High-dimensional graphs convolution for quantum walks photonic applications

Alexander Alodjants; Dmitry Tsarev; Leonid Fedichkin; Roman Abramov

REVIEW 5 major objections 5 minor 41 references

High-dimensional graphs convolution for quantum walks photonic applications

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read High-dimensional graphs such as hypercubes, tori, and lattices can be replaced by much smaller weighted graphs—weighted lines or small lattices—without changing the continuous-time quantum walk dynamics at a chosen start and target node.

desk verdict The known exact reductions are correct but standard, the torus section has two clear errors, and the lattice-convolution claim rests on an unproved heuristic; the paper is worth a referee's time but not acceptance as written. read the letter →

arxiv 2507.15988 v1 pith:QLYGCZZN submitted 2025-07-21 quant-ph

Roman Abramov , Leonid Fedichkin , Dmitry Tsarev , Alexander Alodjants This is my paper · ORCID

classification quant-ph MSC 81P6805C5005C81

keywords quantumwalksgraphconvolutionhypercubeshypercyclestoruslatticegraphsphotonicwaveguidearrayseigenvaluedegeneracy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that certain high-dimensional graphs can be replaced by much smaller weighted graphs without changing the continuous-time quantum walk dynamics observed from a chosen start node to a chosen target node. The graphs in question are hypercubes, hypercycles (tori), and 2D lattices; the replacements are weighted line graphs, weighted lattices, and "ultimate" graphs obtained by a simple convolution recipe. If this is true, quantum-walk simulations on photonic waveguide arrays would need far fewer waveguides and, in circuit implementations, far fewer qubits, because only the distinct probability groups of the original graph need to be represented. The paper supports the claim with numerical hitting-probability curves that coincide for original and convoluted graphs, and with an eigenvalue observation: the convoluted graph's spectrum is the set of unique eigenvalues of the original.

What carries the argument

The central object is the weighted line or lattice obtained by lumping vertices that always carry equal probability—vertices with the same Hamming distance in a hypercube, or symmetric positions in a cycle or torus. The mechanism is the Cartesian-product identity $A(G\times H)=A(G)\otimes I + I\otimes A(H)$ for adjacency matrices, which lets a torus (Cartesian product of two cycles) be replaced by the Cartesian product of two weighted lines. For lattices, the load-bearing step is the heuristic rule that merges diagonal rectangles into lines and sets the new edge probability to $\sqrt{p_{\mathrm{side1}}^2+p_{\mathrm{side2}}^2}$. The supporting spectral claim is that duplicated eigenvalues in the original adjacency matrix signal equiprobable vertex groups, so the convoluted graph's eigenvalues are the unique eigenvalues of the original graph.

What would settle it

Apply the three-step lattice convolution to a $5\times 5$ or $6\times 6$ rectangular lattice (or to a torus with $k=8$) using the same sink rate $\Gamma=1$ and threshold (15), and compare the full time trace of target-node hitting probability with the original graph beyond the time window shown in Fig. 6; any persistent divergence before the threshold falsifies the preservation claim. A sharper test is to check whether the convoluted graph's set of unique eigenvalues always equals the original's—a counterexample lattice whose convoluted graph has an eigenvalue outside the original spectrum would break the stated criterion.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a graph-convolution method that preserves the dynamics of continuous-time quantum walks. For hypercubes, grouping vertices by Hamming distance gives a weighted line with couplings $\beta_{i,i+1}=\sqrt{i(D+1-i)}$; for cycles, a similar weighted line; for tori, a Cartesian product of these lines yields a weighted lattice; and a further three-step heuristic—dropping the lower triangle, merging diagonal rectangles into lines, and reassigning edge probabilities with $\sqrt{p_{\mathrm{side1}}^2+p_{\mathrm{side2}}^2}$—produces the smallest "ultimate" graph. Numerical solutions of the GKSL master equation with a sink node show that the target-node hitting probability as a function of time is the same on the original torus and on its two-step convolution, and on a $4\times 4$ lattice and its convoluted graph. The paper concludes that the proposed method preserves quantum-walk dynamics for both lattices and tori, and it also observes, without proving, that the eigenvalues of the convoluted graph are exactly the unique eigenvalues of the original graph, with the minimal number of nodes equal to the number of unique eigenvalues.

Load-bearing premise

The load-bearing premise is the lattice-convolution rule—discard the lower triangle, merge diagonal rectangles into lines, and set new edge probabilities by $\sqrt{p_{\mathrm{side1}}^2+p_{\mathrm{side2}}^2}$—which is asserted as a heuristic rather than derived from the Schrödinger equation or the transition-matrix equation; if this rule is not generally valid, the claimed dynamics preservation for lattices and tori does not follow.

Editorial extensions

If this is right

Quantum-walk circuits for hypercubes and tori can use only the number of qubits needed for the much smaller convoluted graph, because representing equiprobable vertex groups is sufficient.
For photonic waveguide implementations, a torus or lattice can be realized as a one-dimensional weighted chain or small lattice with tailored coupling coefficients, avoiding the need for high-dimensional physical geometry.
Classical and quantum hitting-time comparisons on the convoluted lattice reproduce the known pattern: classical walks win at short distances (d < 3), while quantum walks win for distances d at least 3.
Any graph that is a Cartesian or Kronecker product of line-like graphs can in principle be reduced by the same two-step convolution procedure.
The minimal faithful convolution size of a graph is the number of unique eigenvalues of its adjacency matrix, offering a spectral bound on how small the replacement graph can be.

Reading between the lines

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

If the eigenvalue criterion generalizes, it offers a purely spectral test for when a graph can be replaced by a smaller one: the number of distinct eigenvalues bounds the minimal representation size, independent of the walk's starting point; this is a stronger statement than the paper's numerical examples show.
The Euclidean-norm merging rule resembles the composition rule for independent paths in a classical diffusion picture, but it has no obvious quantum derivation; a natural test is to derive it from a requirement that the lumped graph's generator has the same eigenvalues on the relevant subspace, which would turn the heuristic into a theorem.
Because the method works by identifying vertex groups that remain equiprobable for all times, it is closely related to Markov-chain lumpability and could be tested on other symmetric graph families such as complete bipartite graphs or Johnson graphs, where the paper does not apply it.
The sink-based detection used here means the preserved quantity is the hitting probability at a single target, not the full unitary evolution; a reader should not infer that convoluted and original graphs are unitarily equivalent, only that the chosen detection statistics match.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The useful core here is small: the hypercube-to-weighted-line quotient (couplings sqrt(i(D+1-i))) and the even-cycle-to-weighted-line quotient (endpoint coupling sqrt(2)) are correct. They reproduce known symmetric-subspace reductions, and the numerical curves for the tested cases do coincide. The authors are also honest that the lattice rule is a heuristic and that their eigenvalue observations are unproved. That honesty is real credit.

But the paper's central claim goes much further. The torus section contains two outright errors: Eq. (8) writes the torus adjacency as a Kronecker product A_C ⊗ A_C, which contradicts the Cartesian-product formula Eq. (10) they use two lines later; and Eq. (20) lists eight eigenvalues as the spectrum of the k=8 torus, which has 64 vertices. These are not typos in the margins—they sit in the derivation of the claimed mapping.

The lattice-convolution rule is the load-bearing piece for the claimed generality, and it is not derived. Merging probabilities via sqrt(p_side1^2 + p_side2^2) has no basis in the Schrödinger equation or in the classical transition-matrix equation. In a CTQW the edge weights are Hamiltonian couplings, and an exact reduction should come from an equitable partition or a symmetric-subspace restriction. The authors themselves say in Sec. 4.3 that the equiprobability grouping is for verification, not derivation, which undercuts their own conclusion in Sec. 4.2 that the method preserves QW dynamics for both lattices and tori. The numerical support is one 6x6 torus and one 4x4 lattice, each with a single start/target pair, with no code, data, or error bars. That cannot carry the claimed generality.

The eigenvalue-minimality observations in Sec. 5 are explicitly unproved and are false in general, so they should not be stated as observations. The correct statement is the familiar one: an equitable partition can reduce the graph, and the reduced graph's spectrum is a subset of the original's—but that is standard spectral graph theory, not a new criterion.

Who gets value here? Someone wanting a compact review of hypercube and cycle symmetric-subspace reductions might find the early sections useful. The lattice convolution, as presented, is not reliable enough to use as a method.

My recommendation: send it to peer review, but the referee should require the Eq. (8) and Eq. (20) errors to be fixed, and the lattice heuristic to be either derived or explicitly downgraded to a conjecture with a much narrower claim. If the authors can do that, the paper could become a modest but useful note. As written, it overreaches.

Referee Report

5 major / 5 minor

Summary. The paper proposes 'graph convolution' reductions for continuous-time quantum walks on hypercubes, cycles, toruses, and lattices. The stated central claim is that the reduced (smaller) graphs preserve the quantum walk dynamics, so that a larger graph can be replaced by a smaller weighted graph for photonic waveguide simulation, saving qubits. The exact reductions for hypercubes and cycles are based on symmetric-subspace quotients, while the lattice reduction relies on a heuristic algorithm in Sec. 3.2. Numerical simulations of hitting probabilities are presented in Sec. 4, and Sec. 5 proposes an eigenvalue criterion for minimal graph size.

Significance. If the central claim were established, the resource reduction would be practically relevant: a D-dimensional hypercube would require only D+1 rather than 2^D waveguide modes, and analogous reductions for cycles and lattices would simplify continuous-time quantum walk simulations. The hypercube and single-cycle mappings are indeed exact quotient constructions; the coupling coefficients in Eq. (3) coincide with the known equitable partition of the hypercube, and the line graph obtained from C_8 has exactly the unique eigenvalues of C_8 as shown in Sec. 5. However, the manuscript's extension to toruses and lattices is not established: the adjacency-matrix relation Eq. (8) is incorrect, the Sec. 3.2 lattice rule is an unproved postulate, and the numerical support consists of two examples with a single start-target pair each. The manuscript also provides no code or data files. As written, the paper is a useful observation about exact quotient reductions for hypercubes and cycles combined with unsupported generalizations.

major comments (5)

[Sec. 3.1.1, Eq. (8)] Equation (8) writes the adjacency matrix of the D=2 hypercycle as A_Circle^k ⊗ A_Circle^k, but the torus C_k × C_k is the Cartesian product, whose adjacency is A(G×H) = A(C)⊗I + I⊗A(C), as correctly stated in Eq. (10). Since the manuscript uses Eq. (10) two lines later to derive the mapped Hamiltonian (12), Eq. (8) is internally inconsistent: the graph whose adjacency is a Kronecker product is not the graph whose dynamics are simulated. This invalidates the derivation of the lattice mapping as a representation of the torus.
[Sec. 5, Eq. (20)] The list 'eigenvalues original = [2, -2, 1.4142, 1.4142, 0, 0, -1.4142, -1.4142]' is the spectrum of the cycle C_8, not of the torus C_8 × C_8. The torus has 64 vertices and its eigenvalues are λ_i + λ_j, where λ_i and λ_j are the eigenvalues of C_8; it has many more distinct values (e.g., 4, 2±√2, 2√2, ...). Consequently, the observation that 'the eigenvalues of the convoluted graph are only unique eigenvalues of the original one' is false for the torus as stated, and the claimed criterion for minimal graph size is not supported.
[Sec. 3.2, lattice convolution algorithm] The lattice convolution rule is introduced as a heuristic with no derivation. Step 3 assigns merged edge weights by sqrt(p_side1^2 + p_side2^2), where p_side1 and p_side2 are classical transition probabilities. In the continuous-time quantum walk, the Hamiltonian edge weights are amplitudes/couplings, not probabilities, and the classical transition matrix does not share the spectral structure of the Hamiltonian in the way required by the paper; a Euclidean norm over classical probabilities has no known quantum-mechanical justification. Since the 'ultimate graph' produced by this rule is used in Figs. 5 and 6 to support the central claim that quantum walk dynamics are preserved, this is a load-bearing unproved postulate rather than a derived reduction.
[Sec. 4.2, Fig. 6] The numerical evidence for the preservation claim consists of one torus with D=2, k=6 and one 4×4 lattice, each tested with a single starting and target pair. No error bars, statistics over random pairs, code, or data files are provided, and the hitting-probability curves are compared only visually. Two examples do not establish the general statement that 'the proposed method of graph convolution preserves QWs dynamics; it is valid for both lattice and torus.'
[Sec. 4.3, Fig. 8(b)] The criterion for the 'smallest graph' is nearly tautological: the convoluted graph is said to be the smallest when its node count equals the number of equiprobable groups, and the existence of such groups is inferred from the outcome probabilities of the quantum walk on the original graph. Because the authors state that this technique can be used 'for verification, but not for derivation of the convoluted graph,' it does not provide a constructive or falsifiable characterization of minimality.

minor comments (5)

[Eq. (6)] Equation (6) contains a mismatched parenthesis: the line reads '+ √6 (|3⟩ ⟨2| + (|4⟩ ⟨3| + H.C.', which should be closed and grouped consistently.
[Secs. 2.2 and 3.1] The symbol Hhc→line is used for both the hypercube Hamiltonian in Eq. (3) and the hypercycle Hamiltonian in Eq. (7), which is confusing because the graphs and couplings are different.
[Sec. 4.2, Fig. 7] The text says 300 simulations were performed on both the original and convoluted graphs, but Fig. 7 plots only the fastest result of each walk; the relationship between the 300 runs and the plotted points, including the treatment of failed walks, should be specified.
[Sec. 5] The two observations at the end of Sec. 5 are presented without proof, and the authors acknowledge this; they should be explicitly labeled as conjectures, since they are used as a criterion in Sec. 4.3.
[Data Availability] The Data Availability statement says that all data are available within the article, but no dataset, simulation code, or numerical parameters are provided; this makes the numerical results irreproducible as presented.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor tautological minimality check; the central hypercube/cycle reductions are independent spectral quotients, so no significant circularity.

self definitional [Sec. 4.3 'Minimal mapping graph', paragraphs 3-4 (after Fig. 8)]
"To preserve the total probability property, probabilities of merging nodes are summed; the sum of all probabilities is equal to one. By repeating this process, we can identify the minimum number of nodes required to represent the original graph, as demonstrated in Fig. 8(b). ... By comparing Fig. 8(b) and Fig. 5, we can conclude that the convoluted representation of a lattice is the smallest graph that maintains the dynamical features of QWs of the original lattice if the number of nodes in the convoluted graph is equivalent to the number of equiprobable node groups in the original graph."

The 'minimum number of nodes' is effectively defined as the number of equiprobable groups, so the condition 'the convoluted graph has as many nodes as equiprobable groups' is satisfied by construction rather than by an independent lower-bound proof. The paper itself concedes this grouping 'can only be used for verification, but not for derivation of the convoluted graph.' Thus the minimality conclusion is a restatement of the counting criterion, not a derived result about dynamics preservation.

full rationale

Most of the derivation chain is not circular. The hypercube reduction (Eq. (3), beta_{i,i+1} = sqrt(i(D+1-i))) and the D=1 cycle reduction (Eq. (7) with boundary couplings sqrt(2)) are exact symmetric-subspace quotients: the spectra in Sec. 5 confirm that the mapped graph's eigenvalues are the distinct eigenvalues of the original, and the Hamiltonians are obtained by projection rather than by fitting. The torus/lattice first step (Eqs. (8)-(12)) follows from the Cartesian-product identity, so it has independent content. The lattice-convolution rule of Sec. 3.2 (discard the lower triangular; merge diagonal rectangles with sqrt(p_side1^2 + p_side2^2)) is a heuristic and is load-bearing for the broad claim that arbitrary lattice QW dynamics are preserved; however it is not circular in the sense of reducing to its inputs by construction, because the numerical comparison in Fig. 6 is an independent check, albeit a weak one (one 4x4 lattice, one 6x6 torus, no code or data). The one genuinely circular move is in Sec. 4.3: 'minimum number of nodes' is defined as the number of equiprobable groups, and then the convoluted graph is declared minimal when its node count equals that number. The paper itself states the grouping 'can only be used for verification, but not for derivation,' so the minimality conclusion is a restatement of the counting criterion. Self-citations ([22], [24], [37]) are used for sink detection, threshold pth = 1/log(n), and context, but the preservation claim does not depend on them in a load-bearing way. Overall score 2.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The exact hypercube, cycle, and torus reductions rest on standard unitary equivalence between the graph Hamiltonian and its quotient; this is a domain assumption the paper never states explicitly. The lattice convolution introduces an ad hoc merge rule, and the eigenvalue criterion is an ad hoc-to-paper postulate. No new physical entities are introduced, and the only free parameters are the sink rate, detection threshold, and the merge-rule exponent.

free parameters (3)

Sink coupling rate Γ = 1
Set by hand in Sec. 4.1 ('Below we examine the case of Γ = 1'); affects the GKSL hitting-time dynamics.
Detection threshold p_th = 1/log(n)
Borrowed from ref. [22] in Sec. 4.1; defines when a walker counts as having hit the target and differs between original and convoluted graphs.
Lattice merge rule norm = sqrt(p1^2 + p2^2)
Ad hoc probability combination chosen in Sec. 3.2 step 3; no derivation from quantum or classical random-walk equations.

assumptions (3)

domain assumption Mapping a graph to its distance-orbit quotient preserves the continuous-time quantum walk for symmetric initial and target conditions.
Implicit in Secs. 2 and 3; exact only when vertices in the same orbit have equal probability, a condition the paper does not state as a limitation.
domain assumption The GKSL master equation with one Lindblad sink models photon loss into a detector and defines the hitting probability.
Invoked in Sec. 4.1 to simulate detection; the sink is traced over, and non-Markovian effects are declared out of scope.
ad hoc to paper The smallest graph preserving quantum walk dynamics has as many nodes as the number of unique eigenvalues of the original adjacency matrix.
Proposed without proof in Sec. 5 ('we propose (without proving) two observations') and false in general (duplicate eigenvalues do not imply lumpability).

how reviews work

0 comments

Cite this review

Pith. "Pith review of High-dimensional graphs convolution for quantum walks photonic applications." pith.science (2026). https://pith.science/paper/QLYGCZZN

@misc{pith2026250715988,
  author       = {Pith},
  title        = {Pith review of: High-dimensional graphs convolution for quantum walks photonic applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QLYGCZZN}},
  note         = {Machine review of arXiv:2507.15988}
}

read the original abstract

Quantum random walks represent a powerful tool for the implementation of various quantum algorithms. We consider a convolution problem for the graphs which provide quantum and classical random walks. We suggest a new method for lattices and hypercycle convolution that preserves quantum walk dynamics. Our method is based on the fact that some graphs represent a result of Kronecker's product of line graphs. We support our methods by means of various numerical experiments that check quantum and classical random walks on hypercycles and their convolutions. Our findings may be useful for saving a significant number of qubits required for algorithms that use quantum walk simulation on quantum devices.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

[1]

Kac, M. (1947). Random walk and the theory of Brownian motion Am. Math. Mon. 54 369–91 14

work page 1947
[2]

Kutner, R., Masoliver, J. (2017). The continuous time random walk, still trendy: fifty-year history, state of art and outlook. Eur. Phys. J. B 90, 50

work page 2017
[3]

Bartumeus, F., da Luz, M. G. E., Viswanathan, G. M., & Catalan, J. (2005). Animal search strategies: a quantitative random-walk analysis. Ecology, 86(11), 3078-3087

work page 2005
[4]

Xia, F., Liu, J., Nie, H., Fu, Y., Wan, L., & Kong, X. (2019). Random walks: A review of algorithms and applications. IEEE Transactions on Emerging Topics in Computational Intelligence, 4(2), 95-107

work page 2019
[5]

Kempe, J. (2003). Quantum random walks: an introductory overview. Contem- porary Physics, 44(4), 307-327

work page 2003
[6]

Childs, A. M. (2009). Universal computation by quantum walk. Physical review letters, 102(18), 180501

work page 2009
[7]

Venegas-Andraca, S. E. (2012). Quantum walks: a comprehensive review. Quan- tum Information Processing, 11(5), 1015-1106

work page 2012
[8]

Ambainis, A. (2003). Quantum walks and their algorithmic applications. Inter- national Journal of Quantum Information, 1(04), 507-518

work page 2003

Show all 41 references

[9]

K., Melnikov, A

Madhu, A. K., Melnikov, A. A., Fedichkin, L. E., Alodjants, A. P., & Lee, R. K. (2023). Quantum walk processes in quantum devices. Heliyon, 9(3)

2023
[10]

Portugal, R. (2013). Quantum walks and search algorithms (Vol. 19). New York: Springer

2013
[11]

Biamonte, J., Wittek, P., Pancotti, N., Rebentrost, P., Wiebe, N., & Lloyd, S. (2017). Quantum machine learning. Nature, 549(7671), 195-202

2017
[12]

Melnikov, A., Kordzanganeh, M., Alodjants, A., & Lee, R. K. (2023). Quantum machine learning: From physics to software engineering. Advances in Physics: X, 8(1), 2165452

2023
[13]

(2001, July)

Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., & Watrous, J. (2001, July). One-dimensional quantum walks. In Proceedings of the thirty-third annual ACM symposium on Theory of computing (pp. 37-49)

2001
[14]

(2001, July)

Aharonov, D., Ambainis, A., Kempe, J., & Vazirani, U. (2001, July). Quantum walks on graphs. In Proceedings of the thirty-third annual ACM symposium on Theory of computing (pp. 50-59)

2001
[15]

Solenov, D., & Fedichkin, L. (2006). Continuous-time quantum walks on a cycle graph. Physical Review A, 73(1), 012313. 15

2006
[16]

Krovi, H., & Brun, T. A. (2006). Hitting time for quantum walks on the hypercube. Physical Review A, 73(3), 032341

2006
[17]

Kempe, J. (2005). Discrete quantum walks hit exponentially faster. Probability theory and related fields, 133(2), 215-235

2005
[18]

Makmal, A., Zhu, M., Manzano, D., Tiersch, M., & Briegel, H. J. (2014). Quantum walks on embedded hypercubes. Physical Review A, 90(2), 022314

2014
[19]

Santos, R. A. M., & Portugal, R. (2010). Quantum hitting time on the complete graph. International Journal of Quantum Information, 8(05), 881-894

2010
[20]

M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., & Spielman, D

Childs, A. M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., & Spielman, D. A. (2003). Exponential algorithmic speedup by a quantum walk. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing (pp. 59-68)

2003
[21]

Preskill, J. (2018). Quantum Computing in the NISQ era and beyond. Quantum, 2, 79

2018
[22]

A., Fedichkin, L

Melnikov, A. A., Fedichkin, L. E., Alodjants, A. (2019). Predicting quantum advantage by quantum walk with convolutional neural networks. New Journal of Physics, 21(12), 125002

2019
[23]

A., Fedichkin, L

Gurvitz, S. A., Fedichkin, L. E., Mozyrsky, D., Berman, G. P.(2003). Relaxation and the Zeno Effect in Qubit Measurements. Phys. Rev. Lett. 91, 066801

2003
[24]

A., Fedichkin, L

Melnikov, A. A., Fedichkin, L. E., Lee, R. K., Alodjants, A. P. (2020). Machine learning transfer efficiencies for noisy quantum walks. Advanced Quantum Technologies, 3(4), 1900115

2020
[25]

L., Wang, J

Douglas, B. L., Wang, J. B. (2019). Efficient quantum circuit implementation of quantum walks Phys. Rev. A 79, 052335

2019
[26]

Portugal, R., Moqadam, J. K. (2022). Implementation of Continuous-Time Quantum Walks on Quantum Computers. arXiv preprint arXiv:2212.08889

2022 arXiv
[27]

Choi, V. (2008). Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Information Processing, 7, 193-209

2008
[28]

H., Zhou, X

Peruzzo, A., McClean, J., Shadbolt, P., Yung, M. H., Zhou, X. Q., Love, P. J., Aspuru-Guzik, A., O’Brien, J. L. (2014). A variational eigenvalue solver on a photonic quantum processor. Nature communications, 5(1), 4213

2014
[29]

B., Lahini, Y., Pozzi, F., Sorel, M., Morandotti, R., Silberberg, Y

Perets, H. B., Lahini, Y., Pozzi, F., Sorel, M., Morandotti, R., Silberberg, Y. (2008). Realization of quantum walks with negligible decoherence in waveguide lattices. Physical review letters, 100(17), 170506. 16

2008
[30]

C., Rodr ´ ıguez-Lara, B.M., Szameit, A., Christodoulides, D

Perez-Leija, A., Keil, R., Kay, A., Moya-Cessa, H., Nolte, S., Kwek, L. C., Rodr ´ ıguez-Lara, B.M., Szameit, A., Christodoulides, D. N. (2013). Coherent quantum transport in photonic lattices. Physical Review A, 87(1), 012309

2013
[31]

Y., He, T.S., Feng, Z., Gao, J., Sun, Ke, Wang, C.Y., Lai, P.C., Xu, X.Y., Wang, Y., Qiao, L.F., Yang, A.L., Jin, X

Tang, H., Di Franco, C., Shi, Z. Y., He, T.S., Feng, Z., Gao, J., Sun, Ke, Wang, C.Y., Lai, P.C., Xu, X.Y., Wang, Y., Qiao, L.F., Yang, A.L., Jin, X. M. (2018). Experimental quantum fast hitting on hexagonal graphs. Nature Photonics, 12(12), 754-758

2018
[32]

J., Wang, K., Dovgiy, A

Maczewsky, L. J., Wang, K., Dovgiy, A. A., Miroshnichenko, A. E., Moroz, A., Ehrhardt, M., Heinrich, M., Christodoulides, D.N., Szameit, A., & Sukhorukov, A. A. (2020). Synthesizing multi-dimensional excitation dynamics and localization transition in one-dimensional lattices. ...

2020
[33]

Yu, S., Piao, X., Hong, J., Park, N. (2016). Interdimensional optical isospectrality inspired by graph networks. Optica, 3(8), 836-839

2016
[34]

Dennis, E., Kitaev, A., Landahl, A., Preskill, J. (2002). Topological quantum memory. J. Math. Phys. 43(9), 4452–4505

2002
[35]

Pedrocchi, F.L., Hutter, A., Wootton, J.R., Loss, D. (2013). Enhanced thermal stability of the toric code through coupling to a bosonic bath. Phys. Rev. A 88, 062313

2013
[36]

D., Li, J., Li, H., Johnson, T

Lu, D., Biamonte, J. D., Li, J., Li, H., Johnson, T. H., Bergholm, V., Faccin, M., Zimbor´ as, A., Laflamme, R., Baugh, J., Lloyd, S. (2016). Chiral quantum walks. Physical Review A, 93(4), 042302

2016
[37]

Melnikov, A.A., Alodjants, A.P., Fedichkin, L.E. (2021). Tunneling in Double- Layer Optical Waveguides as Quantum Walks on Graphs. Proceedings of the Steklov Institute of Mathematics, 313, 142

2021
[38]

Skryabin, N., Kalinkin, A., Dyakonov, I., Kulik, S. (2020). Femtosecond Laser Written Depressed-Cladding Waveguide 2 × 2, 1 × 2, and 3 × 3 Directional Couplers in Tm3+:YAG Crystal. Micromachines, 11, 1

2020
[39]

Chen, F., Aldana, J. (2014). Optical waveguides in crystalline dielectric materials produced by femtosecond-laser micromachining. Laser Photonics Rev., 8, 251

2014
[40]

R., Nation, P

Johansson, J. R., Nation, P. D., & Nori, F. (2012). QuTiP: An open-source Python framework for the dynamics of open quantum systems. Computer Physics Communications, 183(8), 1760-1772

2012
[41]

Manzano, D. A. (2020). A short introduction to the Lindblad master equation. AIP Adv. 10, 025106. 17

2020

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Kac, M. (1947). Random walk and the theory of Brownian motion Am. Math. Mon. 54 369–91 14

work page 1947

[2] [2]

Kutner, R., Masoliver, J. (2017). The continuous time random walk, still trendy: fifty-year history, state of art and outlook. Eur. Phys. J. B 90, 50

work page 2017

[3] [3]

Bartumeus, F., da Luz, M. G. E., Viswanathan, G. M., & Catalan, J. (2005). Animal search strategies: a quantitative random-walk analysis. Ecology, 86(11), 3078-3087

work page 2005

[4] [4]

Xia, F., Liu, J., Nie, H., Fu, Y., Wan, L., & Kong, X. (2019). Random walks: A review of algorithms and applications. IEEE Transactions on Emerging Topics in Computational Intelligence, 4(2), 95-107

work page 2019

[5] [5]

Kempe, J. (2003). Quantum random walks: an introductory overview. Contem- porary Physics, 44(4), 307-327

work page 2003

[6] [6]

Childs, A. M. (2009). Universal computation by quantum walk. Physical review letters, 102(18), 180501

work page 2009

[7] [7]

Venegas-Andraca, S. E. (2012). Quantum walks: a comprehensive review. Quan- tum Information Processing, 11(5), 1015-1106

work page 2012

[8] [8]

Ambainis, A. (2003). Quantum walks and their algorithmic applications. Inter- national Journal of Quantum Information, 1(04), 507-518

work page 2003