the eight possible fixup cases. Alternatively, it should
be possible to save volume by carefully integrating the delayed choice CZ
fixups directly into the CCZ factory. We leave this task as future work.

Figure 4:  A circuit diagram and ZX calculus graph simplification showing how
to create and consume an AutoCCZ magic state, powered by delayed choice CZs,
to perform a CCZ gate. The bottom left part of the circuit is producing the
AutoCCZ magic state, then the middle left does parity measurements vs the
target qubits, then the middle right uses those measurements to determine the
basis of measurements that determine whether fixup operations occur, then
finally on the right side all the measurement results are used to update the
Pauli frame tracked in the control software. The circuit can be opened in the
online simulator Quirk by following this link.

Figure 5:  Lattice surgery implementation of the CZ fixups bubble from Figure
4. The 3d model is stored in ancillary file “ccz-fixup.skp”; it can be opened
online using Sketchup. The tee-junctions in the 3d diagram have been numbered,
so that it is easier to see how they correspond to the cycle in the ZX
calculus graph. The pink box on the right, with routing qubit “chimneys”, is
the simplified abstract representation that we will use in larger diagrams.

Figure 6:  Performing a Toffoli gate by consuming an AutoCCZ state.

Figure 7:  Performing a series of Toffoli gates using reaction limited quantum
computation [8]. The Toffolis can be laid out in a spacelike fashion, so that
they are “performed simultaneously”, but doing so will produce routing qubits
that the control software will still process serially. The rate at which the
control software can cycle between computing Pauli frames and applying fixups,
i.e. the time it takes to measure a set of routing qubits and determine the
basis to measure the next set in, determines the speed of the quantum
computation.

##  4 Improvements to CCZ distillation

Operations that are not native to the surface code can be performed using
magic state distillation [3] and gate teleportation [12]. A particularly
useful magic state is the CCZ state $|CCZ\rangle=CCZ|+\rangle^{\otimes
3}=\sum_{a,b,c\in\\{0,1\\}}(-1)^{abc}|abc\rangle$. The reason this state is
useful is because the quantum equivalent of the AND gate, the Toffoli gate, is
not native to the surface code but can be performed by consuming one CCZ state
[13, 6, 11] (see Figure 6). Algorithms with a lot of arithmetic, such as
Grover’s algorithm and Shor’s algorithm, perform many Toffoli gates and
benefit from using a state specialized to this task.

It is also possible to perform Toffoli gates using T states, but four states
are required instead of one [13]. Whether this is better or worse than using a
CCZ state, or some other technique, depends on the relative spacetime volumes
of the different types of magic state factories, their error rates, and the
number of operations that need to be performed.

In this paper we will be using the CCZ factory from [11]. It produces level 0
T states using the state injection technique of Li [14], then distills level 1
T states using one round of 15-to-1 distillation based on the Reed-Muller code
[3], then uses those T states to perform an error-detecting Toffoli [13] to
produce the final CCZ states.

We make several small improvements to this factory, in order to improve its
depth:

  * •

We choose code distances based on the target error rate, instead of based on
what packs most neatly.

  * •

We use six level 1 T factories instead of five.

  * •

We improve the depth of the T teleportation/injection construction.

  * •

We interleave the level 1 T factories better.

Figure 8:  Improved method for gate teleporting a T state into a qubit. This
3d model is stored in ancillary file “inject-t.skp”; it can be opened online
using Sketchup. The new method (right) saves $0.5d$ surface code cycles, where
$d$ is the code distance of the factory, compared to the old method (left)
[9].

Figure 9:  The CCZ factory from [11], after adjusting the level 1 distance to
17 and the level 2 distance to $d=27$, has a footprint that fits inside a
$15\times 8$ rectangular patch of distance $d=27$ logical qubits.

Figure 10:  3D diagram of the core of the CCZ factory factory we are using,
derived from the factory in [11]. This 3d model is stored in ancillary file
“ccz-factory.skp”; it can be opened online using Sketchup. The triplets of
open-ended pipes are the CCZ state output. Each small open-ended pipe is for a
T state input, coming from the six level 1 T factories (not shown here, see
Figure 11). Due to how the factory is interleaved with itself, a CCZ state of
sufficient quality is produced every $d\cdot 5$ surface code cycles where $d$
is the code distance.

Figure 11:  Different views of a 3D diagram of the level 1 T factory we are
using to produce $|T_{1}\rangle$ states, derived from the factory in [9, 11].
This 3d model is stored in ancillary file “t1-factory.skp”; it can be opened
online using Sketchup. Due to how the factory is interleaved with itself, a
$|T_{1}\rangle$ state is produced every ${d_{1}}\cdot 5.75$ surface code
cycles (where ${d_{1}}$ is the level 1 code distance).

The exact code distances that we choose depends on the number of operations
that have to be performed. Loosely speaking, the level 2 code distance will
almost always be lower (e.g. 27 instead of the original 31) and the level 1
code distance will sometimes be larger (e.g. 17 instead of the original 15).
Changing the code distances requires recomputing the footprint and depth of
the factory. Depending on the ratio between the level 1 and level 2 code
distances, the output rate may be limited by either the production rate of
level 1 T states or by the height of the level 2 part of the factory. The
footprint of the factory is similarly lower bounded by either one code
distance or the other, depending on their relative size (e.g. see Figure 9).

Since writing [11], we have checked that it is possible to slightly lower the
locations where T states are injected into our factories. Basically, because
the T states are being injected on the outside of the vertical poles storing
the factory’s qubits, while the stabilizer measurements are happening on the
insides of the poles, they are not constrained into waiting for each other and
it is possible to overlap the two. We show this new injection style, which
reduces the depth of the CCZ factory from $5.5d$ to $5d$, in Figure 8.

We also used this new injection style in the level 1 T factories within the
CCZ factory. Additionally, we found a slightly better way to interleave
successive instantiations of the T factories. In total, as shown in Figure 11,
we achieved a depth of $5.75d_{1}$ (compared to $6.25d_{1}$ in [11]).

For extremely large problem sizes, where the number of Toffoli operations
exceeds the capabilities of the CCZ state factory from [11], we can fall back
to the T state factory from [9]. According to the spreadsheet included in
[11], and accounting for the ability to increase the code distances, this
transition becomes necessary when performing on the order of ten trillion
Toffoli operations.

In order to perform a reaction limited computation, more than one factory is
needed. The exact number depends on the reaction time of the control system,
the cycle time of the surface code, and the depth of the factory. In this
paper we are assuming a reaction time of 10 microseconds and a cycle time of 1
microsecond. For the sake of example, we will assume a level 1 code distance
of 17 and a level 2 code distance of 27.

The production rate of the factory can be limited by either the level 1 or
level 2 distances. At level 2 the production rate of the factory is limited by
the factory’s depth times the cycle time times the level 2 code distance. So
the level 2 part of the factory is technically capable of producing states at
a rate of $(5\cdot 1\mu s\cdot 27)^{-1}\approx 7.4$ kHz. The level 1 part of
the factory needs to produce 8 level 1 T states for each CCZ state that will
be output. There are six level 1 T factories, and they have a depth of
$5.75d_{1}$, which means the output rate of the entire factory cannot be
larger than $(5.75\cdot 17\cdot 1\mu s\cdot 8/6)^{-1}\approx 7.7$ kHz.
Therefore the level 2 code distance is the limiting factor, and the factory
runs at $7.4$ kHz.

In a reaction limited computation, one CCZ state will be needed per reaction
time of the classical control system. That is to say, CCZ states are consumed
at a rate of 100 kHz. Therefore, given our assumptions, a reaction limited
computation requires $\lceil 100/7.4\rceil=14$ CCZ factories running in
parallel.

Note that the combined footprint of these 14 CCZ factories would be
approximately 2.6 million physical qubits. This suggests that quantum
computations covering less than five million physical qubits will see no gains
from circuit-level parallelism, such as using carry-lookahead adders instead
of ripple-carry adders, because there simply isn’t the space to fit the data
qubits, the factories, and the routing needed to run multiple threads of
execution faster than a single thread of execution.

##  5 Reaction limited ripple-carry addition

Our goal in this section is to perform a reaction limited addition using the
ripple-carry adder described by Cuccaro et al [4] (see Figure 5). There are
two problems we need to solve to make this possible. First, we need to figure
out how to lay out the operations of the adder in a spacelike fashion. Second,
we need to efficiently route the CCZ states being produced by the many CCZ
factories into the ripple-carry computation.

Thanks to the flexibility of the AutoCCZ state, laying out the ripple-carry
process in a spacelike fashion is straightforward. We just need
implementations of the MAJ and UMA carry rippling operations that accept a CCZ
state and propagate the involved bits horizontally across space, instead of
vertically through time. Figure 13 shows a 3d topological diagram of a MAJ
implementation with this property. It consumes one AutoCCZ state, propagates
carry bits left to right, propagates the qubits being operated on front to
back, and propagates nothing past-to-future. It is possible to tile multiple
copies of this block across space, as shown in Figure 14. The control system
will then analyze the dependencies between the AutoCCZ states, and propagate
the carry signal in a reaction limited fashion (as in Figure 7).

Figure 12:  Cuccaro’s ripple-carry adder [4]. Uses local operations travelling
in a “V” shaped wave. If $m$ is the length of the target register $t$, then
the Toffoli count and measurement depth of this construction is $2m-3$.

Figure 13:  Lattice surgery implementation of the MAJ operation. Top-left is
the MAJ circuit from Cuccaro’s paper [4]. Top-center is a ZX calculus graph
with function equivalent to the circuit. Center-right is a lattice surgery
implementation of the ZX graph (stored in ancillary file “maj.skp” which can
be opened online using Sketchup). Bottom-center is a simplified representation
of the 3d model that shows only the ports and the 3x3x5 bounding box.

Figure 14:  Lattice surgery implementation of a series of MAJ operations. Each
light blue box is an instance of Figure 13. To enable parallelization, the
carry propagation is done left to right, through space, instead of forward
through time. Dark and light coloring indicates boundary type, but otherwise
coloring is for labelling. Red bars are injecting three qubits from an AutoCCZ
state, green bars are passing qubits from the target register through the
block (front to back), and yellow bars are passing qubits from the offset
register through the block (front to back).

Note that we can’t lay out the _entire_ addition in a spacelike fashion. That
would require a number of factories proportional to the size of the addition,
instead of proportional to the reaction time of the control system. We instead
zig-zag the addition back and forth across space, performing just enough carry
rippling to keep the factories and the control system busy.

We now move on to the routing question. Each MAJ box has four inputs and three
outputs. One of the inputs, and also one of the outputs, is the carry qubit.
Another two of the inputs (and outputs) are the data qubits, one from the
target register and one from the offset register. The remaining input is the
three qubits making up the CCZ part of an AutoCCZ state. We must route all
these input and output qubits in a way that causes them to intersect the MAJ
box at the right place and at the right time.

Our solution to this problem is to zig-zag the carry qubit back and forth
along the X axis (right/left through space), while running data qubits through
along the Y axis (forward/back through space). We place CCZ factories in front
of and behind the zig-zagging area, so that their outputs are produced
directly adjacent to where they are needed making routing trivial. We then
leave small gaps between adjacent factories, so that data qubits from outside
the zig-zagging area can be routed through those gaps as needed.

As more and more data qubits are routed from behind the zig-zagging area to in
front (or vice versa), we gradually shift the zig-zagging area backward (or
forward). We interleave the two data registers into alternating rows, so that
qubits that need to reach the same MAJ box at the same time are adjacent.
Within each row we do additional interleaving, spacing out qubits that are
technically sequential in the register. This prevents traffic jams as the data
qubits are routed through the gaps between the factories.

The layout we are describing can be seen in 2d in Figure 15 and in 3d in
Figure 16. We also provide an “across time” view in Figure 18. Note how there
is some additional space for the CCZ fixup boxes from Figure 5 and for
left/right routing of data qubits. There are two CCZ fixup boxes per CCZ
factory because the routing qubits (the “chimneys”) emerging from a CCZ fixup
box can extend vertically into the next “zig-zag layer” before the control
system determines how to measure said qubits.

This layout is capable of performing ripple-carry additions at the reaction
limited rate, propagating the carry information from qubit to qubit at 100kHz.
Given our physical assumptions, this layout would add a pair of thousand-qubit
registers in approximately 20 milliseconds.

Figure 15:  Layout of a reaction limited ripple-carry addition, assuming a
level 2 code distance of 27 and a level 1 code distance of 17. The CCZ
factories (red boxes) and CCZ fixups (pink boxes) are producing AutoCCZ states
fed into the MAJ computations (light blue region). As the carry qubit zig-zags
back and forth, qubits from the offset register and target register (green and
yellow rows) are routed through gaps between the factories.

Figure 16:  Surface code activity during an addition, assuming a level 2 code
distance of 27 and a level 1 code distance of 17. This 3d model is stored in
ancillary file “adder-layout.skp”, which can be opened online using Sketchup.
Yellow and green rows are qubits from the target and input registers of the
addition. Dark blue rows are qubits from an idle register. Red boxes are CCZ
magic state factories, auto-corrected by the pink boxes (see Figure 3). Light
blue boxes are the MAJ operation of Cuccaro’s adder (see Figure 13), arranged
into a spacelike sweep to keep ahead of the control software (see Section 3).
The full adder is formed by repeating this pattern, with the operating area
gradually sweeping up and then down through the green/yellow data (see Figure
18).

##  6 Clifford limited table lookups

A ripple-carry adder is ideal for reaction limited computation because it has
only a small amount of Clifford operations per Toffoli operation. A table
lookup (also called a QROM read [1]) is exactly the opposite. For each Toffoli
performed in a table lookup, there is a huge amount of Clifford work to do. In
particular, each Toffoli triggers a gigantic multi-target CNOT potentially
touching all of the lookup’s output qubits. Because of this, the limiting
factor during a table lookup is not the control system’s reaction time but
rather access to the output qubits.

In order to target a logical qubit with a CNOT, there must be an unused
logical-qubit sized patch of surface code adjacent to that logical qubit. The
CNOT operation will then need to use that patch for $d$ cycles, where $d$ is
the code distance. For qubits where only one side is accessible, only one CNOT
can be performed per $d$ cycles. Given our assumption of a surface code cycle
time of 1 microsecond, and using a code distance of 27 just for example, this
suggests a maximum CNOT rate of 37kHz (instead of the 100kHz of a reaction
limited computation).

It is often possible to work around this CNOT rate limitation. For example, if
there are multiple single-control single-target CNOTs all targeting the same
qubit, it is possible to fuse the many CNOTs into a single generalized CNOT
where the control is a Pauli product of all the individual controls. But this
doesn’t work in the case of table lookups, because the set of relevant control
qubits differs from output qubit to output qubit.

One thing we can do is to make two sides of each qubit accessible, instead of
just one. The gigantic multi-target CNOTs can then alternate between using one
side, and using the other side. This doubles the achievable Toffoli rate from
37kHz to 74kHz, which is much closer to 100kHz. One can go further, e.g.
making temporary copies of qubits in order to allow more and more parallel
access, but the space tradeoffs start to become problematic. In this paper we
will limit ourselves to two-sided access.

We focus on the case where we are performing a large lookup that prepares a
register whose rows are interleaved between rows of another register. This
specific case is interesting to us because it is the kind of lookup that is
needed when performing windowed arithmetic [10], where groups of addition
operations are replaced by single lookup+addition operations.

While performing the lookup, we use a tiled row interleaving pattern of
$R\\_L\\_L\\_R$ where an $L$ is a lookup data row, an $R$ is an existing data
row not involved in the lookup, and an underscore is a empty access row that
can be used when performing the multi-target CNOTs. The multi-target CNOTs
alternate between using the single inner access row and both of the outer
access rows. In order to access the access rows, we place cross-row access
corridors on opposing sides of the layout. The multi-target CNOTs alternate
between using the two access corridors, so that they can branch into
individual access rows as needed.

Figure 17 has a 2d floor plan of where factories and access hallways are
located during a lookup. We also provide an “across time” view in Figure 18 of
a lookup followed by an addition.

Figure 17:  Data layout during a double-access lookup, assuming a level 2 code
distance of 27 and a level 1 code distance of 17. Output qubits (yellow) are
interleaved between the rows of an existing register (green). The vertical
access corridors and horizontal access rows provide two distinct ways to
simultaneously access all output qubits when performing many-target CNOTs. The
dark gray area in the bottom region is sufficient space to run a unary
iteration [1]. The number of factories shown is slightly lower than what is
needed to saturate the access rate of the double-access hallways.

Figure 18:  Layout of data (top to bottom) over time (left to right) during a
lookup addition. During lookup, the target register is spread out to make room
for the temporary register that will hold the lookup’s output. During addition
the target register and lookup output register are squeezed through a moving
operating area that sweeps up then down, applying the MAJ and UMA sweeps of
Cuccaro’s adder. Uncomputing the lookup is done using measurement based
uncomputation [2], and has negligible cost compared to the other steps.

##  7 Conclusion

In this paper we defined a self-correcting AutoCCZ state, improved the
distillation factory and delayed choice CZ constructions underlying it, and
demonstrated the basics of using it to lay out low-depth surface code
computations. We laid out a reaction limited ripple-carry adder, a Clifford
limited table lookup, and ensured they fit together in a fashion that would be
useful for performing windowed arithmetic.

We estimated, under plausible physical assumptions for future large-scale
superconducting qubit platforms, that running a serial circuit at the reaction
limited rate requires 14 CCZ factories. Because this number of factories
covers several million physical qubits, we believe that (under our physical
assumptions) computations involving fewer than five million physical qubits
will not benefit from using circuits that perform Toffolis in parallel (as in
carry-lookahead adders) instead of serially (as in ripple-carry adders).

We hope readers find the techniques we’ve described, and the example layouts
we’ve presented, helpful when laying out their own large scale computations.

##  8 Contributions

Craig Gidney constructed the AutoCCZ state and produced the adder and lookup
layouts using it. Austin Fowler helped iterate on ideas for laying out CCZ
fixups, found the more efficient factory interleavings, and played a
supervisory role guiding the content of the paper.

##  9 Acknowledgements

We thank Hartmut Neven for creating an environment where this research was
possible in the first place.

## References

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# The impact of nuclear deformation on relativistic heavy-ion collisions:
assessing consistency in nuclear physics across energy scales

Giuliano Giacalone  Institut für Theoretische Physik, Universität Heidelberg,
Philosophenweg 16, 69120 Heidelberg, Germany  Jiangyong Jia
<EMAIL_ADDRESS>Department of Chemistry, Stony Brook University,
Stony Brook, NY 11794, USA  Physics Department, Brookhaven National
Laboratory, Upton, NY 11976, USA  Chunjian Zhang  Department of Chemistry,
Stony Brook University, Stony Brook, NY 11794, USA

###### Abstract

In the hydrodynamic framework of heavy-ion collisions, elliptic flow, $v_{2}$,
is sensitive to the quadrupole deformation, $\beta$, of the colliding ions.
This enables one to test whether the established knowledge on the low-energy
structure of nuclei is consistent with collider data from high-energy
experiments. We derive a formula based on generic scaling laws of
hydrodynamics to relate the difference in $v_{2}$ measured between collision
systems that are close in size to the value of $\beta$ of the respective
species. We validate our formula in simulations of 238U+238U and 197Au+197Au
collisions at top Relativistic Heavy Ion Collider (RHIC) energy, and
subsequently apply it to experimental data. Using the deformation of 238U from
low-energy experiments, we find that RHIC $v_{2}$ data implies
$0.16\lesssim|\beta|\lesssim 0.20$ for 197Au nuclei, i.e., significantly more
deformed than reported in the literature, posing an interesting issue in
nuclear phenomenology.

###### pacs:

25.75.Gz, 25.75.Ld, 25.75.-1

## Introduction.

The hydrodynamic modeling of the quark-gluon plasma (QGP) formed in
relativistic heavy-ion collisions is a precision tool to understand the wealth
of measurements obtained at the BNL Relativistic Heavy Ion collider (RHIC) and
at the CERN Large Hadron Collider (LHC) Bernhard:2019bmu ; Gardim:2019xjs ;
Devetak:2019lsk ; Nijs:2020ors ; Nijs:2020roc ; Everett:2020yty ;
Everett:2020xug . The success of this framework is largely based upon a
correct description of the initial condition of the QGP prior to its dynamical
expansion Giacalone:2017uqx . One does in general expect that such initial
condition is impacted by the quadrupole deformation of the colliding ions
Heinz:2004ir ; Filip:2007tj ; Filip:2009zz ; Giacalone:2018apa . This has been
demonstrated in particular by recent flow data in 238U+238U collisions at RHIC
Adamczyk:2015obl . In principle, the uncertainty brought by this observation
to the overall picture should be under control, as the structure of nuclear
ground states is well constrained by nuclear experiments at low energy, and
one may assume that the structure probed at colliders on ultra-short time
scales of order $10^{-24}s$ is the same. For an unbiased interpretation of
high-energy data, it is crucial to check whether this is indeed the case,
i.e., that the manifestations of nuclear deformation at high energy are
consistent with the expectations from low-energy physics.

The majority of nuclei are deformed in their ground state, presenting an
intrinsic quadrupole moment in their mass distribution, $\int|{\bf
r}|^{2}Y_{20}\rho({\bf r})\neq 0$. Experimentally Raman:1201zz ;
Pritychenko:2013gwa , the deformation of an (even-even) nucleus of mass number
$A$ and charge $Ze$ is quantified by
$\beta=\frac{4\pi}{3ZeR_{0}^{2}}\sqrt{B(E2)\\!\uparrow}$, where
$R_{0}=1.2A^{1/3}$, and B(E2)$\uparrow$ is the measured transition probability
of the electric quadrupole operator from the ground state to the first $2^{+}$
state. Nearly spherical nuclei, such as 208Pb, have $\beta\approx 0$, while
well-deformed nuclei, like 238U, have $\beta\approx 0.3$.

In heavy-ion collisions, deformed nuclei are modeled through 2-parameter Fermi
(2pF) mass densities: $\rho({\bf r})\propto\left(1+\exp\left[|{\bf
r}|-R_{0}(1+\beta Y_{20})\right]/a_{0}\right)^{-1}$, with the value of $\beta$
taken (up to small corrections Shou:2014eya ) from low-energy experiments.
Colliding randomly oriented deformed nuclei impacts the initial state of the
QGP, enhancing in particular the fluctuations of its ellipticity
Giacalone:2018apa , $\varepsilon_{2}$, determined by the transverse positions
$(r,\phi)$ of the participant nucleons $\varepsilon_{2}=|\sum
r^{2}e^{i2\phi}/\sum r^{2}|$ Teaney:2010vd . In hydrodynamics,
$\varepsilon_{2}\neq 0$ yields an elliptical imbalance in the pressure-
gradient forces Ollitrault:1992bk  that drive the expansion of the QGP. This
pressure imbalance results in a $\cos(2\phi)$ modulation of the azimuthal
distribution of detected hadrons, $dN/d\phi\propto 1+2v_{2}\cos(2\phi)$, where
$v_{2}$ is the elliptic flow coefficient Heinz:2013th . In hydrodynamic
calculations Niemi:2015qia , $v_{2}$ emerges indeed as a response to the
initial eccentricity, $v_{2}=k_{2}\varepsilon_{2}$, so that $\beta\neq 0$ in
the colliding nuclei leads to enhanced fluctuations of the observed $v_{2}$.

In this Letter, we address the question of whether the values of $\beta$ found
in low-energy literature are consistent with $v_{2}$ data at high energy. We
introduce a simple method to do so, and argue that, at present, the sought
consistency of nuclear experiments across energy scales is not achieved.

## Relating $\boldsymbol{v_{2}^{2}}$ to the quadrupole deformation.

The idea is to compare systems that are close in size. As we show in the next
section, the dependence of the mean squared (ms) elliptic flow on $\beta$ is
the following:

$v_{2}\\{2\\}^{2}\equiv\left\langle v_{2}^{2}\right\rangle=a+b\beta^{2}.$ (1)

where averages are performed over events in a narrow centrality class, and the
physical meaning of the coefficients $a$ and $b$ will be clarified below.
Given two collision systems X+X and Y+Y, we introduce the following quantities
[the subscript X(Y) indicates a quantity evaluated in X+X(Y+Y) collisions]:

$r_{v_{2}^{2}}\equiv\frac{\left\langle v_{2}^{2}\right\rangle_{\rm
Y}}{\left\langle v_{2}^{2}\right\rangle_{\rm X}},\hskip
10.0ptr_{b}=\frac{b_{\rm X}}{b_{\rm Y}},\hskip 10.0ptr_{a}=\frac{a_{\rm
X}}{a_{\rm Y}},\hskip 10.0ptr_{\rm Y}=\frac{b_{\rm Y}}{a_{\rm Y}}.$ (2)

With these definitions and Eq. (1), we can express the quadrupole deformation
parameter of species Y, $\beta^{2}_{\rm Y}$, as a linear function of
$\beta^{2}_{\rm X}$,

$\beta^{2}_{\mathrm{Y}}=\left(\frac{r_{v_{2}^{2}}r_{a}-1}{r_{\mathrm{Y}}}\right)+\left(r_{v_{2}^{2}}r_{b}\right)\beta^{2}_{\mathrm{X}}\;.$
(3)

The ratios $r_{b}$, $r_{a}$, $r_{\rm Y}$ can be reliably predicted in
hydrodynamics, so that Eq. (3) can be used to verify the consistency of data
from nuclear experiments at different energy scales. Specifically, in
hydrodynamics we expect:

  * •

The coefficient $b$ in Eq. (1) quantifies how efficiently the fluctuations in
the global geometry due to the deformed, randomly oriented nuclear shapes are
converted into fluctuations of elliptic flow. The contribution of the term
$b\beta^{2}$ to the ms $v_{2}$ in Eq. (1) is thus of the same nature as the
contribution from the so-called elliptic flow in the reaction plane
$v_{2,\mathrm{RP}}$ to the ms $v_{2}$ in collisions of spherical nuclei. At a
given collision centrality, the relative contribution of $v_{2,\mathrm{RP}}$
to the ms $v_{2}$ varies very slowly with the mass number, therefore, if X and
Y are large systems, we expect $r_{b}\simeq 1$.

  * •

The coefficient $a$ in Eq. (1) corresponds to ms $v_{2}$ in the absence of
deformation. Therefore, $a$ in central collisions is the $v_{2}$ originating
solely from fluctuations, e.g., in the positions of the participant nucleons.
This quantity scales with the inverse mass number, $1/A$, and an additional
factor from viscous damping. Considering $\beta_{\rm X}=\beta_{\rm Y}=0$ and
$\left\langle
v_{2}^{2}\right\rangle=k_{2}^{2}\left\langle\varepsilon_{2}^{2}\right\rangle$,
if X and Y are close in size the relative difference in elliptic flow between
X+X and Y+Y collisions, $\Delta\left\langle
v_{2}^{2}\right\rangle/\left\langle v_{2}^{2}\right\rangle=\left(\left\langle
v_{2}^{2}\right\rangle_{\rm X}-\left\langle v_{2}^{2}\right\rangle_{\rm
Y}\right)/\left\langle v_{2}^{2}\right\rangle_{\rm Y}$, has an intuitive
decomposition:

$\displaystyle\frac{\Delta\left\langle v_{2}^{2}\right\rangle}{\left\langle
v_{2}^{2}\right\rangle}=\frac{\Delta
k_{2}^{2}}{k_{2}^{2}}+\frac{\Delta\left\langle\varepsilon_{2}^{2}\right\rangle}{\left\langle\varepsilon_{2}^{2}\right\rangle}\;.$
(4)

The contribution coming from the variation of $\varepsilon_{2}$ is related to
the variation in the mass number,
$\Delta\left\langle\varepsilon_{n}^{2}\right\rangle/\left\langle\varepsilon_{n}^{2}\right\rangle=\Delta\frac{1}{A}/\frac{1}{A}$
(up to corrections of few percents Giacalone:2017dud ), while viscous damping
drives the variation of the response coefficient, $\Delta k_{2}/k_{2}$, which
we estimate as follows. The response, $k_{n}$, to the $n^{\mathrm{th}}$
harmonic in viscous hydrodynamics is damped with the respect to the ideal
hydrodynamic value, $k_{n,{\rm ih}}$. The damping is linear in the viscosity,
and larger for higher harmonics. In the simplified scenario of Ref.
Gubser:2010ui , for instance, one has $k_{n}/k_{n,\mathrm{ih}}\approx
1-Kn^{2}$, where $K$ encodes the viscous correction Teaney:2012ke . This leads
to $\Delta k_{n}/k_{n}\approx-\Delta Kn^{2}k_{n,\mathrm{ih}}/k_{n}$. For large
systems Alver:2010dn , $k_{3,\mathrm{ih}}/k_{3}\approx
k_{2,\mathrm{ih}}/k_{2}$, so that:

$\Delta k_{2}/k_{2}=4/9~{}\Delta k_{3}/k_{3}.$ (5)

Recent state-of-the-art hydrodynamic simulations Gardim:2020mmy  report
however a slightly smaller damping, reflected by a larger coefficient,
$0.57\sim 5/9$, in the rhs of Eq. (5). Now, since $v_{3}$ is not affected by
the deformation of the colliding ions [see Fig. 1(b)], we can estimate the
variation of the $a$ coefficient in Eq. (1) (i.e., the variation of $v_{2}$ in
the case $\beta=0$) from the variation of the mass number and the
experimentally measured variation of $\left\langle v_{3}^{2}\right\rangle$:

$r_{a}-1=\frac{\Delta
a}{a}=(1-x)\frac{\Delta(1/A)}{1/A}+x\frac{\Delta\left\langle
v_{3}^{2}\right\rangle}{\left\langle
v_{3}^{2}\right\rangle},\;x\approx\frac{4}{9}\;.$ (6)

  * •

The ratio $r_{\rm Y}$ is a property of a single collision system, and has to
be evaluated through an explicit calculation. Its value is however largely
model-independent, as we explain in the next section.

Wrapping up, Eq. (3) relates the deformation parameters of two ions close in
size to the ratio of elliptic flow coefficients. The ratios $r_{b}$, $r_{a}$,
and $r_{\rm Y}$ are properties of the hydrodynamic description, and can be
predicted by generic scaling laws, as we now demonstrate through numerical
calculations.

## Numerical validation.

To gather the huge statistics of events required to constrain observables in
central collisions, we employ the multi-phase transport model (AMPT) as a
proxy for hydrodynamics. This model has proven successful in describing
collective flow data in small and large collision systems at RHIC and LHC
Adare:2015cpn ; Ma:2014pva ; Bzdak:2014dia ; Nie:2018xog . AMPT starts with a
Glauber Monte Carlo calculation Loizides:2014vua , which determines event-to-
event the collision impact parameter and participant nucleons, $N_{\rm part}$.
The system evolution is modeled with strings that first melt into partons,
followed by elastic partonic scatterings, which engender the hydrodynamic
collectivity, followed by parton coalescence and hadronic rescattering. We use
AMPT v2.26t5 in string-melting mode, and a partonic cross section of 3.0 $m$b
Ma:2014pva ; Bzdak:2014dia , which gives a reasonable description of
197Au+197Au $v_{2}$ data at RHIC. We simulate 238U+238U collisions at
$\mbox{$\sqrt{s_{\mathrm{NN}}}$}=193$ GeV with $\beta$=0, 0.15, 0.22,
$\pm$0.28, 0.34, 0.4, as well as $\mbox{$\sqrt{s_{\mathrm{NN}}}$}=200$ GeV
197Au+197Au collisions implementing $\beta$=0, -0.13. We emphasize that this
is the first such calculation, where one systematically scans over several
$\beta$ values, ever performed. The 2pF parameters for the colliding ions are
taken from the fits of their nuclear charge densities DeJager:1987qc . We use
hadrons with $0.2<p_{\mathrm{T}}<2$ GeV and $|\eta|<2$, and define the event
centrality from either $N_{\mathrm{part}}$ or the charged hadron multiplicity,
$N_{\mathrm{ch}}$, in the window $|\eta|<1$.

Figure 1:  Mean-squared $v_{2}$, $\left\langle v_{2}^{2}\right\rangle$ (left),
and $v_{3}$, $\left\langle v_{3}^{2}\right\rangle$ (right), as a function of
$N_{\mathrm{part}}/2A$ in 197Au+197Au and 238U+238U collisions, for different
values of $\beta$. The insets show the corresponding
$\left\langle\varepsilon_{2}^{2}\right\rangle$ and
$\left\langle\varepsilon_{3}^{2}\right\rangle$. The dashed boxes indicate
roughly the 0-1% range, on which our analysis is focused.

Figure 1 shows $\left\langle v_{2}^{2}\right\rangle$ and $\left\langle
v_{3}^{2}\right\rangle$ as functions of $N_{\mathrm{part}}/2A$, as well as the
corresponding $\left\langle\varepsilon_{2}^{2}\right\rangle$ and
$\left\langle\varepsilon_{3}^{2}\right\rangle$ in the inset panels. We note a
strong dependence of $\left\langle v_{2}^{2}\right\rangle$ on the value of
$\beta$ in central collisions, whereas $\left\langle v_{3}^{2}\right\rangle$
is independent of $\beta$. The $v_{2}$ values are similar between $\beta=0.28$
and $\beta=-0.28$, confirming that $v_{2}$ is an even function of $\beta$
111Although in extremely central collisions, say 0–0.2%, we found
$\varepsilon_{2}(\beta=-0.28)>\varepsilon_{2}(\beta=0.28)$ and
$v_{2}(\beta=-0.28)>v_{2}(\beta=0.28)$. These features are present as well in
the curves of the $\left\langle\varepsilon_{n}^{2}\right\rangle$,
demonstrating the geometric origin of $v_{n}$ in our simulations. We note that
the value of $\left\langle v_{3}^{2}\right\rangle$ is larger in 197Au+197Au
collisions, due to the smaller $A$.

Figure 2 shows that $\left\langle v_{2}^{2}\right\rangle$ is indeed linear in
$\beta^{2}$, in agreement with Eq. (1). The calculation is performed for
different centrality classes (defined from the distribution of $N_{\rm ch}$),
showing that the linear relation is valid even in non-central collisions. To
emphasize the geometric origin of this result, we show that
$\left\langle\varepsilon_{2}^{2}\right\rangle$ is also linear in $\beta^{2}$:

$\left\langle\varepsilon_{2}^{2}\right\rangle=a^{\prime}+b^{\prime}\beta^{2}\;.$
(7)

Figure 3 shows the centrality dependence of $r_{b}$, $r_{a}$, and $r_{\rm Y}$
(${\rm X}=^{197}$Au, ${\rm Y}=^{238}$U), and demonstrates explicitly the
points made in the previous section.

Figure 3(a) shows $r_{b}=b_{\rm X}/b_{\rm Y}$, and confirms the expectation
that this quantity should be close to unity. This is a geometric effect. We
find indeed that plotting $r^{\prime}_{b}$, obtained from the linear fits of
$\left\langle\varepsilon_{2}^{2}\right\rangle$ across centrality, yields a
dependence which is essentially identical to that of $r_{b}$, implying a minor
role of the hydrodynamic response.

Figure 3(b) shows $r_{a}=a_{\rm X}/a_{\rm Y}$ as a function of centrality. The
hydrodynamic expectation given by Eq. (6), is shown as a line for the most
central bins. The agreement with the calculated $r_{a}$ is excellent in the
0–1% bin, confirming our arguments. The result is $r_{a}=1.18$ with
${\Delta(1/A)}/{1/A}=0.21$, $x=4/9$ in Eq. (6) and ${\Delta\left\langle
v_{3}^{2}\right\rangle}/\left\langle v_{3}^{2}\right\rangle=0.136$ from AMPT
in the 0–1% bin. We have checked that the estimated value of $r_{a}$ reduces
only by $\sim 1\%$ if a larger coefficient $x=5/9$ is used, showing that the
uncertainty on the precise magnitude of the viscous correction does not affect
our analysis. Once more, $r^{\prime}_{a}$ in Fig. 3(b) calculated from the
eccentricity shows the same centrality dependence, indicating that the
hydrodynamic response yields only a global rescaling factor close to unity.

Figure 3(c) shows $r_{\rm U}=a_{\rm U}/b_{\rm U}$, which is 25.6 in the 0–1%
bin. We argue that this is a generic prediction of hydrodynamics, and not of
our specific setup. From Eq. (1), and considering
$v_{2}\\{2\\}^{2}=\kappa_{2}^{2}\varepsilon_{2}\\{2\\}^{2}$ at fixed
centrality, one can write

$\displaystyle r_{\rm U}=\frac{d\ln
v_{2}\\{2\\}^{2}}{d\beta^{2}}\biggr{|}_{\beta^{2}=0}=$
$\displaystyle\frac{1}{\kappa_{2}^{2}(\beta^{2}=0)}\frac{d\kappa_{2}^{2}}{d\beta^{2}}$
$\displaystyle+$
$\displaystyle\frac{1}{\varepsilon_{2}\\{2\\}^{2}(\beta^{2}=0)}\frac{d\varepsilon_{2}\\{2\\}^{2}}{d\beta^{2}}\biggr{.}$
(8)

Now, $d\kappa_{2}^{2}/d\beta^{2}$ is determined by how an increase in system
size due to $\beta$ modifies the hydrodynamic response. This is dictated by
generic scaling laws, irrespective of the chosen setup. Similarly, from
explicit calculations within different initial-state models we find that the
variation $d\varepsilon_{2}\\{2\\}^{2}/d\beta^{2}$ is essentially model-
independent. The values of $\kappa_{2}^{2}$ and $\varepsilon_{2}\\{2\\}^{2}$
evaluated at $\beta^{2}=0$ are, on the other hand, model-dependent. However,
if models are tuned such to return the same $v_{2}\\{2\\}^{2}$, i.e., the same
product $\kappa_{2}^{2}\varepsilon_{2}\\{2\\}^{2}$ after hydrodynamics, then
any such model dependence would disappear in Eq. (Numerical validation.). The
value of $r_{\rm U}$ appears to be, hence, a solid prediction. That said, Eq.
(Numerical validation.) contains
$1/\varepsilon_{2}\\{2\\}^{2}|_{\beta^{2}=0}$, therefore, one expects $r_{\rm
U}$ to present a strong centrality dependence, confirmed by the trends in Fig.
3(c). This engenders an uncertainty from the centrality definition. In
particular, repeating these calculations with the centrality defined according
to $N_{\rm part}$ instead of $N_{\rm ch}$, we find that $r_{\rm U}$ increases
by about 20%.

Figure 2: ${\left\langle v_{2}^{2}(\beta)\right\rangle/\left\langle
v_{2}^{2}(0)\right\rangle-1}=b/a~{}\beta^{2}$ (empty symbols) and
${\left\langle\varepsilon_{2}^{2}(\beta)\right\rangle/\left\langle\varepsilon_{2}^{2}(0)\right\rangle-1}=b^{\prime}/a^{\prime}~{}\beta^{2}$
(full symbols) as a function of $\beta^{2}$ in 238U+238U collisions. Different
symbols correspond to different centrality classes based on $N_{\mathrm{ch}}$.

In summary, the hydrodynamic expectations on the ratios $r_{b}$, $r_{a}$, and
$r_{\rm Y}$ pointed out in the previous section are confirmed by our numerical
results. We can thus move on and apply Eq. (3) to existing data from nuclear
structure and heavy ion experiments.

Figure 3: Empty symbols: $r_{b}=b_{\rm Au}/b_{\rm U}$ (a), $r_{a}=a_{\rm
Au}/a_{\rm U}$ (b), $r_{\rm U}=a_{\rm U}/b_{\rm U}$ (c), as functions of the
centrality percentile. Full symbols: same, but with $a\rightarrow a^{\prime}$,
$b\rightarrow b^{\prime}$. The lines represent the hydrodynamic expectation,
and should match the empty symbols in the ultra-central limit (0–1%).

## Application to RHIC data.

We apply Eq. (3), with ${\rm X}=^{197}$Au and ${\rm Y}=^{238}$U, to the 0–1%
most central $v_{2}$ data collected by the STAR Collaboration, where one has
$r_{v_{2}^{2}}=1.49\pm 0.05$ Adamczyk:2015obl . For the value of $r_{\rm U}$,
we employ the estimate of the AMPT model. We take into account the fact that
this quantity is smeared by the centrality definition, and allow for an
asymmetric 20% uncertainty, i.e., $r_{\rm U}=25.6^{+0}_{-5.1}$. For the value
of $r_{a}$, from Eq. (6) we obtain $r_{a}=1.18$, with no expected sizable
uncertainty. Lastly, for the value of $r_{b}$ we use the large-system limit,
but allow for a small asymmetric error, $r_{b}=1^{+0}_{-0.05}$. Using these
parameters from high-energy heavy-ion collisions, we plot in Fig. 4, as a
solid line, how $\beta^{2}_{\rm U}$ depends on $\beta^{2}_{\rm Au}$, following
Eq. (3). The dashed curves represent the estimated total uncertainty.

Next, we couple this plot to the expectations from low-energy nuclear
structure physics. The quadrupole deformation of 238U is known well.
Experimental determinations from transition probabilities give $\beta=0.286$
Raman:1201zz . On the theoretical side, we look at the state-of-art
tabulations of even-even ground-state deformations by Delaroche et al.
Delaroche:2009fa  (5DCH) and by Bender et al. Bender:2005ri , which give
$\beta_{\rm U}=0.29$ and 0.292, respectively. We consider, then,
$0.28<\beta_{\rm U}<0.29$ as the estimate from low-energy physics. The
situation for the odd-even 197Au is less transparent, as there is no
experimental determination of its value of $\beta$ in the literature. Further,
197Au is in a transition region between well-deformed rare-earth nuclei and
the spherical 208Pb, and as such it is triaxial in the ground state. Such
feature is included in the comprehensive 5DCH calculation, which can be used
to reliably estimate the deformation of 197Au from the deformation of its
neighbors, leading to $0.10<|\beta_{\rm Au}|<0.14$.

Adding this knowledge to Fig. 4 shapes a region in the upper-left corner of
the $(\beta_{\rm Au}^{2},\beta_{\rm U}^{2})$ plane preferred by low-energy
nuclear data. This region lies well outside the constraint defined by the
elliptic flow data, whose allowed ranges of $\beta_{\rm U}$ and $\beta_{\rm
Au}$ are highlighted as grey areas. The recent preliminary observation jia  of
a large anti-correlation between $v_{2}$ and the average transverse momentum,
$\langle p_{\mathrm{T}}\rangle$, in central 238U+238U collisions points to
$\beta_{\rm U}\approx 0.3$ at high energy Giacalone:2020awm , in agreement
with the low-energy estimates. Therefore, our result is likely an issue
related to 197Au. Using $\beta_{\rm U}\simeq 0.29$, the $v_{2}$ data implies
$0.16\lesssim|\beta_{\rm Au}|\lesssim 0.20$, which is significantly more
deformed than suggested by nuclear structure calculations. This may be viewed
as the first experimental constraint on the deformation of this nuclide.

Figure 4: $\beta_{\rm U}^{2}$ as a function of $\beta_{\rm Au}^{2}$. The
region between the dashed lines is consistent with the hydrodynamic
expectation based on Eq. (3) and high-energy elliptic flow data in 0–1%
centrality. The region highlighted in red in the upper-left corner is allowed
by low-energy nuclear data. The chosen intervals for $\beta_{\rm U}$ and
$\beta_{\rm Au}$ are motivated in the text.

Before concluding, we note that additional evidence that $\beta_{\rm Au}\sim
0.2$ can be found in preliminary STAR data on the above mentioned correlation
between $v_{2}$ and $\langle p_{\mathrm{T}}\rangle$. Preliminary results jia
indicate that such correlation in mid-central 197Au+197Au collisions is $i)$
bigger than in 238U+238U collisions, $ii)$ lower than measured in 208Pb+208Pb
collisions (at the same centralities) by the ALICE collaboration emil  with
nearly identical kinematic cuts. This ordering among systems is naturally
explained by 197Au being more deformed than 208Pb and less deformed than 238U.
However, taking $\beta_{\rm U}=0.29$ and $\beta_{\rm Pb}=0.06$ from low-energy
data Raman:1201zz , we have checked in initial-state calculations that capture
well the ratios of $v_{n}$-$\langle p_{t}\rangle$ correlations between
different systems Bally:2021qys  that the magnitude of the deviations observed
experimentally can not be explained with $|\beta_{\rm Au}|=0.1$, while
implementing $|\beta_{\rm Au}|\approx 0.2$ improves dramatically the agreement
with data. These results will be reported in future work.

## Conclusion and outlook.

Anisotropic flow in high-energy nuclear collisions emerges as a dynamical
response of the QGP to its initial spatial anisotropy. The latter is affected
by the geometric shape of the colliding nuclei, leading to an intrinsic
connection between the phenomenology of heavy-ion collisions and the structure
of atomic nuclei. Matching high-energy data to low-energy expectations, we
assess if our knowledge of nuclear physics across energy scales leads to
consistent results. Equation (3) allows one to do so in a way that is robust
against the details of the hydrodynamic modeling. Using the low-energy
estimates of $\beta_{\rm U}$, our analysis of 197Au+197Au data points to a
deformation of 197Au larger than found in low-energy literature.

Further efforts are required to elucidate this issue. At low energy, what is
missing is the evaluation of the structure properties of 197Au in a state-of-
art theoretical framework, such as that of Ref. Bally:2014jsa . This would
reduce the spread of the $\beta_{\rm Au}$ interval in Fig. 4. At high energy,
one should repeat our analysis on more collision systems. We have attempted to
do so for 0–1% 129Xe+129Xe and 208Pb+208Pb collisions. The ALICE Collaboration
reports Acharya:2018ihu  a large ratio $r_{v_{2}^{2}}=\left\langle
v_{2}^{2}\right\rangle_{\rm Xe}/\left\langle v_{2}^{2}\right\rangle_{\rm
Pb}\simeq 2.56$. We do not have yet AMPT results for 129Xe+129Xe collisions,
however, we have checked via the initial-state calculations of Ref.
Giacalone:2018apa  that $b^{\prime}_{\rm Xe}\approx b^{\prime}_{\rm U}$ and
$a^{\prime}_{\rm Xe}\simeq(238/129)a^{\prime}_{\rm U}$. The logic of the
present discussion should apply, i.e., $b_{\rm Xe}\approx b_{\rm U}$, $a_{\rm
Xe}\approx(238/129)a_{\rm U}$, so that $r_{\rm Xe}\approx(129/238)r_{\rm U}$.
Using $v_{3}\\{2\\}_{\rm Xe}/v_{3}\\{2\\}_{\rm Pb}=1.22$ from the CMS
Collaboration Sirunyan:2019wqp , we obtain $r_{a}=a_{\rm Pb}/a_{\rm Xe}\approx
0.64$ from Eq. (6) (increases to 0.65 if $x=5/9$ is used). With $r_{b}=1$ and
$\beta_{\rm Pb}=0.06$ Raman:1201zz  this leads to $\beta_{\rm Xe}\approx 0.24$
via Eq. (3). Much as for 197Au, this value is larger than found at low energy
models, where $\beta_{\rm Xe}\approx 0.2$ Bally:2021qys . The same analysis
should be performed on collisions of 96Ru, 96Zr, and possibly 63Cu nuclei, to
assess whether the discrepancy between low-energy and high-energy data are
systematic.

Arguably, though, the most efficient way to do so would be collecting data
from collisions of even-even species that are close in size but have different
and experimentally measured deformation. The ideal candidates for such a study
are the stable samarium isotopes, which present a remarkable transition from
spherical to well-deformed shapes Mustonen:2018ody . One could collide, for
instance, 144Sm, which is essentially as spherical as 208Pb, 148Sm, mildly
deformed with a triaxial ground state, much as 129Xe and 197Au, and 154Sm,
which is a well-deformed nucleus like 238U. With the ideas introduced in this
paper, it would thus be possible to assess from high-energy data if the
evolution of $\beta$ along the isotope chain is consistent with the low-energy
expectations. Systematic deviations would eventually open deeper physics
questions.

## Acknowledgements.

We thank Benjamin Bally, Michael Bender, Shengli Huang and Jean-Yves
Ollitrault for useful discussions. The work of G.G. is supported by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg
STRUCTURES Excellence Cluster), SFB 1225 (ISOQUANT) and FL 736/3-1. The work
of J.J and C.Z is supported by DOE DEFG0287ER40331 and NSF PHY-1913138.

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