Quantum-Inspired Hamiltonian Optimization, Stochastic Tensor Networks and Adaptive Congestion Routing for Large-Scale QKD Networks

Jose Luis Rosales

arxiv: 2605.27425 · v1 · pith:NTEUNEOQnew · submitted 2026-05-19 · 🪐 quant-ph

Quantum-Inspired Hamiltonian Optimization, Stochastic Tensor Networks and Adaptive Congestion Routing for Large-Scale QKD Networks

Jose Luis Rosales This is my paper

Pith reviewed 2026-06-30 17:48 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum key distributionnetwork routingHamiltonian optimizationtensor networksstochastic annealingadaptive congestion controlquantum-inspired algorithms

0 comments

The pith

QKD network routing can be optimized by evolving configurations under an effective Hamiltonian using stochastic annealing and tensor networks.

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

This paper develops a quantum-inspired framework for adaptive routing in Quantum Key Distribution networks. It models the network as a stochastic graph evolving under a Hamiltonian that includes terms for latency, key generation rate, congestion, risk, and capacity. Optimization is performed using Metropolis annealing with local updates and compressed via stochastic boundary matrix product states. The approach seeks to handle dynamic traffic while satisfying security and performance constraints in large-scale setups. If successful, it would allow efficient orchestration of QKD networks without exhaustive search of routing options.

Core claim

The communication network is represented as a stochastic interacting graph whose routing configurations evolve under an effective Hamiltonian containing latency, keyrate, congestion, risk and capacity terms. The resulting optimization landscape is explored through a stochastic Metropolis annealer based on incremental local Hamiltonian updates, and a stochastic boundary-MPS tensor-network approximation that compresses the low-energy routing sector through thermal branch selection.

What carries the argument

Effective Hamiltonian on a stochastic interacting graph, explored via Quantum Monte Carlo inspired annealing and stochastic Tensor-Network State compression using boundary-MPS.

If this is right

Joint optimization of latency, secret key rate, congestion, capacity and security constraints under dynamic traffic.
Scalable exploration of routing configurations for large QKD networks.
Compression of low-energy routing states using tensor networks.
Bridge to quantum-native routing systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method might generalize to other constrained network optimization problems in classical communications.
Implementing the annealer on quantum hardware could further speed up the process for very large instances.
Validation on real QKD testbeds would be needed to confirm practical gains over existing routing protocols.

Load-bearing premise

The communication network can be represented as a stochastic interacting graph whose routing configurations evolve under an effective Hamiltonian containing latency, keyrate, congestion, risk and capacity terms.

What would settle it

Running the proposed Metropolis annealer and tensor network method on a small simulated QKD network and checking if the found routes achieve higher average key rates or lower congestion than standard shortest-path or load-balancing algorithms would test the claim; failure to do so would falsify it.

read the original abstract

Quantum Key Distribution (QKD) networks require routing methodologies capable of jointly optimizing latency, secret key generation rate, congestion, finite capacity and operational security constraints under dynamically evolving traffic conditions. In this work we introduce a quantum-inspired optimization framework for adaptive multi-demand routing in QKD communication networks based on effective Hamiltonian modelling, Quantum Monte Carlo inspired annealing and stochastic Tensor-Network State (TNS) compression. The communication network is represented as a stochastic interacting graph whose routing configurations evolve under an effective Hamiltonian containing latency, keyrate, congestion, risk and capacity terms. The resulting optimization landscape is explored through two complementary approaches: a stochastic Metropolis annealer based on incremental local Hamiltonian updates, and a stochastic boundary-MPS tensor-network approximation that compresses the low-energy routing sector through thermal branch selection. The resulting framework establishes a scalable bridge between QKD network orchestration, statistical-physics-inspired optimization, tensor-network compression and future quantum-native routing systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Proposal for Hamiltonian-based routing in QKD nets using annealing and boundary MPS, but no validation or topology analysis provided.

read the letter

The main thing to know is that this paper maps QKD routing to an effective Hamiltonian whose terms cover latency, key rate, congestion, risk and capacity, then proposes to sample good configurations with a Metropolis-style annealer plus stochastic boundary-MPS compression of the low-energy states. That is the entire contribution.

The framing itself is reasonable. QKD networks really do face a multi-objective routing problem under changing load, and folding the objectives into one energy function gives a compact way to state the task. The two solvers are sketched at the level of local Hamiltonian updates and thermal branch selection in the tensor network. Those are established techniques from statistical mechanics and tensor networks, so the combination is a legitimate extension to this application.

The soft spots are substantial and central. The abstract supplies no derivations, no small examples, no bond-dimension scaling, and no comparison against standard routing heuristics. The stress-test concern holds: boundary MPS (even in its thermal form) stays efficient only on graphs with low treewidth or limited entanglement range. Real QKD networks are general meshes or arbitrary topologies, so nothing in the text shows why the compression will remain cheap or why the method scales. Without that, the claim of a "scalable bridge" is unsupported.

Because the materials contain only the abstract, there is also no code, no data, and no independent check. The work is therefore a modeling proposal rather than a result.

This is for the narrow set of people already doing optimization for quantum communication networks who might want to experiment with physics-inspired heuristics. A reader looking for a theorem, a benchmark, or a reproducible method will find nothing to cite or build on.

I would not send it to peer review. The idea is internally coherent, but there is no evidence that the proposed solvers deliver on the scalability promise, so referees would have nothing concrete to assess.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces a quantum-inspired framework for adaptive multi-demand routing in QKD networks. The network is modeled as a stochastic interacting graph whose routing configurations evolve under an effective Hamiltonian that includes latency, keyrate, congestion, risk and capacity terms. Two complementary solvers are proposed: a stochastic Metropolis annealer based on local Hamiltonian updates, and a stochastic boundary-MPS tensor-network approximation that compresses the low-energy routing sector. The central claim is that this combination establishes a scalable bridge between QKD orchestration, statistical-physics optimization, tensor-network compression and future quantum-native routing systems.

Significance. If the claimed scalability and correctness of the boundary-MPS compression for general QKD topologies can be established, the work would supply a concrete statistical-physics route to joint optimization of latency, key rate and congestion under dynamic traffic, an area where conventional integer-linear-programming approaches scale poorly. The explicit mapping to an effective Hamiltonian and the use of thermal tensor-network methods are potentially reusable beyond QKD.

major comments (1)

[Abstract] Abstract, paragraph 2: the central scalability claim rests on the assertion that a stochastic boundary-MPS approximation 'compresses the low-energy routing sector' for an arbitrary 'stochastic interacting graph.' Boundary-MPS (and its thermal variants) are known to remain polynomial-cost only when the underlying interaction graph has low treewidth or can be arranged as a chain/ladder with limited entanglement range. The manuscript provides no restriction of the network topology, no bond-dimension scaling analysis, and no numerical evidence that metropolitan-scale QKD meshes (typical 2-D or higher treewidth graphs) remain compressible. This assumption is load-bearing for the 'scalable bridge' claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below and will revise the manuscript accordingly to strengthen the presentation of the boundary-MPS scalability claim.

read point-by-point responses

Referee: [Abstract] Abstract, paragraph 2: the central scalability claim rests on the assertion that a stochastic boundary-MPS approximation 'compresses the low-energy routing sector' for an arbitrary 'stochastic interacting graph.' Boundary-MPS (and its thermal variants) are known to remain polynomial-cost only when the underlying interaction graph has low treewidth or can be arranged as a chain/ladder with limited entanglement range. The manuscript provides no restriction of the network topology, no bond-dimension scaling analysis, and no numerical evidence that metropolitan-scale QKD meshes (typical 2-D or higher treewidth graphs) remain compressible. This assumption is load-bearing for the 'scalable bridge' claim.

Authors: We agree that the current manuscript does not supply an explicit treewidth analysis, bond-dimension scaling study, or numerical benchmarks on 2-D metropolitan meshes, and that the abstract presents the method for arbitrary stochastic interacting graphs. In the revised version we will add a dedicated subsection (in Methods) that (i) discusses the effective treewidth arising from the locality of QKD link constraints and demand routing variables, (ii) reports preliminary bond-dimension scaling results on small 2-D grid instances, and (iii) qualifies the abstract and introduction to note that the approach is expected to remain efficient when the routing configuration graph admits moderate treewidth or can be suitably embedded. These additions will be supported by new figures showing compression ratios versus bond dimension for representative QKD topologies. We view this as a clarification and strengthening rather than a change to the core framework. revision: yes

Circularity Check

0 steps flagged

No circularity: framework proposal is self-contained modeling choice

full rationale

The provided text introduces a modeling choice (network as stochastic interacting graph with effective Hamiltonian) and two optimization methods (Metropolis annealer, boundary-MPS compression) without any derivation chain, equations, fitted parameters, or self-citations that reduce a claimed result to its inputs by construction. The statement that the framework 'establishes a scalable bridge' is a direct description of the proposed approach itself rather than a prediction or theorem derived from prior steps. No load-bearing self-citation, ansatz smuggling, or renaming of known results is present. This is the normal case of a modeling paper whose central content does not collapse to tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; therefore the ledger is necessarily incomplete and records only the modeling premises explicitly stated in the abstract.

free parameters (1)

weights of latency, keyrate, congestion, risk and capacity terms
The effective Hamiltonian is stated to contain these five classes of terms; their relative strengths are not given numerical values or fitting procedures in the abstract.

axioms (1)

domain assumption The communication network can be represented as a stochastic interacting graph whose routing configurations evolve under an effective Hamiltonian
Explicitly invoked in the second paragraph of the abstract as the starting point for both the annealing and tensor-network approaches.

pith-pipeline@v0.9.1-grok · 5688 in / 1372 out tokens · 27046 ms · 2026-06-30T17:48:35.709346+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

41 extracted references · 33 canonical work pages · 2 internal anchors

[1]

Quantum cryptogra- phy: Public key distribution and coin tossing,

C. H. Bennett and G. Brassard, “Quantum cryptogra- phy: Public key distribution and coin tossing,” Proceed- ings IEEE International Conference on Computers, Sys- tems and Signal Processing, Bangalore, India, pp. 175– 179 (1984)

1984
[2]

Quantum cryptography based on Bell’s theorem,

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett.67, 661–663 (1991). DOI: 10.1103/PhysRevLett.67.661

work page doi:10.1103/physrevlett.67.661 1991
[3]

Quan- tum cryptography,

N. Gisin, G. Ribordy, W. Tittel and H. Zbinden, “Quan- tum cryptography,” Rev. Mod. Phys.74, 145–195 (2002). DOI: 10.1103/RevModPhys.74.145

work page doi:10.1103/revmodphys.74.145 2002
[4]

Advances in Quantum Cryptogra- phy,

S. Pirandolaet al., “Advances in Quantum Cryptogra- phy,” Adv. Opt. Photon.12, 1012–1236 (2020). DOI: 10.1364/AOP.361502

work page doi:10.1364/aop.361502 2020
[5]

The engineer- ing of software-defined quantum key distribution net- works,

A. Aguado, V. López, D. R. López, M. Peev, A. Pastor- Perales, J. Folgueira and V. Martín, “The engineer- ing of software-defined quantum key distribution net- works,” IEEE Commun. Mag.57, 20–26 (2019). DOI: 10.1109/MCOM.2019.1800763

work page doi:10.1109/mcom.2019.1800763 2019
[6]

The Madrid Quantum Network: A Quantum-Classical Integrated Infrastructure,

V. Martin, A. Aguado, P. Salas, A. L. Sanz, J. P. Brito, D. R. Lopez, V. Lopez, A. Pastor-Perales, A. Poppe and M. Peev, “The Madrid Quantum Network: A Quantum-Classical Integrated Infrastructure,” OSA Ad- vanced Photonics Congress (2019). DOI: 10.1364/NET- WORKS.2019.QTW3E.5

work page doi:10.1364/net- 2019
[7]

Quantum Aware SDN Nodes in the Madrid Quantum Network,

V. Martin, A. Aguado, J. P. Brito, A. L. Sanz, P. Salas, D. R. Lopez, V. Lopez, A. Pastor-Perales, A. Poppe and M. Peev, “Quantum Aware SDN Nodes in the Madrid Quantum Network,” ICTON 2019. DOI: 10.1109/IC- TON.2019.8840338

work page doi:10.1109/ic- 2019
[8]

A Components Based Framework for Quan- tum Key Distribution Networks,

V. Martin, D. Lopez, A. Aguado, J. P. Brito, J. Setien, P. Salas, C. Escribano, V. Lopez, A. Pastor-Perales and M. Peev, “A Components Based Framework for Quan- tum Key Distribution Networks,” ICTON 2020. DOI: 10.1109/ICTON51198.2020.9203181

work page doi:10.1109/icton51198.2020.9203181 2020
[9]

Enabling Quantum Key Distribution Networks via Software-Defined Networking,

A. Aguado, V. López, J. P. Brito, A. Pastor, D. R. López and V. Martin, “Enabling Quantum Key Distribution Networks via Software-Defined Networking,” ONDM
[10]

DOI: 10.23919/ONDM48393.2020.9133024

work page doi:10.23919/ondm48393.2020.9133024 2020
[11]

Quantum Technologies in the Telecommunica- tions Industry,

V. Martin, J. P. Brito, C. Escribano, M. Menchetti, C. White, A. Lord, F. Wissel, M. Gunkel, P. Gavignet, N. Genay, O. Le Moult, C. Abellán, A. Manza- lini, A. Pastor-Perales, V. López and D. López, “Quantum Technologies in the Telecommunica- tions Industry,” arXiv:2107.13360 (2021). DOI: 10.48550/arXiv.2107.13360

work page doi:10.48550/arxiv.2107.13360 2021
[12]

Madrid Quantum Network: A First Step to Quantum Internet,

M. I. García Cid, L. Ortiz Martín and V. Martín Ayuso, “Madrid Quantum Network: A First Step to Quantum Internet,” ACM (2021). DOI: 10.1145/3465481.3470056

work page doi:10.1145/3465481.3470056 2021
[13]

ETSI Quantum Key Distribution(QKD):OrchestrationInterfaceforSoftware Defined Networks v1.0,

V. Martin Ayuso and J. P. Brito, “ETSI Quantum Key Distribution(QKD):OrchestrationInterfaceforSoftware Defined Networks v1.0,” ETSI Technical Specification (2022)

2022
[14]

Long-range QKD without trusted nodes is not possible with current technology,

B. Huttner, R. Alléaume, E. Diamanti, F. Fröwis, P. Grangier, H. Hübel, V. Martin, A. Poppe, J. A. Slater, T. Spiller, W. Tittel, B. Tranier, A. Wonfor and H. Zbinden, “Long-range QKD without trusted nodes is not possible with current technology,” npj Quantum Information8, 108 (2022). DOI: 10.1038/s41534-022- 00613-4

work page doi:10.1038/s41534-022- 2022
[15]

A Brief Introduction to Standardiza- tion Activities in Quantum Communications,

V. Martin Ayuso, “A Brief Introduction to Standardiza- tion Activities in Quantum Communications,” (2023)

2023
[16]

A Feasible Hy- brid Quantum-Assisted Digital Signature for Arbi- trary Message Length,

M. I. García Cid, L. Ortiz Martín, D. Domingo Martín, R. Martín Sánchez-Ledesma, J. P. Brito Méndez and V. Martín Ayuso, “A Feasible Hy- brid Quantum-Assisted Digital Signature for Arbi- trary Message Length,” arXiv:2303.00767 (2023). DOI: 10.48550/arXiv.2303.00767

work page doi:10.48550/arxiv.2303.00767 2023
[17]

Teaching quantum communications through a hands-on laboratory,

A. Sebastián-Lombraña, L. Ortiz, J. P. Brito, J. Sáez de Buruaga, R. Vicente, R. B. Méndez, R. Artiñano and V. Martín, “Teaching quantum communications through a hands-on laboratory,” arXiv:2303.01251 (2023). DOI: 10.48550/arXiv.2303.01251

work page doi:10.48550/arxiv.2303.01251 2023
[18]

Towards Large Scale QKD Networks,

V. Martin, M. Peev, J. P. Brito, L. Ortiz, R. Brito- Méndez, R. Vicente, J. Sáez-Buruaga, A. Sebastián- Lombraña, M. García-Cid, J. Faba, J. Setién, P. Salas, C.Escribano, L.Mengual, F.Fung, J.Morales, A.Muñiz, A. Pastor-Perales and D. López, “Towards Large Scale QKD Networks,” Proc. ICTON 2024 (2024)

2024
[19]

Current Status, Gaps, and Future Directions in Quantum Key Distribution Standards: Implications for Industry,

J. Morales Sáez, A. Pastor Perales, R. Cantó Palan- car, D. R. López, J. Folgueira Chavarria and V. Martin Ayuso, “Current Status, Gaps, and Future Directions in Quantum Key Distribution Standards: Implications for Industry,” Proc. QCNC 2024 (2024). 9

2024
[20]

Towards Holistic Quantum Communications In- frastructures,

A. Sebastián-Lombraña, L. Ortiz, V. Martín and collabo- rators, “Towards Holistic Quantum Communications In- frastructures,” (2025)

2025
[21]

Mézard, G

M. Mézard, G. Parisi and M. A. Virasoro,Spin Glass Theory and Beyond, World Scientific (1987). DOI: 10.1142/0271

work page doi:10.1142/0271 1987
[23]

Nishimori,Statistical Physics of Spin Glasses and In- formation Processing, Oxford University Press (2001)

H. Nishimori,Statistical Physics of Spin Glasses and In- formation Processing, Oxford University Press (2001). DOI: 10.1093/acprof:oso/9780198509417.001.0001

work page doi:10.1093/acprof:oso/9780198509417.001.0001 2001
[24]

Equation of State Calcula- tions by Fast Computing Machines,

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, “Equation of State Calcula- tions by Fast Computing Machines,” J. Chem. Phys.21, 1087–1092 (1953). DOI: 10.1063/1.1699114

work page doi:10.1063/1.1699114 1953
[25]

Monte Carlo sampling methods using Markov chains and their applications,

W. K. Hastings, “Monte Carlo sampling methods using Markov chains and their applications,” Biometrika57, 97–109 (1970). DOI: 10.1093/biomet/57.1.97

work page doi:10.1093/biomet/57.1.97 1970
[26]

Opti- mization by Simulated Annealing,

S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, “Opti- mization by Simulated Annealing,” Science220, 671–680 (1983). DOI: 10.1126/science.220.4598.671

work page doi:10.1126/science.220.4598.671 1983
[27]

Density matrix formulation for quantum renormalization groups,

S. R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett.69, 2863–2866 (1992). DOI: 10.1103/PhysRevLett.69.2863

work page doi:10.1103/physrevlett.69.2863 1992
[28]

Density-matrix algorithms for quantum renormalization groups,

S. R. White, “Density-matrix algorithms for quantum renormalization groups,” Phys. Rev. B48, 10345–10356 (1993). DOI: 10.1103/PhysRevB.48.10345

work page doi:10.1103/physrevb.48.10345 1993
[29]

Schollw \"o ck, The density-matrix renormalization group in the age of matrix product states, Ann

U. Schollwöck, “The density-matrix renormalization group in the age of matrix product states,” Ann. Phys. 326, 96–192 (2011). DOI: 10.1016/j.aop.2010.09.012

work page doi:10.1016/j.aop.2010.09.012 2011
[30]

A practical introduction to tensor net- works: Matrix product states and projected entangled pair states,

R. Orús, “A practical introduction to tensor net- works: Matrix product states and projected entangled pair states,” Ann. Phys.349, 117–158 (2014). DOI: 10.1016/j.aop.2014.06.013

work page doi:10.1016/j.aop.2014.06.013 2014
[31]

Verstraete, J

F. Verstraete, V. Murg and J. I. Cirac, “Matrix prod- uct states, projected entangled pair states, and vari- ational renormalization group methods for quantum spin systems,” Adv. Phys.57, 143–224 (2008). DOI: 10.1080/14789940801912366

work page doi:10.1080/14789940801912366 2008
[32]

The quantum internet,

H. J. Kimble, “The quantum internet,” Nature453, 1023–1030 (2008). DOI: 10.1038/nature07127

work page doi:10.1038/nature07127 2008
[33]

Quantum inter- net: A vision for the road ahead,

S. Wehner, D. Elkouss and R. Hanson, “Quantum inter- net: A vision for the road ahead,” Science362, eaam9288 (2018). DOI: 10.1126/science.aam9288

work page doi:10.1126/science.aam9288 2018
[34]

A quantum network stack and protocols for reliable entanglement-based networks,

A. Pirker and W. Dür, “A quantum network stack and protocols for reliable entanglement-based networks,” New J. Phys.21, 033003 (2019). DOI: 10.1088/1367- 2630/ab05f7

work page doi:10.1088/1367- 2019
[35]

Ising formulations of many NP problems

A. Lucas, “Ising formulations of many NP problems,” Front. Phys.2, 5 (2014). DOI: 10.3389/fphy.2014.00005

work page doi:10.3389/fphy.2014.00005 2014
[36]

Quantum annealing in the transverse Ising model,

T. Kadowaki and H. Nishimori, “Quantum annealing in the transverse Ising model,” Phys. Rev. E58, 5355–5363 (1998). DOI: 10.1103/PhysRevE.58.5355

work page doi:10.1103/physreve.58.5355 1998
[37]

Colloquium: Quantum annealing and analog quantum computation,

A. Das and B. K. Chakrabarti, “Colloquium: Quantum annealing and analog quantum computation,” Rev. Mod. Phys.80, 1061–1081 (2008). DOI: 10.1103/RevMod- Phys.80.1061

work page doi:10.1103/revmod- 2008
[38]

A Quantum Approximate Optimization Algorithm

E. Farhi, J. Goldstone and S. Gutmann, “A Quantum Approximate Optimization Algorithm,” arXiv:1411.4028 (2014). DOI: 10.48550/arXiv.1411.4028

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1411.4028 2014
[39]

Perfect sampling with unitary tensor networks,

A. J. Ferris and G. Vidal, “Perfect sampling with unitary tensor networks,” Phys. Rev. B85, 165146 (2012). DOI: 10.1103/PhysRevB.85.165146

work page doi:10.1103/physrevb.85.165146 2012
[40]

Unbiased Monte Carlo for the age of tensor networks

A. J. Ferris, “Unbiased Monte Carlo for the age of tensor networks,” arXiv:1507.00767 (2015). DOI: 10.48550/arXiv.1507.00767

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1507.00767 2015
[41]

Quantum Computing in the NISQ era and beyond,

J. Preskill, “Quantum Computing in the NISQ era and beyond,” Quantum2, 79 (2018). DOI: 10.22331/q-2018- 08-06-79

work page doi:10.22331/q-2018- 2018
[42]

Technical Description of the D- Wave Quantum Processing Unit,

D-Wave Systems Inc., “Technical Description of the D- Wave Quantum Processing Unit,” Technical Report, D- Wave Systems (2020). 10 Appendix A: Meta-code of the QKD Routing QMC–Metropolis Optimizer Meta-code: Stochastic Boundary Truncation 1:Input:number of nodesn, average connec- tivityk, number of demandsM, number of candidate pathsq, number of Monte Carl...

2020

[1] [1]

Quantum cryptogra- phy: Public key distribution and coin tossing,

C. H. Bennett and G. Brassard, “Quantum cryptogra- phy: Public key distribution and coin tossing,” Proceed- ings IEEE International Conference on Computers, Sys- tems and Signal Processing, Bangalore, India, pp. 175– 179 (1984)

1984

[2] [2]

Quantum cryptography based on Bell’s theorem,

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett.67, 661–663 (1991). DOI: 10.1103/PhysRevLett.67.661

work page doi:10.1103/physrevlett.67.661 1991

[3] [3]

Quan- tum cryptography,

N. Gisin, G. Ribordy, W. Tittel and H. Zbinden, “Quan- tum cryptography,” Rev. Mod. Phys.74, 145–195 (2002). DOI: 10.1103/RevModPhys.74.145

work page doi:10.1103/revmodphys.74.145 2002

[4] [4]

Advances in Quantum Cryptogra- phy,

S. Pirandolaet al., “Advances in Quantum Cryptogra- phy,” Adv. Opt. Photon.12, 1012–1236 (2020). DOI: 10.1364/AOP.361502

work page doi:10.1364/aop.361502 2020

[5] [5]

The engineer- ing of software-defined quantum key distribution net- works,

A. Aguado, V. López, D. R. López, M. Peev, A. Pastor- Perales, J. Folgueira and V. Martín, “The engineer- ing of software-defined quantum key distribution net- works,” IEEE Commun. Mag.57, 20–26 (2019). DOI: 10.1109/MCOM.2019.1800763

work page doi:10.1109/mcom.2019.1800763 2019

[6] [6]

The Madrid Quantum Network: A Quantum-Classical Integrated Infrastructure,

V. Martin, A. Aguado, P. Salas, A. L. Sanz, J. P. Brito, D. R. Lopez, V. Lopez, A. Pastor-Perales, A. Poppe and M. Peev, “The Madrid Quantum Network: A Quantum-Classical Integrated Infrastructure,” OSA Ad- vanced Photonics Congress (2019). DOI: 10.1364/NET- WORKS.2019.QTW3E.5

work page doi:10.1364/net- 2019

[7] [7]

Quantum Aware SDN Nodes in the Madrid Quantum Network,

V. Martin, A. Aguado, J. P. Brito, A. L. Sanz, P. Salas, D. R. Lopez, V. Lopez, A. Pastor-Perales, A. Poppe and M. Peev, “Quantum Aware SDN Nodes in the Madrid Quantum Network,” ICTON 2019. DOI: 10.1109/IC- TON.2019.8840338

work page doi:10.1109/ic- 2019

[8] [8]

A Components Based Framework for Quan- tum Key Distribution Networks,

V. Martin, D. Lopez, A. Aguado, J. P. Brito, J. Setien, P. Salas, C. Escribano, V. Lopez, A. Pastor-Perales and M. Peev, “A Components Based Framework for Quan- tum Key Distribution Networks,” ICTON 2020. DOI: 10.1109/ICTON51198.2020.9203181

work page doi:10.1109/icton51198.2020.9203181 2020

[9] [9]

Enabling Quantum Key Distribution Networks via Software-Defined Networking,

A. Aguado, V. López, J. P. Brito, A. Pastor, D. R. López and V. Martin, “Enabling Quantum Key Distribution Networks via Software-Defined Networking,” ONDM

[10] [10]

DOI: 10.23919/ONDM48393.2020.9133024

work page doi:10.23919/ondm48393.2020.9133024 2020

[11] [11]

Quantum Technologies in the Telecommunica- tions Industry,

V. Martin, J. P. Brito, C. Escribano, M. Menchetti, C. White, A. Lord, F. Wissel, M. Gunkel, P. Gavignet, N. Genay, O. Le Moult, C. Abellán, A. Manza- lini, A. Pastor-Perales, V. López and D. López, “Quantum Technologies in the Telecommunica- tions Industry,” arXiv:2107.13360 (2021). DOI: 10.48550/arXiv.2107.13360

work page doi:10.48550/arxiv.2107.13360 2021

[12] [12]

Madrid Quantum Network: A First Step to Quantum Internet,

M. I. García Cid, L. Ortiz Martín and V. Martín Ayuso, “Madrid Quantum Network: A First Step to Quantum Internet,” ACM (2021). DOI: 10.1145/3465481.3470056

work page doi:10.1145/3465481.3470056 2021

[13] [13]

ETSI Quantum Key Distribution(QKD):OrchestrationInterfaceforSoftware Defined Networks v1.0,

V. Martin Ayuso and J. P. Brito, “ETSI Quantum Key Distribution(QKD):OrchestrationInterfaceforSoftware Defined Networks v1.0,” ETSI Technical Specification (2022)

2022

[14] [14]

Long-range QKD without trusted nodes is not possible with current technology,

B. Huttner, R. Alléaume, E. Diamanti, F. Fröwis, P. Grangier, H. Hübel, V. Martin, A. Poppe, J. A. Slater, T. Spiller, W. Tittel, B. Tranier, A. Wonfor and H. Zbinden, “Long-range QKD without trusted nodes is not possible with current technology,” npj Quantum Information8, 108 (2022). DOI: 10.1038/s41534-022- 00613-4

work page doi:10.1038/s41534-022- 2022

[15] [15]

A Brief Introduction to Standardiza- tion Activities in Quantum Communications,

V. Martin Ayuso, “A Brief Introduction to Standardiza- tion Activities in Quantum Communications,” (2023)

2023

[16] [16]

A Feasible Hy- brid Quantum-Assisted Digital Signature for Arbi- trary Message Length,

M. I. García Cid, L. Ortiz Martín, D. Domingo Martín, R. Martín Sánchez-Ledesma, J. P. Brito Méndez and V. Martín Ayuso, “A Feasible Hy- brid Quantum-Assisted Digital Signature for Arbi- trary Message Length,” arXiv:2303.00767 (2023). DOI: 10.48550/arXiv.2303.00767

work page doi:10.48550/arxiv.2303.00767 2023

[17] [17]

Teaching quantum communications through a hands-on laboratory,

A. Sebastián-Lombraña, L. Ortiz, J. P. Brito, J. Sáez de Buruaga, R. Vicente, R. B. Méndez, R. Artiñano and V. Martín, “Teaching quantum communications through a hands-on laboratory,” arXiv:2303.01251 (2023). DOI: 10.48550/arXiv.2303.01251

work page doi:10.48550/arxiv.2303.01251 2023

[18] [18]

Towards Large Scale QKD Networks,

V. Martin, M. Peev, J. P. Brito, L. Ortiz, R. Brito- Méndez, R. Vicente, J. Sáez-Buruaga, A. Sebastián- Lombraña, M. García-Cid, J. Faba, J. Setién, P. Salas, C.Escribano, L.Mengual, F.Fung, J.Morales, A.Muñiz, A. Pastor-Perales and D. López, “Towards Large Scale QKD Networks,” Proc. ICTON 2024 (2024)

2024

[19] [19]

Current Status, Gaps, and Future Directions in Quantum Key Distribution Standards: Implications for Industry,

J. Morales Sáez, A. Pastor Perales, R. Cantó Palan- car, D. R. López, J. Folgueira Chavarria and V. Martin Ayuso, “Current Status, Gaps, and Future Directions in Quantum Key Distribution Standards: Implications for Industry,” Proc. QCNC 2024 (2024). 9

2024

[20] [20]

Towards Holistic Quantum Communications In- frastructures,

A. Sebastián-Lombraña, L. Ortiz, V. Martín and collabo- rators, “Towards Holistic Quantum Communications In- frastructures,” (2025)

2025

[21] [21]

Mézard, G

M. Mézard, G. Parisi and M. A. Virasoro,Spin Glass Theory and Beyond, World Scientific (1987). DOI: 10.1142/0271

work page doi:10.1142/0271 1987

[22] [23]

Nishimori,Statistical Physics of Spin Glasses and In- formation Processing, Oxford University Press (2001)

H. Nishimori,Statistical Physics of Spin Glasses and In- formation Processing, Oxford University Press (2001). DOI: 10.1093/acprof:oso/9780198509417.001.0001

work page doi:10.1093/acprof:oso/9780198509417.001.0001 2001

[23] [24]

Equation of State Calcula- tions by Fast Computing Machines,

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, “Equation of State Calcula- tions by Fast Computing Machines,” J. Chem. Phys.21, 1087–1092 (1953). DOI: 10.1063/1.1699114

work page doi:10.1063/1.1699114 1953

[24] [25]

Monte Carlo sampling methods using Markov chains and their applications,

W. K. Hastings, “Monte Carlo sampling methods using Markov chains and their applications,” Biometrika57, 97–109 (1970). DOI: 10.1093/biomet/57.1.97

work page doi:10.1093/biomet/57.1.97 1970

[25] [26]

Opti- mization by Simulated Annealing,

S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, “Opti- mization by Simulated Annealing,” Science220, 671–680 (1983). DOI: 10.1126/science.220.4598.671

work page doi:10.1126/science.220.4598.671 1983

[26] [27]

Density matrix formulation for quantum renormalization groups,

S. R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett.69, 2863–2866 (1992). DOI: 10.1103/PhysRevLett.69.2863

work page doi:10.1103/physrevlett.69.2863 1992

[27] [28]

Density-matrix algorithms for quantum renormalization groups,

S. R. White, “Density-matrix algorithms for quantum renormalization groups,” Phys. Rev. B48, 10345–10356 (1993). DOI: 10.1103/PhysRevB.48.10345

work page doi:10.1103/physrevb.48.10345 1993

[28] [29]

Schollw \"o ck, The density-matrix renormalization group in the age of matrix product states, Ann

U. Schollwöck, “The density-matrix renormalization group in the age of matrix product states,” Ann. Phys. 326, 96–192 (2011). DOI: 10.1016/j.aop.2010.09.012

work page doi:10.1016/j.aop.2010.09.012 2011

[29] [30]

A practical introduction to tensor net- works: Matrix product states and projected entangled pair states,

R. Orús, “A practical introduction to tensor net- works: Matrix product states and projected entangled pair states,” Ann. Phys.349, 117–158 (2014). DOI: 10.1016/j.aop.2014.06.013

work page doi:10.1016/j.aop.2014.06.013 2014

[30] [31]

Verstraete, J

F. Verstraete, V. Murg and J. I. Cirac, “Matrix prod- uct states, projected entangled pair states, and vari- ational renormalization group methods for quantum spin systems,” Adv. Phys.57, 143–224 (2008). DOI: 10.1080/14789940801912366

work page doi:10.1080/14789940801912366 2008

[31] [32]

The quantum internet,

H. J. Kimble, “The quantum internet,” Nature453, 1023–1030 (2008). DOI: 10.1038/nature07127

work page doi:10.1038/nature07127 2008

[32] [33]

Quantum inter- net: A vision for the road ahead,

S. Wehner, D. Elkouss and R. Hanson, “Quantum inter- net: A vision for the road ahead,” Science362, eaam9288 (2018). DOI: 10.1126/science.aam9288

work page doi:10.1126/science.aam9288 2018

[33] [34]

A quantum network stack and protocols for reliable entanglement-based networks,

A. Pirker and W. Dür, “A quantum network stack and protocols for reliable entanglement-based networks,” New J. Phys.21, 033003 (2019). DOI: 10.1088/1367- 2630/ab05f7

work page doi:10.1088/1367- 2019

[34] [35]

Ising formulations of many NP problems

A. Lucas, “Ising formulations of many NP problems,” Front. Phys.2, 5 (2014). DOI: 10.3389/fphy.2014.00005

work page doi:10.3389/fphy.2014.00005 2014

[35] [36]

Quantum annealing in the transverse Ising model,

T. Kadowaki and H. Nishimori, “Quantum annealing in the transverse Ising model,” Phys. Rev. E58, 5355–5363 (1998). DOI: 10.1103/PhysRevE.58.5355

work page doi:10.1103/physreve.58.5355 1998

[36] [37]

Colloquium: Quantum annealing and analog quantum computation,

A. Das and B. K. Chakrabarti, “Colloquium: Quantum annealing and analog quantum computation,” Rev. Mod. Phys.80, 1061–1081 (2008). DOI: 10.1103/RevMod- Phys.80.1061

work page doi:10.1103/revmod- 2008

[37] [38]

A Quantum Approximate Optimization Algorithm

E. Farhi, J. Goldstone and S. Gutmann, “A Quantum Approximate Optimization Algorithm,” arXiv:1411.4028 (2014). DOI: 10.48550/arXiv.1411.4028

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1411.4028 2014

[38] [39]

Perfect sampling with unitary tensor networks,

A. J. Ferris and G. Vidal, “Perfect sampling with unitary tensor networks,” Phys. Rev. B85, 165146 (2012). DOI: 10.1103/PhysRevB.85.165146

work page doi:10.1103/physrevb.85.165146 2012

[39] [40]

Unbiased Monte Carlo for the age of tensor networks

A. J. Ferris, “Unbiased Monte Carlo for the age of tensor networks,” arXiv:1507.00767 (2015). DOI: 10.48550/arXiv.1507.00767

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1507.00767 2015

[40] [41]

Quantum Computing in the NISQ era and beyond,

J. Preskill, “Quantum Computing in the NISQ era and beyond,” Quantum2, 79 (2018). DOI: 10.22331/q-2018- 08-06-79

work page doi:10.22331/q-2018- 2018

[41] [42]

Technical Description of the D- Wave Quantum Processing Unit,

D-Wave Systems Inc., “Technical Description of the D- Wave Quantum Processing Unit,” Technical Report, D- Wave Systems (2020). 10 Appendix A: Meta-code of the QKD Routing QMC–Metropolis Optimizer Meta-code: Stochastic Boundary Truncation 1:Input:number of nodesn, average connec- tivityk, number of demandsM, number of candidate pathsq, number of Monte Carl...

2020