Permanents in linear optical networks

Stefan Scheel

REVIEW

Permanents in linear optical networks

Stefan Scheel

Quantum Physics · Submitted 18 Jun 2004 · 13 references

Pith's verdict: Eq. (10) is correct and is the reason to cite this paper; the complexity sentence in Sec. III is a promissory note, and it names the wrong scaling variable.

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arXiv IDquant-ph/0406127 (version 1)

Pith IDFLDKCODN, Pith's permanent record of this paper and its review

CitationStefan Scheel. "Permanents in linear optical networks." arXiv:quant-ph/0406127 (2004). https://pith.science/paper/FLDKCODN

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Editor's note

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Short version: the permanent identity is right and is the reason to keep this paper. The complexity claim in the abstract and Sec. III is not established, and one clause of it is simply wrong on its own terms. Sec. IV has real algebra errors but they are secondary and don't touch the main result. This is a short 2004 note that reads as a precursor to boson sampling, and it should be read that way rather than as a hardness paper, which it isn't and doesn't claim to be in the body — only in the abstract's second sentence [abstract:s2]. The stress-test's read lands: the soft spot is the complexity sentence, not Eq. (10). What's actually new and good: the multiplicity counting in going from Eq. (8) to Eq. (10) is done carefully, the sum-over-sequences form in Eq. (14) is clean, and Eq. (13) turning |per Λ| ≤ 1 into a one-line probability argument is a genuinely nice derivation of Marcus–Newman. That last bit is the kind of thing you steal for a lecture. Where it's soft: the sentence in Sec. III that the calculation 'cannot be performed in polynomial time (polynomial in the number of modes)' has two problems. No hardness result is cited or proved — Valiant is absent, and hardness for the restricted class that actually arises (repeated-row/column submatrices of unitaries) is exactly what Aaronson and Arkhipov had to argue seven years later. And the scaling variable is wrong: the permanent in Eq. (10) is n×n with n the total photon number, so for fixed n the cost is polynomial in the mode count N. As written the sentence is false at fixed n and unsupported at growing n. That doesn't damage Eq. (10), but it means the abstract's headline implication is a promissory note. Sec. IV.B: Eq. (23) has a sign and a missing factor in the |0⟩⟨2| term and a missing TR in the |2⟩ coefficient; Eq. (28) is garbled; Eq. (34) uses the real-sphere measure, giving 3/8 for a qubit where Haar on CP¹ gives 1/3. So Eq. (24) and the higher-dimensional numbers shouldn't be quoted without redoing them. Sec. IV is an illustration, though, not load-bearing. Who it's for: anyone working on linear-optical quantum information or boson sampling who wants the Fock-basis/permanent dictionary in one place. Worth a serious referee, with the brief being to cut the complexity claim back to what's proved and to fix or drop Sec. IV.B.

Referee report

Major objections (4)

[Sec. III, sentence following Eq. (14); abstract:s2] The claim that computing Fock-basis matrix elements 'cannot be performed in polynomial time' and that such calculations are 'generically computationally hard' is asserted without proof and without citation. The relevant hardness result is Valiant's #P-completeness of the permanent, which is nowhere cited; Sec. II, where the n! cost of Eq. (1) is noted, is the natural place for it. More importantly, the reduction as set up does not by itself give hardness: Eq. (10) shows that a matrix element reduces to a permanent of a submatrix of a unitary, so an efficient algorithm for matrix elements would give an efficient algorithm for permanents of unitaries only, not for general matrices. Establishing hardness requires the converse embedding, and no argument is given that permanents of the repeated-row/repeated-column submatrices that actually arise are hard. As written, Eq. (10) is a rewriting, not a complexity result, and abstract:s2 overstates what has been shown.
[Sec. III, parenthetical after Eq. (14)] The parenthetical '(polynomial in the number of modes)' identifies the wrong scaling variable. The permanent in Eq. (10) is an n×n object where n is the total photon number, not the mode count N; for fixed n the cost of evaluating the sum in Eq. (1) is polynomial in N (indeed constant for n = 2, where the permanent is a 2×2 expression). The exponential growth is in n, the total photon number, and the sentence as printed is false in the fixed-photon-number regime. This should be corrected and the two variables N and n kept distinct throughout, including in the statement of the complexity conclusion.
[Sec. IV.A, Eq. (23)] Eq. (23) as printed cannot be a reduced density matrix. The first term is written with an overall minus sign, −[√2 c_1 d_1 T*R*|0⟩] ⊗ h.c., which contributes a negative diagonal entry −2|c_1 d_1|^2|TR|^2 to the |0⟩⟨0| element. In addition, the |2⟩ component of the third term is written as √2 c_1 d_1|2⟩, whereas Eq. (22) gives the |20⟩ amplitude as √2 c_1 d_1 TR, so the factor TR appears to be missing. Since Eq. (24) is obtained from Eq. (23), the reader cannot verify the coefficient 3/64 from the printed intermediate expression. Please correct Eq. (23) and, if possible, show one intermediate step connecting it to Eq. (24).
[Sec. IV.B, Eqs. (34)–(36)] The integration measure in Eq. (34) does not appear to be the invariant measure on the state space, and the resulting moments in Eqs. (35)–(36) do not agree with the Haar averages. For N = 1 (a single qubit, Eq. (32) with Θ_2 = 0), Eq. (34) reduces to flat dΘ_1 and Eq. (35) gives ⟨|c_1|^4⟩ = 3/8, whereas the invariant average over CP^1 is 1/3; correspondingly Eq. (36) gives 1/8 where the Haar value is 1/6. The general Haar results for a state of dimension d = N+1 are 2/(d(d+1)) and 1/(d(d+1)), not 3/((N+1)(N+3)) and 1/((N+1)(N+3)). This matters because the appeal to Schur's Lemma to remove phase dependence presumes invariance of the measure. Please recheck Eqs. (34)–(37) and state explicitly which group and which measure are being averaged over.

Minor objections (8)

[Sec. III, Eq. (8)] As printed, the multinomial factorials in Eq. (8) appear in the numerator; the coefficient of the term labelled by {n_ij} should be (∏_i n_i!)^{1/2}/∏_{i,j} n_ij!. The same comment applies to the index structure of Eq. (9), where the product ∏_{k,l} Λ_{n_kl lk} is hard to reconcile with the surrounding notation.
[Sec. III, text before Eq. (9)] The text states that 'the n_ij form a n × n-matrix'; the matrix is N × N, with n the total photon number. Please correct, and likewise the phrase 'the valuei' in the sentence introducing Eq. (14).
[Sec. IV.B, Eq. (28)] The index structure of Eq. (28) does not close: the second factor ⟨ψ_1,ψ_2|Û†|k_2⟩⟨k_2|Û|ψ_1,ψ_2⟩ carries only k_2, while the outer sum runs over both k_1 and k_2. Please check the labelling of the traced-out mode and rewrite the expression so that the trace over subsystem 1 is manifest.
[Sec. IV.B, Eq. (31)] Eq. (31) has unbalanced brackets in the arguments of the permanents (for example 'per Λ[(1^{k_1},2^{k_2})|(1^{q_1},2^{q_2})' is missing a closing bracket) and the four permanents are run together without separators. Please reformat.
[Sec. II, after Eq. (2)] The statement that Eq. (2) is 'the only known simplification for permanents' is too strong; for instance per Λ = per Λ^T, permanents of block-diagonal matrices factorize, and permanents of rank-one matrices are elementary. Please soften the wording.
[Abstract and Sec. III, Eq. (13)] The phrase 'takes its values across the unit disk' should read 'within the unit disk'. Also, the abstract's 'generically computationally hard' should be replaced by a precise statement once the complexity discussion is fixed.
[Sec. III, Eqs. (15)–(16)] The notation in which the partial matrix element carries a subscript '|n − Σ_j m_j⟩' outside the ket is unclear; please define it in words or rewrite the projection in a more standard form.
[Sec. II and Sec. III] The paper would benefit from a sentence relating Eq. (10) to where this Fock-basis permanent formula has appeared previously in the quantum optics literature, beyond the general Caianiello references [5,6], so that the reader can see what is new here and what is a restatement.

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The paper

The paper claims

Fock-basis amplitudes of passive linear optical networks are permanents of submatrices of the network's unitary, so the natural bookkeeping for photon redistribution is a hard matrix function.

The reading

The paper asks what the amplitude is for a passive linear optical network — a fixed array of beam splitters and phase shifters described by a unitary matrix $U$ — to turn one Fock state $|n_1,n_2,...,n_N⟩$ into another $|m_1,m_2,...,m_N⟩$. The answer it derives is a permanent: the sum over all diagonals, with no permutation signs, of the submatrix of $U$ whose rows and columns are repeated according to the output and input photon counts. The author's point is that this is not a curiosity but the natural language for the problem, because the redistribution of indistinguishable photons among modes is a purely combinatorial counting exercise. Two consequences are drawn: any calculation built on Fock-basis matrix elements — optimizing success probabilities of linear-optics gates, averaging over unitaries — inherits the cost of computing permanents, and the probability interpretation of quantum mechanics yields short derivations of known matrix facts. The worked example is the entanglement power of a single beam splitter, which collapses to a polynomial in $per U$ alone.

What carries the argument

The permanent, $per Λ = ∑_{σ∈S_n}∏_i Λ_{iσ(i)}$, together with the repeated-index submatrix notation $Λ[Ω'|Ω]$. The permanent is the determinant without permutation signs, so no row or column operations simplify it; the only surviving rule is $per PΛD = per P · per Λ · per D$ for permutation and diagonal matrices. That absence of algebraic shortcuts is what carries the author's complexity moral: if the amplitude is a permanent and permanents admit no linear reduction to determinants (the Marcus–Minc result the paper cites), then there is no structural shortcut through linear algebra. The machinery is then applied to the linear entropy of one arm of a beam splitter, where group-averaging over the input states kills all phases and leaves an expression depending only on $per Λ$; for two qubit inputs the entanglement power is $P_L(U) = (3/64)[1-(per Λ)^2][13+9(per Λ)^2]$, maximal at $39/64$ when $per Λ = 0$, i.e. for a balanced beam splitter.

Extended reading notes

Core claim

The central result is an explicit formula for the matrix element of a unitary network between two Fock states. Writing $Ω=(1^{n_1},2^{n_2},...,N^{n_N})$ and $Ω'=(1^{m_1},...,N^{m_N})$ for the input and output mode lists with multiplicities, and $Λ[Ω'|Ω]$ for the submatrix of the network unitary obtained by repeating row $i$ exactly $m_i$ times and column $j$ exactly $n_j$ times, the paper obtains $$⟨m_1,...,m_N|U|n_1,...,n_N⟩ = (∏_i n_i!)^{-1/2}(∏_j m_j!)^{-1/2} per Λ[Ω'|Ω].$$ The multiplicities matter: many permutations of columns give the same diagonal, and the factorial prefactors are exactly the correction for that overcounting. The same construction gives the full state transformation as a sum over non-decreasing index sequences, and a projection formula when only some output modes are specified. From the special case of one photon in each of $N$ modes the author reads off $per Λ = ⟨1|...⟨1|U|1⟩...|1⟩$, so the permanent of a unitary is a probability amplitude and therefore obeys $|per Λ| ≤ 1$ — a quantum-mechanical derivation of a bound usually obtained from the Marcus–Newman theorem.

Load-bearing premise

The step from an identity to a hardness claim assumes that computing the permanent of the particular submatrices arising here — repeated rows and columns drawn from a unitary — admits no polynomial-time algorithm; the paper asserts this but does not cite the standard #P-completeness result for the permanent, nor prove hardness for this restricted class of matrices.

Editorial extensions

If this is right

Every Fock-basis matrix element of a passive linear optical network can be written as a permanent of a submatrix of the network unitary, so the amplitude table of such a device is a table of permanents.
Optimization problems over linear optical networks — maximizing gate success probabilities, averaging entanglement measures over input states — reduce to expressions in permanents and therefore share their computational cost.
For a single beam splitter acting on two $(N+1)$-dimensional input states, the group-averaged linear entropy depends only on absolute values of permanents of the beam splitter matrix, at most quartically.
The permanent of any unitary matrix has modulus at most one, because it equals the amplitude for an all-single-photon state to remain an all-single-photon state after the transformation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The identity makes the amplitude structure of linear optical networks the natural starting point for asking whether sampling from such a device can be simulated classically; the paper stops at matrix elements and does not push to sampling.
Because the amplitude depends on the input and output counts only through the repeated-index submatrix, one could try to classify networks by which families of permanents they can realize — an inverse problem the paper does not pose.
The single-beam-splitter result suggests that for a fixed interferometer the entanglement power is a low-degree function of a single scalar, which invites the guess that more general networks have entanglement power governed by a small number of permanent invariants.
Reading $|per Λ| ≤ 1$ as a statement about interference of $N$ indistinguishable photons suggests the bound is tight only in degenerate limits, which is a checkable numerical question about unitary families.

Original abstract

We develop an abstract look at linear optical networks from the viewpoint of combinatorics and permanents. In particular we show that calculation of matrix elements of unitarily transformed photonic multi-mode states is intimately linked to the computation of permanents. An implication of this remarkable fact is that all calculations that are based on evaluating matrix elements are generically computationally hard. Moreover, quantum mechanics provides simpler derivations of certain matrix analysis results which we exemplify by showing that the permanent of any unitary matrix takes its values across the unit disk in the complex plane.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,586 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.25 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.28 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We develop an abstract look at linear optical networks from the viewpoint of combinatorics and permanents. In particular we show that calculation of matrix elements of unitarily transformed photonic multi-mode states is intimately linked to the computation of permanents. An implication of this remarkable fact is that all calculations that are based on evaluating matrix elements are generically computationally hard. Moreover, quantum mechanics provides simpler derivations of certain matrix analysis results which we exemplify by showing that the permanent of any unitary matrix takes its values across the unit disk in the complex plane.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We develop an abstract look at linear optical networks from the viewpoint of combinatorics and permanents.	contribution 0.97	0.93	0.06	asserted 0.96
s1	In particular we show that calculation of matrix elements of unitarily transformed photonic multi-mode states is intimately linked to the computation of permanents.	result 0.80	0.88	0.29	asserted 1.00
s2	An implication of this remarkable fact is that all calculations that are based on evaluating matrix elements are generically computationally hard.	interpretation 0.99	0.61	0.66	asserted 0.98
s3	Moreover, quantum mechanics provides simpler derivations of certain matrix analysis results which we exemplify by showing that the permanent of any unitary matrix takes its values across the unit disk in the complex plane.	result 0.94	0.92	0.08	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
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references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
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R cross-paper relations

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References

Works cited (13)

[1]

P.W. Shor, in Proceedings of the Symposium on the Foun- dations of Computer Science, 1994, Los Alamitos (IEEE Computer Society Press, New York, 1994), pp. 124–134

View on Pith 1994
[2]

E. Knill, R. Laﬂamme, and G.J. Milburn, Nature (Lon- don) 409, 46 (2001); M. Koashi, T. Yamamoto, and N. Imoto, Phys. Rev. A 63, 030301 (2001)

View on Pith 2001
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S. Scheel, K. Nemoto, W.J. Munro, and P.L. Knight, Phys. Rev. A 68, 032310 (2003)

View on Pith 2003
[4]

S. Scheel, J. Pachos, E.A. Hinds, and P.L. Knight, Quan- tum gates and decoherence , to appear in Lecture Notes in Physics (Springer, Heidelberg, 2004), Preprint quant- ph/0403152

View on Pith 2004
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E.R. Caianiello, Nuovo Cimento 10, 1634 (1953); 13, 637 (1959); 14, 177 (1959)

View on Pith 1953
[6]

E.R. Caianiello, Combinatorics and Renormalization in Quantum Field Theory , Frontiers in Physics, Lecture Note Series (W.A. Benjamin, Reading, MA, 1973)

View on Pith 1973
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H. Minc, Permanents, Encyclopedia of Mathematics and its applications, Vol. 6 (Addison-Wesley, Reading, MA, 1978)

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[8]

M. Marcus and H. Minc, Illinois J. Math. 5, 376 (1961)

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[9]

W. Vogel, D.-G. Welsch, and S. Wallentowitz, Quantum Optics, An Introduction (Wiley-VCH, Berlin, 2001)

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M. Marcus and M. Newman, Ann. Math. 675, 47 (1962)

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J. Eisert and M.B. Plenio, J. Quant. Inf. and Comp. 1, 479 (2003)

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J. Eisert, K. Jacobs, P. Papadopoulos, and M.B. Ple- nio, Phys. Rev. A 62, 052317 (2000); P. Zanardi, C. Za- lka, and L. Faoro, Phys. Rev. A 62, 030301(R) (2000); M.A. Nielsen, C.M. Dawson, J.L. Dodd, A. Gilchrist, D. Mortimer, T.J. Osborne, M.J. Bremner, A.W. Har- row, and A. Hines, Phys. Rev. A 67, 052301 (2003); C.H. Bennett, A.W. Harrow, D.W. Leung, J...

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S. Scheel, L. Kn¨ oll, T. Opatrn´ y, and D.-G. Welsch, Phys. Rev. A 62, 043803 (2000); M.M. Wolf, J. Eisert, and M.B. Plenio, Phys. Rev. Lett. 90, 047904 (2003)

View on Pith 2000

Forward citations

Cited by 15 Pith papers

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cond-mat.dis-nn 2026-10 conditional novelty 7.0 of 10

The 2k-th moments of permanents and Hafnians of random matrices are computed by a replica-symmetric saddle point for k>2, including prefactors and first 1/N corrections.
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Derives a boson-fermion complementarity identity in linear interferometers that implies a previously unknown relation between |perm(A)|^2 and |det(A)|^2 for complex matrices, extending Muir's 19th-century identity.
Symmetry-resolved tree tensor network analysis of Bell-state discrimination with ancilla-assisted passive linear optics
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A new algorithmic upper bound for permanents via a permanental Schur inequality, but the proof has an off-by-one recurrence error and a flawed PSD lemma.
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For spin-S sampling, the number of sites m needed to match the hardness of Fock-state boson sampling scales as m ~ n^(1+3/(2S)), a large reduction from the spin-1/2 quartic scaling.
Unveiling Hierarchical Invariants in Multiphoton Linear Optics
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Purity in linear optics is shown to decompose into three conserved subspace contributions, experimentally confirmed for two-photon two-mode states.

How later papers used it

80 later papers cite it

Evidence: substantial. 79 of the citing papers share no author with it, and 6 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 108 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 5 papers with this label

The distance between the unitaries experimentally realized and the target ones can be estimated as $∑_i √(P_i^theo ·P_i^exp)$, where $P_i^exp$ is the experimental detection frequency for the $i$-th output configuration, while $P_i^theo$ is the one estimated based on the theory [ref] .

Experimental quantum-enhanced kernels on a photonic processor (2024) · Results · 0.93

How the literature uses it

77 arXiv papers cite this in 105 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 77 sentences without a confident label

Uses its method or tool 4 papers

Having obtained $U$, Alice calculates the probability of bosonic scattering events in the following way [ref] .

Photonic Boson Sampling in a Tunable Circuit · Introduction · 0.80
The distance between the unitaries experimentally realized and the target ones can be estimated as $∑_i √(P_i^theo ·P_i^exp)$, where $P_i^exp$ is the experimental detection frequency for the $i$-th output configuration, while $P_i^theo$ is the one estimated based on the theory [ref] .

Experimental quantum-enhanced kernels on a photonic processor · Results · 0.93
To this end, we have to evaluate the transition probabilities between many-particle eigenstates [ref] , first.

Interference of Identical Particles and the Quantum Work Distribution · Quantum work distribution for identical particles · 0.88

Compares with it 1 paper

Comparing the experimental results presented in Figs. [ref](b) and [ref](d) with the theoretical predictions in Figs. [ref](c) and [ref](e), we can see that the presented results agree well with the theoretical predictions [ref] .

Large Reconfigurable Quantum Circuits with SPAD Arrays and Multimode Fibers · Introduction · 0.88

Cites it as background 21 papers

Random quantum circuit sampling [ref] and Boson sampling [ref] are considered good computational tasks for demonstrating quantum advantage with current noisy quantum computers.

Race for Quantum Advantage using Random Circuit Sampling · Introduction · 0.89
Recall that the scattering amplitudes of Fock states evolved through a lossless interferometer are given by the permanent matrix function [ref] [equation] where $U_,$ is constructed from $U$ by repeating its $j$th row $m_j$ times and its $j$th column $n_j$ times for all $j ∈[M]$.

Threshold detection statistics of bosonic states · Fock state inputs · 0.81
Interestingly, the permanent appears in the expression of the output amplitudes of linear optical quantum computations with noninteracting bosons [ref] , as in the Boson Sampling model of quantum computation [ref] .

Quantum-inspired permanent identities · Introduction · 0.81

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

P.W. Shor, in Proceedings of the Symposium on the Foun- dations of Computer Science, 1994, Los Alamitos (IEEE Computer Society Press, New York, 1994), pp. 124–134

View on Pith 1994

[2] [2]

E. Knill, R. Laﬂamme, and G.J. Milburn, Nature (Lon- don) 409, 46 (2001); M. Koashi, T. Yamamoto, and N. Imoto, Phys. Rev. A 63, 030301 (2001)

View on Pith 2001

[3] [3]

S. Scheel, K. Nemoto, W.J. Munro, and P.L. Knight, Phys. Rev. A 68, 032310 (2003)

View on Pith 2003

[4] [4]

S. Scheel, J. Pachos, E.A. Hinds, and P.L. Knight, Quan- tum gates and decoherence , to appear in Lecture Notes in Physics (Springer, Heidelberg, 2004), Preprint quant- ph/0403152

View on Pith 2004

[5] [5]

E.R. Caianiello, Nuovo Cimento 10, 1634 (1953); 13, 637 (1959); 14, 177 (1959)

View on Pith 1953

[6] [6]

E.R. Caianiello, Combinatorics and Renormalization in Quantum Field Theory , Frontiers in Physics, Lecture Note Series (W.A. Benjamin, Reading, MA, 1973)

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