# Sample complexity of matrix product states at finite temperature

Atsushi Iwaki<EMAIL_ADDRESS>Chisa Hotta  Department of
Basic Science, University of Tokyo, Meguro-ku, Tokyo 153-8902, Japan

###### Abstract

For quantum many-body systems in one dimension, computational complexity
theory reveals that the evaluation of ground-state energy remains elusive on
quantum computers, contrasting the existence of a classical algorithm for
temperatures higher than the inverse logarithm of the system size. This
highlights a qualitative difference between low- and high-temperature states
in terms of computational complexity. Here, we describe finite-temperature
states using the matrix product state formalism. Within the framework of
random samplings, we derive an analytical formula for the required number of
samples, which provides both quantitative and qualitative measures of
computational complexity. At high and low temperatures, its scaling behavior
with system size is linear and quadratic, respectively, demonstrating a
distinct crossover between these numerically difficult regimes of quantitative
difference.

##  I Introduction

The realization of quantum many-body states at finite temperature in classical
or quantum computers is a pivotal topic as it pertains to modern research
areas of thermalization [1, 2], many-body localization [3, 4], quantum many-
body scars [5, 6], and so on whose clarification often relies heavily on
numerical tools. Recently, quantum simulators have proved themselves as
powerful platforms for realizing such states in a large system size in
laboratories [7, 8, 9, 10, 11, 12], allowing for comparisons with algorithmic
approaches in numerical simulations.

Computational complexity of physical states studies the degree of difficulty
of computational problems in relation to the complexity classes, and serves as
a guide to such problems. One important class is Quantum Merlin-Arthur (QMA)
that tests whether a polynomial-size of a quantum state can be verified within
a required polynomial time in a quantum computer. For one-dimensional (1D)
quantum many-body states, it is known that the evaluation of exact ground-
state energy is QMA-complete, indicating that the problem is difficult even
for quantum computers [13, 14, 15, 16, 17]. However, empirically, we can
efficiently calculate their ground states very accurately for large system
sizes even in classical computers using e.g. the density matrix
renormalization group (DMRG) method [18, 19, 20]. To be precise, such
calculations are more established for gapped systems [21, 22, 23, 24] where
the entanglement area law holds [25, 26], and the degrees of difficulty of
calculation are closely related to whether the system is gapped or gapless.
However, to know whether a given Hamiltonian has a gap or not is already an
undecidable problem in general [27, 28].

At finite temperature, preparing thermal states in classical computers is
relatively easier than the ground state for almost all cases, just as it is
easy to prepare finite-temperature states in experiments. We schematically
show in Fig. 1(a) the classification known so far for the computational
complexity of states over different temperature regimes and different spatial
dimensions.

Figure 1:  (a) Schematic illustration of the computational complexity of thermal and ground state. Regions highlighted in red have efficient classical algorithms, whereas in the blue regions, the ground-state energy cannot be efficiently calculated even for quantum computers in general. (b) Temperature dependence of the required number of samples in MPS-based random sampling methods we derive in this paper. It scales linearly with the system size $N$ at high temperatures, and it is proportional to $N^{2}$ at low temperatures, in leading order where the initial bond dimension $\chi$ is large. The crossover is characterized by the inverse temperature $\beta_{c}$ where $\Delta E$ is the spectral gap.  Table 1:  Classification of finite-temperature methods. Physical memory to realize in a computer, and the required number of sample averages, as functions of the system size $N$ and the bond dimension $\chi$, available $N$ and the applicable spatial dimensions are shown.  method | memory | sample $\\#$ | system size | spatial dimension | types of state
---|---|---|---|---|---
full density op. | $D^{2}=e^{\mathcal{O}(N)}$ | 1 | $\lesssim 15$ | any | Gibbs
MPO (density op.) | $Nd\chi^{2}$ | 1 | $\sim 100$ | 1D or 2D | Gibbs
TPQ / FTL | $D=e^{\mathcal{O}(N)}$ | $\gtrsim e^{-\mathcal{O}(N)}$ | $\lesssim 30$ | any | pure
METTS | $Nd\chi^{2}$ | $\sim 100?$ | $\sim 100$ | 1D or 2D | mixed
TPQ-MPS | $Nd\chi^{2}$ | $\mathcal{O}(N/\chi^{2})$ | $\sim 100$ | 1D or 2D | nearly pure

Recently, rigorous analysis is progressing in the field of quantum
information. For 1D systems, the approximate tensor network representation of
the Gibbs state is obtained when the inverse temperature $\beta$ is smaller
than the logarithm of the system size $N$, as $\beta\lesssim\log N$ [29, 30].
For higher-dimensional systems, it was proved that the partition function can
be classically simulated in polynomial time with $N$ at temperatures higher
than a constant independent of $N$ [31, 32]. Phase transitions can only occur
in higher spatial dimensions at lower temperature than discussed here, but
this region is not well understood. It is considered to be related to an open
problem known as the quantum PCP conjecture in the context of computational
complexity theory [33].

Practical numerical methods have a long history in condensed matter physics.
The stochastic quantum Monte Carlo (QMC) method allows large system size $N$,
but faces a sign problem [34, 35]. High-temperature expansions [36, 37] and
numerical linked-cluster expansions [38, 39, 40], are established, while the
former has a temperature bound of $\beta^{-1}\lesssim J$ for typical energy
scale $J$, and the latter fails when the correlation length diverges.

Methods directly describing thermal states have more variants (see Table 1).
The most conventional Gibbs state requires the computational memory of $D^{2}$
where $D$ is the dimension of the Hilbert space because it is a maximally
mixed state that is represent by the full-rank density operator. Thermal
equilibrium can be represented by a single pure state as well. The finite-
temperature Lanczos (FTL) methods can handle it by an appropriate choice of
basis with a computational memory of $D$ [41, 42, 43], and some are called the
thermal pure quantum (TPQ) method [44, 45]. Supported by the concept of
typicality [46, 47, 48, 49], it is guaranteed that the required number of
samples is very small, whereas the available $N$ is limited by the growth of
$D=d^{N}$ where $d$ is the local dimension. To access large systems, tensor
network methods have been actively studied. Particularly, the matrix product
states (MPS) are commonly used in 1D [50] taking advantage of the time-
evolving block decimation (TEBD) algorithm [51, 52, 53]. For the Gibbs state,
with the doubled Hilbert space, there are matrix product operator (MPO) [54,
55], the purification represented by MPS [56] and its analogues [57, 58, 59].
Besides the Gibbs and TPQ states, we have many thermal states in between; we
previously developed the TPQ-MPS method for the nearly pure state (see
§.II.3), that belongs to the family of MPS-based random sampling methods [60,
61, 62, 63], but provides the purest thermal state. The minimally entangled
typical thermal states (METTS) method [64, 65] has less purity than TPQ-MPS,
while its purity is being improved by different devices [66, 67, 68, 69, 70,
71].

In principle, all these finite-temperature numerical approaches start from
high temperature and in approaching the low temperature, suffer numerical
difficulties. Many of them belong to the random sampling method, which
prepares the initial random states at high temperature and cools them down by
the imaginary time evolution. In that framework, the numerical difficulty can
be measured by the required number of samples, which we call “sample
complexity”. The sample complexity crucially depends on the expression power
of the method for the sampled states. Based on this idea, we have recently
classified the Gibbs state, the TPQ state and the states in between them as
thermal mixed quantum (TMQ) states [72]. The Gibbs state method that can
basically express (semiclassical) product state requires a large sampling
number, while the TPQ methods that can express quantum states with large
enough entanglement require only a few samples. In Table 1, we show the
classification of methods in this context; the expression power about the
quantum state is the ability to store an entropy in the form of quantum
entanglement in a single sample, because the thermal entropy (stored as the
number of samples) and the quantum entanglement entropy are the two faces of
the same coin.

In this paper, we investigate the required number of samples in MPS-based
methods, focusing on the TPQ-MPS method which utilizes a natural description
of a TMQ state very close to the TPQ state. There is a convenient and
measurable quantity that has the amplitude proportional to the required number
of samples, called the normalized fluctuation of partition function (NFPF)
[72]. Here, we propose an analytical formula of the NFPF which applies from
zero to the highest temperature. Importantly, the leading orders of the NFPF
or the number of random samples in terms of $N$ are different between low- and
high-temperature limits (see Fig. 1(b)). With the numerical demonstrations
given for several quantum spin models, we verify this formula and demonstrate
for the first time the explicit crossover between the two temperature regions
which are qualitatively different in the language of computational complexity.

##  II Random sampling methods and NFPF

The physical quantity in thermal equilibrium is described as

$\langle O\rangle_{\beta}=\operatorname{Tr}[\rho_{\mathrm{G}}(\beta)O],$ (1)

where a density operator $\rho_{\mathrm{G}}(\beta)$ represents the Gibbs state
and is given as

$\rho_{\mathrm{G}}(\beta)=\frac{e^{-\beta H}}{Z(\beta)},\quad
Z(\beta)=\operatorname{Tr}e^{-\beta H}.$ (2)

Here, we introduce the framework of the random sampling method for finite-
temperature calculations, and the way to evaluate their efficiency [62] based
on Ref. [72]. This framework applies to the Monte Carlo methods without using
a Markov chain, but does not include well-established Markov-chain Monte Carlo
methods such as QMC and METTS. However, its applicability is wide, and is not
only limited to finite-temperature calculations but to other quantum states
whose realization follows the distributions other than the Boltzmann
distribution.

###  II.1 Random sampling methods

We initially prepare a random state $\ket{\psi_{0}}$ that satisfies

$\overline{\ket{\psi_{0}}\bra{\psi_{0}}}=cI,$ (3)

where $\overline{\cdots}$ denotes taking the random average and $c$ is a
system-size-dependent constant. This equation implies that $\ket{\psi_{0}}$
does not favor any specific energy, and it represents one of the states
realized in the high-temperature limit (In practice, what temperature it
targets depends on the quality of the random sampling and the approximations
used for the representation of $\ket{\psi_{0}}$). We perform an imaginary time
evolution to cool down the temperature to $\beta^{-1}$, to obtain the
corresponding state,

$\displaystyle\ket{\psi_{\beta}}=e^{-\beta H/2}\ket{\psi_{0}},$ (4)

where $H$ is the system Hamiltonian. The partition function or certain
physical quantities are obtained by taking the random average.

$\displaystyle
Z(\beta)=\frac{1}{c}\overline{\braket{\psi_{\beta}}{\psi_{\beta}}}$ (5)
$\displaystyle\langle
O\rangle_{\beta}=\frac{\overline{\braket{\psi_{\beta}}{O}{\psi_{\beta}}}}{\overline{\braket{\psi_{\beta}}{\psi_{\beta}}}}$
(6)

As the ideal random averages are not accessible, we approximate them by using
sample averages in the numerical simulations as

$\displaystyle
Z^{\mathrm{samp}}_{M}(\beta)=\frac{1}{cM}\sum_{i=1}^{M}\braket{\psi_{\beta}^{(i)}}{\psi_{\beta}^{(i)}},$
(7) $\displaystyle\langle
O\rangle^{\mathrm{samp}}_{\beta,M}=\frac{\sum_{i=1}^{M}\braket{\psi_{\beta}^{(i)}}{O}{\psi_{\beta}^{(i)}}}{\sum_{j=1}^{M}\braket{\psi_{\beta}^{(j)}}{\psi_{\beta}^{(j)}}},$
(8)

where $\\{\ket{\psi_{\beta}^{(i)}}\\}_{i=1}^{M}$ are $M$ independent
realizations of $\ket{\psi_{\beta}}$. If we take a sufficiently large value of
$M$, the law of large numbers guarantees that the sample average matches the
random average.

Conventionally, the sufficient sample number $M$ required was empirically
determined depending on the physical quantities we want to obtain so as to
sufficiently reduce the variance according to the objective. However, this
ambiguity can be removed by using a quantity we recently introduced, which is
named the NFPF [72]. By using the NFPF, we can decisively discuss $M$ as not
only the sufficient but also the necessary value to qualify the thermal
quantum state on equal footing across different numerical methods we apply in
operating the random sampling method. The value of $M$ or the NFPF depends on
how the initial random states are prepared and on the approximations of the
method.

The NFPF is given as

$\displaystyle\delta
z^{2}=\frac{\mathrm{Var}\left(\braket{\psi_{\beta}}{\psi_{\beta}}\right)}{\left(\overline{\braket{\psi_{\beta}}{\psi_{\beta}}}\right)^{2}}.$
(9)

It quantifies the random fluctuation of the partition function,
$Z(\beta)=\operatorname{Tr}e^{-\beta H}$, as

$\displaystyle\frac{1}{Z(\beta)}\overline{\left(Z^{\mathrm{samp}}_{M}(\beta)-Z(\beta)\right)^{2}}=\frac{\delta
z^{2}}{M}.$ (10)

Therefore, to obtain the partition function with a relative error $\epsilon$,
we require $M_{\epsilon}$ samples given by

$\displaystyle M_{\epsilon}=\frac{\delta z^{2}}{\epsilon^{2}}.$ (11)

With Chebyshev’s inequality, we can ascribe a strictly probabilistic
interpretation to the value of $M_{\epsilon}$.

$\displaystyle\mathrm{Prob}\left[|Z^{\mathrm{samp}}_{M}(\beta)-Z(\beta)|\geq\epsilon
Z(\beta)\right]\leq\frac{M_{\epsilon}}{M}$ (12)

Under the condition,

$\displaystyle\mathrm{Var}\left(\braket{\psi_{\beta}}{O}{\psi_{\beta}}\right)\leq\mathrm{(const.)}\times\|O\|^{2}\mathrm{Var}(\braket{\psi_{\beta}}{\psi_{\beta}}),$
(13)

the NFPF gives a bound of the fluctuation of the physical quantity $O$ as

$\displaystyle\overline{\left(\langle
O\rangle^{\mathrm{samp}}_{\beta,M}-\langle
O\rangle_{\beta}\right)^{2}}\lesssim\mathrm{(const.)}\times\|O\|^{2}\frac{\delta
z^{2}}{M}.$ (14)

This condition typically holds except for a few specific cases, and indeed the
required sample size for the evaluation of energy is proportional to
$M_{\epsilon}$ [72].

###  II.2 NFPF of TPQ state

As an ideal reference system, we consider the TPQ method which uses the entire
Hilbert space with the dimension $D$ to describe the thermal state. There are
several different ways to prepare initial random states in the TPQ method
[73]. We choose to generate the initial state so as to have the coefficients
of the basis following an independent complex Gaussian distribution.
Importantly, this random state is independent of the choice of the basis, and
thus can be expressed as $\ket{\psi_{0}}=\sum_{n}a_{n}\ket{n}$ where
$\\{\ket{n}\\}_{n}$ is the energy eigenstates. With this setting, we can
calculate the NFPF of the TPQ method analytically as

$\displaystyle\delta z^{2}_{\mathrm{TPQ}}=e^{-Ns_{2}(\beta)},$ (15)

where $s_{2}(\beta)$ is the rescaled Rényi-2 entropy of the Gibbs state as

$\displaystyle
s_{2}(\beta)=-\frac{1}{N}\log\operatorname{Tr}[\rho_{\mathrm{G}}(\beta)^{2}],$
(16)

which is called the thermal Rényi-2 entropy. Since the NFPF decreases
exponentially with the system size, the TPQ method proves to be highly
efficient from the perspective of sample complexity. Moreover, the inequality
(13) holds for all physical quantities in the TPQ state. As a consequence,
random fluctuations of physical quantities in the TPQ method are significantly
reduced, and for appropriately large systems, only a few samples are
necessary.

###  II.3 TPQ-MPS method

We briefly explain the TPQ-MPS method [74]. Unlike the standard MPS form with
open edges where the interactions are absent, we apply the specific form of
MPS that hosts two auxiliary sites located on both edges. The advantage of
this construction has already proved in two previous works; in 1D [74], the
number of sample averages $M$ decreases significantly in lowering the
temperature, and quite significantly, the variance of measured quantities is
suppressed at lower temperatures, contrarily to all the other known methods
including QMC or the standard TPQ method (or the FTL method) using the full
Hilbert space. The reason is as follows; in usual MPS, the bond dimension is 1
at both edges, and increases as $d^{n}$ as we approach $n=1,2,\cdots$ toward
the center site, which is known to be reflected in the Page curve of the
entanglement entropy. It means that due to the boundary condition, the amount
of entanglement entropy stored in the system is significantly suppressed
toward the edge sites. However, in our construction, the aid of auxiliaries
helps to avoid such suppression, and the bond dimension and the entanglement
entropy stays almost constant regardless of the location of the bipartition of
the system. The amount of entropy stored thus follows the volume law that
accounts for that of the actual thermal entropy of the system, allowing the
system to be nearly pure.

For applications in 2D [75] which is usually considered to be much difficult,
the Kitaev honeycomb model is chosen, where the MPS is constructed in the same
manner as 1D wrapping the cylinder in a spiral construction, for which part of
the nearest neighbor interactions on the original lattice is converted as
longer range interactions. However, it was shown that the method works fairly
well, reproducing two peak structures of the specific heat that are reported
in the previous studies using the almost non-interacting framework taking
advantage of the Majorana description of the model.

The initial state is represented as a random MPS with auxiliaries as

$\displaystyle\ket{\psi_{0}}=\sum_{\\{\alpha\\}}^{\chi}\sum_{\\{i\\}}^{d}$
$\displaystyle
A_{\alpha_{0}\alpha_{1}}^{[1]i_{1}}A_{\alpha_{1}\alpha_{2}}^{[2]i_{2}}\cdots
A_{\alpha_{N-1}\alpha_{N}}^{[N]i_{N}}\times$
$\displaystyle\times\ket{\alpha_{0},i_{1},i_{2},\dots,i_{N},\alpha_{N}}.$ (17)

The auxiliary systems have the same degrees of freedom as the initial bond
dimension $\chi$. We then apply an imaginary time evolution, Eq. (4) and
obtain $\ket{\psi_{\beta}}$ for this construction and evaluate the physical
quantities following Eqs. (7) and (8). The bond dimension may increase through
imaginary time evolution using the TEBD technique or applying the MPO.
Previously, we have shown that the information of higher temperature states is
properly discarded through the truncation process in the TPQ-MPS method [75].

##  III Sample complexity of matrix product states

In this section, we analytically derive a formula for the NFPF of the TPQ-MPS.
By using the formula for the ideal random average distribution, we are able to
perform the analytical evaluation of the related quantities.

To simplify the formula, we utilize diagram notations of tensor networks. We
first focus on Eq. (17) depicted as

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where the circle represents a matrix $A$, and thin/bold lines mean
physical/auxiliary degrees of freedom, respectively. We colored the matrices
(circles) differently implying that each matrix follows an independent
probability distribution. We assume that all elements of matrices obey an
independent complex Gaussian distribution. A tensor $a_{i}$ which is taken
from an independent complex Gaussian distribution satisfies

$\overline{a_{i}a_{j}^{*}}=\delta_{ij},\quad\overline{a_{i}a_{j}^{*}a_{k}a_{l}^{*}}=\delta_{ij}\delta_{kl}+\delta_{il}\delta_{jk}.$
(19)

These equations are represented in the following tensor network diagrams.

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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{}
{}{}{}\pgfsys@moveto{5.69055pt}{0.0pt}\pgfsys@lineto{5.69055pt}{11.3811pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{5.69055pt}{28.45276pt}\pgfsys@lineto{5.69055pt}{39.83386pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\right]}=\raisebox{-0.5pt}{ \leavevmode\hbox to0.4pt{\vbox
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},\quad\overline{\left[\raisebox{-0.45pt}{ \leavevmode\hbox to14.75pt{\vbox
to40.23pt{\pgfpicture\makeatletter\hbox{\hskip 1.6854pt\lower-0.2pt\hbox
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{}{}{}\pgfsys@moveto{5.69055pt}{28.45276pt}\pgfsys@lineto{5.69055pt}{39.83386pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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to14.75pt{\vbox to40.23pt{\pgfpicture\makeatletter\hbox{\hskip
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} {}{{}}{}
{}{}{}\pgfsys@moveto{5.69055pt}{28.45276pt}\pgfsys@lineto{5.69055pt}{39.83386pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{5.69055pt}{11.3811pt}\pgfsys@moveto{12.5192pt}{11.3811pt}\pgfsys@curveto{12.5192pt}{15.1525pt}{9.46194pt}{18.20975pt}{5.69055pt}{18.20975pt}\pgfsys@curveto{1.91916pt}{18.20975pt}{-1.13809pt}{15.1525pt}{-1.13809pt}{11.3811pt}\pgfsys@curveto{-1.13809pt}{7.60971pt}{1.91916pt}{4.55246pt}{5.69055pt}{4.55246pt}\pgfsys@curveto{9.46194pt}{4.55246pt}{12.5192pt}{7.60971pt}{12.5192pt}{11.3811pt}\pgfsys@closepath\pgfsys@moveto{5.69055pt}{11.3811pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
}\right]}=\raisebox{-0.45pt}{ \leavevmode\hbox to17.47pt{\vbox
to40.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{}
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{}{}{}\pgfsys@moveto{17.07166pt}{0.0pt}\pgfsys@lineto{17.07166pt}{39.83386pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }+\raisebox{-0.45pt}{
\leavevmode\hbox to17.47pt{\vbox
to40.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{17.07166pt}{39.83386pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{17.07166pt}{0.0pt}\pgfsys@lineto{0.0pt}{39.83386pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }.$ (20)

We will calculate the NFPF of the TPQ-MPS by using the above relations.

###  III.1 Non-interacting Hamiltonian

Here, we consider a Hamiltonian which is the sum of single-site local
operators with translational invariance.

$\displaystyle H=\sum_{i=1}^{N}h_{i},\quad
h_{i}=\overset{1}{I}\otimes\cdots\otimes\overset{i}{h}\otimes\cdots\otimes\overset{N}{I}$
(21)

The partition function is defined at each local site independently of each
other as $Z(\beta)=z(\beta)^{N}$ with $z(\beta)=\operatorname{Tr}e^{-\beta
h}$. To evaluate the NFPF, we initially calculate the norm of the finite-
temperature state which is expressed as

$\displaystyle\braket{\psi_{\beta}}{\psi_{\beta}}=\raisebox{-0.45pt}{
\leavevmode\hbox to112.57pt{\vbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{}{{}}{} {}{} {}{}
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}{{{}}
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}{{{}}
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}{{{}}
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} \pgfsys@invoke{\lxSVG@closescope
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}\definecolor{pgffillcolor}{rgb}{1,0.7,0.7}\pgfsys@color@rgb@fill{1}{0.7}{0.7}\pgfsys@invoke{
}{}\pgfsys@moveto{17.07166pt}{8.53583pt}\pgfsys@moveto{25.60748pt}{8.53583pt}\pgfsys@curveto{25.60748pt}{13.25009pt}{21.78592pt}{17.07166pt}{17.07166pt}{17.07166pt}\pgfsys@curveto{12.35739pt}{17.07166pt}{8.53583pt}{13.25009pt}{8.53583pt}{8.53583pt}\pgfsys@curveto{8.53583pt}{3.82156pt}{12.35739pt}{0.0pt}{17.07166pt}{0.0pt}\pgfsys@curveto{21.78592pt}{0.0pt}{25.60748pt}{3.82156pt}{25.60748pt}{8.53583pt}\pgfsys@closepath\pgfsys@moveto{17.07166pt}{8.53583pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.57166pt}{6.20944pt}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{91.3941pt}{6.20944pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{8.53583pt}{25.60748pt}\pgfsys@moveto{8.53583pt}{25.60748pt}\pgfsys@lineto{8.53583pt}{42.67914pt}\pgfsys@lineto{25.60748pt}{42.67914pt}\pgfsys@lineto{25.60748pt}{25.60748pt}\pgfsys@closepath\pgfsys@moveto{25.60748pt}{42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{34.14331pt}{25.60748pt}\pgfsys@moveto{34.14331pt}{25.60748pt}\pgfsys@lineto{34.14331pt}{42.67914pt}\pgfsys@lineto{51.21497pt}{42.67914pt}\pgfsys@lineto{51.21497pt}{25.60748pt}\pgfsys@closepath\pgfsys@moveto{51.21497pt}{42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{85.35828pt}{25.60748pt}\pgfsys@moveto{85.35828pt}{25.60748pt}\pgfsys@lineto{85.35828pt}{42.67914pt}\pgfsys@lineto{102.42993pt}{42.67914pt}\pgfsys@lineto{102.42993pt}{25.60748pt}\pgfsys@closepath\pgfsys@moveto{102.42993pt}{42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }$ (22)

where  represents the operator $e^{-\beta h}$. By applying Eq. (20) to each
matrix, we can calculate the random average of the norm as

$\displaystyle\overline{\braket{\psi_{\beta}}{\psi_{\beta}}}$
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}\cdots\raisebox{-0.45pt}{ \leavevmode\hbox to26.01pt{\vbox
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$\displaystyle=\chi^{N+1}z(\beta)^{N}=\chi^{N+1}Z(\beta).$ (23)

This calculation verifies Eq. (5). In the TPQ-MPS method, it is found that
$c=\chi^{N+1}$. Next, we calculate the random average of the square of the
norm.

$\displaystyle\overline{\braket{\psi_{\beta}}{\psi_{\beta}}^{2}}$
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}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{39.83386pt}{0.0pt}\pgfsys@lineto{39.14331pt}{0.0pt}\pgfsys@curveto{36.38185pt}{0.0pt}{34.14331pt}{2.23854pt}{34.14331pt}{5.0pt}\pgfsys@lineto{34.14331pt}{17.7622pt}\pgfsys@curveto{34.14331pt}{20.52367pt}{36.38185pt}{22.7622pt}{39.14331pt}{22.7622pt}\pgfsys@lineto{39.83386pt}{22.7622pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{39.83386pt}{28.45276pt}\pgfsys@lineto{39.14331pt}{28.45276pt}\pgfsys@curveto{36.38185pt}{28.45276pt}{34.14331pt}{30.6913pt}{34.14331pt}{33.45276pt}\pgfsys@lineto{34.14331pt}{46.21497pt}\pgfsys@curveto{34.14331pt}{48.97643pt}{36.38185pt}{51.21497pt}{39.14331pt}{51.21497pt}\pgfsys@lineto{39.83386pt}{51.21497pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }+\raisebox{-0.45pt}{
\leavevmode\hbox to7.29pt{\vbox to52.81pt{\pgfpicture\makeatletter\hbox{\hskip
0.8pt\lower-0.8pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.69055pt}{0.0pt}\pgfsys@curveto{3.45201pt}{0.0pt}{5.69055pt}{2.23854pt}{5.69055pt}{5.0pt}\pgfsys@lineto{5.69055pt}{46.21497pt}\pgfsys@curveto{5.69055pt}{48.97643pt}{3.45201pt}{51.21497pt}{0.69055pt}{51.21497pt}\pgfsys@lineto{0.0pt}{51.21497pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{22.7622pt}\pgfsys@lineto{1.84528pt}{22.7622pt}\pgfsys@curveto{2.39757pt}{22.7622pt}{2.84528pt}{23.20992pt}{2.84528pt}{23.7622pt}\pgfsys@lineto{2.84528pt}{27.45276pt}\pgfsys@curveto{2.84528pt}{28.00505pt}{2.39757pt}{28.45276pt}{1.84528pt}{28.45276pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} \leavevmode\hbox
to17.47pt{\vbox
to51.61pt{\pgfpicture\makeatletter\hbox{\hskip-11.1811pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{}{{}}{} {}{} {}{} {}{}
{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{17.07166pt}{5.0pt}\pgfsys@lineto{17.07166pt}{46.21497pt}\pgfsys@curveto{17.07166pt}{48.97643pt}{19.3102pt}{51.21497pt}{22.07166pt}{51.21497pt}\pgfsys@lineto{23.45276pt}{51.21497pt}\pgfsys@curveto{26.21422pt}{51.21497pt}{28.45276pt}{48.97643pt}{28.45276pt}{46.21497pt}\pgfsys@lineto{28.45276pt}{5.0pt}\pgfsys@curveto{28.45276pt}{2.23854pt}{26.21422pt}{0.0pt}{23.45276pt}{0.0pt}\pgfsys@lineto{22.07166pt}{0.0pt}\pgfsys@curveto{19.3102pt}{0.0pt}{17.07166pt}{2.23854pt}{17.07166pt}{5.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{11.3811pt}{5.69055pt}\pgfsys@moveto{11.3811pt}{5.69055pt}\pgfsys@lineto{11.3811pt}{17.07166pt}\pgfsys@lineto{22.7622pt}{17.07166pt}\pgfsys@lineto{22.7622pt}{5.69055pt}\pgfsys@closepath\pgfsys@moveto{22.7622pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{11.3811pt}{34.14331pt}\pgfsys@moveto{11.3811pt}{34.14331pt}\pgfsys@lineto{11.3811pt}{45.52441pt}\pgfsys@lineto{22.7622pt}{45.52441pt}\pgfsys@lineto{22.7622pt}{34.14331pt}\pgfsys@closepath\pgfsys@moveto{22.7622pt}{45.52441pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} \leavevmode\hbox
to7.29pt{\vbox to52.81pt{\pgfpicture\makeatletter\hbox{\hskip
0.8pt\lower-0.8pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{5.69055pt}{0.0pt}\pgfsys@lineto{5.0pt}{0.0pt}\pgfsys@curveto{2.23854pt}{0.0pt}{0.0pt}{2.23854pt}{0.0pt}{5.0pt}\pgfsys@lineto{0.0pt}{46.21497pt}\pgfsys@curveto{0.0pt}{48.97643pt}{2.23854pt}{51.21497pt}{5.0pt}{51.21497pt}\pgfsys@lineto{5.69055pt}{51.21497pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{5.69055pt}{22.7622pt}\pgfsys@lineto{3.84528pt}{22.7622pt}\pgfsys@curveto{3.29298pt}{22.7622pt}{2.84528pt}{23.20992pt}{2.84528pt}{23.7622pt}\pgfsys@lineto{2.84528pt}{27.45276pt}\pgfsys@curveto{2.84528pt}{28.00505pt}{3.29298pt}{28.45276pt}{3.84528pt}{28.45276pt}\pgfsys@lineto{5.69055pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
}\right]\raisebox{-0.45pt}{ \leavevmode\hbox to7.29pt{\vbox
to52.81pt{\pgfpicture\makeatletter\hbox{\hskip 0.8pt\lower-0.8pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.69055pt}{0.0pt}\pgfsys@curveto{3.45201pt}{0.0pt}{5.69055pt}{2.23854pt}{5.69055pt}{5.0pt}\pgfsys@lineto{5.69055pt}{17.7622pt}\pgfsys@curveto{5.69055pt}{20.52367pt}{3.45201pt}{22.7622pt}{0.69055pt}{22.7622pt}\pgfsys@lineto{0.0pt}{22.7622pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{0.69055pt}{28.45276pt}\pgfsys@curveto{3.45201pt}{28.45276pt}{5.69055pt}{30.6913pt}{5.69055pt}{33.45276pt}\pgfsys@lineto{5.69055pt}{46.21497pt}\pgfsys@curveto{5.69055pt}{48.97643pt}{3.45201pt}{51.21497pt}{0.69055pt}{51.21497pt}\pgfsys@lineto{0.0pt}{51.21497pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }$ (24)

This calculation can be recognized within Temperley-Lieb algebra [76]. For
simplicity, we define two symbol $\mathcal{A}$ and $\mathcal{B}$ as

$\mathcal{A}=\raisebox{-0.45pt}{ \leavevmode\hbox to7.29pt{\vbox
to52.81pt{\pgfpicture\makeatletter\hbox{\hskip 0.8pt\lower-0.8pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.69055pt}{0.0pt}\pgfsys@curveto{3.45201pt}{0.0pt}{5.69055pt}{2.23854pt}{5.69055pt}{5.0pt}\pgfsys@lineto{5.69055pt}{17.7622pt}\pgfsys@curveto{5.69055pt}{20.52367pt}{3.45201pt}{22.7622pt}{0.69055pt}{22.7622pt}\pgfsys@lineto{0.0pt}{22.7622pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{0.69055pt}{28.45276pt}\pgfsys@curveto{3.45201pt}{28.45276pt}{5.69055pt}{30.6913pt}{5.69055pt}{33.45276pt}\pgfsys@lineto{5.69055pt}{46.21497pt}\pgfsys@curveto{5.69055pt}{48.97643pt}{3.45201pt}{51.21497pt}{0.69055pt}{51.21497pt}\pgfsys@lineto{0.0pt}{51.21497pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} \leavevmode\hbox
to17.47pt{\vbox
to51.61pt{\pgfpicture\makeatletter\hbox{\hskip-11.1811pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{}{{}}{} {}{} {}{} {}{}
{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{17.07166pt}{5.0pt}\pgfsys@lineto{17.07166pt}{17.7622pt}\pgfsys@curveto{17.07166pt}{20.52367pt}{19.3102pt}{22.7622pt}{22.07166pt}{22.7622pt}\pgfsys@lineto{23.45276pt}{22.7622pt}\pgfsys@curveto{26.21422pt}{22.7622pt}{28.45276pt}{20.52367pt}{28.45276pt}{17.7622pt}\pgfsys@lineto{28.45276pt}{5.0pt}\pgfsys@curveto{28.45276pt}{2.23854pt}{26.21422pt}{0.0pt}{23.45276pt}{0.0pt}\pgfsys@lineto{22.07166pt}{0.0pt}\pgfsys@curveto{19.3102pt}{0.0pt}{17.07166pt}{2.23854pt}{17.07166pt}{5.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{11.3811pt}{5.69055pt}\pgfsys@moveto{11.3811pt}{5.69055pt}\pgfsys@lineto{11.3811pt}{17.07166pt}\pgfsys@lineto{22.7622pt}{17.07166pt}\pgfsys@lineto{22.7622pt}{5.69055pt}\pgfsys@closepath\pgfsys@moveto{22.7622pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{{}}{} {}{} {}{} {}{}
{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{17.07166pt}{33.45276pt}\pgfsys@lineto{17.07166pt}{46.21497pt}\pgfsys@curveto{17.07166pt}{48.97643pt}{19.3102pt}{51.21497pt}{22.07166pt}{51.21497pt}\pgfsys@lineto{23.45276pt}{51.21497pt}\pgfsys@curveto{26.21422pt}{51.21497pt}{28.45276pt}{48.97643pt}{28.45276pt}{46.21497pt}\pgfsys@lineto{28.45276pt}{33.45276pt}\pgfsys@curveto{28.45276pt}{30.6913pt}{26.21422pt}{28.45276pt}{23.45276pt}{28.45276pt}\pgfsys@lineto{22.07166pt}{28.45276pt}\pgfsys@curveto{19.3102pt}{28.45276pt}{17.07166pt}{30.6913pt}{17.07166pt}{33.45276pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{11.3811pt}{34.14331pt}\pgfsys@moveto{11.3811pt}{34.14331pt}\pgfsys@lineto{11.3811pt}{45.52441pt}\pgfsys@lineto{22.7622pt}{45.52441pt}\pgfsys@lineto{22.7622pt}{34.14331pt}\pgfsys@closepath\pgfsys@moveto{22.7622pt}{45.52441pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} \leavevmode\hbox
to7.29pt{\vbox
to52.81pt{\pgfpicture\makeatletter\hbox{\hskip-33.3433pt\lower-0.8pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{39.83386pt}{0.0pt}\pgfsys@lineto{39.14331pt}{0.0pt}\pgfsys@curveto{36.38185pt}{0.0pt}{34.14331pt}{2.23854pt}{34.14331pt}{5.0pt}\pgfsys@lineto{34.14331pt}{17.7622pt}\pgfsys@curveto{34.14331pt}{20.52367pt}{36.38185pt}{22.7622pt}{39.14331pt}{22.7622pt}\pgfsys@lineto{39.83386pt}{22.7622pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{39.83386pt}{28.45276pt}\pgfsys@lineto{39.14331pt}{28.45276pt}\pgfsys@curveto{36.38185pt}{28.45276pt}{34.14331pt}{30.6913pt}{34.14331pt}{33.45276pt}\pgfsys@lineto{34.14331pt}{46.21497pt}\pgfsys@curveto{34.14331pt}{48.97643pt}{36.38185pt}{51.21497pt}{39.14331pt}{51.21497pt}\pgfsys@lineto{39.83386pt}{51.21497pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
},\quad\mathcal{B}=\raisebox{-0.45pt}{ \leavevmode\hbox to7.29pt{\vbox
to52.81pt{\pgfpicture\makeatletter\hbox{\hskip 0.8pt\lower-0.8pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.69055pt}{0.0pt}\pgfsys@curveto{3.45201pt}{0.0pt}{5.69055pt}{2.23854pt}{5.69055pt}{5.0pt}\pgfsys@lineto{5.69055pt}{46.21497pt}\pgfsys@curveto{5.69055pt}{48.97643pt}{3.45201pt}{51.21497pt}{0.69055pt}{51.21497pt}\pgfsys@lineto{0.0pt}{51.21497pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{22.7622pt}\pgfsys@lineto{1.84528pt}{22.7622pt}\pgfsys@curveto{2.39757pt}{22.7622pt}{2.84528pt}{23.20992pt}{2.84528pt}{23.7622pt}\pgfsys@lineto{2.84528pt}{27.45276pt}\pgfsys@curveto{2.84528pt}{28.00505pt}{2.39757pt}{28.45276pt}{1.84528pt}{28.45276pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} \leavevmode\hbox
to17.47pt{\vbox
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{{}}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{5.69055pt}{22.7622pt}\pgfsys@lineto{3.84528pt}{22.7622pt}\pgfsys@curveto{3.29298pt}{22.7622pt}{2.84528pt}{23.20992pt}{2.84528pt}{23.7622pt}\pgfsys@lineto{2.84528pt}{27.45276pt}\pgfsys@curveto{2.84528pt}{28.00505pt}{3.29298pt}{28.45276pt}{3.84528pt}{28.45276pt}\pgfsys@lineto{5.69055pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }.$ (25)

which are not commutable to each other. Subsequently, the random average of
the square of the norm can be expressed as the sum of all arrangement patterns
of the non-commutative symbols $\mathcal{A}$ and $\mathcal{B}$.

$\displaystyle\overline{\braket{\psi_{\beta}}{\psi_{\beta}}^{2}}=\mathcal{A}\mathcal{A}\cdots\mathcal{A}+\mathcal{B}\mathcal{A}\cdots\mathcal{A}+\mathcal{A}\mathcal{B}\cdots\mathcal{A}+\cdots$
(26)

Since $\mathcal{A}^{N}$ is identical to
$\left(\overline{\braket{\psi_{\beta}}{\psi_{\beta}}}\right)^{2}$, the
variance of the norm is the sum of arrangements that include at least one
$\mathcal{B}$.

$\displaystyle\mathrm{Var}\left(\braket{\psi_{\beta}}{\psi_{\beta}}\right)=\mathcal{B}\mathcal{A}\cdots\mathcal{A}+\mathcal{A}\mathcal{B}\cdots\mathcal{A}+\cdots+\mathcal{B}\mathcal{B}\cdots\mathcal{B}$
(27)