Universal Complex Quantum-Like Bits from Hermitian Weighted Graphs

Ethan Dickey; Sabre Kais

arxiv: 2604.23991 · v1 · submitted 2026-04-27 · 🪐 quant-ph

Universal Complex Quantum-Like Bits from Hermitian Weighted Graphs

Ethan Dickey , Sabre Kais This is my paper

Pith reviewed 2026-05-08 04:26 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Hermitian weighted graphsquantum-like bitscomplex amplitudesspectral reductioneigenstatesbipartite couplingsuniversal realizationsdiscrete graphs

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The pith

Hermitian weighted graphs realize any prescribed complex two-level state as an exact eigenstate with real spectrum.

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

The paper establishes that Hermitian weighted couplings between two regular subgraphs allow any non-basis complex quantum-like bit state to be realized exactly as an eigenstate. This construction works for any chosen real eigenvalue and any nonzero signed spectral gap. A sympathetic reader would care because the approach overcomes phase restrictions that appear in complex-symmetric or real-symmetric alternatives and supplies both continuous-parameter and discrete finite-graph realizations. The work also proves that discrete realizations using only entries from the set of zero and fourth roots of unity are dense in the space of such states.

Core claim

For any non-basis target state, any prescribed real eigenvalue, and any prescribed nonzero signed spectral gap, a Hermitian weighted coupling realizes the target exactly. Algebraically regular bipartite couplings reduce the full graph-supported operator exactly to a 2x2 effective block on span{|0>, |1>}, with the subgraphs supplying normalized all-ones eigenvectors; Hermitian conjugate pairing removes the phase obstructions that confine the other symmetry classes.

What carries the argument

Hermitian weighted coupling between algebraically regular subgraphs that reduces the full operator exactly to a controllable 2x2 block via normalized all-ones eigenvectors.

If this is right

Hermitian conjugate pairing supports arbitrary complex amplitudes while preserving real two-level spectra.
An independently tuned directed-coupling model supplies a second route to the same universality.
Exact discrete realizations exist using only matrix entries from {0, ±1, ±i}.
Such discrete realizations are dense in the space of synchronized pure states.

Where Pith is reading between the lines

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

The density of discrete realizations suggests that finite small graphs can approximate any target state to arbitrary precision.
The same reduction technique could be applied to embed effective two-level dynamics inside larger graph-based models for simulation.
Varying the choice of regular subgraphs might yield additional constraints or freedoms on the achievable spectral gaps.

Load-bearing premise

Algebraically regular bipartite couplings reduce the full graph-supported operator exactly to a 2x2 effective block on span{|0>, |1>}, with the subgraphs supplying normalized all-ones eigenvectors.

What would settle it

Build a Hermitian weighted graph for a chosen target state whose amplitude ratio r satisfies neither r squared real nor r plus one over r real, compute the spectrum of the full operator, and check whether the eigenvector restricted to the two reference vertices matches the target with the prescribed real eigenvalue and gap.

Figures

Figures reproduced from arXiv: 2604.23991 by Ethan Dickey, Sabre Kais.

**Figure 1.** Figure 1: A QL-bit built from two regular subgraphs coupled by a bipartite block. In the real symmetric bench view at source ↗

read the original abstract

We study when block-coupled regular graphs can realize prescribed complex quantum-like bit states as exact synchronized eigenstates. Two regular subgraphs $G_A$ and $G_B$ supply normalized all-ones eigenvectors $V_A$ and $V_B$, and algebraically regular bipartite couplings reduce the full graph-supported operator exactly to a $2\times 2$ effective block on $\mathcal S=\operatorname{span} \{ \lvert 0\rangle, \lvert 1\rangle \}$. Within this reduction we prove that two natural symmetric complexifications are not universal under a real-spectrum requirement: complex symmetric coupling with real diagonal regularities forces the target computational basis amplitude ratio $r=\omega_2/\omega_1$, for $\lvert \psi\rangle = \omega_1\lvert 0\rangle + \omega_2\lvert 1\rangle$, to satisfy $r^2\in\mathbb{R}$, while real symmetric coupling with complex diagonal regularities forces $r+1/r\in\mathbb{R}$. Replacing complex symmetry by Hermitian coupling removes this phase obstruction. For any nonbasis target state, any prescribed real eigenvalue, and any prescribed nonzero signed spectral gap, a Hermitian weighted coupling realizes the target exactly. Additionally, an independently tuned directed-coupling model gives a second universality mechanism. We then pass from continuous effective parameters to finite weighted graphs with entries in $\{0, \pm1, \pm i\}$ (the fourth roots of unity and zero), characterize the balanced discrete coupling lattice by perfect matchings, and show that exact discrete Hermitian realizations are dense in the synchronized pure-state space. These results give a universality taxonomy for complex QL-bits and identify Hermitian conjugate pairing as the robust structural mechanism that supports arbitrary complex amplitudes with real two-level spectra.

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

Hermitian couplings on regular graphs let you realize any complex two-level state exactly as an eigenstate with real spectrum and controllable gap, and discrete fourth-root versions are dense.

read the letter

The main takeaway is that Hermitian weighted bipartite couplings between two regular subgraphs reduce the operator exactly to a 2x2 block and remove the phase obstructions that block symmetric couplings. Complex symmetric weights force the amplitude ratio r to satisfy r squared real, while real symmetric weights with complex diagonals force r plus one over r real. Hermitian weights avoid both restrictions, so for any non-basis target state, any real eigenvalue, and any nonzero signed gap you can match exactly. They also sketch a directed-coupling route as a second mechanism. The discrete part shows that weights restricted to {0, ±1, ±i} can be arranged via balanced perfect matchings to keep regularity, and these exact realizations are dense in the space of synchronized pure states. That density claim is the clearest new piece relative to earlier graph-walk literature. The reduction step itself is clean once the subgraphs supply normalized all-ones eigenvectors and the bipartite edges are algebraically regular; both are standard and the algebra checks out on the surface. The taxonomy of why Hermitian works where symmetric does not is useful for anyone trying to engineer states this way. Soft spots are limited. The constructions assume you can always find sufficiently large regular subgraphs and matchings for a given approximation tolerance, but the paper does not spell out explicit size bounds or construction algorithms, so practical scaling remains a bit open. The directed-coupling section is mentioned only briefly and not developed to the same level. Overall the algebra is self-contained and the claims are falsifiable. This is for people working on quantum walks on graphs or on graph-based models for quantum information. A reader who cares about universality results or about Hermitian structures in discrete quantum systems will find the taxonomy and density argument worth checking. It is worth sending to peer review; the central reduction and the Hermitian advantage are specific enough that referees can verify them directly.

Referee Report

3 major / 2 minor

Summary. The manuscript claims that block-coupled regular graphs with algebraically regular bipartite couplings reduce exactly to a 2×2 effective Hermitian block on span{|0⟩, |1⟩} via the all-ones eigenvectors of the regular subgraphs. Symmetric couplings restrict the target amplitude ratio r=ω₂/ω₁ to r²∈ℝ or r+1/r∈ℝ, but Hermitian weighted couplings achieve universality: for any non-basis complex state, any real eigenvalue, and any nonzero signed spectral gap, an exact realization exists. Discrete realizations with weights in {0, ±1, ±i} are dense via balanced perfect-matching decompositions that preserve regularity.

Significance. If the algebraic reductions and existence arguments hold, this supplies a constructive universality taxonomy for complex quantum-like bits with real two-level spectra, identifying Hermitian conjugate pairing as the mechanism that removes phase obstructions present in symmetric cases. Strengths include the exact (parameter-free in the continuous limit) reduction to the effective 2×2 block and the density proof for finite discrete graphs; these could support graph-based models in quantum simulation.

major comments (3)

The central reduction step (algebraically regular bipartite couplings yielding an exact 2×2 block on the all-ones subspace): the manuscript must supply the explicit block-matrix form of the full operator and verify that off-block terms vanish identically under the stated regularity conditions, as this underpins every subsequent universality claim.
Hermitian universality construction: the argument that the free parameters (real diagonal regularities plus complex off-diagonal coupling) suffice to match arbitrary complex r, real eigenvalue, and prescribed nonzero gap requires an explicit surjective map or closed-form solution; without it the existence claim remains an assertion rather than a verified construction.
Discrete density via perfect-matchings decomposition: the proof that {0, ±1, ±i}-weighted balanced decompositions preserve both algebraic regularity and the exact 2×2 reduction must be checked for all target states, including those near the computational basis where the gap condition is tight.

minor comments (2)

Define 'synchronized eigenstates' and 'algebraically regular' at first use with a short equation or example.
Ensure consistent use of 'QL-bits' versus 'quantum-like bits' and expand the abstract's parenthetical on fourth roots of unity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the presentation of our results. We address each major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

Referee: The central reduction step (algebraically regular bipartite couplings yielding an exact 2×2 block on the all-ones subspace): the manuscript must supply the explicit block-matrix form of the full operator and verify that off-block terms vanish identically under the stated regularity conditions, as this underpins every subsequent universality claim.

Authors: We agree that an explicit block-matrix representation will improve clarity. The reduction follows from the fact that the all-ones vectors of the regular subgraphs are eigenvectors, and algebraic regularity of the bipartite couplings ensures that the operator maps the orthogonal complement to itself while projecting exactly onto span{|0⟩, |1⟩}. In the revision we will add the full block-matrix form of the graph-supported operator together with a direct verification that all off-block entries vanish identically under the regularity hypotheses. revision: yes
Referee: Hermitian universality construction: the argument that the free parameters (real diagonal regularities plus complex off-diagonal coupling) suffice to match arbitrary complex r, real eigenvalue, and prescribed nonzero gap requires an explicit surjective map or closed-form solution; without it the existence claim remains an assertion rather than a verified construction.

Authors: The manuscript solves the 2×2 eigenvalue problem explicitly: given target ratio r (non-basis), real eigenvalue λ, and signed gap δ > 0, the Hermitian matrix entries are obtained by equating the eigenvector and eigenvalue conditions, yielding a unique solution for the diagonal regularities and off-diagonal coupling. This map is surjective onto the admissible parameter space. We will insert the closed-form expressions for these parameters in the revised text so that the construction is fully constructive rather than existential. revision: yes
Referee: Discrete density via perfect-matchings decomposition: the proof that {0, ±1, ±i}-weighted balanced decompositions preserve both algebraic regularity and the exact 2×2 reduction must be checked for all target states, including those near the computational basis where the gap condition is tight.

Authors: The density argument relies on the fact that any Hermitian coupling with nonzero gap admits a balanced perfect-matching decomposition that preserves algebraic regularity and the 2×2 reduction; the gap condition is maintained by construction as long as the target gap is nonzero (even if small). We will add an explicit remark and a brief limiting-case verification for states arbitrarily close to the computational basis (with correspondingly small but positive gap) to confirm that the decomposition remains valid. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is algebraically self-contained

full rationale

The paper establishes its universality claims through explicit algebraic reductions: regular subgraphs supply normalized all-ones eigenvectors, algebraically regular bipartite couplings reduce the operator exactly to a 2x2 block on span{|0>, |1>}, and Hermitian weighting supplies free real diagonal and complex off-diagonal parameters that are solved directly to match arbitrary complex amplitude ratios, real eigenvalues, and nonzero gaps. The discrete case follows from an independent construction via balanced perfect-matchings decompositions with {0, ±1, ±i} entries. No step equates a derived quantity to its own input by definition, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose content is unverified outside the present work; all arguments are direct existence proofs internal to the stated graph-algebraic assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard linear algebra (eigenvector properties, block reductions) and graph theory (regularity, bipartite couplings). No free parameters are introduced to fit the target states; the constructions are existence proofs for prescribed targets. No new physical entities are postulated.

axioms (2)

domain assumption Algebraically regular bipartite couplings reduce the full operator exactly to the 2x2 effective block
Invoked in the opening setup to justify the effective 2x2 description.
standard math Subgraphs supply normalized all-ones eigenvectors
Standard property of regular graphs used to define the basis states.

pith-pipeline@v0.9.0 · 5611 in / 1379 out tokens · 19408 ms · 2026-05-08T04:26:38.888958+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 29 canonical work pages

[1]

Manuela A. D. Aguiar and Ana Paula S. Dias. 2018. Synchronization and equitable partitions in weighted networks.Chaos: An Interdisciplinary Journal of Nonlinear Science28, 7 (07 2018), 073105. doi:10.1063/ 1.4997385

2018
[2]

Graziano Amati and Gregory D. Scholes. 2025. Quantum information with quantumlike bits.Phys. Rev. A 111 (Jun 2025), 062203. Issue 6. doi:10.1103/PhysRevA.111.062203

work page doi:10.1103/physreva.111.062203 2025
[3]

Yuto Ashida, Zongping Gong, and Masahito Ueda. 2020. Non-Hermitian physics.Advances in Physics 69, 3 (2020), 249–435. arXiv:https://doi.org/10.1080/00018732.2021.1876991 doi:10.1080/00018732.2021. 1876991

work page doi:10.1080/00018732.2021.1876991 2020
[4]

Opperman, Christino Tamon, and Andrew Tollefson

Rachel Bachman, Eric Fredette, Jessica Fuller, Michael Landry, Michael C. Opperman, Christino Tamon, and Andrew Tollefson. 2012. Perfect state transfer on quotient graphs.Quantum Information & Computation 12, 3-4 (2012), 293–313. Published version URL unavailable; arXiv:https://arxiv.org/abs/1108.0339

work page arXiv 2012
[5]

Bender and Stefan Boettcher

Carl M. Bender and Stefan Boettcher. 1998. Real Spectra in Non-Hermitian Hamiltonians HavingPT Symmetry.Phys. Rev. Lett.80 (Jun 1998), 5243–5246. Issue 24. doi:10.1103/PhysRevLett.80.5243

work page doi:10.1103/physrevlett.80.5243 1998
[6]

Bender, Dorje C

Carl M. Bender, Dorje C. Brody, and Hugh F. Jones. 2002. Complex Extension of Quantum Mechanics. Phys. Rev. Lett.89 (Dec 2002), 270401. Issue 27. doi:10.1103/PhysRevLett.89.270401

work page doi:10.1103/physrevlett.89.270401 2002
[7]

Bergholtz, Jan Carl Budich, and Flore K

Emil J. Bergholtz, Jan Carl Budich, and Flore K. Kunst. 2021. Exceptional topology of non-Hermitian systems.Rev. Mod. Phys.93 (Feb 2021), 015005. Issue 1. doi:10.1103/RevModPhys.93.015005

work page doi:10.1103/revmodphys.93.015005 2021
[8]

Sougato Bose. 2003. Quantum Communication through an Unmodulated Spin Chain.Phys. Rev. Lett.91 (Nov 2003), 207901. Issue 20. doi:10.1103/PhysRevLett.91.207901

work page doi:10.1103/physrevlett.91.207901 2003
[9]

DiVincenzo, and Daniel Loss

Sergey Bravyi, David P. DiVincenzo, and Daniel Loss. 2011. Schrieffer–Wolff transformation for quantum many-body systems.Annals of Physics326, 10 (2011), 2793–2826. doi:10.1016/j.aop.2011.06.004

work page doi:10.1016/j.aop.2011.06.004 2011
[10]

Sergey Bravyi and Jeongwan Haah. 2012. Magic-state distillation with low overhead.Phys. Rev. A86 (Nov 2012), 052329. Issue 5. doi:10.1103/PhysRevA.86.052329

work page doi:10.1103/physreva.86.052329 2012
[11]

Sergey Bravyi and Alexei Kitaev. 2005. Universal quantum computation with ideal Clifford gates and noisy ancillas.Phys. Rev. A71 (Feb 2005), 022316. Issue 2. doi:10.1103/PhysRevA.71.022316

work page doi:10.1103/physreva.71.022316 2005
[12]

Sergey Bravyi, Graeme Smith, and John A. Smolin. 2016. Trading Classical and Quantum Computational Resources.Phys. Rev. X6 (Jun 2016), 021043. Issue 2. doi:10.1103/PhysRevX.6.021043

work page doi:10.1103/physrevx.6.021043 2016
[13]

Roads towards fault- tolerant universal quantum computation

Earl T. Campbell, Barbara M. Terhal, and Christophe Vuillot. 2017. Roads towards fault-tolerant universal quantum computation.Nature549, 7671 (Sept. 2017), 172–179. doi:10.1038/nature23460

work page doi:10.1038/nature23460 2017
[14]

Andrew M. Childs. 2009. Universal Computation by Quantum Walk.Phys. Rev. Lett.102 (May 2009), 180501. Issue 18. doi:10.1103/PhysRevLett.102.180501

work page doi:10.1103/physrevlett.102.180501 2009
[15]

Dorlas, Artur Ekert, Alastair Kay, and Andrew J

Matthias Christandl, Nilanjana Datta, Tony C. Dorlas, Artur Ekert, Alastair Kay, and Andrew J. Landahl
[16]

Perfect transfer of arbitrary states in quantum spin networks.Phys. Rev. A71 (Mar 2005), 032312. Issue 3. doi:10.1103/PhysRevA.71.032312

work page doi:10.1103/physreva.71.032312 2005
[17]

Matthias Christandl, Nilanjana Datta, Artur Ekert, and Andrew J. Landahl. 2004. Perfect State Transfer in Quantum Spin Networks.Phys. Rev. Lett.92 (May 2004), 187902. Issue 18. doi:10.1103/PhysRevLett. 92.187902

work page doi:10.1103/physrevlett 2004
[18]

Chu and Gene H

Moody T. Chu and Gene H. Golub. 2002. Structured inverse eigenvalue problems.Acta Numerica11 (2002), 1–71. doi:10.1017/S0962492902000016

work page doi:10.1017/s0962492902000016 2002
[19]

Ethan Dickey, Abhijeet Vyas, and Sabre Kais. 2025. Construction and Rigorous Analysis of Quantum-Like States. arXiv:2507.21289 [quant-ph]https://arxiv.org/abs/2507.21289

work page arXiv 2025
[20]

El-Ganainy , author K

Ramy El-Ganainy, Konstantinos G. Makris, Mercedeh Khajavikhan, Ziad H. Musslimani, Stefan Rotter, and Demetrios N. Christodoulides. 2018. Non-Hermitian physics and PT symmetry.Nature Physics14, 1 (2018), 11–19. doi:10.1038/nphys4323

work page doi:10.1038/nphys4323 2018
[21]

Edward Farhi and Sam Gutmann. 1998. Quantum computation and decision trees.Phys. Rev. A58 (Aug 1998), 915–928. Issue 2. doi:10.1103/PhysRevA.58.915 22

work page doi:10.1103/physreva.58.915 1998
[22]

Chris Godsil. 2012. State transfer on graphs.Discrete Mathematics312, 1 (2012), 129–147. doi:10.1016/ j.disc.2011.06.032Algebraic Graph Theory — A Volume Dedicated to Gert Sabidussi on the Occasion of His 80th Birthday

2012
[23]

Mark Howard, Joel Wallman, Victor Veitch, and Joseph Emerson. 2014. Contextuality supplies the ‘magic’ for quantum computation.Nature510, 7505 (June 2014), 351–355. doi:10.1038/nature13460

work page doi:10.1038/nature13460 2014
[24]

Cody Jones. 2013. Multilevel distillation of magic states for quantum computing.Phys. Rev. A87 (Apr 2013), 042305. Issue 4. doi:10.1103/PhysRevA.87.042305

work page doi:10.1103/physreva.87.042305 2013
[25]

Julia Kempe. 2003. Quantum random walks: An introductory overview.Contemporary Physics44, 4 (2003), 307–327. arXiv:https://doi.org/10.1080/00107151031000110776 doi:10.1080/00107151031000110776

work page doi:10.1080/00107151031000110776 2003
[26]

Hari Krovi and Todd A. Brun. 2007. Quantum walks on quotient graphs.Phys. Rev. A75 (Jun 2007), 062332. Issue 6. doi:10.1103/PhysRevA.75.062332

work page doi:10.1103/physreva.75.062332 2007
[27]

Ali Mostafazadeh. 2002. Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian.J. Math. Phys.43, 1 (01 2002), 205–214. doi:10.1063/ 1.1418246

2002
[28]

S Unnikrishna Pillai, Torsten Suel, and Seunghun Cha. 2005. The Perron-Frobenius theorem: some of its applications.IEEE Signal Processing Magazine22, 2 (2005), 62–75. doi:10.1109/MSP.2005.1406483

work page doi:10.1109/msp.2005.1406483 2005
[29]

Gregory D. Scholes. 2024. Quantum-like states on complex synchronized networks. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 480, 2295 (08 2024), 20240209. arXiv:https://royalsocietypublishing.org/rspa/article- pdf/doi/10.1098/rspa.2024.0209/512921/rspa.2024.0209.pdf doi:10.1098/rspa.2024.0209

work page doi:10.1098/rspa.2024.0209/512921/rspa.2024.0209.pdf 2024
[30]

Scholes and Graziano Amati

Gregory D. Scholes and Graziano Amati. 2025. Quantumlike Product States Constructed from Classical Networks.Phys. Rev. Lett.134 (Feb 2025), 060202. Issue 6. doi:10.1103/PhysRevLett.134.060202

work page doi:10.1103/physrevlett.134.060202 2025
[31]

Seddon and Earl T

James R. Seddon and Earl T. Campbell. 2019. Quantifying magic for multi-qubit oper- ations.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sci- ences475, 2227 (07 2019), 20190251. arXiv:https://royalsocietypublishing.org/rspa/article- pdf/doi/10.1098/rspa.2019.0251/1221292/rspa.2019.0251.pdf doi:10.1098/rspa.2019.0251

work page doi:10.1098/rspa.2019.0251/1221292/rspa.2019.0251.pdf 2019
[32]

Victor Veitch, S A Hamed Mousavian, Daniel Gottesman, and Joseph Emerson. 2014. The resource theory of stabilizer quantum computation.New Journal of Physics16, 1 (jan 2014), 013009. doi:10.1088/1367-2630/ 16/1/013009

work page doi:10.1088/1367-2630/ 2014
[33]

Victor Veitch, Nathan Wiebe, Christopher Ferrie, and Joseph Emerson. 2013. Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation.New Journal of Physics 15, 1 (jan 2013), 013037. doi:10.1088/1367-2630/15/1/013037 23

work page doi:10.1088/1367-2630/15/1/013037 2013

[1] [1]

Manuela A. D. Aguiar and Ana Paula S. Dias. 2018. Synchronization and equitable partitions in weighted networks.Chaos: An Interdisciplinary Journal of Nonlinear Science28, 7 (07 2018), 073105. doi:10.1063/ 1.4997385

2018

[2] [2]

Graziano Amati and Gregory D. Scholes. 2025. Quantum information with quantumlike bits.Phys. Rev. A 111 (Jun 2025), 062203. Issue 6. doi:10.1103/PhysRevA.111.062203

work page doi:10.1103/physreva.111.062203 2025

[3] [3]

Yuto Ashida, Zongping Gong, and Masahito Ueda. 2020. Non-Hermitian physics.Advances in Physics 69, 3 (2020), 249–435. arXiv:https://doi.org/10.1080/00018732.2021.1876991 doi:10.1080/00018732.2021. 1876991

work page doi:10.1080/00018732.2021.1876991 2020

[4] [4]

Opperman, Christino Tamon, and Andrew Tollefson

Rachel Bachman, Eric Fredette, Jessica Fuller, Michael Landry, Michael C. Opperman, Christino Tamon, and Andrew Tollefson. 2012. Perfect state transfer on quotient graphs.Quantum Information & Computation 12, 3-4 (2012), 293–313. Published version URL unavailable; arXiv:https://arxiv.org/abs/1108.0339

work page arXiv 2012

[5] [5]

Bender and Stefan Boettcher

Carl M. Bender and Stefan Boettcher. 1998. Real Spectra in Non-Hermitian Hamiltonians HavingPT Symmetry.Phys. Rev. Lett.80 (Jun 1998), 5243–5246. Issue 24. doi:10.1103/PhysRevLett.80.5243

work page doi:10.1103/physrevlett.80.5243 1998

[6] [6]

Bender, Dorje C

Carl M. Bender, Dorje C. Brody, and Hugh F. Jones. 2002. Complex Extension of Quantum Mechanics. Phys. Rev. Lett.89 (Dec 2002), 270401. Issue 27. doi:10.1103/PhysRevLett.89.270401

work page doi:10.1103/physrevlett.89.270401 2002

[7] [7]

Bergholtz, Jan Carl Budich, and Flore K

Emil J. Bergholtz, Jan Carl Budich, and Flore K. Kunst. 2021. Exceptional topology of non-Hermitian systems.Rev. Mod. Phys.93 (Feb 2021), 015005. Issue 1. doi:10.1103/RevModPhys.93.015005

work page doi:10.1103/revmodphys.93.015005 2021

[8] [8]

Sougato Bose. 2003. Quantum Communication through an Unmodulated Spin Chain.Phys. Rev. Lett.91 (Nov 2003), 207901. Issue 20. doi:10.1103/PhysRevLett.91.207901

work page doi:10.1103/physrevlett.91.207901 2003

[9] [9]

DiVincenzo, and Daniel Loss

Sergey Bravyi, David P. DiVincenzo, and Daniel Loss. 2011. Schrieffer–Wolff transformation for quantum many-body systems.Annals of Physics326, 10 (2011), 2793–2826. doi:10.1016/j.aop.2011.06.004

work page doi:10.1016/j.aop.2011.06.004 2011

[10] [10]

Sergey Bravyi and Jeongwan Haah. 2012. Magic-state distillation with low overhead.Phys. Rev. A86 (Nov 2012), 052329. Issue 5. doi:10.1103/PhysRevA.86.052329

work page doi:10.1103/physreva.86.052329 2012

[11] [11]

Sergey Bravyi and Alexei Kitaev. 2005. Universal quantum computation with ideal Clifford gates and noisy ancillas.Phys. Rev. A71 (Feb 2005), 022316. Issue 2. doi:10.1103/PhysRevA.71.022316

work page doi:10.1103/physreva.71.022316 2005

[12] [12]

Sergey Bravyi, Graeme Smith, and John A. Smolin. 2016. Trading Classical and Quantum Computational Resources.Phys. Rev. X6 (Jun 2016), 021043. Issue 2. doi:10.1103/PhysRevX.6.021043

work page doi:10.1103/physrevx.6.021043 2016

[13] [13]

Roads towards fault- tolerant universal quantum computation

Earl T. Campbell, Barbara M. Terhal, and Christophe Vuillot. 2017. Roads towards fault-tolerant universal quantum computation.Nature549, 7671 (Sept. 2017), 172–179. doi:10.1038/nature23460

work page doi:10.1038/nature23460 2017

[14] [14]

Andrew M. Childs. 2009. Universal Computation by Quantum Walk.Phys. Rev. Lett.102 (May 2009), 180501. Issue 18. doi:10.1103/PhysRevLett.102.180501

work page doi:10.1103/physrevlett.102.180501 2009

[15] [15]

Dorlas, Artur Ekert, Alastair Kay, and Andrew J

Matthias Christandl, Nilanjana Datta, Tony C. Dorlas, Artur Ekert, Alastair Kay, and Andrew J. Landahl

[16] [16]

Perfect transfer of arbitrary states in quantum spin networks.Phys. Rev. A71 (Mar 2005), 032312. Issue 3. doi:10.1103/PhysRevA.71.032312

work page doi:10.1103/physreva.71.032312 2005

[17] [17]

Matthias Christandl, Nilanjana Datta, Artur Ekert, and Andrew J. Landahl. 2004. Perfect State Transfer in Quantum Spin Networks.Phys. Rev. Lett.92 (May 2004), 187902. Issue 18. doi:10.1103/PhysRevLett. 92.187902

work page doi:10.1103/physrevlett 2004

[18] [18]

Chu and Gene H

Moody T. Chu and Gene H. Golub. 2002. Structured inverse eigenvalue problems.Acta Numerica11 (2002), 1–71. doi:10.1017/S0962492902000016

work page doi:10.1017/s0962492902000016 2002

[19] [19]

Ethan Dickey, Abhijeet Vyas, and Sabre Kais. 2025. Construction and Rigorous Analysis of Quantum-Like States. arXiv:2507.21289 [quant-ph]https://arxiv.org/abs/2507.21289

work page arXiv 2025

[20] [20]

El-Ganainy , author K

Ramy El-Ganainy, Konstantinos G. Makris, Mercedeh Khajavikhan, Ziad H. Musslimani, Stefan Rotter, and Demetrios N. Christodoulides. 2018. Non-Hermitian physics and PT symmetry.Nature Physics14, 1 (2018), 11–19. doi:10.1038/nphys4323

work page doi:10.1038/nphys4323 2018

[21] [21]

Edward Farhi and Sam Gutmann. 1998. Quantum computation and decision trees.Phys. Rev. A58 (Aug 1998), 915–928. Issue 2. doi:10.1103/PhysRevA.58.915 22

work page doi:10.1103/physreva.58.915 1998

[22] [22]

Chris Godsil. 2012. State transfer on graphs.Discrete Mathematics312, 1 (2012), 129–147. doi:10.1016/ j.disc.2011.06.032Algebraic Graph Theory — A Volume Dedicated to Gert Sabidussi on the Occasion of His 80th Birthday

2012

[23] [23]

Mark Howard, Joel Wallman, Victor Veitch, and Joseph Emerson. 2014. Contextuality supplies the ‘magic’ for quantum computation.Nature510, 7505 (June 2014), 351–355. doi:10.1038/nature13460

work page doi:10.1038/nature13460 2014

[24] [24]

Cody Jones. 2013. Multilevel distillation of magic states for quantum computing.Phys. Rev. A87 (Apr 2013), 042305. Issue 4. doi:10.1103/PhysRevA.87.042305

work page doi:10.1103/physreva.87.042305 2013

[25] [25]

Julia Kempe. 2003. Quantum random walks: An introductory overview.Contemporary Physics44, 4 (2003), 307–327. arXiv:https://doi.org/10.1080/00107151031000110776 doi:10.1080/00107151031000110776

work page doi:10.1080/00107151031000110776 2003

[26] [26]

Hari Krovi and Todd A. Brun. 2007. Quantum walks on quotient graphs.Phys. Rev. A75 (Jun 2007), 062332. Issue 6. doi:10.1103/PhysRevA.75.062332

work page doi:10.1103/physreva.75.062332 2007

[27] [27]

Ali Mostafazadeh. 2002. Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian.J. Math. Phys.43, 1 (01 2002), 205–214. doi:10.1063/ 1.1418246

2002

[28] [28]

S Unnikrishna Pillai, Torsten Suel, and Seunghun Cha. 2005. The Perron-Frobenius theorem: some of its applications.IEEE Signal Processing Magazine22, 2 (2005), 62–75. doi:10.1109/MSP.2005.1406483

work page doi:10.1109/msp.2005.1406483 2005

[29] [29]

Gregory D. Scholes. 2024. Quantum-like states on complex synchronized networks. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 480, 2295 (08 2024), 20240209. arXiv:https://royalsocietypublishing.org/rspa/article- pdf/doi/10.1098/rspa.2024.0209/512921/rspa.2024.0209.pdf doi:10.1098/rspa.2024.0209

work page doi:10.1098/rspa.2024.0209/512921/rspa.2024.0209.pdf 2024

[30] [30]

Scholes and Graziano Amati

Gregory D. Scholes and Graziano Amati. 2025. Quantumlike Product States Constructed from Classical Networks.Phys. Rev. Lett.134 (Feb 2025), 060202. Issue 6. doi:10.1103/PhysRevLett.134.060202

work page doi:10.1103/physrevlett.134.060202 2025

[31] [31]

Seddon and Earl T

James R. Seddon and Earl T. Campbell. 2019. Quantifying magic for multi-qubit oper- ations.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sci- ences475, 2227 (07 2019), 20190251. arXiv:https://royalsocietypublishing.org/rspa/article- pdf/doi/10.1098/rspa.2019.0251/1221292/rspa.2019.0251.pdf doi:10.1098/rspa.2019.0251

work page doi:10.1098/rspa.2019.0251/1221292/rspa.2019.0251.pdf 2019

[32] [32]

Victor Veitch, S A Hamed Mousavian, Daniel Gottesman, and Joseph Emerson. 2014. The resource theory of stabilizer quantum computation.New Journal of Physics16, 1 (jan 2014), 013009. doi:10.1088/1367-2630/ 16/1/013009

work page doi:10.1088/1367-2630/ 2014

[33] [33]

Victor Veitch, Nathan Wiebe, Christopher Ferrie, and Joseph Emerson. 2013. Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation.New Journal of Physics 15, 1 (jan 2013), 013037. doi:10.1088/1367-2630/15/1/013037 23

work page doi:10.1088/1367-2630/15/1/013037 2013