Pith. sign in

REVIEW 2 major objections 5 minor 52 references

The Impact of Architecture and Cost Function on Dissipative Quantum Neural Networks

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Adding an ancilla layer to each perceptron lets even a three-node dissipative quantum neural network represent any quantum channel, and the paper identifies which of eight cost functions trains it best.

desk verdict The isometry parametrization and channel-universal architecture are solid and worth citing, but the p-fidelity formula has a real error and the cost-function comparison lacks the reproducibility details to support its ranking. read the letter →

arxiv 2502.09526 v1 pith:7INKYWHU submitted 2025-02-13 quant-ph

Tobias C. Sutter , Christopher Popp , Beatrix C. Hiesmayr This is my paper · ORCID
classification quant-ph PACS 03.67.-a03.67.Lx
keywords dissipativequantumneuralnetworkschanneluniversalityisometryparametrizationcompositecostfunctioncomparisonChoistatetrainingdiamonddistanceWerner
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 extends dissipative quantum neural networks (DQNNs) so that each building block can implement any quantum channel, a property it calls quantum channel universality. The extension is simple: after every input or hidden layer the network adds an ancilla layer that is later traced out, turning each perceptron into a general isometry rather than a unitary. With that change, a minimal network of three qubits -- one input, one ancilla, one output -- can represent every completely positive trace-preserving map from input to output, provided the ancilla and output dimensions match. Using a derived one-to-one parametrization of isometries, the authors train this minimal network on random qubit channels and on the Werner channel under eight candidate cost functions. Under Choi-state training the Bures distance and Hilbert-Schmidt distance reach channel distances below $10^{-3}$, while hypothesis-testing-based costs leave the network much further from the target.

What carries the argument

The central object is the extended perceptron isometry, $V_k^{(\ell,\ell+1,\ell+2)}\in\mathrm{Iso}(\mathcal{H}_\ell,\mathcal{H}_{\ell+2}^{(k)}\otimes\mathcal{H}_{\ell+1}^{(k)}\otimes\mathcal{H}_\ell)$, which maps the current layer into the next layer plus an ancilla that will be discarded. The paper parametrizes every such isometry by the composite parametrization $V = \left[\prod_{m=0}^{d_1-1}\prod_{n=m+1}^{d_2-1}\Lambda_{m,n}\right]\left[\prod_{l=0}^{d_1-1}e^{iP_l\lambda_{ll}}\right]\mathbb{1}_{d_2\times d_1}$, with $\Lambda_{m,n}=e^{iP_n\lambda_{n,m}}e^{iY_{m,n}\lambda_{m,n}}$. This one-to-one parametrization is what carries the argument: it guarantees that the minimal network's single perceptron can reach every Stinespring isometry, and it provides the parameter count and the analytical gradients used in the numerical training.

What would settle it

Re-run the same 100-channel comparison with multiple random initializations per cost function and compute the diamond distance with an exact semidefinite-programming solver instead of the Monte Carlo estimate; if the gap between the Bures and Hilbert-Schmidt costs and the next-best cost function falls within the run-to-run variance, the reported ranking is not established.

Watch

Extended reading notes

Core claim

The paper's central claim is that the conventional DQNN ansatz, whose unitary perceptrons cannot always be factored into a single Stinespring isometry, becomes quantum channel universal when each perceptron is replaced by an isometry that grows the Hilbert space by one ancilla neuron. Concretely, for any CPTP map $\mathcal{E}: \mathcal{D}(\mathcal{H}_1)\to\mathcal{D}(\mathcal{H}_3)$ with $\dim(\mathcal{H}_2)=\dim(\mathcal{H}_3)$, there is an isometry $V\in\mathrm{Iso}(\mathcal{H}_1,\mathcal{H}_2\otimes\mathcal{H}_3)$ such that $\mathcal{E}(\rho)=\mathrm{Tr}_2[V\rho V^\dagger]$, and the minimal three-neuron extended network realizes exactly this map. The composite parametrization implemented in the training loop realizes $V$ with $2d_1d_2-d_1^2$ real parameters, fewer than the $d^4$ needed for the unitary formulation. Numerically, the paper finds the claimed universality in action: with Choi training, the best cost functions drive the mean diamond distance to a few $10^{-4}$ after 1000 iterations, whereas random-state training stalls around $5\times10^{-2}$.

Load-bearing premise

The numerical ranking of the eight cost functions assumes that the implementation details -- the size of the random initialization perturbation, the ADAM settings, the batch-regeneration rule, and the Monte Carlo algorithm used for the diamond distance -- do not change the relative ordering, and that mean values without error bars are reliable indicators.

Editorial extensions

If this is right

  • Each perceptron of a larger extended DQNN is itself quantum channel universal, so deep networks are assembled from universally expressive building blocks.
  • The switch from unitaries to isometries cuts the number of variational parameters per perceptron from $d^4$ to $d^2(2d-1)$, reducing the optimization cost for the same architecture.
  • Choi-state training avoids the non-unique choice of sampling geometry and gives an objective benchmark; under it the Bures and Hilbert-Schmidt distances outperform the other six cost functions by at least an order of magnitude in final diamond distance.
  • With random-state training, almost all eight cost functions converge to roughly the same channel distance near $5\times10^{-2}$, so cost-function choice matters less when training on sampled inputs.
  • For the Werner channel, the learning speed correlates with the channel parameter $\alpha$: higher $\alpha$ (including entanglement-breaking channels) trains faster, while the completely depolarizing channel $\alpha=0$ shows delayed convergence.

Reading between the lines

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

  • The paper proves universality for the minimal single-perceptron network but leaves open whether composing these universal perceptrons preserves channel universality for the whole deep network; a natural next step is to check that closure under composition.
  • Since the Hilbert-Schmidt distance performs best despite violating the data-processing inequality, the relevant feature for trainability may be gradient behavior near the optimum rather than monotone contraction under channels; this could be tested by comparing gradient norms of the eight cost functions along the same training trajectory.
  • Because the Hilbert-Schmidt distance is measurable with SWAP tests or shadow tomography, the paper's preferred cost function is a concrete candidate for a hardware demonstration of the same Werner-channel learning task.
  • The Werner-parameter dependence suggests a testable conjecture: the entanglement-breaking region of a channel family controls Choi-training speed, which one could probe with other one-parameter families such as Pauli or amplitude-damping channels.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes an extended dissipative quantum neural network architecture in which each perceptron is realized as an isometry acting on the input layer, an ancilla layer, and the next hidden/output layer. The authors argue that a minimal network consisting of one input, one ancilla, and one output qudit is quantum channel universal when the ancilla and output dimensions coincide, because every CPTP map admits a Stinespring isometry of the required dimension. They derive a one-to-one composite parametrization of isometries from the composite parametrization of U(d), count the resulting parameters, and compare eight cost functions under Choi training and random state training on 100 random qubit channels and on the Werner channel, using the diamond distance as an independent performance metric.

Significance. The universality claim is a meaningful conceptual contribution: it gives a precise Stinespring-based notion of what it means for a DQNN building block to be able to implement arbitrary quantum channels, and it is well supported by the stated dimension counting. The composite parametrization in Appendix A has the correct real dimension 2d1d2−d1^2, and the analytic gradient formulas in Appendix B are a useful technical resource. The numerical comparison of cost functions is potentially valuable, especially the observation that F_QCB and D_QRE fail to converge under Choi training, but the numerical section is not yet reproducible from the information given. In addition, the paper contains a concrete mathematical error in the claimed reduction of the p-fidelity family to the Uhlmann-Jozsa and Hilbert-Schmidt fidelities.

major comments (2)
  1. [Sec. 3, Eq. (20)] The claimed special cases of the p-fidelity are incorrect as written. Substituting p=1 into Eq. (20) gives numerator ||sqrt(sigma)sqrt(rho)||_1^2 = F_Uhlmann(rho,sigma), but the denominator is max{||sigma||_2, ||rho||_2} = max{sqrt(Tr(rho^2)), sqrt(Tr(sigma^2))}, which is not identically 1. Hence Eq. (20) does not reduce to the Uhlmann-Jozsa fidelity in Eq. (21). Similarly, p=2 gives numerator Tr(rho sigma) with denominator max{(Tr(rho^4))^(1/4), (Tr(sigma^4))^(1/4)}, not max{Tr(rho^2), Tr(sigma^2)}, so Eq. (23) is not obtained either. The statement that Eq. (20) 'covers' the Uhlmann-Jozsa and Hilbert-Schmidt fidelities therefore needs correction, either by using the correct p-fidelity definition from Ref. [29] or by presenting F1 and F2 independently rather than as special cases of Eq. (20).
  2. [Sec. 4.1, Fig. 4a] The central numerical conclusion is that D1 and DHS outperform the other cost functions, with mean diamond distances of 3.43e-4 and 4.55e-4, respectively, while F_QCB and D_QRE fail to converge. The manuscript reports no confidence intervals or standard deviations, no random seeds, no ADAM learning rate or other optimizer hyperparameters, no initialization perturbation scale, and no details of the Monte Carlo diamond-norm estimator from Ref. [41]. Since the gap between the two best cost functions is only about 1.1e-4, this ranking could be within the noise of these choices. The batch-regeneration rule for random state training ('once a cost optimum is reached for a batch') is not operationalized, and the order of arguments of the asymmetric DQRE in the cost function C(rho_tar, rho_out) is not specified. These details should be supplied, ideally together with code or data, before the cost-function comparison can be assessed as a reproducible experimental claim.
minor comments (5)
  1. [Sec. 4.2] The sentence 'using Choi training and the Hilbert-Schmidt distance cost function (19)' should refer to Eq. (18), since Eq. (19) is the trace distance.
  2. [Fig. 3 caption] The caption says 'An extended version of the network in Fig. 3' but should refer to Fig. 1.
  3. [Sec. 4.2, Fig. 5] The statement that 'other cost functions show similar behavior' is not supported by a displayed figure or table; the authors should either specify which cost functions were tested for the Werner channel or soften the claim.
  4. [Sec. 2.2] The universality argument would be easier to verify if the Stinespring dimension bound were stated explicitly: a channel from H1 to H3 has a Stinespring isometry with environment dimension at most d1*d3, which is exactly what the condition dim(H2)=dim(H3) supplies.
  5. [Sec. 5] The phrase 'at the prize of increasing their size' contains a typo: 'prize' should be 'price'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the universality construction is a direct Stinespring dilation, the isometry parametrization is derived from an external unitary-group theorem, and the numerical cost-function comparison is benchmarked by an independent diamond-distance metric.

full rationale

The central universality claim (Sec. 2.2) is not equivalent to its inputs. A minimal extended DQNN with one isometric perceptron V ∈ Iso(H1, H3⊗H2⊗H1) produces ρout = Tr_{1,2}[V ρin V†]; with dim(H2)=dim(H3) this is exactly the Stinespring form of an arbitrary CPTP map from H1 to H3, so the claim rests on the standard Stinespring theorem rather than on a redefinition. The isometry parametrization (App. A, Eq. (6)) is derived from the composite parametrization of U(d) in Refs. [14,15]; that cited result is an externally published, parameter-free theorem whose assumptions do not include the isometry parametrization, so the derivation is not circular even though one author is shared. The numerical trainability study (Sec. 4) trains with various cost functions but evaluates all trained networks with the diamond norm, an independent channel-distance measure from Ref. [41], so the cost-function ranking is not a fitted parameter renamed as a prediction. The only self-citations (Refs. [7], [14,15]) supply a derivative identity and a unitary-group parameterization; neither is load-bearing in the sense of being equivalent to the paper's claims. No circular step can be exhibited with a specific equation-to-equation reduction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the completeness of the prior unitary parametrization, the standard Stinespring and Choi formalisms, and an unverified numerical diamond norm estimator. No new physical entities are introduced.

free parameters (3)
  • Random initialization perturbation scale
    Training uses isometry parameters initialized randomly but close to zero with a small numerical perturbation; the scale is not specified and affects convergence speed (Sec. 4).
  • ADAM optimizer hyperparameters
    ADAM is used for 1000 iterations, but learning rate and decay settings are not reported; these affect the relative performance of cost functions.
  • Monte Carlo diamond norm parameters
    The diamond distance is computed with the Monte Carlo algorithm of Ref. [41], but the number of samples or convergence criteria are not reported; reported values near 1e-4 may be sensitive to this.
assumptions (4)
  • standard math The composite parametrization of U(d2) from Refs. [14,15] covers all unitary matrices, up to possible boundary ambiguities.
    Used in Appendix A to derive the isometry parametrization; the paper does not prove this result and relies on the cited prior work.
  • standard math Stinespring dilation theorem and Choi-Jamiolkowski isomorphism.
    Used to define quantum channel universality in Sec. 2.2 and Choi training in Sec. 2.1.3.
  • domain assumption The Monte Carlo diamond norm estimate of Ref. [41] is accurate at the reported precision.
    All numerical benchmarks rely on diamond distance estimates; no convergence or error analysis is reported.
  • standard math The p-fidelity definitions and fidelity axioms from Refs. [21,29] are valid.
    The cost function discussion in Sec. 3 relies on these properties, although Eq. (20) does not reduce to the claimed Uhlmann fidelity for p=1.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Impact of Architecture and Cost Function on Dissipative Quantum Neural Networks." pith.science (2026). https://pith.science/paper/7INKYWHU

@misc{pith2026250209526,
  author       = {Pith},
  title        = {Pith review of: The Impact of Architecture and Cost Function on Dissipative Quantum Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7INKYWHU}},
  note         = {Machine review of arXiv:2502.09526}
}
read the original abstract

Combining machine learning and quantum computation is a potential path towards powerful applications on quantum devices. Regarding this, quantum neural networks are a prominent approach. In this work, we present a novel architecture for dissipative quantum neural networks (DQNNs) in which each building block can implement any quantum channel, thus introducing a clear notion of universality suitable for the quantum framework. To this end, we reformulate DQNNs using isometries instead of conventionally used unitaries, thereby reducing the number of parameters in these models. We furthermore derive a versatile one-to-one parametrization of isometries, allowing for an efficient implementation of the proposed structure. Focusing on the impact of different cost functions on the optimization process, we numerically investigate the trainability of extended DQNNs. This unveils significant training differences among the cost functions considered. Our findings facilitate both the theoretical understanding and the experimental implementability of quantum neural networks.

Figures

Figures reproduced from arXiv: 2502.09526 by the authors.

Figure 1
Figure 1. Conventional dissipative quantum neural network (DQNN) with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The minimal version of an extended DQNN consists of three neurons and can be viewed as the prototype [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. An extended version of the network in Fig. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The plots show the mean diamond distance [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Learning the Werner channel EW,α using Choi training with the Hilbert-Schmidt distance (18) as the cost function. The colors indicate the value of α ∈ [−1, 1]. The completely depolarizing channel EW,0 is highlighted in black [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

52 extracted references · 47 canonical work pages

  1. [29]

    Quantum fidelity measures for mixed states

    Yeong-Cherng Liang, Yu-Hao Yeh, Paulo E M F Mendonça, Run Yan Teh, Margaret D Reid, and Peter D Drummond. “Quantum fidelity measures for mixed states”. Reports on Progress in Physics82, 076001 (2019)

    work page 2019

  2. [41]

    Com- puting the distance between quantum chan- nels: usefulness of the Fano representation

    Giuliano Benenti and Giuliano Strini. “Com- puting the distance between quantum chan- nels: usefulness of the Fano representation”. Journal of Physics B: Atomic, Molecular and Optical Physics43, 215508 (2010)

    work page 2010

  3. [1]

    Quantum computing and the entanglement frontier

    John Preskill. “Quantum computing and the entanglement frontier” (2012). arXiv:1203.5813 [quant-ph]

    arXiv 2012

  4. [2]

    Quantum Computing in the NISQ era and beyond

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

    work page 2018

  5. [3]

    Challenges and opportunities in quantum machine learning

    M. Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J. Coles. “Challenges and opportunities in quantum machine learning”. Nature Computational Science 2, 567–576 (2022)

    work page 2022

  6. [4]

    Training deep quantum neural networks

    Kerstin Beer, Dmytro Bondarenko, Terry Farrelly, Tobias J. Osborne, Robert Salz- mann, Daniel Scheiermann, and Ramona Wolf. “Training deep quantum neural networks”. Nature Communications 11, 808 (2020)

    work page 2020

  7. [5]

    Approximation capabilities of multilayer feedforward networks

    Kurt Hornik. “Approximation capabilities of multilayer feedforward networks”. Neural Networks4, 251–257 (1991)

    work page 1991

  8. [6]

    Is Quantum Advantage the Right Goal for Quantum Machine Learning?

    Maria Schuld and Nathan Killoran. “Is Quantum Advantage the Right Goal for Quantum Machine Learning?”. PRX Quan- tum 3, 030101 (2022)

    work page 2022

Show all 52 references
  1. [7]

    A quantum informa- tion theoretic view on a deep quantum neu- ral network

    Beatrix C. Hiesmayr. “A quantum informa- tion theoretic view on a deep quantum neu- ral network”. AIP Conference Proceedings 3061, 020001 (2024)

    2024

  2. [8]

    Effect of data encod- ing on the expressive power of variational quantum-machine-learning models

    Maria Schuld, Ryan Sweke, and Jo- hannes Jakob Meyer. “Effect of data encod- ing on the expressive power of variational quantum-machine-learning models”. Physi- cal Review A103, 032430 (2021)

    2021

  3. [9]

    Cost func- tion dependent barren plateaus in shallow parametrized quantum circuits

    M.Cerezo, AkiraSone, TylerVolkoff, Lukasz Cincio, and Patrick J. Coles. “Cost func- tion dependent barren plateaus in shallow parametrized quantum circuits”. Nature Communications 12, 1791 (2021)

    2021

  4. [10]

    Trainability of Dis- sipative Perceptron-Based Quantum Neural Networks

    Kunal Sharma, M. Cerezo, Lukasz Cincio, and Patrick J. Coles. “Trainability of Dis- sipative Perceptron-Based Quantum Neural Networks”. Physical Review Letters 128, 180505 (2022)

    2022

  5. [11]

    Entanglement-Induced Barren Plateaus

    Carlos Ortiz Marrero, Mária Kieferová, and Nathan Wiebe. “Entanglement-Induced Barren Plateaus”. PRX Quantum 2, 040316 (2021)

    2021

  6. [12]

    Entanglement de- vised barren plateau mitigation

    Taylor L. Patti, Khadijeh Najafi, Xun Gao, and Susanne F. Yelin. “Entanglement de- vised barren plateau mitigation”. Physical Review Research3, 033090 (2021)

    2021

  7. [13]

    Entanglement of Formation of an Arbitrary State of Two Qubits

    William K. Wootters. “Entanglement of Formation of an Arbitrary State of Two Qubits”. Physical Review Letters80, 2245– 2248 (1998)

    1998

  8. [14]

    A composite pa- rameterization of unitary groups, den- sity matrices and subspaces

    Christoph Spengler, Marcus Huber, and Beatrix C Hiesmayr. “A composite pa- rameterization of unitary groups, den- sity matrices and subspaces”. Journal of Physics A: Mathematical and Theoretical 43, 385306 (2010)

    2010

  9. [15]

    Composite param- eterization and Haar measure for all uni- tary and special unitary groups

    Christoph Spengler, Marcus Huber, and Beatrix C. Hiesmayr. “Composite param- eterization and Haar measure for all uni- tary and special unitary groups”. Journal of Mathematical Physics53, 013501 (2012)

    2012

  10. [16]

    The Theory of Quantum Information

    John Watrous. “The Theory of Quantum Information”. Cambridge University Press. (2018). 1 edition

    2018

  11. [17]

    GeometryofQuantumStates: AnIntroduc- tion to Quantum Entanglement

    Ingemar Bengtsson and Karol Zyczkowski. “GeometryofQuantumStates: AnIntroduc- tion to Quantum Entanglement”. Cambridge University Press. Cambridge (2006)

    2006

  12. [18]

    Adam: A Method for Stochastic Optimiza- tion

    Diederik P. Kingma and Jimmy Ba. “Adam: A Method for Stochastic Optimiza- tion” (2017). arXiv:1412.6980 [cs]

    2017 arXiv

  13. [19]

    Completely positive linear maps on complex matrices

    Man-Duen Choi. “Completely positive linear maps on complex matrices”. Linear Algebra and its Applications10, 285–290 (1975)

    1975

  14. [20]

    Linear transformations which preserve trace and positive semidefi- niteness of operators

    A. Jamiołkowski. “Linear transformations which preserve trace and positive semidefi- niteness of operators”. Reports on Mathe- matical Physics3, 275–278 (1972)

    1972

  15. [21]

    Quantifying Entangle- ment

    V. Vedral, M. B. Plenio, M. A. Rippin, 11 and P. L. Knight. “Quantifying Entangle- ment”. Physical Review Letters 78, 2275– 2279 (1997)

    1997

  16. [22]

    Fidelity for Mixed Quantum States

    Richard Jozsa. “Fidelity for Mixed Quantum States”. Journal of Modern Optics41, 2315– 2323 (1994)

    1994

  17. [23]

    Operationally Invariant Measure of the Distance between Quantum States by Complementary Measurements

    Jinhyoung Lee, M. S. Kim, and Časlav Brukner. “Operationally Invariant Measure of the Distance between Quantum States by Complementary Measurements”. Physi- cal Review Letters91, 087902 (2003)

    2003

  18. [24]

    Stabilization of Quantum Computations by Symmetriza- tion

    Adriano Barenco, André Berthiaume, David Deutsch, Artur Ekert, Richard Jozsa, and Chiara Macchiavello. “Stabilization of Quantum Computations by Symmetriza- tion”. SIAM Journal on Computing 26, 1541–1557 (1997)

    1997

  19. [25]

    Quantum Fin- gerprinting

    Harry Buhrman, Richard Cleve, John Wa- trous, and Ronald De Wolf. “Quantum Fin- gerprinting”. Physical Review Letters 87, 167902 (2001)

    2001

  20. [26]

    Entanglement measures and the Hilbert–Schmidt distance

    Masanao Ozawa. “Entanglement measures and the Hilbert–Schmidt distance”. Physics Letters A268, 158–160 (2000)

    2000

  21. [27]

    Distance measures to compare real and ideal quantum processes

    Alexei Gilchrist, Nathan K. Langford, and Michael A. Nielsen. “Distance measures to compare real and ideal quantum processes”. Physical Review A71, 062310 (2005)

    2005

  22. [28]

    Beyond strong subaddi- tivity? improved bounds on the contraction of generalized relative entropy

    Mary Beth Ruskai. “Beyond strong subaddi- tivity? improved bounds on the contraction of generalized relative entropy”. Reviews in MathematicalPhysics 06, 1147–1161 (1994)

    1994

  23. [30]

    The “transition probability

    A. Uhlmann. “The “transition probability” in the state space of a *-algebra”. Reports on Mathematical Physics9, 273–279 (1976)

    1976

  24. [31]

    Noncommuting Mixed States Cannot Be Broadcast

    HowardBarnum, CarltonM.Caves, Christo- pher A. Fuchs, Richard Jozsa, and Benjamin Schumacher. “Noncommuting Mixed States Cannot Be Broadcast”. Physical Review Let- ters 76, 2818–2821 (1996)

    1996

  25. [32]

    Cross-Platform Verification of Intermediate Scale Quantum Devices

    Andreas Elben, Benoît Vermersch, Rick Van Bijnen, Christian Kokail, Tiff Brydges, Christine Maier, Manoj K. Joshi, Rainer Blatt, Christian F. Roos, and Peter Zoller. “Cross-Platform Verification of Intermediate Scale Quantum Devices”. Physical Review Letters 124, 010504 (2020)

    2020

  26. [33]

    Principles of Quantum Communication Theory: A Modern Approach

    Sumeet Khatri and Mark M. Wilde. “Principles of Quantum Communication Theory: A Modern Approach” (2024). arXiv:2011.04672 [quant-ph]

    2024 arXiv

  27. [34]

    Discriminat- ing States: The Quantum Chernoff Bound

    K. M. R. Audenaert, J. Calsamiglia, R. Muñoz-Tapia, E. Bagan, Ll. Masanes, A. Acin, and F. Verstraete. “Discriminat- ing States: The Quantum Chernoff Bound”. Physical Review Letters98, 160501 (2007)

    2007

  28. [35]

    The Chernoff lower bound for symmetric quan- tum hypothesis testing

    Michael Nussbaum and Arleta Szkoła. “The Chernoff lower bound for symmetric quan- tum hypothesis testing”. The Annals of Statistics 37, 1040–1057 (2009)

    2009

  29. [36]

    Zur quantenmechanischen Be- gründung des zweiten Hauptsatzes der Wärmelehre

    O. Klein. “Zur quantenmechanischen Be- gründung des zweiten Hauptsatzes der Wärmelehre”. ZeitschriftfürPhysik 72, 767– 775 (1931)

    1931

  30. [37]

    Completely positive maps and entropy inequalities

    Göran Lindblad. “Completely positive maps and entropy inequalities”. Communications inMathematicalPhysics 40, 147–151(1975)

    1975

  31. [38]

    An initialization strategy for addressing barren plateaus in parametrized quantum circuits

    Edward Grant, Leonard Wossnig, Mateusz Ostaszewski, and Marcello Benedetti. “An initialization strategy for addressing barren plateaus in parametrized quantum circuits”. Quantum 3, 214 (2019)

    2019

  32. [39]

    On the hardness of distinguishing mixed-state quantum com- putations

    B. Rosgen and J. Watrous. “On the hardness of distinguishing mixed-state quantum com- putations”. In 20th Annual IEEE Conference on Computational Complexity (CCC’05). Pages 344–354. (2005)

    2005

  33. [40]

    Quantum computations: algorithms and error correction

    A Yu Kitaev. “Quantum computations: algorithms and error correction”. Rus- sian Mathematical Surveys 52, 1191– 1249 (1997)

    1997

  34. [42]

    QuantumInformation.jl—A Julia package for numerical computation in quan- tum information theory

    Piotr Gawron, Dariusz Kurzyk, and Łukasz Pawela. “QuantumInformation.jl—A Julia package for numerical computation in quan- tum information theory”. PLOS ONE 13, e0209358 (2018)

    2018

  35. [43]

    Random quantum operations

    Wojciech Bruzda, Valerio Cappellini, Hans- Jürgen Sommers, and Karol Życzkowski. “Random quantum operations”. Physics Let- ters A373, 320–324 (2009)

    2009

  36. [44]

    Induced measures in the space of mixed quantum states

    Karol Zyczkowski and Hans-Jürgen Som- 12 mers. “Induced measures in the space of mixed quantum states”. Journal of Physics A: Mathematical and General 34, 7111 (2001)

    2001

  37. [45]

    Quantum states with Einstein-Podolsky-Rosencorrelationsadmit- ting a hidden-variable model

    Reinhard F. Werner. “Quantum states with Einstein-Podolsky-Rosencorrelationsadmit- ting a hidden-variable model”. Physical Re- view A40, 4277–4281 (1989)

    1989

  38. [46]

    Ap- proximating quantum channels by com- pletely positive maps with small Kraus rank

    Cécilia Lancien and Andreas Winter. “Ap- proximating quantum channels by com- pletely positive maps with small Kraus rank”. Quantum8, 1320 (2024)

    2024

  39. [47]

    Shadow tomography of quantum states

    Scott Aaronson. “Shadow tomography of quantum states”. In Proceedings of the 50th Annual ACM SIGACT Symposium on The- ory of Computing. Pages 325–338. Los An- geles CA USA (2018). ACM

    2018

  40. [48]

    Predicting many proper- ties of a quantum system from very few measurements

    Hsin-Yuan Huang, Richard Kueng, and John Preskill. “Predicting many proper- ties of a quantum system from very few measurements”. Nature Physics 16, 1050– 1057 (2020)

    2020

  41. [49]

    Quantum Tomog- raphy

    G.MauroD’Ariano, MatteoG.A.Paris, and Massimiliano F. Sacchi. “Quantum Tomog- raphy”. In Peter W. Hawkes, editor, Ad- vancesinImagingandElectronPhysics. Vol- ume 128, pages 205–308. Elsevier (2003)

    2003

  42. [50]

    Fast state tomography with optimal error bounds

    M Guţă, J Kahn, R Kueng, and J A Tropp. “Fast state tomography with optimal error bounds”. Journal of Physics A: Mathemati- cal and Theoretical53, 204001 (2020)

    2020

  43. [51]

    Effect of barren plateaus on gradient- free optimization

    Andrew Arrasmith, M. Cerezo, Piotr Czarnik, Lukasz Cincio, and Patrick J. Coles. “Effect of barren plateaus on gradient- free optimization”. Quantum5, 558 (2021)

    2021

  44. [52]

    A generalized inverse for ma- trices

    R. Penrose. “A generalized inverse for ma- trices”. Mathematical Proceedings of the Cambridge Philosophical Society 51, 406– 413 (1955). A The Composite Parametrization of Isometries In this section, we derive the composite parametrization for isometries. As the name suggests,...

    1955

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.