Qubit seriation: Improving data-model alignment
using spectral ordering

Atithi Acharya 1,3Jing Chen 3

Jacob Miller 2

Manuel Rudolph 2,4
Alejandro Perdomo-Ortiz 2

1

Department of Physics, Rutgers University, Piscataway, NJ 08854, USA
Zapata Computing Canada Inc., 325 Front St W, Toronto, ON, M5V 2Y1
3
Zapata Computing Inc., 100 Federal Street, Boston, MA 02110, USA
4
École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland
2

Abstract
With the advent of quantum and quantum-inspired machine learning, adapting
the structure of learning models to match the structure of target datasets has beenshown to be crucial for obtaining high performance. Probabilistic models basedon tensor networks (TNs) are prime candidates to benefit from data-dependent
design considerations, owing to their bias towards correlations which are local
with respect to the topology of the model. In this work, we use methods fromspectral graph theory to search for optimal permutations of model sites which areadapted to the structure of an input dataset. Our method uses pair-wise mutual
information estimates from the target dataset to ensure that strongly correlated bits
are placed closer to each other relative to the model’s topology. We demonstrate
the effectiveness of such prepossessing for probabilistic modeling tasks, finding
substantial improvements in the performance of generative models based on matrix
product states (MPS) across a variety of datasets. We also show how spectral
embedding, a dimensionality reduction technique from spectral graph theory, canbe used to gain further insights into the structure of datasets of interest.

Introduction
The development of increasingly powerful quantum computers has placed renewed focus on the nearterm potential of quantum models and algorithms for solving problems of significant real-world value.
Probabilistic modeling, where a model is trained to learn the structure of an unknown distribution
from an unlabeled dataset of samples, has emerged as an application of particular promise for quantum
methods [29], owing to provable advantages in expressivity [12] and generalization [1] arising fromthe distinct properties of quantum state spaces. Within this domain, the use of quantum-inspired
tensor networks (TNs) has allowed many of the advantages of fully-quantum probabilistic models to
be enjoyed in a simulated classical setting [14, 5], while also permitting the development of hybrid
quantum-classical models that exploit the complementary properties of both model families forpractical benefit [13, 18, 36, 32].
Although promising, the relative newness of quantum and quantum-inspired machine learningalgorithms means that best practices for ensuring optimal performance remain unsettled. While alarge amount of attention has been dedicated to overcoming the phenomenon of barren plateaus in
optimization landscapes [25, 6, 17], we focus here on a less well-understood issue, namely, unveilingthe dataset’s geometry on the performance of the machine-learning (ML) models. The importance of
optimally matching model geometry to the structure of probabilistic modeling problems are wellunderstood in the TN community [9, 23, 7], but prior proposals for ensuring this optimal matching
Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

Figure 1: Schematic representation of our qubit seriation framework: Training data is frequently
structured such that the most correlated variables in the data are associated with distant sites. Our
spectral re-ordering method is able to ensure locality of strongly-correlated variables, as seen in the
pairwise mutual information plots, where larger correlations are placed closer to the diagonal. This
helps in the training of probabilistic models, notably those based on tensor networks.

have tended to rely on ad-hoc search through various model configurations [21, 16, 22], entailing ahigh cost due to repeated model retraining, while also failing to make use of insights present in the
structure of the classical data itself. A notable exception is the problem-specific solution of Barcza etal. [4], where spectral ordering methods were shown to be capable of improving the performanceof density matrix renormalization group (DMRG) calculations within quantum chemistry problems.
Recent efforts to incorporate geometric considerations into quantum machine learning (QML),
although of a different flavor than considered here, include the works of [26, 20, 30, 27] where theauthors incorporate geometric priors arising from problem-specific symmetries into quantum models.
In this work, we introduce a simple method for ordering the variables of a classical dataset to ensure
an optimal match between the correlations present in the data and the connectivity of 1D quantum or
quantum-inspired models. We refer to this problem as qubit seriation, in recognition of its similarity
with the seriation problem of linearly-ordering sequential data, which has proven important in domains
as diverse as archaeology [31], DNA sequencing [11, 3], and natural language processing [34]. Our
approach makes use of spectral graph theory techniques to efficiently compute an ordering directlyfrom the pairwise mutual information between variables in a classical dataset of interest, thus
guaranteeing that strongly correlated variables are mapped to nearby qubits, and weakly correlated
variables to more distant qubits. We demonstrate the effectiveness of our procedure in probabilistic
modeling experiments utilizing matrix product states (MPS), where reordering the variables of aclassical dataset prior to optimization is shown to significantly boost the performance of the trainedmodel. We show how spectral embedding tools can be used to extend these methods to models
with more complex connectivities, and develop heuristics for understanding the impact of noise or
small dataset size on the output ordering. Overall, our work emphasizes the practical importance ofgeometric considerations in quantum machine learning, and demonstrates the performance benefits
that are possible with the use of principled approaches to solving these issues.

MethodWe begin by outlining the proof of the distance-sensitive ordering and show convex relaxations of
permutation vectors into eigenvectors of the Laplacian of the similarity matrix. Then we use thereordering techniques to substantially improve the performance of the TN models. Given a dataset,
we construct a similarity graph Laplacian and find that its lowest eigenvectors, particularly the Fiedler
vector, can be used to help in improving the learnability of TNML and QML models. We use spectral
embedding, a powerful dimensionality reduction tool to enable model design in more than one
dimension. We also study the limitations of our method when it is applied to the realistic case of
noisy data. We build indicators for the spectral ordering solutions to analyze their applicability.
We begin by introducing spectral ordering which generates a distance sensitive ordering. It ensuresthe more strongly correlated sites are kept closer, whereas distant sites are less correlated. The spectral
ordering solution is based on a spectral graph theory [35] approach, which uses eigenvectors of thegraph Laplacian L coming from the data similarity matrix. As a measure of similarity, we use themutual information I(Xi ; Xj ) = DKL (P (Xi , Xj )||P (Xi )P (Xj )) where Xi,j are data variables onsites i, j and P denotes their marginal probability distributions
over the dataset. The graph LaplacianP
is then defined as L = D − W with D := (dii ) = ( j I(Xi ; Xj )) representing the diagonal degree
matrix and W := (wij )i̸=j = (I(Xi ; Xj ))i̸=j the weight matrix. As such, the MI statistics of2

the data can be understood as an undirected graph over the data sites with non-negative edges, asI(Xi ; Xj ) ≥ 0 is always positive by definition. We are trying to ensure that after the data has beenordered, the adjacent sites have high MI, i.e., they are more strongly correlated, whereas distant sites
are less correlated. This can be seen in Fig. 1, where, after ordering, next neighbor sites tend to havelarge MI values. For data with n sites, and an n-qubit learning problem, the ordering can be definedas the index permutation π(1, 2, ..., n) = (π(1), π(2), ..., π(n)). The permuted MI weight matrix
is (πW π T )ij = wπ(i),π(j) . One choice of a cost function for qubit seriation i.e. distance sensitive
ordering, is
1X
1 X
C(π) =(i − j)2 wπ(i),π(j) =(i − j)2 wπ(i),π(j) .
(1)
2 i,j
2
π(i),π(j)While this combinatorial cost function is likely to have no polynomial-time optimal algorithms [8], it
can be tackled directly to find good orderings or model topologies. Fascinatingly, it was shown thatthe second eigenvector of L, commonly called the Fiedler vector, provides an optimal 1D solution in
the noiseless case [3]. More precisely, the Fiedler vector solves a convex relaxation of C in Eq. 1
with continuous variables
P xi ∈ [−1, 1]. Additionally, the shifting necessary in the discrete variables
introduces a constraint i xi = 0.
X
X
1X
C(π) =(xi − xj )2 wi,j =x2i di −xi xj wi,j = xT (D − W )x = xT Lx.
(2)
2 i,j
i
i,j
See Refs. [11, 3] for a similar analysis that was intended for applications in DNA sequencing. Thisrelaxation requires that xT x = 1, which can be added a lagrange multiplier λ. The stationary pointsof this cost function occurs when x is the eigenvector of the positive semidefinite operator L. Now
the cost function reads as C(π) = xT Lx − λ(xT x − 1), which is identical to writing the RayleighT
quotient R(L, x) = xxTLx
for L and x, and then minimizing it. We note that the lowest eigenvalue ofxT
the Laplacian is always λ0 = 0 with the eigenvector being the trivial solution
P x0 = (1, 1, ..., 1) . The
other n − 1 eigenvectors are orthogonal to x0 which then satisfies the i xi = 0 condition naturally.
This helps us in identifying that x1 minimizes the cost function while satisfying the constraints. By
sorting the elements in x1 in ascending order and re-arranging the indices accordingly, we achieve
the optimal ordering for the original qubit seriation problem. Other ways of relaxing the cost function(1) exist and can be exploited for some noisier datasets.

Results and discussions
To put our qubit seriation method based on spectral graph theory to the test, we train a matrix productstate (MPS) generative model [15, 18] to fit various datasets and demonstrate advantages gained by
qubit seriation. We demonstrate the success of the solution to qubit seriation with 1D, 2D and othertree structured data. After training the MPS based generative model using the negative log likelihood(NLL), we monitor the Kullback-Leibler (KL) divergence between the training data distribution pT
and MPS distribution qθ , given by
pT (x)1 X
L (θ) = KL (pT ||qθ ) = Ex∼pT (x) log= − log (|T |) −log (qθ (x)) .
(3)qθ (x)|T |x∈T

The numerics suggest that the qubit seriation leads to a better solution in approximately 99% of thetotal 1000 experiments for all three chosen datasets. The solution is accepted to be better if Lseriation
is lower than Lrandom by at least 1% of Lrandom . We applied seriation algorithms to samples comingfrom a random MPS implemented in the ITensor library [10]. It is worth noting that the improvement
for random MPS is not as significant as other datasets. This can be seen as a consequence of the lackof significant gaps in the spectrum of the graph Laplacian. We also studied data samples generated
from a 1D Markov chain to better understand the role of noise in the stability of spectral solutions.
We infer that it is generally better to have a significant spectral gap to have a significant and robust
improvement. We also tested the seriation algorithm on the bars and stripes dataset [24], groundstate of the Toric code Hamiltonian [19, 33] and samples coming from a random PEPS model [28].
The KL divergence values and their variance are vastly improved using spectral ordering of the data,
and for bars and stripes, we recover in all instances that a low bond dimension such as 23 = 8 issufficient to perfectly fit the dataset. See Fig. 2. We also observed similar improvements in learning
the ground state of the Toric code Hamiltonian. These improvements can be qualitatively attributed
3

Figure 2: We plot the final KL divergence between the true data distribution and the MPS model
distribution. We have data generated by a random MPS to the left, followed by samples from an
Ising tree Hamiltonian in the center and finally a dataset of 4 × 3 bars and stripes images to the right.
Results from training on data with random site ordering are marked with gray, while those fromtraining on data which has been rearranged using spectral ordering are marked with blue. Each curveshows the median over 1000 different shufflings of the original dataset, while the fluctuations overthese runs are shown as shaded regions around the curve. In all three cases, the spectral orderingsolution leads to lower training error with greater than 99% confidence. In the bottom row, we plot thecorresponding Laplacian eigenvalues to better understand the different types of training advantages(see text for more details).

Figure 3: Demonstrating spectral embedding on an Ising tree dataset. (left): The undirected weighted
graph is depicted with weights corresponding to the coupling strength Jij of the Ising Hamiltonian.(right): Spectral embedding of the sites in the data using the second and the third eigenvectors of
the MI graph Laplacian. We use the MI statistics of 1000 bit-strings corresponding to the groundstate configurations of the Ising Hamiltonian on Sz basis. Data sites are colored for convenience.
The spectral embedding correctly identifies the more strongly-correlated clusters of qubits 7, 8, 9,
5, and 11, and of qubits 2, 6, 4, 0, and 10. It is important to note that distance in this eigenvector
subspace scales inversely with correlation.

to the eigenspectrum of the graph Laplacian associated with the pairwise mutual information matrix.
As we see in Fig. 2, there are large gaps between the low and the intermediate eigenvalues, indicatingthat if we use the eigenvector corresponding to a smaller eigenvalue for spectral ordering, we are
guaranteed to ensure an ordering that preserves locality. Lastly, we studied the samples coming froman Ising Tree Hamiltonian using two different methods. We used the best 1D spectral ordering todemonstrate an improvement. We also recovered the tree-like structure where the eigenvectors of thegraph Laplacian ensure that the tightly coupled pairs are placed closer than the loosely coupled ones.
4The stability of the spectral ordering by the Fiedler vector can be indicated by the spectral gap λ2 − λ1of the Laplacian L(W ). In general the spectral gap (λk − λk−1 ) is used for stability analysis when
the algorithm uses the subspace formed by k eigenvectors. If there is degeneracy in the system,
then there might be more than one optimal solutions since the solutions form a degenerate subspace.
It does not matter which direction is picked within the subspace or if a more general algorithm is
employed. However, if there is no gap between the bands of eigenvalues, then it is indicative that
there will be no benefit to the algorithm via seriation.
The spectral ordering solution is stable when the magnitude of unstructured noise is less than the
spectral gap [11].
√||∆L||F ≤ (λk − λk−1 )/ 2
We also demonstrate that the sampling noise in the estimation of mutual information also leads to thelessening of the spectral gap. Thus further leading to an unstable or less useful solution to seriation(see Fig. 4). Further details can be found in [2], where the authors perform a similar analysis ofstability but for closely related spectral clustering problems. We present a proof and interpretation ofthese results in more detail while also noting the special cases which lead to stronger guarantees on
the spectral solution to seriation.

Figure 4: We see the lessening of spectral gap (λ1 − λ2 ) of the normalized Laplacian if we constructnoisier Laplacians by using fewer samples to estimate the pairwise mutual information matrix. This
also means that the Fiedler vector is not a unique vector that will seriate the qubits, meaning that slightvariations in the data will generate very different permutations as answers to the seriation problem.
The data here is taken from a Markov chain, with the legend showing the number of samples used toestimate the pairwise mutual information.
.

Conclusion
In our work, we apply spectral graph theory methods for qubit seriation in tensor network based
generative algorithms. Given a dataset, we constructed a similarity graph Laplacian and found thatits lowest eigenvectors, and particularly the Fiedler vector, can be used to help TNML and QML
in learning better. We also outlined the proof of the distance sensitive ordering and showed convex
relaxations of permutation vectors into eigenvectors of the Laplacian of the similarity matrix. While
we demonstrated improved training performance using MPS, other models can also benefit fromthis. Concretely, clustering algorithms on the spectral embedding of data can be used to design TNarchitectures and quantum circuit ansätze for machine learning tasks that are specific to each given
dataset. Given that most negative results are derived using uninformed and generic architectures, weare optimistic that our work can lead to improve the model-data compatibility. We can also leverage
the benefits of seriation in generative modeling tasks by classical machine learning models such asrecurrent neural networks and its variants, using the fact that sequential learning is sensitive to data
ordering [34].

5Broader Impact
Overall, our results show the practical benefit of spectral graph theory methods for solving theseriation problem in probabilistic modeling, which we anticipate to be of even greater utility inproblems involving larger numbers of random variables and model sizes. While our results were
demonstrated with matrix product state models, we expect that these methods can additionallybenefit sequential classical models, such as recurrent neural networks, as well as other graph-based
probabilistic models, such as probabilistic graphical models and more general tensor networks. For
these latter cases, the question of generalizing our methods for the optimal placement of randomvariables on a general graph (rather than on a line) remains an open question of interest.

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