# Coarse-grained quantum state tomography with optimal POVM construction

Donghun Jung  Center for Quantum Information, Korea Institute of Science and
Technology, Seoul, 02792, Republic of Korea  Department of Physics,
Sungkyunkwan University, Suwon 16419, Republic of Korea  Young-Wook Cho
A*STAR Quantum Innovation Centre (Q.Inc), Institute of Materials Research and
Engineering (IMRE), Agency for Science, Technology and Research (A*STAR),
138634 Singapore  Yosep Kim  yosep9201 gmail.com Department of Physics, Korea
University, Seoul, 02841, Republic of Korea  Junghyun Lee<EMAIL_ADDRESS>Center for Quantum Information, Korea Institute of Science and Technology,
Seoul, 02792, Republic of Korea

###### Abstract

Constructing an integrated large-scale qubit system of realistic size requires
addressing the challenge of physical crowding among qubits. This constraint
poses an issue of coarse-grained (CG) measurement, wherein information from
the multi-qubit system is collectively gathered. In this work, we introduce a
novel approach to reconstruct the target density matrix from a comprehensive
set of Positive Operator-Valued Measures (POVM) using a Parameterized Quantum
Circuit (PQC) under the constraint of CG measurement. We improve the
robustness and stability of CG quantum state tomography (QST) by optimizing
the POVM set to achieve a generalized symmetric informationally complete
(GSIC) POVM through maximization of the von Neumann entropy. This optimized
construction of CG-POVMs is scalable to an $N$-qubit system. We further
discuss a more efficient construction of $N$-qubit CG-QST without exponential
increases in two-qubit gates or circuit depth per measurement. Our scheme
offers a viable pathway towards a detector-efficient large-scale solid-state
embedded qubit platform by reconstructing crucial quantum information from
collective measurements.

##  .1 Introduction

Quantum technology relies on precise measurement and characterization of
quantum systems, employing methodologies like quantum state tomography
(QST)[1, 2], quantum process tomography (QPT)[3, 4, 5, 6, 7], and randomized
benchmarking[8, 9]. Despite its perceived inefficiency in large-scale qubit
systems, QST still plays a crucial role in validating quantum devices,
evaluating the performance of specific quantum gates, and pinpointing errors.
Its widespread adoption reflects its practical utility as a common technique
for inferring the state of a quantum system through strategic measurements and
analysis[10, 11, 12, 13, 14].

Figure 1:  (a) Schematic illustration of a collective CG measurement for a
multi-qubit system. A single detector collectively captures the total qubit
state information, distinguishing only binary outcomes, indicative of the
presence or absence of a signal. The qubits are closely located, so that two-
qubit interaction is present. (b) A brick-wall-shaped parameterized quantum
circuit used to establish CG-POVM operators for an $N$-qubit system. The
circuit comprises layers of blue-colored gate blocks, consisting of
parametrized single and two-qubit gates.

The QST necessitates the preparation of an ensemble of identical quantum
systems, each subjected to an informationally complete set of positive
operator valued measures (IC-POVMs). However, the probabilistic outcomes of
individual POVM measurement inherently yield limited information, often
overshadowed by statistical errors originating from the measuring devices[15,
16]. Therefore, there has been a great interest to find the most noise-
resilient and efficient set of measurements for QST among various choice of
IC-POVM sets under specifically given conditions. In the class of projective
measurements, mutually unbiased bases (MUB)[17, 18] provide the most efficient
information extraction per measurement, facilitating efficient and accurate
QST for certain classes of states. However, MUBs for QST are often
overcomplete. On the other hand, in the class of generalized POVM
measurements, symmetric, informationally complete (SIC) POVMs[19, 20] are
known as optimal QST measurements with minimal set of POVMs, providing
efficient and noise resilient QST especially under noisy environments.

While SIC-POVM based QST is known to be optimal in general cases, the
existence of SIC POVMs for all dimensions is still an open question.
Furthermore, in practice, stringent conditions for SIC-POVMs pose difficulties
to implement SIC-POVM based QST. For instance, every SIC-POVM element must be
rank one, which may not be straightforward to be implemented with noisy
measurement apparatus. Recently, a generalized concept of SIC POVM has been
introduced by relaxing the rank condition. Notably, the general SIC-POVMs
(GSIC-POVMs)[21, 22] are shown to exist in all dimensions. And various
properties for GSIC-POVMs have been studied, including uncertainty
relations[23, 24, 25], entanglement detections[26, 27, 28], and the quantum
state tomography[22]. Especially, GSIC POVMs are known to be optimal for QST
under the given arbitrary rank of POVM elements.

Although optimal QST strategies for different measurement constraints have
been extensively studied, a common assumption is that the physical number of
detectors is at least great or equal to the number of qubits to be
characterized assuming the ability of individual measurements on each qubit. A
totally different scenario should be considered in a highly integrated large-
scale qubit system, where the individual state detection is not available
anymore. In this paper, we address the problem by introducing a coarse-grained
(CG) measurement. In our scenario, we assume that only a single physical
detector and adjacent qubit interactions are available as shown in Figure 1a.
Under this stringent but realistic CG measurement constraint, we propose a
strategic method to characterize a multi-qubit quantum system by optimizing a
parameterized quantum circuit. Interestingly, we find that our strategy
naturally leads us to the implementation of GSIC POVMs, providing noise-
resilient and efficient QST. Note that approaches to find GSIC POVMs were
theoretically introduced, but the implementation of GSIC POVMs under specific
conditions was not considered yet.

The main part of this letter is organized as follows: First, we propose a
scheme for constructing CG-POVM operators with maximized QST fidelity, aiming
to approach a GSIC POVM set[21, 22]. We adopt von Neumann entropy as a metric
to maximize the unbiasedness of the set of CG-POVM bases, thereby enhancing
resilience against statistical noise during the density matrix
reconstruction[13, 21, 22]. Next, we present a circuit resource-efficient
method for constructing CG-POVM operators, utilizing a two-qubit system as a
toy model, while relaxing the GSIC condition. The strategy to minimize the use
of two-qubit gates and the circuit depth in the context of CG measurement-
based QST is discussed. This is pertinent for the practical implementation of
our scheme, as the fidelity of CG-POVM operators is particularly sensitive to
the number of two-qubit gates as well as the circuit depth employed. Finally,
to access the scalability, we extend our optimal CG-POVM construction
methodology to an $N$-qubit system with reduced circuit layer and depth. By
employing a brick-wall-shaped parameterized quantum circuit ansatz with gate
blocks (Fig.1b), we demonstrate the feasibility of $N$-qubit CG-QST without
encountering an exponential rise in the number of two-qubit gates or circuit
depth per CG-POVM. Here, we utilize the parameterized Ising coupling gate as a
two-qubit gate, where only nearest-neighbor qubits can be coupled[29, 30].

##  .2 Coarse-Grained Measurement Background

In the general QST protocol designed for a multi-qubit system, the assumption
is based on the individual readout of each qubit state, projected onto a set
of POVM bases formulated from the POVM operators. This involves varying the
measurement projection basis of each qubit, often using the Pauli basis, and
measure their correlations. The expectation values of these POVM bases are
then extracted to reconstruct the system’s complete density matrix. However,
in the context of coarse-grained (CG) measurements, a single detector captures
the aggregate information of the entire system, distinguishing only between
two states, such as the presence or absence of signals.(Fig.1a). CG
measurement becomes prevalent in scenarios involving multi-qubits embedded
within a solid-state system, where the proximity between qubits is essential
for constructing efficient two-qubit interactions[31, 32]. Individual readout
of each qubit presents significant challenges, attributed to constraints like
the optical diffraction limit[31] or the collective nature of electrical
conductance measurements[33]. Even with super-resolution imaging
techniques[34, 35, 36], scalability remains limited due to system
complexities.

In CG measurement, information about the entire multi-qubit system is
collectively detected and the detailed information about their correlations
for each qubit’s distinct bases is now embedded in the total event count of
correlation-independent projection bases. The mathematical representation of
the CG measurement operator $M_{CG}$ for the entire $N$-qubit system is as
follows:

$M_{CG}=\frac{1}{N}\left(\sum_{i=1}^{N}I_{2}^{\otimes
i-1}\otimes\ket{0}\bra{0}\otimes I_{2}^{\otimes N-i}\right)$ (1)

Here, $I_{2}$ denotes the $2\times 2$ identity operator. The CG measurement
process can be represented by a CG-POVM comprising two positive semi-definite
observable operators $\left\\{M_{CG},I_{2^{N}}-M_{CG}\right\\}$. Due to the
dilution of correlative information under CG measurement, it becomes
impossible to contruct an informationally complete set of CG-POVM operators
generated solely by using single-qubit gates. Meanwhile, additional
construction of CG-POVM operators $\Omega^{k}=G^{k\dagger}M_{CG}G^{k}$,
forming the set of CG-POVM bases $\\{\Omega^{k},I_{2^{N}}-\Omega^{k}\\}$, can
be achieved by implementing a unitary gates $G^{k}$, involving two-qubit
gates. This approach effectively facilitates the reconstruction of correlated
measurements, even under the condition of incoherent collective summation of
multi-qubit state information. For a specific example of CG-POVM for two-qubit
case for QST, please see Supplementary Information.

##  .3 Construction of CG-POVM towards GSIC

We approach the problem of CG-QST as solving a linear equation problem.
Consider a simplified model of a two-qubit system. The density matrix $\rho$
can be projected onto a 16-dimensional vector $\Theta$ by selecting a complete
set of orthonormal basis in the corresponding Hilbert-Liouville space
$\\{\Gamma_{i}\\}_{i=1}^{16}$, satisfying
$\Tr(\Gamma_{i}\Gamma_{j})=\delta_{ij}$. Without loss of generality, the basis
can be selected as
$\left\\{\Gamma\right\\}=\left\\{I_{2},\sigma_{x},\sigma_{y},\sigma_{z}\right\\}^{\otimes
2}$, where $\sigma_{x,y,z}$ are the Pauli matrices.

Therefore, $\rho$ can be expressed as a linear combination of $\Theta_{i}$
with $\Theta=(\theta_{1},\theta_{2},\cdots,\theta_{16})$.

$\rho=\sum_{i=1}^{16}\theta_{i}\Gamma_{i}$ (2)

For a given set of CG-POVM bases
$\\{\Omega^{k},I_{4}-\Omega^{k}\\}_{k=1}^{n}$, each CG-POVM operator
$\Omega^{k}$ can also be expressed as a 16-dimensional vector under the same
bases $\\{\Gamma_{i}\\}_{i=1}^{16}$:

$\Omega^{k}=\sum_{i=1}^{16}\omega^{k}_{i}\Gamma_{i}$ (3)

where $\omega^{k}_{i}=\Tr(\Omega^{k}\Gamma_{i})$. When measurements are
performed using the CG-POVM bases, the expectation values are given as:

$\hat{p}_{k}=\Tr(\rho\Omega^{k})=\sum_{i=1}^{16}\theta_{i}\omega^{k}_{i}$ (4)

leading to the linear equation,

$\begin{pmatrix}\omega^{1}_{1}&\omega^{1}_{2}&\cdots&\omega^{1}_{16}\\\
\omega^{2}_{1}&\omega^{2}_{2}&\cdots&\omega^{2}_{16}\\\
\vdots&\vdots&\ddots&\vdots\\\
\omega^{n}_{1}&\omega^{n}_{2}&\cdots&\omega^{n}_{16}\\\
\end{pmatrix}\begin{pmatrix}\theta_{1}\\\ \theta_{2}\\\ \vdots\\\
\theta_{16}\\\ \end{pmatrix}=\begin{pmatrix}\hat{p}_{0}\\\ \hat{p}_{1}\\\
\vdots\\\ \hat{p}_{n}\end{pmatrix}$ (5)

or simply, $X\Theta=\hat{P}$. To solve above linear equation for two-qubit
system, a minimum of 16 CG-POVM bases should be selected to determine all
elements of $\Theta_{i}$, that is $n=16$. The chosen set of CG-POVM bases
$\\{\Omega^{k},I_{4}-\Omega^{k}\\}$ must ensure $\rank(X)=16$, corresponding
to the full rank of the two-qubit $X$ matrix, enabling the solution of the
linear equation. By utilizing the Maximum Likelihood Estimation (MLE)
technique to meet the physical constraints for the density matrix, including
hermiticity, positivity, and normalization, we can determine the optimal
solution for $\Theta$ that satisfies these conditions[37, 38].

After constructing the CG-POVM basis set satisfying the full-rank condition,
we demonstrate the optimization of the CG-POVM basis set to minimize the
impact of noise during measurement, such as the shot noise which follows a
Poisson distribution. Let $\epsilon_{k}=\hat{p}_{k}-p_{k}$ represent the
statistical error arising from the estimator $\hat{p}_{k}$, which is
proportional to $\sqrt{N_{p}}$, and $N_{p}$ is the total number of detected
photons or electrons. From Equation (4), we derive the following relation:

$\hat{p}_{k}=\sum_{i=1}^{16}\Theta_{i}\omega^{k}_{i}+\epsilon_{k}$ (6)

From Equation (5), the reconstructed density matrix now incorporates the
statistical error:

$\hat{\Theta}=X^{-1}\hat{P}=\Theta+X^{-1}\epsilon$ (7)

The magnitude of this error is contingent on $X$ and, thus, on the selection
of the set of CG-POVM bases $\\{\Omega^{k}\\}_{k=1}^{16}$. Let
$X=UDU^{\dagger}$ represent the eigen-decomposition of $X$ where
$D=\mathrm{diag}\\{e_{1},e_{2},\cdots,e_{16}\\}$ and $U$ is a unitary matrix.
The magnitude of the error is then:

$\left|\Theta_{\text{err}}\right|^{2}=\epsilon^{\dagger}(X^{\dagger}X)^{-1}\epsilon=\epsilon^{\dagger}UD^{-2}U^{\dagger}\epsilon.$
(8)

where $D^{-1}=\mathrm{diag}\\{e_{1}^{-1},e_{2}^{-1},\cdots,e_{16}^{-1}\\}$. We
assume homogeneous error $\epsilon_{k}$ accross all measurement to pursue
symmetric and information complete measurements. This harmonic mean of the
eigenvalues of $X$ is minimized when the square of each eigenvalue of $X$ is
uniform.

Figure 2:  (a) Top: The CG-POVM QST process for a two-qubit system is
depicted. Bottom: An individual readout scheme using conventional POVM basis.
CG-QST and QST process is performed after the gate operation $\mathcal{O}$.
(b) The graph presents the average infidelity $1-F$ in relation to the maximum
photon count for each measurement, highlighting a discernible trend where
increasing von Neumann entropy of the POVM set correlates with a decrease in
infidelity. This indicates that higher entropy POVM sets tend to be more
effective in QST, demonstrating a more robust performance against noise and
improved state reconstruction accuracy. (c) Standard deviation of infidelity
$1-F$ across target density matrices for various CG-POVM sets. The trend
suggests that POVM sets with higher entropy tend to yield more consistent
reconstruction results, reflecting their unbiased nature across a range of
quantum states. (d) Comparison of the standard deviation of infidelity between
the conventional POVM and the optimized CG-POVM sets. Despite the higher
entropy of the conventional set, the optimized CG-POVM set closely approaches
the performance of the GSIC POVM bases, potentially offering better resilience
to noise and an unbiased estimation of quantum states.

Our optimization approach involves initially deriving the Gram matrix from the
POVM basis set, followed by determining the eigenvalues, and subsequently
computing the von Neumann entropy based on the extracted eigenvalues. This
strategy aims to mitigate the impact of statistical error on density matrix
reconstruction by minimizing $\Tr\left(D^{-1}\right)^{2}$ (in Eq. (8)), as
quantified through the Gram matrix representation. The Gram matrix $\Pi$ is
defined as $\Pi=X^{\dagger}X$, where each element is the inner product of the
POVM bases:

$\Pi_{ij}=\Tr\Omega^{i}\Omega^{j}$ (9)

The eigenvalues of the Gram matrix represent the influence of each measurement
on a certain basis[39, 40]. Next, the von Neumann entropy of the Gram matrix
is introduced as a measure of its eigenvalue uniformity:

$S=-\Tr(\Pi^{\prime}\ln\Pi^{\prime})=-\sum_{i=1}^{16}e_{i}^{\prime}\ln
e_{i}^{\prime}$ (10)

where $\Pi^{\prime}$ is a normalized Gram matrix, $\Pi^{\prime}=\Pi/\Tr\Pi$ so
that $\sum_{i=1}^{16}e_{i}^{\prime}=1$. By computing the von Neumann entropy
of the Gram matrix, we gauge the impartiality of the constructed CG-POVM basis
set. Opting for a CG-POVM basis set with greater von Neumann entropy
facilitates a more uniform measurement over the entire Hilbert space,
contributing to a decrease in the statistical error of the estimated density
matrix. When the entropy reaches its maximum, the vector $\omega_{i}^{k}$ is
uniformly spread throughout the Hilbert space, ensuring that the basis
elements of the CG-POVM are uniformly correlated. Consequently, the CG-POVM
basis set becomes close to the status of a GSIC POVM set (Supplementary
Information). Please note that our optimal CG-POVM construction argument can
be generalized to an N-qubit system with dimension $4^{N}$.

##  .4 Optimal CG-QST Simulation Results

Figure 2a illustrates the schematic diagram of the CG-QST process (upper) and
the conventional individual readout QST process (lower) for two-qubit system.
Applying proposed method to determine the optimal CG-POVM basis set, we
compare the infidelity of QST using our approach with the conventional
individual readout scheme. We conduct 100 Monte Carlo (MC) simulations of the
QST process using the CG-POVM basis set for each 36 symmetrically chosen
target density matrices of a two-qubit system where the target states are
eigenstates of Pauli matrices
$\left\\{\outerproduct{\psi_{i}}{\psi_{i}}\otimes\outerproduct{\psi_{j}}{\psi_{j}}\right\\}$
and
$\ket{\psi_{i}}\in\left\\{\ket{0},\ket{1},\ket{+},\ket{-},\ket{i},\ket{-i}\right\\}$,
incorporating randomly generated Poisson noise. The same number of MC
simulations were conducted using the conventional Pauli matrix-based POVM
basis set with selective readout. We used Pauli measurement basis
$\\{\outerproduct{\psi_{i}}{\psi_{i}}\otimes\outerproduct{\psi_{j}}{\psi_{j}}\\}$,
where $\ket{\psi_{i}}$ is one of the following states
$\left\\{\ket{0},\ket{1},\ket{+},\ket{i}\right\\}$. The infidelity $1-F$ in
QST is expressed as $1-\Tr\sqrt{\sqrt{\rho}\hat{\rho}\sqrt{\rho}}$, with
$\rho$ representing the target density matrix and $\hat{\rho}$ denoting the
reconstructed density matrix. The mean infidelity value represents the each
method’s robustness against noise, while the standard deviation of the
infidelity gauges the method’s independence from the target state.

As shown in Figure 2b, in general, the POVM basis set with higher von Neumann
entropy exhibits superior performance in terms of mean infidelity, indicating
greater robustness against statistical errors. In light of this, the
conventional measurement operator set, having a von Neumann entropy of 2.17,
still outperforms the CG-POVM basis set, which achieves a maximum entropy of
1.54. Meanwhile, the difference in infidelity is sufficiently close,
suggesting that a comparable order of infidelity magnitude could be attained
by just increasing the number of measurement shots. Furthermore, the trend
within CG-POVM basis sets with different entropy values suggests that higher
entropy correspond to lower standard deviations of infidelities. This
indicates that a more uniform CG-POVM basis set leads to more unbiased
reconstructions across different target density matrices (Fig.2c). One
important remark is that the optimal CG-POVM basis set, achieved by maximizing
the von Neumann entropy to 1.54, coincides the GSIC basis under the coarse-
grained measurement condition (Supplementary Information). This highlights the
effectiveness of our scheme in constructing a nearly unbiased operator set.
For instance, although the conventional Pauli matrix-based POVM basis set,
with higher entropy, exhibits greater resilience to Poisson noise compared to
the CG-POVM basis set, it lacks optimization for proximity to its GSIC.
Consequently, the unbiased nature of the optimized CG-POVM basis set might
outperform the conventional POVM basis set (Fig.2d).

Figure 3:  (a) Gate-efficient construction strategy for CG-POVM basis in a
two-qubit system. The figure illustrates a two-tiered approach where gate
blocks $G^{k}$ are layered: the first seven $\Omega^{1}$ to $\Omega^{7}$ basis
consist only of single-qubit gates (class-I), while the subsequent nine
$\Omega^{8}$ to $\Omega^{16}$ involve a paramatrized Ising coupling gate
$e^{-i\frac{\theta}{2}\sigma_{z}\otimes\sigma_{z}}$ (ZZ) encapsulated by
single-qubit gates (class-II). (b) general multi-qubit quantum circuit with a
brick-wall gate block structure for CG-POVM. The circuit depth contraction
scheme enables full Pauli basis measurement capability with complexity scaling
that is non-exponential.

##  .5 Resource-Efficient CG-POVM Construction

We propose a resource-efficient strategy for constructing a set of CG-POVM
bases, aiming to minimize the utilization of two-qubit gates and the circuit
depth.

First, we compose each gate block $G^{k}$ with two different classes: a
class-I consists solely of single-qubit gates, and class-II contains one Ising
coupling gate encapsulated by single-qubit gates (Fig.3a). For the example of
two qubit system, the resultant unitary gate $G^{k}$ is then in
$\mathbb{SU}(2)\otimes\mathbb{SU}(2)$ for class-I and $\mathbb{SU}(4)$ for
class-II. Then, we initiate the construction of a set of CG-POVM bases by
layering $G^{k}$ exclusively with class-I, sequentially transitioning each
$G^{k}$ block from class-I to II until meeting the full-rank condition.
Although the GSIC condition is not satisfied now, this strategy guarantees the
full-rank condition of the $X$ matrix with suitable parameters for the quantum
circuit ansatz, while minimizing the number of required two-qubit gates.
Minimizing the number of two-qubit gates is crucial, as they tend to introduce
more noise compared to single-qubit gates, affecting the overall fidelity of
generating CG-POVM bases. For the example of two-qubit system, our strategy
yields 7 CG-POVM bases, $1\leq k\leq 7$, without utilizing any two-qubit gate
and the remaining 9 CG-POVM bases, $8\leq k\leq 16$, with the inclusion of one
two-qubit gate, thus constructing fully ranked CG-POVM bases (Fig.3a). Minimum
required 9 class-II $G^{k}$ gate blocks corresponds to the number of joint
Pauli bases in the conventional selective readout case. Following our strategy
of minimizing the number of two-qubit gates, we can further optimize CG-POVM
basis set by maximing the von Neumann entropy of the Gram matrix, using only 9
class-II $G^{k}$ gate blocks out of 16 bases. Under these conditions, the
highest entropy achieved is 1.516, slightly lower than the maximum entropy
achievable with CG-POVM GSIC bases constructed using all 16 bases with class-
II $G^{k}$ gate blocks. Nevertheless, comparable QST performance is expected
for the gate efficient CG-POVM basis set.

Next, for the $N>2$ multi-qubit system, we contract the number of layers from
the large circuit layer until the CG-POVM operator can no longer measure the
entire set of Pauli bases. In general, gate blocks in quantum circuit can be
arranged in a brick-wall structure with all gate types of class-II,
encompassing two-qubit gates (Fig.3b). At first, a sufficiently large number
of CG-POVM operators are generated for a given circuit ansatz and the number
of circuit layers, employing random variables for the circuit parameters, to
ensure that the vector $\omega_{i}^{k}$ is distributed throughout the entire
Hilbert space. We then sequentially contract the circuit layer, verifying if
the CG-POVM operator can measure a specific Pauli basis by computing
$\omega_{i}^{k}$ and checking for nonzero values in $\omega_{i}^{k}$. If
$\omega_{i}^{k}=0$ for any $i$ across all $k$, it indicates an inability to
extract information for that particular Pauli basis, resulting in the failure
of QST. This confirmation approach is more feasible for contracting an N-qubit
system compared to directly determining the rank of the CG-POVM set, which
requires significantly more computational intensity. Thus, our approach
enables us to determine the minimal number of layers required for the CG-POVM
within this quantum circuit framework to measure all Pauli bases.

##  .6 Multi qubit CG-QST Simulation Results

Figure 4:  (a) The relation between the minimum circuit layers and the number
of qubits, showing non-exponential scaling for full Pauli basis measurement.
(b) The relationship between the number of two-qubit gates and the rank of the
$X$ matrix, indicating efficient scaling of CG-POVM respect to total qubit
number N. (c) Minimum circuit depth as a function of qubit number, affirming
the scalability of CG-POVM construction without exponential growth in depth.

Utilizing the aforementioned resource-efficient strategy, we demonstrate that
our CG-POVM construction scheme for QST can be extended to an $N$-qubit system
without exponential increases in the number of circuit layers, the number of
two-qubit gates per measurement, and the circuit depth. The number of circuit
layers corresponds to the quantity of gate block columns in the quantum
circuit ansatz, while the circuit depth is defined as the maximum path length
within the circuit. Specifically, the $G^{k}$ class-I gate block has a circuit
depth of 1, whereas class-II has a circuit depth of 3.

First, Figure 4a illustrates the minimum number of layers required as the
number of qubits increases, using the circuit layer contraction strategy
introduced in the previous section. The result indicates that the minimum
number of layers does not increase exponentially with the number of qubits.
Next, it is important to note that the specific Pauli bases can be constructed
with the CG-POVM basis set without necessitating the inclusion of all gate
block types with class-II. The rank calculation is conducted after determining
the minimum circuit layer for various numbers of qubits. Figure 4b depicts the
maximum achievable rank of the $X$ matrix by varying the number of two-qubit
gates per CG-POVM operator. The full-rank condition of the $X$ matrix is
satisfied when the number of two-qubit gates per measurement is $2N-3$. Thus,
we confirm that the number of two-qubit gates per measurement also does not
grow exponentially with the number of qubits. Finally, as shown in Figure 4c,
the minimum required circuit depth is computed by varying the number of
qubits. For a fixed number of qubits, we utilize a minimized number of circuit
layers with the minimum number of $2N-3$ two-qubit gates per CG-POVM operator,
constructing all possible quantum circuits for the minimum depth calculation.
The circuit depth minimization process follows the same strategy, validating
whether the CG-POVM operator can measure the entire Pauli basis, as used in
circuit layer minimization. We also confirm that the minimum circuit depth
required does not increase exponentially with the number of qubits. After
constructing CG-POVM basis set with the most resource-efficient circuit, we
can further optimize the set to minimize the effect of noise during
measurement by increasing the von Neumann entropy of the Gram matrix as
illustrated in Figure 2.

##  .7 Conclusion

In summary, the CG-POVM basis set, characterized by a $4^{N}$-rank $X$ matrix,
supports QST for an $N$-qubit system, yielding $4^{N}$ measurement outcomes.
The MLE technique is employed to determine the optimal solution for the target
density matrix $\rho$ while adhering to physical constraints. Additionally,
optimization of the CG-POVM basis set, aimed at minimizing the impact of noise
during measurement, involves maximizing the von Neumann entropy of the Gram
matrix. With an increasing number of qubits, the growth of two-qubit gates per
measurement and circuit depth avoids exponential escalation, making CG-QST an
efficient approach for N-qubit systems. We anticipate the applicability of our
strategy not only in constructing CG-QST but also in designing target set of
POVM bases with CG constraints, leveraging the universality of parameterized
quantum circuit ansatz methodology[41, 42, 43].

Designing an optimal CG-QST scheme can be directly applied to solid-state
embedded defect qubit systems such as the nitrogen-vacancy (NV) centers or the
silicon-vacancy (SiV) centers in diamond[44, 45]. One of the challenges in
achieving a scalable electronic spin defect-based qubit system is the lack of
individual readout capability. The proposed CG-QST technique offers a
promising solution enabled by the individual control of each spin defect
through different quantization axes[31, 32] or the same quantization axes with
frequency encoding techniques[46] and engineered two-qubit gates via dipolar
interactions[32]. Extending beyond solid-state qubits, the CG-QST method
presents an opportunity to create a more resource-efficient multi-qubit
platform by reducing the number of detectors required compared to the total
number of qubits[47]. In our specific implementation, the CG-POVM basis set
necessitated only one detector for an $N$-qubit state reconstruction,
distinguishing only two states, such as signal-in-state and no-signal-state.
In the general scenario, the number of detectors can exceed one, and the count
of distinguishable states can surpass two as well. This leads to a higher von
Neumann entropy of the Gram matrix, thereby enhancing the performance of QST.
Hence, a trade-off emerges between the quantity of detectors and QST
performance. Nevertheless, a comparable range of fidelity can be always
attained through an increase in the measurement shot number. The resource-
efficient optimal CG-POVM construction presents an alternative avenue,
offering minimized detector usage while maximizing measurement performance for
multi-qubit platforms.

##  .8 Acknowledgement

The authors would like to thank Yong Siah Teo, and Conlon Lorcan for helpful
discussions. This material is based upon work supported by, or in part by, the
KIST institutional program (Project. No. 2E32941) and the National Research
Foundation of Korea STEAM program under Grant No. 2N73600, and A*STAR (Project
No. C230917009).

##  .9 Contributions

J.L., Y.C. and Y.K. conceived the idea, J.L. and Y.K. co-supervised the
project. D.J. developed and performed numerical simulations. All authors
analyzed and discussed the data. All authors participated in writing the
manuscript.

##  .10 References

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# Contribution of internal degree of freedom of soft molecules to Soret effect

Takeaki Araki and Natsumi Chikakiyo  Department of Physics, Kyoto University,
Kyoto 606-8502, Japan

###### Abstract

We studied the Soret effect in binary dimer-monomer mixtures using non-
equilibrium molecular dynamics simulations and investigated the pure
contribution of the internal degree of freedom of flexible molecules to the
Soret effect. We observed that the thermal diffusion factor tends to decrease
and change its sign as the molecules become softer. We proposed two possible
mechanisms of our observations: change of the molecule structures with the
temperature, causing bulkier molecules to migrate to the hotter region;
asymmetry of the restitution between rigid and flexible molecules, due to
which flexible molecules show larger restitution when placed at the hotter
region.

##  I introduction

When a fluid mixture is subjected to a temperature gradient, concentration
gradients build up. This phenomenon is the Soret effect or thermal diffusion
[1, 2, 3, 4], which is widely observed in mixtures of small molecules [5, 6],
polymers [7, 8, 9, 10, 11, 12], colloids [13, 10], and biomolecules [14, 15].
Since the 19 th century, many experimental, theoretical and numerical studies
have been reported [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. However, its
physical mechanisms are still not understood completely.

The Soret effect is caused by some factors; its origin is often considered the
equilibrium thermodynamics [16, 22]. If the chemical potential of each species
depends on the temperature, the concentration gradient is induced under the
temperature gradient to homogenize this potential. The isotope effect, in
which the mass difference causes the concentration gradient, is difficult to
explain based on the equilibrium thermodynamics [17, 18, 23, 24, 25, 26]. The
heavier molecules tend to migrate to the cold side, while the lighter ones
move to the hot side. It is also known that differences in diameter, moment of
inertia, interaction between the solute and the solvent, and/or those among
the solutes contribute significantly in the Soret effect [2].

When considering the Soret effect, the molecules are usually treated as rigid
objects, while the internal degrees of freedom of the molecules are ignored.
If flexible, the molecules are thermalized and adapt their structures to the
local environment (the temperature, the pressure, and the concentration).
These changes may influence the Soret effect. Polymers are examples of
molecules that have large internal degrees of freedom [7, 8, 9, 10, 11, 12].
If a polymer is much longer than its persistent length, it behaves as a
flexible chain; it behaves as a rigid rod if it is shorter than the persistent
length. It was reported that the Soret coefficient of the polymer solutions
changes with the chain length when it is short. However, when it is
sufficiently long, the Soret coefficient is saturated to a value that is
independent of the chain length. For the polymers, the mass and the moment of
inertia change with the chain length. Although some theoretical studies on the
Soret effect of the polymer systems have been reported, the pure contribution
of the flexibility of the molecules to the Soret effect is still unclear. The
aim of this work is to investigate the roles of the internal degrees of
freedom in the Soret effect by employing the simplest molecular model.

##  II Molecular dynamics simulation

We perform molecular dynamics simulations of mixtures of two molecular species
in three dimensional rectangular boxes ($V=L_{x}L_{y}L_{z}$). One species is
the simplest molecule, which consists of a single spherical particle; the
other is a dimer, which consists of two identical spherical particles. The
particles in the monomers and dimers interact mutually with the Weeks-
Chandler-Andersen (WCA) potential [27], which is given by

$\displaystyle U_{\rm
WCA}(r)=\left\\{\begin{array}[]{ll}4\epsilon\left[(\sigma/r)^{2n}-(\sigma/r)^{n}+1/4\right]&(r\leq
2^{1/n}\sigma)\\\ 0&(r>2^{1/n}\sigma)\end{array}\right.,$ (3)

where $\epsilon(=1)$ and $\sigma(=1)$ represent the strength and range of the
WCA potential. $n$ is a parameter for characterizing the hardness of the WCA
potential. If we do not mention explicitly, we set $n=6$. The two particles in
each dimer are bounded by a harmonic potential,

$\displaystyle U_{\rm sp}(r)=k(r-r_{0})^{2},$ (5)

where $k$ is the spring constant, while $r_{0}$ is the natural length of the
bond, and we set $r_{0}=2^{1/n}\sigma$. The WCA potential between the
particles in each dimer is not included. The mass of the particles is given by
$m$; thus the mass of the dimer is $2m$. The numbers of the monomer and dimer
molecules are given by $N_{\rm m}$ and $N_{\rm d}$. The particle packing
fraction is defined by $\phi=\pi\sigma^{3}N_{\rm t}/(6V)$, where $N_{\rm t}$
is the total particle number, $N_{\rm t}=N_{\rm m}+2N_{\rm d}$. The mixing
ratio of the dimer is given by $\chi=2N_{\rm d}/N_{\rm t}$.

A temperature gradient is imposed along the $x$-axis by using boundary driven
non-equilibrium molecular dynamics simulation. The thermostatting regions are
set up at the edges ($x=0$ and $x=L_{x}$) and at the center ($x=L_{x}/2$) of
the rectangular cell, while their width is $0.03L_{x}$. The temperatures in
these thermostatting regions are imposed to $T=T_{\rm h}$ at the edges and
$T=T_{\rm c}$ at the center ($T_{\rm h}>T_{\rm c}$) by means of the Langevin
thermostat. In the other bulk regions, the particle position and velocity are
updated without any thermostat. The equations of motion are solved with
velocity Verlet algorithm using LAMMPS [28], in which the time increment is
$\delta t=0.0005\sqrt{\epsilon/(m\sigma^{2})}$. We fix the cell width as
$L_{x}=80\sigma$, while $L_{y}$ and $L_{z}$ are changed to adjust the packing
fraction $\phi$. The total particle number is $N_{\rm t}=64000$, and the
packing fraction is changed from $\phi=0.037$ to $0.234$. If we do not mention
explicitly, we set the mixing ratio to $\chi=1/2$. With the SHAKE algorithm
[29], we also consider dimers, in which the particle separations are fixed to
$r=r_{0}$. To observe the pure effect of the dimers, we also study the Soret
effect in mixtures of two monomers A and B, among which the masses or the
radii differ.

##  III 3D simulation results

Figure 1(a) shows the profiles of the kinetic energies along the $x$-axis. The
temperatures in the thermostatted regions are set to $T_{\rm h}=5\epsilon$ and
$T_{\rm c}=\epsilon$. We divide the cell into 100 slabs along the $x$-axis,
and then, we calculate the local densities of the monomer particle $\rho_{\rm
m}(x)$ and the dimer particle $\rho_{\rm d}(x)$, and the kinetic energies
averaged per particle in each thin slab. Here, $\int{\rm d}x\rho_{\rm
d}(x)=2N_{\rm d}$ and $\int{\rm d}x\rho_{\rm m}(x)=N_{\rm m}$ are held. We
plot the translational kinetic energies of the monomer $K_{\rm m}^{\rm tra}$,
and the three modes (translational $K_{\rm d}^{\rm tra}$, rotational $K_{\rm
d}^{\rm rot}$, and vibrational $K_{\rm d}^{\rm vib}$) of the kinetic energies
of the dimer. In the equilibrium state, $K_{\rm m}^{\rm tra}$, $K_{\rm d}^{\rm
tra}$, $K_{\rm d}^{\rm rot}$, and $K_{\rm d}^{\rm vib}$ should agree with
$3T/2$, $3T/2$, $T$, and $T/2$, respectively. It can be seen that all the
averaged kinetic energies collapse on a master curve after appropriate
scalings (see the inset of Fig. 1(a)). The kinetic energies in the
thermostated regions are consistent with the target temperatures $T_{\rm h}$
and $T_{\rm c}$ and vary linearly with $x$ in the bulk regions. Thus, the
temperature is controlled well in our system.

Figure 1:  (a) Profiles of the local kinetic energies of the monomers ($K_{\rm
m}^{\rm tra}$) and the dimers ($K_{\rm d}^{\rm tra},\,K_{\rm d}^{\rm vib}$,
and $K_{\rm d}^{\rm rot}$) along the temperature gradient. The average packing
fraction and the spring constant are $\phi=0.073$ and
$k=400\epsilon/\sigma^{2}$, respectively. The temperatures at the
thermostatted regions are $T_{\rm h}=5\epsilon$ and $T_{\rm c}=\epsilon$. In
the inset, the scaled kinetic energies are replotted. (b) Profiles of the
local densities of the dimer $\rho_{\rm d}$ and the monomer $\rho_{\rm m}$.
The parameters are the same as those in (a).

Figure 1(b) plots the local density profiles of the dimer $\rho_{\rm d}(x)$
and the monomer $\rho_{\rm m}(x)$. The spring constant is
$k=400\epsilon/\sigma^{2}$, while the packing fraction is $\phi=0.073$. Both
densities are higher in the colder than in the hotter regions. It is also
indicated that the density of the dimers is slightly higher than that of the
monomers in the cold region. The Soret effect is induced in this mixture of
dimers and monomers.

Figure 2(a) illustrates the profiles of the concentration field of the dimer
in a dilute mixture, which is defined as $c(x)=\rho_{\rm d}(x)/(\rho_{\rm
m}(x)+\rho_{\rm d}(x))$. If the concentration is homogeneous, its value agrees
with the mixing ratio, i.e., $c(x)=\chi$. The total packing fraction and the
spring constant are set to $\phi=0.073$ and $k=400\epsilon/\sigma^{2}$,
respectively. In Fig. 2(a), we change the temperature difference $\Delta
T=T_{\rm h}-T_{\rm c}$ by fixing the average temperature to
$\langle{T}\rangle(=(T_{\rm h}+T_{\rm c})/2)=3\epsilon$. In the absence of the
temperature difference ($\Delta T=0$), the concentration is almost constant in
space (not shown here). As $\Delta T$ increases, the concentration near the
cold region increases, while that near the hot region decreases. This means
that the dimers tend to migrate to the colder side. The degree of dimer
migration is almost proportional to the temperature difference. Figure 2(b)
presents the profiles of the concentration in a dense mixture ($\phi=0.234$).
In contrast to the dilute mixture, the dimers migrate to the hotter region in
the dense mixture. The degree of dimer migration is almost proportional to the
temperature difference.

Figure 2:  (a) Profiles of the concentration of the dimers $c=\rho_{\rm
d}/(\rho_{\rm d}+\rho_{\rm m})$ in a dilute mixture of $\phi=0.073$. The
temperature difference $\Delta T=T_{\rm h}-T_{\rm c}$ is changed. (b) Profiles
of the concentration of the dimers in a dense mixture of $\phi=0.234$.

Instead of the Soret coefficient $S_{T}$, we analyze the Soret effect with the
thermal diffusion factor $\alpha_{T}=TS_{T}$. We evaluate it from the local
thermal diffusion factor,

$\displaystyle\alpha(x)=-\frac{T}{c(1-c)}\frac{\partial c/\partial x}{\partial
T/\partial x},$ (6)

and average $\alpha(x)$ over the system to get $\alpha_{T}=\int{\rm
d}x\alpha(x)/L_{x}$. A positive value of $\alpha_{T}$ implies the dimer
migrates to the cold side. Figure 3(a) shows the dependences of $\alpha_{T}$
against the packing fraction $\phi$. The spring constant in the dimers is
changed from $k=10\epsilon/\sigma^{2}$ to $500\epsilon/\sigma^{2}$.
$\alpha_{T}$ for the rigid dimer is also given in the same figure. In Fig.
3(b), we replot $\alpha_{T}$ as functions of the spring constant $k$ for
several densities.

In dilute systems, the thermal diffusion factors are positive for any $k$,
which indicates that the dimers migrate to the colder side, as illustrated in
Fig. 2(a). As the packing fraction $\phi$ increases, the thermal diffusion
factors decrease. Here the decreasing rate of $\alpha_{T}$ against $\phi$ is
larger in the mixture of smaller $k$ than in that of larger $k$. For smaller
$k$, the thermal diffusion factor changes its sign from positive to negative
with $\phi$. In contrast, the thermal diffusion factor for the rigid dimer
remains positive in the simulated range of $\phi$. The molecule becomes more
rigid, and the thermal diffusion factor is likely to converge for the rigid
molecules when the spring constant is increased. In Fig. 3(b), the thermal
diffusion factor changes its sign with the spring constant $k$ in the
intermediated mixtures ($\phi=0.146$). Our results clearly demonstrate that
the internal degrees of freedom in the molecules can influence the Soret
effect, which is usually considered to be an inter-molecular phenomenon.

Figure 3:  (a) Plots of the thermal diffusion factor $\alpha_{T}$ against the
packing fraction $\phi$. The spring constant is changed. $\alpha_{T}$ for the
mixture of the rigid dimer, and monomer is also presented. (b) Plots of
$\alpha_{T}$ with respect to the spring constant $k$. The packing fraction is
changed.

##  IV Discussion

First, we consider the mechanism of the Soret effect in terms of the isotope
effect. In Fig. 4, we plot the thermal diffusion factor in mixtures of two
monomers A and B (black circles). In these mixtures, the interactions among
the particles are given by Eq. (1), while the mass of the A monomer is twice
larger than that of the B monomer, i.e., $\sigma_{\rm AA}=\sigma_{\rm
BB}=\sigma_{\rm AB}=\sigma$, with $m_{\rm A}=2m$ and $m_{\rm B}=m$. In Fig. 4,
the positive values of the thermal diffusion factor agree with the isotope
effect; heavier particles A migrate to the colder region. In contrast to the
decreasing thermal diffusion factor in the dimer-monomer mixtures (Fig. 3(a)),
the thermal diffusion factor is increased with the packing fraction in the
monomer-monomer mixtures. Thus, it is concluded that the Soret effect in the
dimer-monomer mixtures is essentially different from the isotope effect. It is
also known that molecules of larger inertia moments migrate to the colder
region [17, 24, 23]. If the Soret effect observed in this study is due to the
inertia effect, the thermal diffusion factor would be increased by using
longer natural length of the bond interaction in the dimers (Eq. (2)).
However, we confirmed that the thermal diffusion factor is increased when the
natural length in the bond interaction is shortened (not shown here). In
particular, if we set $r_{0}=0$ in Eq. (2) and use larger $k$, the thermal
diffusion factor is close to that due to the isotope effect. Thus, we believe
that the Soret effect in our mixtures is not due to the inertia effect.

Figure 4:  The thermal diffusion factors in the mixtures of two monomers A and
B are plotted against $\phi$. black circle: mixtures of monomers with
different masses; red square: mixtures of monomers with different radii.

Next we consider the contribution of the thermodynamic chemical potential to
the Soret effect in our simulations. The linear non-equilibrium thermodynamic
theory gives the particle flux $\vec{J}_{i}$ ($i=$d or m) as [1]

$\displaystyle\vec{J}_{i}=-\rho_{i}L_{1i}\nabla\frac{\tilde{\mu}_{i}}{T}+\rho_{i}L_{2i}\nabla\frac{1}{T},$
(7)

where $L_{1i}$ and $L_{2i}$ are phenomenological kinetic coefficients, and
$\tilde{\mu}_{i}$ is an effective chemical potential. In the steady state, it
should vanish, i.e., $\vec{J}_{i}=0$. If $L_{2i}=0$, the Soret effect is
determined by the thermodynamic properties of the chemical potential against
$\chi$ and $T$. In Figs. 5(a), (b) and (c), we represent the effective
chemical potential of the dimer $\tilde{\mu}_{\rm d}$ against the mixing ratio
$\chi$, the temperature $T$, and the packing fraction $\phi$, respectively. In
Fig. 5(a), we changed the mixing ratio $\chi$ with fixing the total number
$N_{\rm t}$. The effective chemical potential is given as $\tilde{\mu}_{\rm
d}=\mu_{\rm d}-(v_{\rm d}/v_{\rm m})\mu_{\rm m}$, where $\mu_{i}$ and $v_{i}$
are the chemical potential and partial molecular volume of the $i$-component.
In Fig. 5, we assume $v_{\rm d}/v_{\rm m}=2$, for simplicity. The chemical
potentials $\mu_{\rm d}$ and $\mu_{\rm m}$ are are obtained through the Widom
insertion method [30] in the simulations without the temperature difference.
As indicated in Fig. 5(a), the chemical potential difference is almost
independent of the mixing ratio $\chi$, while $\tilde{\mu}_{\rm d}/T$ is an
increasing function of $T$ in Fig. 5(b). The chemical potential modulation
induced by the temperature gradient cannot be compensated by the chemical
potential change with the mixing ratio. This means that the Soret effect in
our system is not caused by the thermodynamic chemical potentials, and it is
purely a non-equilibrium behavior. The chemical potential difference also
depends on the packing fraction in Fig. 5(c). $\partial(\tilde{\mu}_{\rm
d}/T)/\partial T$ is compensated roughly by $\partial(\tilde{\mu}_{\rm
d}/T)/\partial\phi$ (see Fig. 1(b)).

Figure 5:  The differences of the chemical potentials between the dimers and
monomers $\Delta\mu$ are plotted against the temperature $\langle{T}\rangle$,
the mixing ratio $\chi$, and the packing fraction $\phi$ in (a), (b) and (c),
respectively. In (a) and (c), the temperature is $T=3\epsilon$.

As discussed above, we cannot understand our simulation results with
conventional knowledge of the Soret effect, i.e., the contributions of the
mass, inertia moment and chemical potential. Here, we propose possible
mechanisms of the Soret effect in our dimer-monomer mixtures. In Fig. 6(a), we
present the probability distribution $P(r)$ of the bond length $r$ of the
dimers in the mixtures of $\phi=0.036$ and $0.234$. The spring constant is
$k=10\epsilon/\sigma^{2}$ and the temperature is changed. When we obtain
$P(r)$, the temperature gradients are not imposed. As the temperature is
increased, not only the distribution width is broadened, but also the peak
position is shifted to large $r$. We plot their average lengths
$\langle{r}\rangle$ against the temperature in Fig. 6(b). Those for
$k=500\epsilon/\sigma^{2}$ are also plotted. The dimer molecule is more
stretched at the high temperature. In other words, the dimer becomes bulkier.
The degree of the stretching is large when $k$ is small, so that the
corresponding effective volume of the dimer is increased more largely for
small $k$. In the dilute mixtures ($\phi=0.036$), the probability distribution
agrees well with the Boltzmann distribution ($4\pi r^{2}\times\exp[-U_{\rm
sp}(r)/T]$). In the dense mixtures, on the other hand, the probability
distribution is inconsistent with the Boltzmann distribution. The average bond
length in the dense mixture is shorter than that in the dilute mixture as
shown in Fig. 6(b). The molecules are surrounded by other molecules, all of
which are exerting non-bonded forces of the molecules. The degree of the bond
stretching is suppressed more in the dense system.

In Fig. 4, we also plotted the thermal diffusion factor in other monomer-
monomer mixtures as a function of the packing fraction $\phi$ (red squares).
In these mixtures, the masses of both monomers are the same, while the size of
the A-particle is set to be larger than that of B-particle. We replace
$\sigma$ in the WCA potential (Eq. (1)) to $\sigma_{\rm
AA}=\sigma,\,\sigma_{\rm AB}=0.8\sigma$, and $\sigma_{\rm BB}=0.6\sigma$ for
the A-A, A-B and B-B pairs of the monomers, respectively. The potential
strength $\epsilon$ is not changed. As the packing fraction increases, the
thermal diffusion factor decreases and becomes negative eventually. The bulky
A-monomers migrate to the colder region in the dilute mixtures, while they
move to the hotter region in the dense mixtures. These observations are
consistent with a previous study [23]. Although the physical mechanism of this
Soret effect still remains unclear, we consider that the Soret effect in the
dimer-monomer mixtures is related to that in these monomer-monomer mixtures.
When $k$ is small, the dimer molecules become bulkier. Then the bulky dimers
tend to move to the hotter region as that in the monomer-monomer mixtures,
although these dimers are two times as heavy as the monomers. If $k$ is large,
the dimer size does not change much, making this behavior unremarkable.

##  V 1D simulation

In this section, we propose another possible mechanism of the Soret effect,
with one-dimensional (1D) simulations [21]. The motions of the two particles
in a dimer and a monomer particle are constrained in the $x$-axis. We prepared
initial dimer and monomer, which obey the Maxwell-Boltzmann distributions at
$T=T_{\rm d}$ and $T=T_{\rm m}$, respectively. The 1D dimer has three types of
energies ($K_{\rm d}^{\rm tra}$, $K_{\rm d}^{\rm vib}$, and spring potential
$U_{\rm sp}$), the statistical averages of which are set to $T_{\rm d}/2$. The
translational kinetic energy of the monomer is set to $\langle{K_{\rm m}^{\rm
tra}}\rangle=T_{\rm m}/2$. The monomer particle and one dimer particle, which
we call as the first particle (and the other as the second one), interact via
the WCA potential (Eq. (1)) with $n=6$. If the dimer and the monomer collide
and bounce, we calculate the coefficient of restitution $e$ from the
velocities of the dimer and monomers. The collision between the dimer and the
monomer is inelastic, since the sum of the translational kinetic energies,
$K_{\rm d}^{\rm tra}+K_{\rm m}^{\rm tra}$, is not conserved after the
collision. In particular, when the energy due to the internal degrees of
freedom ($K_{\rm d}^{\rm vib}+U_{\rm sp}$) is transferred to the translational
kinetic energies ($K_{\rm d}^{\rm tra}+K_{\rm m}^{\rm tra})$, the coefficient
of restitution is likely to exceed unity [31]. The coefficient of restitution
is scattered statistically, so that we obtain its average with $10^{7}$
samples.

Figure 6:  (a) Probability distributions $P(r)$ of the bond length $r$ in the
dimer-monomer mixtures of $\phi=0.036$ and $0.234$. The spring constant is
$k=10\epsilon/\sigma^{2}$. The temperature $T$ is changed. (b) Plots of the
averaged bond length $\langle{r}\rangle$ with respect to the temperature
$\Delta T$ in the dimer-monomer mixtures.

In Fig. 7(a), we present the coefficients of restitution with respect to the
spring constant $k$ in two cases, X: $(T_{\rm d},T_{\rm
m})=(5\epsilon,\epsilon)$, and Y: $(T_{\rm d},T_{\rm
m})=(\epsilon,5\epsilon)$. In both cases, the coefficient of restitution is
larger than unity when $k$ is small, and it approaches to unity when $k$ is
large. It can be observed that the dimer and monomer bounce asymmetrically,
i.e., $e_{\rm X}>e_{\rm Y}$ over the entire range of $k$. In case X, the
coefficient of restitution is decreased monotonically to unity as the spring
constant is increased. In case Y, on the other hand, the coefficient of
restitution changes non-monotonically with $k$. When $k$ is small, $e_{\rm Y}$
is decreased with $k$ and becomes smaller than unity. Then, it turns to
increase and approaches unity for large $k$. In case X, the dimer is likely to
have larger internal energy than the sum of the translational kinetic energies
($K_{\rm d}^{\rm vib}+U_{\rm sp}>K_{\rm d}^{\rm tra}+K_{\rm m}^{\rm tra}$)
before the collision. Upon the collision, the internal energy changes to the
translational energy, increasing the coefficient of restitution. In case Y, on
the other hand, the internal energy is comparably small ($K_{\rm d}^{\rm
vib}+U_{\rm sp}<K_{\rm d}^{\rm tra}+K_{\rm m}^{\rm tra}$), allowing the
translational energy transferred to the internal energy. As a result, the
coefficient of restitution in case Y is likely to be smaller than that in case
X, $e_{\rm X}>e_{\rm Y}$. When the dimer is “hotter” than the monomer, the
molecules bounce more, resulting in the negative thermal diffusion factor,
with which the dimers migrate to the hotter region.

In Fig. 7(b), we plot the difference between the coefficients of restitution
in two different cases, $\Delta e=e_{\rm X}-e_{\rm Y}$. Here, we show $\Delta
e$ for several different values of $n(=3,6,12$, and $24)$ in the WCA potential
(Eq. (1)). In any $n$, $\Delta e$ is positive and it decays with $k$. With
small $n$, the particles are soft, so that the collision time increase, and
the monomer and the first particle in the dimer interact via the WCA potential
slowly. If the intra-particle interaction is strong enough (with large $k$),
the energy and the momentum are well transferred to the second particle in the
dimer during the collision. Meanwhile, the collision becomes more elastic when
$n$ is small and $k$ is large, making the coefficients of restitution $e_{\rm
X}$ and $e_{\rm Y}$ converge to unity and $\Delta e$ goes to zero. When $n$ is
large, on the other hand, the collision time is shortened. If the intra-
molecule interaction is weak (with small $k$), the second particle is
negligible upon the collision. Since we set the particle masses to be same,
the kinetic energies and momenta are exchanged between the monomer and the
first particle during the collision. However, after the collision, the
translational and vibrational kinetic energies are redistributed within the
dimer. If $T_{\rm d}=0$ in case Y, in particular, the coefficient of
restitution is $e_{\rm Y}=1/2$ in this limit of large $n$. As a result, the
difference between $e_{\rm X}$ and $e_{\rm Y}$ tends to zero more gradually
when $n$ is large.

Figure 7:  (a) The coefficients of restitution between a dimer and a monomer
are shown against the spring constant $k$, which are obtained in one-
dimensional simulations for two cases. X: $(T_{\rm d},T_{\rm
m})=(5\epsilon,\epsilon)$, Y: $(T_{\rm d},T_{\rm m})=(\epsilon,5\epsilon)$.
(b) The differences of the coefficients of restitution $\Delta e=e_{\rm
X}-e_{\rm Y}$ is shown with the spring constant. The exponent $n$ in the WCA
potential is changed.

Although we consider that this asymmetry of the restitutions can cause the
Soret effect also in the actual systems, the relationship between the Soret
effect in the 1D simulations and that in the above 3D simulations should be
considered carefully. The collisions among the molecules in the 3D system are
mostly oblique and not head-on. Thus, it is considered that the asymmetric
restitution plays more minor roles in the three dimensional mixtures, even if
it is the case. Because of the asymmetric restitution and the energy transfer,
the thermalization of the internal and kinetic degrees of freedom might be
violated. As shown in Fig. 1(a), however, we have not observed any relevant
difference between the kinetic and vibrational temperatures. We consider that
the linear temperature gradient are formed well after the particle
heterogeneity due to the Soret effect is induced, although we have no evidence
supporting this claim.

Here we mention a previous study of Garriga et al.[21], in which the authors
have considered the Soret effect in the 1D binary mixtures of light and heavy
particles. They found that the asymmetric collisions lead to the strong Soret
effect. Although their molecular description is quite different from ours,
their findings would support our conjecture.

##  VI Summary

In this article, we numerically studied the Soret effect in the dimer-monomer
mixtures and found that the internal degrees of the freedom of the molecules
can contribute significantly the Soret effect, which is usually treated as an
inter-molecular phenomenon. In dense system, in particular, the rigid dimers
migrate to the colder side, while the flexible ones move to the hotter side.

To explain the Soret effect in our system, we proposed two possible
mechanisms. One regards to the changes of the molecular shape; flexible
molecules adapt their volume depending on the local temperature, while bulkier
dimer molecules tend to migrate to the hotter side, overcoming the isotope
effect. The other regards the asymmetry of the restitutions. If a molecule
having larger internal degrees of freedom is “hotter”, the coefficient of
restitution with the other “colder” molecule with smaller internal degrees of
freedom becomes larger than that for the opposite case. To determine which of
them (and/or some other mechanisms) is more dominant, we have to investigate
them further through other types of test molecules. The inter-molecular
interactions would be modified by the intra-structures of the molecules in the
non-equilibrium conditions.

In this work, we focused on dimer molecules consisting of two particles.
However, it should be noted that diatomic molecules, such as ${\rm O}_{2}$,
are not considered with this dimer model. The vibrational modes in a real
diatomic molecule are quantised and the higher modes are strongly quenched at
room temperature. In this work, we aimed to consider softer molecules;
polymers are appropriate candidates for such soft molecules. However, the
changes of the degree of the polymerization accompany those of the mass, the
moment of inertia, and the interactions, all of which influence the Soret
effect. Thus, it is difficult to observe the pure effect of the flexibility on
the Soret effect. It was numerically demonstrated that the difference of the
chain stiffness triggers the change of the Soret effect [9]. When the degree
of the polymerization is the same, the Soret coefficient of more flexible
polymers, with short persistent length, tends to be smaller than those of
rigid polymers with long persistent length. We believe that our findings can
help us to further understand such mysterious behaviors of the Soret effect.

## Acknowledgements

This work was supported by KAKENHI (Grants No.17K05612), CREST, JST
(JPMJCR1424, JPMJCR2095).

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