Permutation Matrix Representation

• Permutation Matrix Representation (PMR) is a framework that expresses permutation matrices via block-diagonal (cycle-block) and multiplicative factorizations, revealing their inherent cycle structure.
• It enables the decomposition of operators in many-body physics into diagonal parts and permutation components, offering error-free quantum Monte Carlo estimators and streamlined quantum simulation algorithms.
• PMR also supports permutation-based encoding in coding theory and low-rank approximations, facilitating efficient solutions in combinatorial optimization and structural matrix analysis.

Permutation Matrix Representation (PMR) denotes a family of algebraic and algorithmic constructions organized around permutation matrices and operators built from them. In the strict matrix-theoretic sense, PMR is the statement that any permutation matrix is, up to permutation similarity, a block-diagonal direct sum of standard cycle-blocks and, simultaneously, a product of commuting elementary cycle factors (1803.02199). In computational many-body physics, PMR denotes decompositions of operators or Hamiltonians into diagonal components multiplied by permutation operators on a computational basis, enabling Trotter error-free quantum Monte Carlo estimators and divided-difference-based quantum simulation algorithms (Gupta et al., 2019, Ezzell et al., 9 Apr 2025, Sabharwal et al., 28 May 2026). The same phrase is also used for permutation-based search encodings in coding theory, for low-rank factorizations of large permutation matrices in matching and assignment, and for structural results on permutation-like matrix groups (Cuéllar et al., 2020, Dröge et al., 2023, Deng et al., 2013, Deng et al., 2015, Deng et al., 2016).

1. Core definitions and representational viewpoint

A permutation matrix $P\in\{0,1\}^{n\times n}$ is a $0$–$1$ matrix with exactly one $1$ in each row and each column. Equivalently, $P$ represents a permutation $\sigma\in S_n$ by

$$P_{i,j}= \begin{cases} 1,& j=\sigma(i),\ 0,& \text{otherwise.} \end{cases}$$

Two $n\times n$ matrices $A,B$ are permutationally similar if there exists a permutation matrix $T$ so that $0$0; this is an equivalence relation on permutation matrices of order $0$1 (1803.02199).

In operator-based PMR, permutation matrices arise from basis permutations. Let $0$2 be an orthonormal many-body basis. An off-diagonal term may act as

$0$3

where $0$4 is a permutation of the basis labels. If a finite group $0$5 of permutations acts simply transitively on $0$6, then any operator $0$7 on this basis admits the PMR decomposition

$0$8

with $0$9 diagonal and $1$0; in particular,

$1$1

so that off-diagonal contributions are represented by permutation operators times diagonal amplitudes (Ezzell et al., 9 Apr 2025).

A related coding-theoretic usage identifies a permutation $1$2 with its permutation matrix $1$3 over $1$4, so that right multiplication by $1$5 permutes the columns of a generator matrix $1$6: $1$7 Here PMR is a representation scheme in which permutations, rather than messages, are the search variables (Cuéllar et al., 2020).

These usages share a common structural principle: a transformation is encoded either exactly by permutation matrices, or by diagonal data attached to permutations, or by compact surrogates designed to preserve permutation constraints.

2. Cycle decomposition and canonical classification

The basic classification theorem states that every permutation matrix $1$8 is permutationally similar to a unique block-diagonal matrix of the form

$1$9

where $1$0 accounts for fixed points, each $1$1 is the standard $1$2-cycle matrix, the cycle sizes satisfy $1$3, and

$1$4

Equivalently,

$1$5

The list $1$6 is the cycle-type of the underlying permutation, so permutational similarity classes of $1$7 permutation matrices are in bijection with integer partitions of $1$8 (1803.02199).

The paper also writes the standard $1$9-cycle block as

$P$0

so that the canonical form may be expressed as

$P$1

This provides a complete canonical representative for each permutation-similarity class (1803.02199).

Besides this additive direct-sum decomposition, every permutation matrix also admits a multiplicative cycle factorization. If the disjoint cycle lengths are $P$2, one defines

$P$3

or, equivalently,

$P$4

and obtains

$P$5

These factors commute pairwise because they act on disjoint subsets of coordinates, and each factor satisfies

$P$6

The coexistence of direct-sum and product factorizations is the core algebraic content of PMR in the classical matrix sense (1803.02199).

A concrete $P$7 example with cycle-type $P$8 decomposes into disjoint cycles $P$9, $\sigma\in S_n$0, and the fixed point $\sigma\in S_n$1, and permutation similarity brings it to

$\sigma\in S_n$2

Multiplicatively,

$\sigma\in S_n$3

with each factor acting as the identity on the complement of its cycle support (1803.02199).

3. Permutation-like matrix groups and global permutation realizations

A matrix group is called permutation-like if every element of the group is similar to a permutation matrix. A central problem is to determine when such elementwise similarity can be upgraded to a single global basis in which the whole group becomes a permutation matrix group (Deng et al., 2013, Deng et al., 2015, Deng et al., 2016).

For $\sigma\in S_n$4, with $\sigma\in S_n$5 an odd prime, a principal result shows that if $\sigma\in S_n$6 is permutation-like, contains a maximal cycle $\sigma\in S_n$7 of order $\sigma\in S_n$8, and the cyclic subgroup $\sigma\in S_n$9 is normal in $$P_{i,j}= \begin{cases} 1,& j=\sigma(i),\ 0,& \text{otherwise.} \end{cases}$$0, then $$P_{i,j}= \begin{cases} 1,& j=\sigma(i),\ 0,& \text{otherwise.} \end{cases}$$1 is conjugate in $$P_{i,j}= \begin{cases} 1,& j=\sigma(i),\ 0,& \text{otherwise.} \end{cases}$$2 to a group of permutation matrices: $$P_{i,j}= \begin{cases} 1,& j=\sigma(i),\ 0,& \text{otherwise.} \end{cases}$$3 In this setting, $$P_{i,j}= \begin{cases} 1,& j=\sigma(i),\ 0,& \text{otherwise.} \end{cases}$$4 embeds in $$P_{i,j}= \begin{cases} 1,& j=\sigma(i),\ 0,& \text{otherwise.} \end{cases}$$5, and the resulting groups are realized as semidirect products

$$P_{i,j}= \begin{cases} 1,& j=\sigma(i),\ 0,& \text{otherwise.} \end{cases}$$6

with orbit lengths governed by the action of $$P_{i,j}= \begin{cases} 1,& j=\sigma(i),\ 0,& \text{otherwise.} \end{cases}$$7 on $$P_{i,j}= \begin{cases} 1,& j=\sigma(i),\ 0,& \text{otherwise.} \end{cases}$$8 (Deng et al., 2013).

This theorem was extended from prime-square dimension to arbitrary odd prime powers. If $$P_{i,j}= \begin{cases} 1,& j=\sigma(i),\ 0,& \text{otherwise.} \end{cases}$$9 is a permutation-like matrix group, contains a maximal cycle $n\times n$0 of length $n\times n$1, and $n\times n$2 is normal in $n\times n$3, then there exists an invertible matrix $n\times n$4 such that

$n\times n$5

The proof proceeds by analyzing the normalizer action

$n\times n$6

splitting into the $n\times n$7-group case and the general case, and constructing a common basis in which both $n\times n$8 and a generator $n\times n$9 become permutation matrices (Deng et al., 2015).

The power-of-two case was subsequently established as well. If $A,B$0 is a permutation-like matrix group and contains a matrix $A,B$1 of exact order $A,B$2 such that $A,B$3, then $A,B$4 is similar to a permutation matrix group (Deng et al., 2016). The proof uses the self-centralized maximal cycle property,

$A,B$5

together with the characterization that a matrix is similar to a permutation matrix if and only if it is diagonalizable with eigenvalues roots of unity and its characteristic polynomial factors as

$A,B$6

A basis of $A,B$7-eigenvectors is then chosen so that the remaining generators act by permuting that basis (Deng et al., 2016).

Taken together, these results show that the normal maximal-cycle hypothesis is strong enough to force a global permutation basis in the prime-square, odd prime-power, and power-of-two cases. The same literature identifies the main limitations: the normality hypothesis is essential in known counterexamples, and extension to arbitrary composite cycle lengths remains open (Deng et al., 2015, Deng et al., 2016).

4. PMR in quantum Monte Carlo

In quantum Monte Carlo, PMR rewrites a Hamiltonian as a sum of diagonal terms times permutation operators. One formulation uses

$A,B$8

where $A,B$9 is the diagonal “classical” part, each $T$0 is diagonal, each $T$1 is a classical permutation matrix acting on the computational basis, and the quantum dimension consists of products of elements of a permutation group (Gupta et al., 2019). A related formulation splits

$T$2

chooses a finite group $T$3 of permutations acting simply transitively on the basis, and writes

$T$4

with $T$5 off-diagonal (Ezzell et al., 9 Apr 2025).

The partition function is expanded exactly, without Trotterization, into configurations $T$6 involving a basis state $T$7 and a product of permutations returning to the identity. In the measurement framework,

$T$8

where $T$9 is the divided difference of the exponential at the diagonal energies. This regrouping is exact and Trotter error-free (Gupta et al., 2019, Ezzell et al., 9 Apr 2025).

A major development is the derivation of formal estimators for arbitrary static observables and exact, explicit estimators for general imaginary-time correlation functions and integrated susceptibilities. For a static observable $0$00,

$0$01

while the two-point function is written as

$0$02

Integrated quantities include

$0$03

with the latter leading to a closed form triple-sum estimator of $0$04 (Ezzell et al., 9 Apr 2025).

Implementation is based on updates of $0$05. The measurement paper specifies: start from $0$06; perform local updates by swapping two adjacent permutations in $0$07 or updating the classical state $0$08; insert or delete a pair of off-diagonals using fundamental-cycle moves; and use composite and worm updates for ergodicity. Measurement costs are $0$09 per sample for standard static observables, $0$10 for two-point functions, $0$11 for integrated energy-susceptibilities, and $0$12 for fidelity-susceptibilities (Ezzell et al., 9 Apr 2025).

The framework also gives a transparent sign criterion. In the off-diagonal expansion, a sign problem occurs if there is any closed walk $0$13 whose product of hopping amplitudes has the wrong sign. The formalism therefore shows that non-stoquastic matrix elements are necessary but not sufficient for a sign problem; the decisive object is the sign of closed walks in the permutation group (Gupta et al., 2019). Numerical demonstrations include verification on a $0$14 transverse-field Ising model against exact diagonalization for $0$15 observables, an $0$16 demonstration for energy, specific heat, $0$17, $0$18, nonlocal sums of random Pauli strings, and dynamic correlators, and a comparison to stochastic series expansion in which PMR converged in seconds while SSE often failed to thermalize in a $0$19 wall-clock on certain random 3-regular MAX2SAT-plus-transverse-field instances (Ezzell et al., 9 Apr 2025, Gupta et al., 2019).

5. PMR in quantum simulation algorithms

For digital quantum simulation, PMR again starts from a decomposition of a Hermitian Hamiltonian into diagonal operators and permutation operators: $0$20 The associated off-diagonal norm is

$0$21

The short-time evolution $0$22 is expanded using divided differences of the exponential, and the off-diagonal correction is written as a sum-of-unitaries suitable for an LCU implementation (Sabharwal et al., 28 May 2026).

The time-independent algorithm proceeds by decomposing $0$23, partitioning the total time $0$24 into

$0$25

segments of length $0$26, truncating the off-diagonal series at

$0$27

writing the truncated operator as an LCU whose total weight is approximately $0$28, and repeating the short-time approximation $0$29 times. The overall error is $0$30 (Sabharwal et al., 28 May 2026).

For the Rydberg interaction Hamiltonian

$0$31

the reported asymptotic gate counts are

$0$32

$0$33

and, with a diagonal approximation,

$0$34

All three use $0$35 qubits (Sabharwal et al., 28 May 2026).

For time-dependent Hamiltonians,

$0$36

each $0$37 is expanded into exponentials,

$0$38

and the time-ordered exponential is rewritten through an integral-free divided-difference identity. Applied to the Floquet-driven transverse-field Ising model in $0$39 dimensions,

$0$40

the PMR gate cost is

$0$41

with qubit cost $0$42 (Sabharwal et al., 28 May 2026).

The comparison with qHOP emphasizes three structural features: PMR gate cost is independent of the drive frequency $0$43, independent of the interaction scale $0$44, and linear in $0$45; PMR ancilla overhead is $0$46; and diagonal terms are handled as phase rotations decoupled from off-diagonal costs (Sabharwal et al., 28 May 2026).

6. Permutation-based search for linear-code distance

In coding theory, PMR is a permutation-based representation for computing the minimum distance of a linear code. Given an $0$47-linear code $0$48 with generator matrix $0$49, the key theorem states that there exists a permutation $0$50 such that, if

$0$51

then at least one row of $0$52 has Hamming weight equal to the minimum distance

$0$53

If that row is $0$54, then $0$55 is a nonzero codeword in $0$56 of weight $0$57 (Cuéllar et al., 2020).

This converts minimum-distance computation into the permutation search problem

$0$58

The search space has size $0$59. For a fixed permutation, evaluation requires column reindexing in $0$60, reduced-row-echelon computation in $0$61, and row-weight scanning in $0$62, so the total per-permutation cost is

$0$63

independent of the field size $0$64 except through the cost of each field operation (Cuéllar et al., 2020).

The implemented evolutionary operators are algebraic crossover

$0$65

and transposition mutation, or “2-swap.” Fitness is

$0$66

with standard binary-tournament selection, elitism, and either reinitialization in GGA or incest-threshold and cataclysmic restart in CHC (Cuéllar et al., 2020).

Experiments were carried out on thirty linear codes: twenty codes over $0$67 of lengths $0$68 to $0$69 and dimensions $0$70 to $0$71, and ten codes over $0$72, including BCH and EQR families. Over $0$73, the permutation-based versions found the true distance in $0$74 of $0$75 instances and improved the best-known upper bounds in the remaining two. Over $0$76, both GGA-Order and CHC-Order matched or beat the best heuristics in the literature. One run showed that Magma’s $0$77 code admits a codeword of weight $0$78 (Cuéllar et al., 2020).

The coding-theoretic PMR differs from the cycle-decomposition and operator-decomposition meanings, but it retains the same central idea: the primary search variable is a permutation encoded by a permutation matrix, and the problem is reformulated so that structural information is exposed after permutation.

7. Low-rank, relaxed, and linear-combination variants

A distinct line of work represents large permutation matrices through low-rank factorization followed by a nonlinearity. Let $0$79 be a permutation matrix and $0$80 with $0$81. The approximation is

$0$82

For exact representation, the paper defines

$0$83

and proves that every permutation can be represented exactly whenever

$0$84

where $0$85 is the $0$86-dimensional Kissing number. For approximate smooth representation, it uses row-wise softmax

$0$87

with $0$88 for sufficiently large $0$89 (Dröge et al., 2023).

The minimal admissible rank is

$0$90

with reported values

$0$91

Thus rank $0$92 suffices to represent any permutation on up to $0$93 elements. Storage falls from $0$94 to $0$95, with memory ratio $0$96. For $0$97, the representation uses $0$98 elements in two small matrices instead of a single matrix with $0$99 elements, yielding up to $1$00 orders of magnitude less memory (Dröge et al., 2023).

Reported applications include linear assignment, quadratic assignment, and shape matching. On FAUST with $1$01 and $1$02, PMR+softmax produced $1$03 valid permutations, $1$04 relative error, and $1$05 Hamming distance. On QAPLIB, $1$06 instances were within $1$07 of known optima. In shape matching on FAUST, geodesic errors improved from $1$08 to $1$09 in the probabilistic setting and from $1$10 to $1$11 in the embedding setting (Dröge et al., 2023).

Related work broadens the matrix class itself. A Permutation-Like Matrix (PLM) is a $1$12 $1$13–$1$14 matrix in which each column has exactly one $1$15, but rows may contain multiple $1$16s; $1$17 is therefore a monoid rather than a group. For every $1$18, the spectral radius satisfies $1$19, every eigenvalue is either $1$20 or a root of unity, and a $1$21 matrix is left-stochastic if and only if it is a convex combination of PLMs (Lippold, 2024). This is a strict relaxation of permutation matrices and should not be conflated with exact PMR based on bijective permutations.

Another adjacent usage studies orthogonal matrices that are linear combinations of permutation matrices. For order $1$22, every complex, real, and rational orthogonal permutative matrix can always be expressed as a linear combination of up to four permutation matrices, and the paper gives a complete parametric characterization through three pattern families $1$23, $1$24, and $1$25 (Mandal et al., 2023). This literature uses permutation matrices as a basis for structured linear combinations rather than as exact one-hot representations.

A plausible implication is that PMR now functions less as a single formalism than as a recurrent design pattern: exact cycle-block classification in algebra, diagonal-times-permutation decompositions in many-body methods, permutation-space encodings in combinatorial search, and compact or relaxed surrogates in large-scale optimization.