1

# Lifetime-based Optimization for Simulating
Quantum Circuits on a New Sunway Supercomputer

Yaojian Chen  Tsinghua UniversityBeijing, China ,  Yong Liu  National
Supercomputing Center in Wuxi Zhejiang Lab, Hangzhou, China ,  Xinmin Shi
Information Engineering UniversityZhengzhou, China ,  Jiawei Song  National
Supercomputing Center in Wuxi, China ,  Xin Liu  National Supercomputing
Center in Wuxi Zhejiang Lab, Hangzhou, China ,  Lin Gan  Tsinghua University
National Supercomputing Center in Wuxi, China ,  Chu Guo  Information
Engineering UniversityZhengzhou, China ,  Haohuan Fu  Tsinghua University
National Supercomputing Center in Wuxi, China ,  Jie Gao  National Research
Centre of Parallel Engineering and TechnologyBeijing, China ,  Dexun Chen
National Supercomputing Center in Wuxi, China and  Guangwen Yang  Tsinghua
University National Supercomputing Center in Wuxi Zhejiang Lab, Hangzhou,
China

(2023)

###### Abstract.

High-performance classical simulator for quantum circuits, in particular the
tensor network contraction algorithm, has become an important tool for the
validation of noisy quantum computing. In order to address the memory
limitations, the slicing technique is used to reduce the tensor dimensions,
but it could also lead to additional computation overhead that greatly slows
down the overall performance. This paper proposes novel lifetime-based methods
to reduce the slicing overhead and improve the computing efficiency, including
an interpretation method to deal with slicing overhead, an in-place slicing
strategy to find the smallest slicing set and an adaptive tensor network
contraction path refiner customized for Sunway architecture. Experiments show
that in most cases the slicing overhead with our in-place slicing strategy
would be less than the cotengra , which is the most used graph path
optimization software at present. Finally, the resulting simulation time is
reduced to 96.1s for the Sycamore quantum processor RQC, with a sustainable
single-precision performance of 308.6Pflops using over 41M cores to generate
1M correlated samples, which is more than 5 times performance improvement
compared to 60.4 Pflops in 2021 Gordon Bell Prize work.

quantum computing, circuit simulator tensor network contraction, Sunway
architecture, slicing, direct memory access, fused design

††copyright: rightsretained††journalyear: 2023††conference: The 28th ACM
SIGPLAN Annual Symposium on Principles and Practice of Parallel Programming;
February 25-March 1, 2023; Montreal, QC, Canada††booktitle: The 28th ACM
SIGPLAN Annual Symposium on Principles and Practice of Parallel Programming
(PPoPP ’23), February 25-March 1, 2023, Montreal, QC, Canada††doi:
10.1145/3572848.3577529††isbn: 979-8-4007-0015-6/23/02††ccs: Software and its
engineering General programming languages††ccs: Social and professional topics
History of programming languages

##  1\. Introduction

Quantum computer has the potential to provide exponential speedups over
classical counterparts in specific tasks. The declaration “Quantum Advantage”
refers to those tasks that can only be solved in reasonable time by using
quantum computers(google-nature-2019, )(osti_1259664, )(zhong2020quantum, ).
However, despite such advantages in the computing capability, low fidelity is
still the major challenge for quantum computers (zlokapa2020boundaries, ).
Classical simulators are therefore important to provide validations for
quantum computer design (pan2021simulating, ). Furthermore, scientists and
researchers in areas that heavily rely on reliable computing resources, such
as quantum algorithm, quantum programming language, and quantum compiler, can
work on classical simulators and obtain close performance(liu2021closing, ).
Considering both the exponential complexity and the urgent demand, improving
the efficiency of the state-of-the-art classical simulator is important.

Directly storing an arbitrary quantum state on a classical computer requires
an exponential amount of memory against the number of qubits. So using
traditional state vector method to simulate quantum circuits is currently
limited to less than 50 qubits(de2007massively-11yearsbefore, ). In recent
years the tensor network contraction (TNC)(google-unigraph, ) algorithm has
demonstrated promising potential, especially in simulating random quantum
circuits (RQCs). When combinated with the slicing technique, the TNC algorithm
could efficiently simulate larger quantum circuits using only a limited amount
of memory.

In TNC algorithm, qubits and quantum gates are represented as tensors, and the
whole quantum circuit is treated as a tensor network (biamonte2017tensor-
nutshell, ). The problem of computing amplitudes of the output quantum state
is transformed into contracting the corresponding tensor networks. The
performance of TNC is mainly determined by the TNC path. In a network with
hundreds of tensors, a good contraction path can easily reduce the complexity
by several magnitudes compared to the bad ones. In the meantime, during the
contraction process, huge intermediate tensors could appear. The slicing
technique can reduce the memory requirement(villalonga2020establishing, )
(breaking-49-qubit, ), at the price of some computational overheads. This work
aims to reduce the computational overhead caused by slicing.

Unlike previous efforts that predominantly use heuristic algorithms, this work
introduces a new conception, _lifetime_ , in order to provide better
interpretability for the slicing optimization in tensor network contraction.
_Lifetime_ describes how an edge affects the TNC process by analyzing all the
tensors and contractions it is involved in. By using _lifetime_ , one is able
to transform the computation into an equivalent form with much less memory
requirement. Slicing works on each two adjacent manually controllable levels
on a multi-level storage system. For Sunway architecture (a most advanced
supercomputing system in China), both the data exchange from hard disk to main
memory, and from main memory to local data memory (LDM) (_i.e._ process level
and thread level), can be optimized by slicing. For process level, we slice
tensors for distributed storage and parallelization, and apply _lifetime_
-based slice finder to reduce the overhead of process division. For thread
level, slicing helps design a fused algorithm to reduce memory access, and
transform the memory-intensive kernels into computationally intensive ones in
some cases, thus improving the optimization capability.

Major contributions of this work include:

  * •

A conception, _lifetime_ of graph edges (tensor dimensions), is proposed to
explain the influence of each dimension, how slicing eliminates this
influence, what is the origin of the slicing overhead, and how to avoid this
overhead.

  * •

Guided by _lifetime_ , a dynamic slicing method with a new target function is
proposed, to helps find smaller slicing set with less slicing overhead.

  * •

Indicated by _lifetime_ , a high performance algorithm for TNC that fits well
with the Sunway architecture is presented.

##  2\. Background

###  2.1. Tensor Network Contraction and Slicing

####  2.1.1. Notation

A tensor network can be treated as an undirected graph, where tensors and
dimensions are denoted by vertices and edges, respectively. We define the
notation similar with the work of cotengra(gray2021hyper, ):

We denote a graph by $G=(V,E)$. $V$ is the vertex set, and $E$ is the edge
set. The incident set $V_{e}$ of a edge $e$ contains the two vertices the edge
connects. Map $w:E\rightarrow R$ denotes the edge weight, as well as the size
on each dimension. Particularly in classical simulation, $w(e)$ is the power
of $2$ for all $e\in E$. The incidence set $s_{v}$ of vertex $v$ is defined as
$s_{v}=\\{e:e\in E,v\in V_{e}\\}$. Vertex contraction merges two vertex into
one by removing shared edge and keeping the rest. In the generated graph
$G^{{}^{\prime}}$, the contracted vertices $v_{0},v_{1}$ are replaced by a new
vertex $v_{2}$, and the edges between $v_{0}$ and $v_{1}$ are removed in the
new edge set. The uncontracted edges in $s_{v_{0}}$ and $s_{v_{1}}$ will be
inherited by $s_{v_{2}}$.

If we perform the pairwise contraction sequentially till there is only one
vertex left, the order would form a contraction path. As a matter of fact, a
equivalence class can be made up by reordering some independent steps, and all
equivalent paths can be uniquely described by a tree structure. A contraction
tree $B$ is defined as $B=(N_{B},E_{B})$, with each edge denoted by a vertex
of original graph or an intermediate vertex generated by contraction.
Specially, the edges connecting to leaf nodes are denoted by vertices in $V$.
Nodes expressed as triplet $node=(e_{1},e_{2},e_{3})$ refer to contractions.
In particular, leaf nodes such as $(1,e_{1},e_{1})$ can be treated as a
multiplication with a scalar $1$. Then leaf nodes can represent the reading
process. A certain contraction path determines the direction of each
contraction. By adding the final contraction as the root, as shown in Fig. 1,
a rooted binary tree is acquired.

Figure 1. Tensor network and its contraction tree. When a certain contraction
path is formed, an additional contraction for the final scalar should be
added.

As for a given contraction tree, the time and space complexity of the
corresponding contracting process can be quantitatively evaluated. The space
cost is the biggest intermediate tensor, _i.e_ $\mathop{\max}_{v\in
E_{B}}\prod_{e\in s_{v}}w(e)$. And the time complexity is:

(1) $C(B)=\sum_{(v_{1},v_{2},v_{3})\in N_{B}}\prod_{e\in(s_{v_{1}}\cup
s_{v_{2}}\cup s_{v_{3}})}w(e)$

To minimize the memory and time cost, the two expressions above are usually
treated as the target function.

For large tensor network, memory limitation is always a serious issue. So
after a path is found, it is necessary to decide whether slicing is needed to
reduce the dimension of tensors. Slicing a tensor alongside a dimension (an
edge in the TN) can transform an $n$-dimension tensor into $w$ different
$(n-1)$-dimension sub-tensors. Slicing temporarily cancels the correlation
between these sub-tensors and provides parallelism. Fig. 2 shows a simple
example of slicing. In this work, slicing are performed on the original $G$,
in which there is $w(e)=2$ for all $e\in E$. As a result, $s$ sliced edges
make up $2^{s}$ independent tasks, and the result will be accumulated after
their individual calculations. Total time complexity may increase after
slicing under some circumstance, then slicing overhead is defined as:

(2) $O(B,S)=\frac{C_{slice}(B)\times 2^{|S|}}{C_{original}(B)}$

where $S$ denotes the set of sliced edges. Slicing overhead comes from
redundant calculations due to the split of tasks. From a higher perspective,
slicing provides a distributed storage strategy and natural stream-like design
since large tensors are sliced to stored and calculated in multi-processes,
respectively. Compared to traditional methods based on heavy communication,
slicing shows its advantage through independent subtasks. However, as the
price, slicing overhead determines the effect, so one of the primary targets
of this work is to explain and reduce such overhead.

Figure 2. An intuitive example of slicing. The second half expresses with
graph as a parallel.

####  2.1.2. Related Work

The demand for slicing originates from the considerable memory cost of high-
rank tensors. In some quantum advantage circuits, there are tensors with
dimensions of more than 60(gray2021hyper, ). They can occupy up to 1000 PB,
which exceeds most storage systems. Slicing helps reduce the memory demand to
the level of TBs or even GBs to make simulation feasible.

Cotengra(gray2021hyper, ) integrates several anytime methods and claims over
$10000\times$ speedup compared to the estimation made by Google(google-
nature-2019, ). It introduced a series of effective heuristic algorithms to
search for contraction path, such as community(girvan2002community, ) and
graph partition(akhremtsev2017engineering, ). After the pre-process which is
implemented in (gray2018quimb, ), rank-1 and rank-2 tensors are absorbed, and
the tensor network is largely simplified. A greedy-based slicing strategy is
built in contengra. It repeatedly chooses a dimension that leads to the most
minor overhead to slice, until the memory demand is satisfied. Like most
greedy methods, local minimum exists in this slicing strategy. In our work,
contengra is applied to find a proper contraction path.

Alibaba proposed a simulator based on an observed structure called
‘stem’(huang2020classical, ). A stem is a computationally-intensive region in
the contraction tree, including most of high-rank tensors. The inexpensive
parts are called branches. Within a single stem, a bigger tensor sequentially
absorbs smaller ones, and about $99\%$ computation cost happens during these
contractions. For slicing, with a similar greedy strategy, they perform local
tuning of contraction tree between two steps of slicing picking. This dynamic
design highly reduced the inherent slicing overhead of a contraction tree.
However, if the condition of local tuning is not satisfied, this strategy may
not be able to find an optimal slicing set. This work can provide $10^{18.8}$
times complexity, slicing overhead of $4$, and a $14.7\%$ FLOPS efficiency.
Due to the high efficiency, dynamic design is applied or updated by many other
softwares, like cotengra(gray2021hyper, ).

As for the New Sunway System, several efforts study high-performance
algorithms for TNC. Sw_Qsim(li2021sw_qsim, ) provides a series of methods for
matrix multiplication and tensor permutation, especially for narrow tensors.
The work that won the 2021 ACM Gordon Bell prize(liu2021closing, ) implemented
Transpose Transpose GEMM Transpose (TTGT) algorithm on Sunway architecture,
which is a fused design of tensor permutation and matrix multiplication in
order to reduce memory access. These efforts significantly improve the FLOPS
efficiency, but all concentrate on one step of the whole path instead of
taking a holistic view. Optimization will soon reach the ceiling as a
bandwidth-constrained problem according to Roofline model(osti_1407078, ). For
extremely deep circuits, other tensor based methods like matrix product state
are also implemented on Sunway architecture(shangmps, ). In this work, we
focus on the hardware-motivated and highly entangled circuits, which have a
clear 2D geometry and are relatively shallow.

###  2.2. Sunway 26010Pro Processor

As a heavy time- and memory-consuming part, performing the actual contraction
generally requires support from sophisticated supercomputers. A new Sunway
supercomputer is selected for this work.

The new-generation Sunway supercomputer has a similar architecture as Sunway
TaihuLight(fu2016sunway, ), with the major computing capability provided by
core groups (CGs). A heterogeneous many-core processor, SW26010pro, is
designed for this supercomputer. Fig. 3 shows the structure of the chip. Each
processor chip has 6 CGs. The computing processing elements (CPEs) of each CG
are arranged as an 8 by 8 grid.

Each CG contains a 16GB main memory, and each CPE contains 256KB local data
memory (LDM). Direct memory access (DMA) with a bandwidth of 51.2 GB/s is
provided between LDM and main memory. Due to the enormous arithmetic
intensity, the memory access bottleneck often turns to the vital problem for
optimization. Remote memory access (RMA) with a peak bandwidth of more than
800 GB/s is designed for data exchange between CPEs within one CG.

Figure 3. SW26010pro Processor. The main memories of $6$ CGs are united to
form a cross dump with $16\times 6$ GB in this work, in order to hold large
tensors.

##  3\. Lifetime

###  3.1. Definition

Here is a simple example showing where the slice overhead comes from. Fig. 4
shows a contraction process on a $4\times 2$ tensor network, and the target
rank is limited to $3$. The ideal case is that the time complexity of all the
contractions in one subtask becomes just $2^{-s}$ after slicing $s$ edges, and
the original complexity is kept. However, from the right part, the first
contraction and the last $2$ contractions in a subtask have not changed after
sliced edge $e$, then they will be calculated twice. These redundant
calculations make up the overhead.

Figure 4. An example to show the origin of slicing overhead. The left part is
a tensor network represented by graph, and the arrow gives the contraction
order. The right part shows the complexity before and after slicing. After
slicing, the total complexity is the summation of all subtasks.

Actually, the sliced edge $e$ determines which contraction will be redundantly
performed. Considering the contractions before slicing, edge $e$ involves four
contractions:

(3) $\displaystyle[f,h,j]\times[c,e,f]\rightarrow[c,e,h,j]$
$\displaystyle[c,e,h,j]\times[b,c]\rightarrow[b,e,h,j]$
$\displaystyle[b,e,h,j]\times[a,b]\rightarrow[a,e,h,j]$
$\displaystyle[a,e,h,j]\times[a,d,e]\rightarrow[d,h,j]$

and these contractions are just inside the dotted box in Fig4, which will not
be redundantly computed.

Based on the analysis above, this work proposed a new concept, _lifetime_ , to
describe the scope of influence of a sliced dimension during the slicing. The
notation comes from section 2.1.1.

###### Definition 0.

Given a tensor network $G=(V,E)$ and a contraction tree $B=(N_{B},E_{B})$, the
concept _lifetime_ of a edge index $k\in E$ refers to a set of tensors
$\\{T_{i_{1}},T_{i_{2}},\dots,T_{i_{n}}\\}\subset E_{B}$, if $k\in
s_{T_{i_{j}}}$, for all $1\leq j\leq n$.

With the definition of _lifetime_ , the scope of influence of a sliced edge is
clearly represented. We can conclude that, after slicing an edge $e$ from a
tensor network, the size of tensors on the lifetime of $e$ will be halved
while the size of the others will not change. The time complexity of the
contractions corresponding to these tensors will not be changed, while the
time complexity of the other contractions will be doubled.

Not merely on the whole contraction tree, _lifetime_ can be conveniently
defined recursively on each sub-tree or intermediately generated tree during
the contraction process by simply replacing the contraction tree in the
definition. The advantage here is that we can focus more on the region with
intensive computation.

###  3.2. Lifetime and Slicing Overhead

Dozens of edges need to be sliced for acceptable size in large tensor networks
from big circuits like Sycamore(google-nature-2019, ) , which makes multi-edge
slicing a critical problem for complexity analysis. Considering that
_lifetimes_ of different edges are independent, we can analyze the _lifetime_
of each sliced edge sequentially. Fig. 5 provides an example of the slicing of
two edges with index $A$ and index $B$. From left to right, the contraction
tree is divided into $5$ parts, lifetime of index $A$ goes through part $2$
and $3$, while $B$ goes through part $3$ and $4$. Index $A$ and $B$ double the
time complexity of part $1,4,5$ and part $1,2,5$ respectively, and make the
whole multiple of redundant calculation shown in Fig. 5 as a piecewise
production. Expanding to a more general circumstance, if a tensor contains $s$
sliced indices with $n$ in total, it will be divided into $2^{s}$ smaller
tensors. For all $2^{n}$ subtasks, there will be $2^{n-s}$ times of redundant
data. Then the time complexity will be multiplied by $2^{n-s}$ times. As a
result, the density of _lifetimes_ plays a decisive role in memory and
computation complexity in a region.

Figure 5. The superposition rule of overhead. A contraction order is shown
from left to right before (the upper half) and after (the lower half) slicing.
Each branch denotes a tensor, and the contractions happen at the intersection
points. Lines with different colors in the tensors denote the edges, and the
collections of all the lines with the same color are naturally _lifetime_ of
these edges.

More sliced edges tend to lead to higher overhead. As a simple condition, when
a slicing set is given, if we additionally pick more edges to slice, then the
overhead will grow unless the _lifetimes_ of the added edges go across the
whole contraction tree, which rarely happens. However, for an arbitrary
slicing set with $n$ edges and $n+1$ edges, can we find a partial ordering
relation between their overhead? We can prove the theorem below with a given
target size. It provides interpretation for redundant slicing more profoundly
and casts the theoretical cornerstone for our slicing strategy. It shows that
a valid smaller slicing set indicates a lower slicing overhead in most
conditions.

###### Theorem 2.

For a tensor network $G(V,E)$, a contraction tree $B(V_{B},E_{B})$ and an
$n$-edge slicing set $S_{1}$, if an $(n-1)$-edge slicing set $S_{2}$ is found,
and the intersection of $S_{1}$ and $S_{2}$ is not empty, then there must
exist an $(n-1)$-edge slicing set $S_{3}$, whose overhead is less than
$S_{1}$.

Figure 6. Time complexity and multiple by slicing on stem (Sycamore$(m=20)$ is
used as an example).

Fig. 6 compares the whole time complexity before and after slicing on a common
contraction tree of Sycamore circuit(google-nature-2019, ). From the figure,
we can conclude that the key to a low overhead is that the time complexity of
the main computation-intensive part is kept, which means big tensors are
contained in lifetimes of as many as sliced edges. Ideally, time complexity
and the slicing caused by multiple are negatively correlated.

###  3.3. Route of Slicing Optimization

Due to the memory bound, we have to do slicing and stand with some overhead.
However, on a multi-level storage system, under some circumstances, the
computation overhead can be translated into data movement.

For TNC, we will introduce two main methods to deal with the overhead, _i.e.,_
to reduce overhead by searching for better slicing sets or to replace overhead
by data movement via stacking, in the following sections, respectively.
Stacking is the inverse operation of slicing, and is feasible when the
capacity of the low storage or distributed memory is enough for the whole
memory requirement. For example, we can store the rank-53 tensor in the hard
disk, get a rank-30 slice at a contraction step by IO, and then put it back
after calculation. This process performs slicing by getting and stacking by
putting naturally. If stacking immediately after the _lifetime_ of a sliced
edge ends, the overhead of this edge is naturally eliminated, and the memory
demand is resolved. Nevertheless, there will be vast costs of data exchange
during data access (use low-level storage) or communication (for distributed
memory). Searching for a better slicing set can avoid memory access and
communication, but it introduces overhead from the redundant calculation. The
choice depends on the specific value of bandwidth and overhead. In case of low
bandwidth and low overhead, slicing optimization has better performance, while
stacking is more suitable in case of high bandwidth and high overhead.

Fig. 7 shows a typical overhead distribution for different target sizes. Data
access costs are translated into equal overhead by arithmetic intensities (AI)
of different levels. The line of equal overhead and the storage capacities
divide the graph into several parts. The commonest manually controllable
storage levels include hard disk, main memory and LDM (for Sunway
architecture). For memory access, $BandWidth_{IO}\ll BandWidth_{DMA}\ll
BandWidth_{accessLDM}$. Then, from hard disk to LDM, stacking becomes more
potential, and in contrast, slicing optimization should be applied.

These methods are not customized for a certain architecture. Actually, all we
need is a multi-level storage system.

Figure 7. Overhead distribution for different storage level by cotengra.
(Sycamore$(m=20)$ as an example with original memory cost dozens of PBs. For
Sunway architecture, there is a 96GB main memory and a 256KB LDM for each CPE)

##  4\. Slicing Overhead Reduction

###  4.1. Overview of Slicing Optimization

The Low bandwidth of IO restricts stacking, and the slicing overhead directly
influences the parallel scalability. As shown in Theorem. 2, optimization of
slicing can be divided into two steps: searching for a smaller slicing set,
and tuning the selected slicing set to achieve a lower overhead. The following
two subsections discuss these two steps separately.

Minimizing the overhead while fitting memory bound describes an optimization
problem with constraints. For a contraction tree $B=(N_{B},E_{B})$, the target
function can be described by the total time complexity after slicing.

(4) $C(B,S)=\sum_{V\in N_{B}}{2^{|s_{V}|+|S|-|S\cap s_{V}|}}$

where $S$ is the set of sliced edges, and $s_{V}$ is the set of edges which
involve the contraction of $V$ in the TN.

The target size which does not exceed the memory capacity is denoted as $t$.
For every $e\in E_{B}$, $|S\cap s_{e}|\geq|s_{e}|-t$. The target function and
the formula together can determine the optimization problem.

###  4.2. Slicing Strategy

Here we define _lifetime_ on the stem(huang2020classical, ), while the
definition of stem is somehow different. In this work, stem is defined as the
most computationally intensive path on the contraction tree. According to
graph theory, on a path, each contraction step has data dependence with its
next step. Since most branches have nothing to do with the memory constraints,
they are pre-contracted, and only stems remain for optimization. After
precondition, the stem becomes a new contraction tree, and we can detect
_lifetime_ of every edge on it.

Assuming the optimal size of the slicing set is $s$, the stem is then
separated into two continuous parts: $M=M_{1}+M_{2}$, and let the optimal size
of $M_{1}$ be $s_{1}$. For $M_{1}$, a number of candidate sets share the same
size $s_{1}$. According to the discussion above, the more tensors _lifetimes_
of $s_{1}$ pass through $M_{2}$, the fewer edges shall be sliced in the
updated $M_{2}$.

Strictly speaking, longer _lifetime_ does not necessarily lead to lower
slicing overhead. However, the containment relationship of _lifetime_ does. So
is the effect of reducing memory. In particular, for the leaf nodes of the
contraction tree, there is only one extension direction, then its length
decides the containment relationship. So we start the search from the two ends
of the stem, and choose one tensor as $M_{1}$ each time and the left part as
$M_{2}$. Therefore, an edge with longer _lifetime_ shall be better. If we
perform this process iteratively, a smaller slicing set is expected. We
propose Algorithm. 1 for details.

Algorithm 1 Procedure slice finder

0: Stem of contraction tree $M$, target dim $t$

$N=len(M),S\leftarrow\emptyset,lf=dict()$

for all edge indices in $M$ do

Calculate $lf[index]$

end for

repeat

$sT=(M[0].dim<M[N-1].dim)?M[0]:M[N-1]$

Add $sT.dim-t$ indices with longest lifetime into $S$

for all $T\in M$ do

if $T.dim\leq t$ then

$M.remove(T)$

end if

end for

Update $lf$, $N$

until $N==0$

slicing set $S$

Though the above strategy does not always provide the slicing set with the
lowest overhead, sometimes even worse than the one found by contengra, we have
successfully satisfied all the prerequisites of the theorem above. The method
above can find a slicing set as small as possible for a given contraction
tree. Combined with the dynamic process for path searching, the overhead may
drop lower. Then, the refiner will find a better slicing set to reduce the
overhead.

###  4.3. Slice Refiner based on Simulated Annealing

Our slice finder can not guarantee to find an optimal slicing set, and it
looks for one as small as possible. So, the main task of this section is to
find the best one given a certain number of sliced edges. Considering that the
size of the slicing set is fixed, the permutation from the former to the
latter can be described as edge replacement.

After slicing, there are some tensors whose dimensions are exactly equal to
the target dimension, and they are named ”critical tensors”. If the lifetime
of a sliced edge contains none of the critical tensors, this edge does not
contribute to memory reduction, and can be removed from the slicing set.
Moreover, from another angle of view, to replace a sliced edge with index $a$,
a direct way is to find another unsliced index $b$ whose lifetime contains all
critical tensors in the lifetime of $a$.

Simply replacing indices greedily sometimes leads to a local minimum. For
instance, the optimal set contains index $a$ and index $b$ while the one we
found contains $c$ and $d$, but replacing $c$ from $a$ may increase the
overhead, or fail to match the memory bound. Exhausted searching or dynamic
programming will lead to the optimal solution, but suffers from a combination
explosion of search space. Then, stochastic algorithm becomes a choice to
balance the time and quality of searching. Simulated annealing endures a
temporary complexity increase to avoid local minimum, and provides a practical
speed, as the Algorithm. 2 shows.

Algorithm 2 Procedure slice refiner

0: Target dim $t$, slicing set $S$, contraction tree $M$, final temperature
$tf$, parameter $\alpha$

Initialize temperature $T$

for all edge indices in $S$ do

Calculate $lf[index]$

end for

repeat

Randomly choose $index$ from $S$

$c\\_tensors=find\\_critical\\_tensors(lf[index],t)$

$candidate=find\\_candidate\\_indices(c\\_tensors)$

for all $can\in candidate$ do

Replace $index$ by $can$

Calculate time complexity $C_{new},C_{ori}$

if $C_{new}<C_{ori}$ then

Update $S$

else

Update $S$ with prob. $p=exp((C_{ori}-C_{new})/C_{ori}/T)$

end if

end for

until $T<tf$

slicing set $S$

##  5\. Fused Design by Secondary Slicing

###  5.1. Difficulty for Thread Level Optimization

For thread level, if organizing the whole path on the CPEs, considering the
256-KB LDM, there will be a huge overhead. As a result, previous works
optimize on thread-level step by step(li2021sw_qsim, )(liu2021closing, ) to
avoid it, turning to memory access instead of vast redundant calculation.
However, frequent memory access also limits the FLOPs efficiency a lot.

Contraction between tensors is implemented by matrix
multiplication(li2021sw_qsim, ). Based on this strategy, more than $70\%$ of
the peak performance can be obtained for square-like matrices. However, for
narrow matrix multiplication, in particular, when two of the $m,n,k$ are less
than 16, which dominated many simulation tasks(google-nature-2019, ), we can
easily deduce $\Theta(MNK)\approx\Theta(MN+NK+MK)$, so that GEMM will change
from a computational intensity problem to a bandwidth-constrained problem.
Another constraint comes from the 256-KB LDM on each CPE, which can only hold
a rank-13 tensor for multiplication. Then, calculating a contraction by a
rank-31 tensor at least needs $2^{12}$ times low-efficiency GEMM for a CG.

To take full advantage of the 512-bits single instruction multi-data (SIMD), a
$4\times 4$ complex GEMM kernel is applied. When $k,n$ are both less than $4$
in one thread, the demand of data length will cause a conflict between the
high-performance kernel and the 2D data distribution. In contrast, 1D
distribution avoids most padding, and slices the GEMM process into simply
paralleled subtasks. Most importantly, the implicit slicing embedded in data
distribution, as a method to overcome the memory limitation of LDM, has a
similar problem with slicing on the process level.

We propose a more efficient slice-stack strategy guided by _lifetime_ , which
can heavily reduce the cost of data movement. Specific for thread level, it is
named as secondary slicing (first slicing happens on the process level).
Secondary slicing between LDM and the main memory will provide a fused
operator to organize parts of the contraction path on CPEs, and greatly save
the memory access cost.

###  5.2. Slice-stack design

Figure 8. A fused design for thread-level optimization. The top part shows the
step-by-step strategy in previous works, and the bottom is the fused design.
Our strategy organizes a sub-path on the LDM, and reduces $n-1$ times of DMA-
get and DMA-put for a length-$n$ sub-path. The superscript $T$ denotes
permutation, and the subscript of tensors denotes the sliced dimensions.

If we do slicing on the whole contraction path, a subtask can be performed on
a process without intermediate communication and IO while standing with
computation overhead. In contrast, if we accept the communication or memory
access, the computation overhead can be avoided.

Such a strategy does not work on the process level due to the low bandwidth of
IO and low computation overhead, but is available between LDM and the main
memory. Since LDM on the new Sunway supercomputer is only feasible for a
rank-13 tensor, if the contraction is performed step by step, we should
frequently do DMA-get and DMA-put between every two steps. Considering the
arithmetic intensity above, memory access will be dominant. To reduce the
frequent data exchange between LDMs and the main memory, performing $n$ steps
of the contraction path successively in one computation kernel helps. At the
beginning of those steps, we do DMA-get once to get the bigger tensor, and
after n steps of calculations, the result will be written back by DMA-put.
Then, $n-1$ times of DMA-get and DMA-put are reduced.

As the top part of Fig. 8 shows, CPEs can obtain a lower rank tensor from the
one in the main memory at each time, throwing out some dimensions. In a
particular contraction, commonly, there are parts of dimensions absorbed, and
the others held. According to this observation, at the beginning of the
computation, if we choose all dimensions which will be absorbed in the
following $n$ steps to form a tensor, and get it into LDMs by DMA with stride,
then each CPE can do the calculation independently, like the bottom part of
Fig. 8 shows. As for the memory bound, we tune the parameter $n$ to determine
the rank of the formed tensor.

In this strategy, the chosen indices remain, and the other indices are sliced.
Different CPEs do different subtasks generated by slicing, which are
embarrassingly parallel. The main question is how to choose sliced indices and
find a proper $n$. The fundamental prerequisite for the slicing set is that
the sliced indices will not be contracted in $n$ steps, and fortunately, such
a condition is just the definition of _lifetime_. So with a particular start
position on the path, we slice the indices which have the longest _lifetime_
and traverse the stem until the _lifetime_ of any one of the sliced indices
ends. If the size of the slicing set added by $13$ (the capacity of LDM) is
less than each tensor on the sub-path, the fused calculation is finished. With
this strategy, there’s no slicing overhead, and the last DMA-put plays the
role of stacking with no extra communication.

Figure 9. An example of secondary slicing. There are 5 edges whose _lifetime_
go across the whole. After slicing these edges, the maximum memory demand is
reduced from $2^{8}$ to $2^{3}$, and $2^{5}$ subtasks are generated for
thread-level parallel. Only the computation inside the dotted box will be
performed in each core, and memory access happens at the beginning and the end
of the computation.

###  5.3. Deep Optimization

####  5.3.1. Permutation Map Reduction by Recursion Formula

In situ computing map and the pre-calculated map are two primary methods for
the tensor permutation. The in situ computing map needs an $O(NlogN)$ time
complexity for a size-$N$ tensor with rank $O(logN)$, and an $O(1)$ space
complexity. In contrast, the latter provides $O(NlogN)$ time complexity for
the first computation and $O(N)$ for the next ones, and space complexity of
$O(N)$. In our fused algorithm, permutations before every step of contractions
become one of the hot spots. Even though using short type, $n$ pre-calculated
maps are too big to store. However, for the strategy based on in situ map, the
tensors’ ranks are about 10, which leads to more than 10 times the cost.

A feasible solution is to combine the advantage of both strategies. For a
specific contraction, $A^{T}B^{T}=C$, in which $T$ denotes permutation, a
typical indices order of a rank-9 $A$ after permutation is like
$0,1,2,4,5,7,8,3,6$, the indices need to be absorbed will be organized at the
end, _i.e._ , $3,6$. Contrary to $A$, such indices are placed at the
beginning, like $3,8,0,1,2,4,5,6,7$. For $A$, obviously, the first 3
dimensions will not participate in the permutation, so only a $1/8$ map is
enough. And for $B$, the last 4 dimensions are the same during the
permutation, so the adjacency of the 4 elements shaped by $4,5,6,7$ will be
held, and the size of the map is reduced to $1/16$. With an offset, map of the
corresponding can be calculated by $map[i+k]=map[i]+k*offset$ when $k<stride$
in $O(1)$, the same as pre-calculated map. The space complexity will be
reduced to $O(N/2^{m})$, where m is the number of continuous indices at the
end or beginning.

####  5.3.2. Eliminate Discrete Memory Access by Communication

Though the fused strategy heavily reduces DMA, the efficiency of DMA becomes
an obstacle. Since we have sliced some edges from the original tensors, the
sub-tensors we need are often discretely distributed in the main memory. If we
want to get a sub-tensor without the last index (assume that it has been
sliced), there will be a space interval between every two elements we need. As
a result, the granularity of DMA will be extremely small, which harms the
bandwidth. In practice, the sliced edges are scattered in different positions.
Under this circumstance, the bandwidth of DMA can only achieve less than
$0.1\%$ of the peak performance, and makes negative optimization.

To improve the bandwidth, the cooperation of 64 CPEs becomes essential. Under
our parallel framework, the last 6 sliced indices are organized between 64
CPEs. Data exchange is much faster between CPEs by Remote memory access (RMA),
whose peak performance can achieve 800GB/s for a whole CG. Then, we treat the
sub-tensors of a 64 CPEs as a whole, and let every CPE access the memory
continuously. This strategy guarantees a basic granularity of 512B for DMA,
which provides more than $50\%$ of the peak performance. After memory access,
RMA is applied to rearrange the data between CPEs. An additional permutation
is applied to improve the granularity of RMA and ensure efficiency. For other
architecture, we can implement this strategy by simply partitioning the
communication groups.

##  6\. Result

Related efforts (_e.g._ , Cotengra(gray2021hyper, ),
Alibaba(huang2020classical, ) and (liu2021closing, )) have all simulated
Sycamore RQC(google-nature-2019, ). To make an appropriate comparison, unless
otherwise specified, the contraction trees used in this work come from the
tensor network of Sycamore as well.

###  6.1. Slicing Overhead and Scaling Result

Figure 10. Slicing size and overhead compared with cotengra. The red points
shows the number of extra slicing edges by cotengra compared with ours; and
the green points shows the ratio of overhead. When the difference is $0$ and
the ratio is $100\%$, two method perform equally.

According to the slice strategy in section 4, given a contraction path, we can
find a slicing set with lower overhead in expectation. Fig. 10 shows the
comparison between our work and cotengra. We find 400 contraction paths by
contengra, and apply our method and cotengra respectively to search for
slicing sets. Compared with cotengra, the slicing sets we found are
potentially smaller, and lead to lower overhead. Specific to each path, our
strategy performs better on more than $98\%$ of cases. As for several
exception cases, an observed phenomenon is that the computationally intensive
regions of the paths are dispersed. And also, since SA is a stochastic method,
it can not guarantee to find an optimal solution.

The best overhead result we found is less to 1.05, and we comprehensively
choose a path with low complexity to do the thread level optimization.

Due to slicing, processes are highly independent with each other, and will
only do _allReduce_ once at the end of program. The strong scaling results and
weak scaling results are showed in Fig. 11.

Figure 11. (Strong scaling results (65536 subtasks in total) and weak scaling
results (16 subtasks on each node).

###  6.2. Computing Efficiency

Figure 12. Optimization by secondary slicing at the thread level. Optimization
is done on a single node with $390$ cores, and the performance is tested for
tasks with different size on a contraction path. Use the step-by-step strategy
for comparison.

Fig. 12 shows that our work is able to significantly improve the computing
efficiency. Using 1024 nodes, a perfect sample or 1M correlated samples can be
generated in 10098.5s. Considering about the scaling result, we project that
we can reduce the whole time cost using 107520 nodes (41,932,800 cores) to
96.1s. The sustainable single-precision performance is projected as
308.6Pflops.

According to Fig. 12, the time of memory access is largely reduced by
secondary slicing, and the time of permutation and GEMM keep similar. This
result verifies our prediction that, secondary slicing can reduce memory
access with the price of some python based pre-conditioning, which needs only
1 core and ignorable time. Another conclusion is that, after secondary
slicing, the computation kernel is transformed from a memory intensive one to
a computation intensive one in some cases. According to Roofline
model(osti_1407078, ) and the arithmetic intensity of Sunway architecture, the
number of floating point operations should exceed 42.3 times of number of
bytes of memory access to achieve a peak performance. The average fused steps
of our strategy is about 10, and there is $MNK/(MN+MK+NK)\approx K$ for a
small $K$ with a average of 4, then in some cases the arithmetic intensity of
TNC can break through 42.3. Fig. 13 shows the Roofline model for a particular
case on thread level. Due to permutation, which consists of LDM access, there
is still a gap between our performance and the peak performance. However,
compared with the original arithmetic intensity of $2.6$ (mixed precision) and
$1.22$ (single precision), the optimization limit is largely improved.

Figure 13. Roofline Model of our work. For different cases, the arithmetic
intensities are different from $10x$ to $40x$. In some cases, the problems
turn into computation-bound.

##  7\. Implication

In this paper, we presented novel _lifetime_ -based methods to reduce the
slicing overhead and improve the computing efficiency for parallel
optimization at both the process level and thread level.

Due to the introduction of _lifetime_ definition, A series of lemmas and
theorems based on _lifetime_ intensify our awareness of a tensor network and
the contraction trees, then an interpretable method to deal with slicing
overhead, and an in-place slicing strategy to find the smallest slicing set,
so that we can reduce the slicing overhead. Our in-place strategy can work
independently, and can also optimize the slicing sets found by dynamic
methods. As a result, it becomes a general post-process when searching for
slicing sets.

Furthermore, based on _lifetime_ , we adopted stacking to avoid heavy overhead
from the redundant calculation, and generalize it for arbitrary multi-level
storage systems. Slicing optimization and stacking make up the two main
strategies for TNC. Then, we design a discriminant based on arithmetic
intensity to choose the strategy on different hierarchies of memory. For
Sunway architecture, we applied it at the thread level. At the necessary price
of more affordable memory access by DMA and faster communication among the CGs
by RMA. The most inspiring result is that, in some cases, the computation
kernels are changed from memory-intensive into computation-intensive, which
reshapes our understanding of the problem.

As a widely applied approach, tensor networks can be found in many research
fields like statistical physics(evenbly2015tensor, ), data
science(cichocki2017tensor, ), sociology(porter2009communities, ), and so on.
A series of problems in physics can be reduced to TNC. For large TNC, the
extreme time and space complexity become the major difficulty. Translating the
huge memory demand into a data stream that can be solved as a pipeline,
slicing plays a significant role during these TNC processes. In addition to
TNC, if we need to deal with a series of large matrix multiplications which
have data dependence, slicing can still work for task distribution.

This work proves that _lifetime_ is essentially promising for the tensor
networks of quantum circuits. Moreover, with the simple definition, _lifetime_
can be easily generalized to tensor works with different features and helps
analyze both the time- and space-complexities of the contraction. It is
foreseeable that _lifetime_ can play important roles in more fields.

## Acknowledgment

We would like to thank Zhen Wang, Zhaoqi Sun, Zegang Li and Yuxuan Li for
advice and discussions.

This work is partially supported by National Key R&D Program of China
(2020YFB0204804, 2020YFB0204800), National Natural Science Foundation of China
(Grant No. T2125006, U1839206, Project No. 62102114), Jiangsu Innovation
Capacity Building Program (Project No.
BM2022028) and the Key Research Project of Zhejiang Lab (No. 2021PB0AC01). The
corresponding authors are Yong Liu, Xin Liu, Lin Gan, Dexun Chen and Guangwen
Yang.

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viscoelasticity of biofilms (both predominantly colloidal, i.e. very little
polymer, and polymeric, i.e. very few cells, biofilms can be made). The
average spatial heterogeneity of bacterial cells in biofilms can be quantified
separately using the Ripley K function [314], but pair correlation functions
can also be used.

Recent studies indicate that the motion of some antibiotics in biofilms is
subdiffusive [317] and biofilm/sputum composites have also been considered.
Transport in the composites was studied using a system consisting of two
different media [89, 318]. Thus, the effective action of biofilms against
antibiotics could be due to the introduction of slow subdiffusive transport,
reducing the rate at which high concentrations of antibiotics arrive at the
bacteria. Other factors are also known to be important, such as phenotypic
changes of the bacteria in the biofilms, which adopt very low metabolic states
e.g. persister cells.

###  7.6 Eukaryotic microorganisms

Eukaryotic microorganisms are known to generate non-equilibrium fluctuations
in their surroundings, which have been studied using passive tracer particles
immersed in suspensions of eukaryotic swimmers. The long-time dynamics of the
tracers was described with a normal (Brownian) yet non-Gaussian diffusion
model [319]. Theoretically it was predicted that hydrodynamic interactions
between the tracer and the swimmers lead to the occurrence of a Lévy flight
regime in the tracer dynamics and non-Gaussian distributions of displacements
[320]. Lévy flight models are nonphysical since they imply instantaneous
transport of particles during jumps, so these results should be revisited with
a Lévy walk model. Interestingly, the shape asymmetry of tracers is predicted
to induce ratchet effects that alter fluctuations in cell motion and lead to
asymptotic super-diffusion [321].

The two-dimensional motion of the social amoeba _Dictyostelium discoideum_ (a
species of soil-dwelling amoeba) was found to be super-diffusive with broadly
distributed diffusivities [322] (probably due to cell-to-cell variability) and
was characterized using an exponential probability distribution of
displacements and generalised gamma probability distribution of cell speeds
[323]. The diffusion of the parasitic soil nematode _Phasmarhabditis
hermaphrodita_ was found to be normal but with a significant variation of the
diffusion coefficients among the the population that followed a gamma
probability distribution [124]. Analysis of the motion of four flagellated
Protozoa species revealed super-diffusion with non-Gaussian probability
distributions of displacements together with a generalised gamma probability
distribution for their speeds and a broad probability distribution of Hurst
exponents characterizing the self-similarity of the cells’ velocity time
series [324].

## Conclusions

The questions involved in heterogeneous anomalous transport are of central
importance to quantitative studies of molecular and cellular biology. The main
challenge is to decipher the origin of the underlying distributions of
anomalous exponents and generalized diffusion coefficients. In specific cases
quantitative treatments are possible e.g. for diffusing particles undergoing
fission and fusion, a power-law probability distribution of cluster size was
predicted by Smoluchowski’s coagulation theory, which naturally leads to
polydispersity with diffusion coefficients that scale with the cluster size as
a power law [261]. Alternative theoretical methodologies to consider the
motility of molecules and cells include focusing on the small system
thermodynamics of trajectories [325] or soft matter models [326]. Here we
prioritised the experimental measurement of the statistics of motion and the
development of agnostic models which can directly describe the anomalous
transport observed. The extreme complexity of high-concentration biological
systems in living cells often precludes more fundamental stochastic
thermodynamic or soft matter modelling. The hope is that heterogeneous
anomalous transport will allow the gap to be bridged to more fundamental
molecular models. Furthermore, many classical soft matter models were based on
Gaussian fluctuations and still need to be extended to describe the
heterogeneous anomalous statistics observed experimentally. Machine learning
techniques (e.g. deep learning neural networks) combined with high-resolution
microscope-based tracking allow some of the major challenges in the
characterisation of HAT to be overcome. Continued development of anomalous
agent-based models will allow a fundamental physical framework to understand
collective anomalous motility to be developed e.g. models for the collective
motion of cells in tissue based on HAT.

So, what if there are more complicated models based on HAT that are required
to fit the stochastic motion of particles in biology? There are a number of
advantages to this approach:

$\bullet$ HAT has _diagnostic potential_ in medicine. Models based on
anomalous transport provide more detailed information to differentiate mutants
or diseased states of cells [74]. This could be used for diagnostic purposes
e.g. cancer cells often have altered motile phenotypes in leukemia that can
drive metastasis and could be diagnosed from a blood sample [327]. Although
genetic analysis of the cancer cells would also be a priority for diagnosis,
connecting genotype to phenotype is a crucial step, and making the connection
necessitates motility assays and robust methods to quantify them. Furthermore,
gait analysis (how cells swim or crawl) could be used to classify varieties of
microorganisms, which again has diagnostic potential [328]. Also, diseased
states of eukaryotic cells (e.g. in humans or livestock) are often associated
with impaired intracellular motility e.g. in motor neuron disease where the
motility of neurotransmitters in endosomes can be compromised [329].
Information could be used from HAT analysis for gene therapy targeted to treat
the phenotypes of impaired intracellular motility and to understand the
success of treatments.

$\bullet$ HAT provides _fundamental insights_ into biology. The machinery
inside cells is finely optimised by evolution to provide efficient solutions
to biological challenges. How do cells and molecules make use of anomalous
stochastic phenomena [6, 7, 330]?

$\bullet$ Anomalous transport presents a _fundamental barrier_ in a diverse
range of problems in quantitative cellular and molecular biology e.g.
understanding the action of enzymes inside cells where the kinetics can be
dramatically affected [331], how cells move in developing embryos [74], the
transcription of information from DNA [169, 332], how reaction-diffusion
processes lead to robust patterning during morphogenesis [104] and the finely
orchestrated movement of endosomes inside cells [72, 71]. Without a thorough
quantitative understanding of anomalous transport, we lack tools to handle a
large number of crucial biological problems.

$\bullet$ Numerous _new physical phenomena_ are still left to be discovered
that are related to heterogeneous anomalous transport e.g. the motility of
molecular motors in active matter [104], collective cellular motility in
embryos [74], the motion of wavefronts of electrical signalling in bacterial
biofilms [50] and how bacteria swim [333]. The field provides an interesting
perspective to the subject of soft matter physics, focusing on the statistics
of motility.

$\bullet$ The _applied mathematics_ of HAT is novel and interesting in its own
right e.g. fractional partial differential equations [334], the generalized
central limit theorem, Ito calcululus [335] and multi-fractals [64].

## Acknowledgements

We would like to thank Viki Allan, Sergei Fedotov, Eli Barkai, Victor
Martorelli, Salman Rogers, Ian Roberts, Tom Millard, Jian Lu, Daniel Han, Jack
Hart, Sorasak Phanphak, Joanna Blee, David Kenwright and Andrew Harrison for
their invaluable help over the years.

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