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REVIEW 3 major objections 4 minor 1 cited by

Sequential quantum processes with group symmetries

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Any (G×H)-invariant quantum comb has a streaming circuit built from Clebsch–Gordan transforms, and this yields an exact 7-query qutrit unitary transposition protocol.

desk verdict The structural theorem on symmetric quantum combs is likely correct and genuinely useful; the headline 7-query transposition claim is a numerical result that needs a certificate before it should be taken as proven. read the letter →

arxiv 2510.07100 v2 pith:3J6F6DMS submitted 2025-10-08 quant-ph

Dmitry Grinko , Satoshi Yoshida , Mio Murao , Maris Ozols This is my paper · ORCID
classification quant-ph MSC 81P4522E70
keywords quantumcombsgroupsymmetryClebsch–GordantransformSchurunitarytranspositioninversioncircuitssemidefiniteprogramming
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 proves that a quantum comb whose Choi matrix commutes with the tensor action of a compact group pair G×H can always be implemented as a streaming circuit: a sequence of generalized Clebsch–Gordan transforms connected by isometries that act only on multiplicity spaces. This turns a symmetry condition into an explicit circuit template, and the same construction extends to G-covariant combs. The template is then used to parameterize symmetric combs with dramatically fewer variables than naive circuit parameterizations, making numerical search feasible for larger query numbers. Optimizing these circuits for the tasks of unitary inversion and transposition, the authors find a deterministic, exact protocol that transposes a qutrit unitary using 7 queries, improving the previous best of 13 queries.

What carries the argument

The key mechanism is the generalized Clebsch–Gordan transform, the unitary isomorphism that decomposes the tensor product of an irrep with a representation into a direct sum of irreps, together with its dual, which uses the maximally entangled state between an irrep and its conjugate to decouple the irrep register. These transforms are chained into a streaming circuit in which all free parameters are concentrated in isometries V_i acting on multiplicity spaces; the comb condition is exactly equivalent to the isometry constraints.

What would settle it

A certified SDP upper bound strictly below 1 for qutrit unitary transposition with 7 queries would refute the exact-protocol claim; alternatively, extracting the optimized isometry parameters and running them through a high-precision independent circuit verifier that checks the comb constraints and channel fidelity would settle whether fidelity is truly 1.

Watch

Extended reading notes

Core claim

Theorem 1 states that any quantum comb with (G×H)-invariant Choi matrix can be realized in the form of Fig. 3, composed of generalized Clebsch–Gordan transforms and isometry operators V_i^{λ_i μ_{i-1}} on the multiplicity spaces, with the memory space M_i possibly nontrivial. The proof introduces a dual Schur transform that avoids preparing a maximally mixed state in the middle of the circuit, which had previously prevented streaming implementations. Applied to U(d)×U(d)-invariant combs for unitary transposition and inversion, the resulting parameterization reduces the number of optimization variables by orders of magnitude. Numerically, the authors report an exact deterministic qutrit unita

Load-bearing premise

The load-bearing premise is that the numerical optimizer found a genuine feasible point with fidelity exactly 1 within the dim-M_i=1 ansatz, and that this point can be promoted to an exactly verified circuit; the paper labels the nonlinear results as lower bounds and gives no explicit isometry values or proof.

Editorial extensions

If this is right

  • Every (G×H)-invariant quantum comb can be run as an online, slot-by-slot circuit; symmetry alone guarantees streamability.
  • The parameterization reduces optimization variables by orders of magnitude, allowing searches for larger query numbers than previous SDP approaches.
  • There exists a deterministic exact qutrit unitary transposition protocol using 7 queries, improving the previous 13-query bound.
  • The same circuit template applies to G-covariant combs, giving streaming implementations for covariant channels and simulations.

Reading between the lines

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

  • The 7-query qutrit result suggests that exact transposition in higher dimensions may also be reachable with far fewer than the current bounds; the same parameterization provides a concrete search space to test this.
  • The dual-Schur trick for eliminating the maximally-mixed-state preparation could be a general tool for streaming symmetric circuits beyond combs, such as covariant channels and open-system simulations.
  • Since the memory registers M_i are left arbitrary in Theorem 1, one could probe fidelity–memory trade-offs by constraining their dimension, a direction the numerical ansatz only begins to explore.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a canonical streaming circuit decomposition for quantum combs whose Choi matrix commutes with tensor-product representations of a compact group G on the input side and H on the output side. The construction uses generalized Clebsch–Gordan transforms and isometries acting on multiplicity spaces, and is extended to G-covariant combs. The authors use this parametrization to numerically optimize combs for unitary inversion and transposition. They report an exact deterministic qutrit unitary-transposition protocol using 7 queries, improving on a previously known 13-query protocol.

Significance. If the structural theorem is correct, it is a valuable unification and simplification: it gives a general streaming form for symmetric quantum combs, reduces the number of optimization variables dramatically, and connects the Schur-transform approach with the comb formalism. The proof in Appendix C is detailed and appears plausible. The numerical SDP benchmarks for small n and the matching with known analytical values are useful sanity checks. The headline 7-query exact transposition would be a notable advance over Ref. [52], but that claim is not presently supported with a certificate or explicit circuit parameters.

major comments (3)
  1. [Appendix B, Table II; Abstract; Conclusion] The claim of an exact deterministic 7-query qutrit transposition protocol is not supported. Table II is explicitly captioned 'Lower bounds from nonlinear optimization', and the fidelity 1.000000(0) is a numerical output of a nonconvex solver, not a proof of feasibility or exactness. No explicit isometry parameters, gate decomposition, or code are supplied. The abstract and conclusion state the protocol as established ('we find a deterministic and exact unitary transposition protocol'), which overstates the evidence. Either provide a certified construction (e.g., explicit rational/algebraic isometries or a verifiable circuit) or clearly reframe the result as numerical evidence/conjecture.
  2. [Application section and Appendix B] The optimization restricts all auxiliary memory registers to dim M_i = 1, and the validation of this restriction is only given for n ≤ 5 against SDP/analytical values. For the crucial n = 7 transposition case, there is no evidence that the true optimal or even any exact protocol can be realized with one-dimensional memory. If a larger memory is required, the reported 1.000000(0) may not correspond to a valid comb. The authors should verify the dim M_i = 1 ansatz for n = 7 (e.g., by an SDP with relaxed memory dimension or an independent feasibility check) or state this as an additional assumption.
  3. [Abstract and Section 'Application'] The phrase 'From numerics, we find a deterministic and exact unitary transposition protocol' conflates a numerical lower bound with an existence proof. Appendix B correctly labels all nonlinear optimization results as lower bounds, so the main text should be consistent with that caveat. If the exactness claim is retained, the paper must supply the actual optimized isometry tensors or a certificate that the fidelity is exactly 1 within the chosen ansatz.
minor comments (4)
  1. [Throughout] Several typos: 'Extention' should be 'Extension'; 'togather' should be 'together'; 'Choir' should be 'Choi' in Appendix C; 'represnetations' should be 'representations'; 'optmimization' should be 'optimization'; in Eq. (C107), 'overdλ' should be 'over dλ'.
  2. [Appendix B, Table II] The table caption says 'Lower bounds from nonlinear optimization', but the main text refers to the values as 'optimal fidelities'. Please consistently describe these as lower bounds unless global optimality is certified.
  3. [Appendix A, Table I] The meaning of '?' entries and the exact status of the red value (heuristic SDP lower bound) should be stated directly in the caption for clarity.
  4. [Appendix C, Eq. (C74)] The notation Supp(C_{i-1}^{λ_{i-1} μ_{i-1}})^T uses a transpose/support convention that is not defined; please define it or use a clearer notation for the pseudoinverse on the support.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is a self-contained constructive decomposition, and the 7-query transposition claim is a numerical construction rather than a fitted parameter renamed as a prediction.

full rationale

The paper's central structural claim (Theorem 1) is derived, not assumed. The proof expands a (G × H)-invariant Choi matrix C in the commutant basis (Eq. C58), translates the comb conditions into algebraic conditions on the block matrices (Lemma 4, with Lemma 3 proved self-contained in Appendix C.6), and then explicitly constructs the isometries V_i from the square roots of those block matrices (Eq. C75). The final circuit is shown to reproduce exactly the original Choi matrix C (Eq. C92). No fitted parameter is embedded in this derivation, and no step reduces by definition to the conclusion it is meant to establish. The numerical application is also not circular in the relevant sense. The fidelity objective (Eq. 24) is the actual channel fidelity to the target unitary transformation; optimizing the isometries V_i to maximize it is a standard constructive search. The reported 7-query fidelity 1.000000(0) appears in Table II explicitly as a lower bound from nonlinear optimization, and the paper itself states: 'These numbers should be treated as lower bounds for the optimal values.' Thus the exactness claim is a numerical construction, not a quantity forced by fitting. The dim M_i = 1 ansatz is checked against independent SDP and analytical values for n ≤ 5, so it is not assumed into the result. Self-citations are present but not load-bearing. References [26, 27] supply representation-theoretic tools; Ref. [42] is an external parametrized-comb method; Refs. [48, 52, 55] provide prior protocols and benchmarks. None of these are used as an unverified uniqueness theorem or as a substitute for the proof of Theorem 1. The main caveat is a verification gap, not circularity: the exact 7-query protocol rests on an Ipopt/Gurobi feasible point with no explicit isometry parameters or independent certificate, and Table II labels nonlinear results as lower bounds. If that numerical point is not exactly feasible or promotable to a certified circuit, the headline improvement would collapse. That is a correctness-risk concern and does not make the derivation circular.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central derivation uses standard compact-group representation theory and quantum-comb theory. The only genuinely ad hoc elements are the one-dimensional memory restriction and the unverified numerical global optimality behind the exact 7-query claim.

free parameters (2)
  • Isometry tensor entries V_i^{λ μ} for d=3, n=7 = not reported
    The claimed exact transposition protocol is an assignment of these optimization variables; the paper reports only the resulting fidelity 1.000000(0), not the variables themselves or a certificate.
  • Auxiliary memory dimension dim M_i = 1
    The optimization restricts each memory register to one dimension. The authors justify this assumption only for n≤5 by matching SDP and analytical results, so the n=7 exactness claim inherits an unproven restriction.
assumptions (5)
  • standard math Peter–Weyl theorem and Schur's lemma for compact groups
    Used throughout Appendix C.1 to decompose tensor representations and describe the commutant as spanned by matrix units E^λ_{pq}.
  • domain assumption Quantum comb characterization via Choi matrices and conditions (4)
    The paper relies on the Chiribella–D'Ariano–Perinotti characterization that any matrix satisfying Eq. (4) is the Choi matrix of a quantum comb, cited as Ref. [39].
  • domain assumption Optimal unitary inversion and transposition protocols can be taken U(d)×U(d)-invariant
    The main text states 'whose optimal protocol have the U(d)×U(d)-invariance'; this is a standard twirling argument but is not proven in the paper.
  • ad hoc to paper Numerical optimizer converges to a global optimum and fidelity exactly 1
    The 7-query exact protocol rests on Ipopt/Gurobi returning 1.000000(0) for a non-convex optimization problem, with no global-optimality certificate.
  • ad hoc to paper Memory registers can be restricted to dim M_i=1 in the n=7 search
    Appendix B adopts one-dimensional memory registers; agreement with SDP/analytic values for n≤5 is used as indirect justification, but the n=7 claim is not independently certified.

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Cite this review

Pith. "Pith review of Sequential quantum processes with group symmetries." pith.science (2026). https://pith.science/paper/3J6F6DMS

@misc{pith2026251007100,
  author       = {Pith},
  title        = {Pith review of: Sequential quantum processes with group symmetries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3J6F6DMS}},
  note         = {Machine review of arXiv:2510.07100}
}
abstract

Symmetry plays a crucial role in the design and analysis of quantum protocols. This result shows a canonical circuit decomposition of a $(G\times H)$-invariant quantum comb for compact groups $G$ and $H$ using the corresponding Clebsch--Gordan transforms, which naturally extends to the $G$-covariant quantum comb. By using this circuit decomposition, we propose a parametrized quantum comb with group symmetry, and derive the optimal quantum comb which transforms an unknown unitary operation $U\in \mathrm{SU}(d)$ into its inverse $U^\dagger$ or transpose $U^\top$. From numerics, we find a deterministic and exact unitary transposition protocol for $d=3$ with $7$ queries to $U$. This protocol improves upon the protocol shown in the previous work, which requires $13$ queries to $U$.

Figures

Figures reproduced from arXiv: 2510.07100 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Quantum comb with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Implementation of the quantum comb with the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Comparison of the number of variables in the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Example of a Bratteli diagram [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Bratteli diagram for symmetric group (Young lattice). Example for [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. An example of the Bratteli diagram for representations [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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  1. Probabilistic Storage and Retrieval of Quantum Superchannels for "Retrospective'' Intervention

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Reference graph

Works this paper leans on

73 extracted references · 39 linked inside Pith · cited by 1 Pith paper

  1. [57]

    Grinko and S

    D. Grinko and S. Yoshida, Quantum Simulation of Ran- dom Unitaries from Clebsch-Gordan Transforms (2025), arXiv:2509.26623 [quant-ph]

    arXiv 2025

  2. [58]

    Yoshida, A

    S. Yoshida, A. Soeda, and M. Murao, Universal adjointa- tion of isometry operations using conversion of quantum 7 supermaps, Quantum9, 1750 (2025), arXiv:2401.10137

    arXiv 2025

  3. [52]

    Y.-A. Chen, Y. Mo, Y. Liu, L. Zhang, and X. Wang, Quantum Algorithm for Reversing Unknown Unitary Evolutions, arXiv:2403.04704 (2024)

    arXiv 2024

  4. [1]

    Noether, Invariant variation problems, Trans- port theory and statistical physics1, 186 (1971), arXiv:physics/0503066

    E. Noether, Invariant variation problems, Trans- port theory and statistical physics1, 186 (1971), arXiv:physics/0503066

    arXiv 1971

  5. [2]

    Itzykson and M

    C. Itzykson and M. Nauenberg, Unitary Groups: Repre- sentations and Decompositions, Rev. Mod. Phys.38, 95 (1966)

    1966

  6. [3]

    A. W. Harrow,Applications of coherent classical com- munication and the Schur transform to quantum infor- mation theory, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, USA (2005), arXiv:quant- ph/0512255

    arXiv 2005

  7. [4]

    Hayashi,A Group Theoretic Approach to Quantum Information(Springer, 2017)

    M. Hayashi,A Group Theoretic Approach to Quantum Information(Springer, 2017)

    2017

  8. [5]

    Hayashi,Group Representation for Quantum Theory (Springer, 2017)

    M. Hayashi,Group Representation for Quantum Theory (Springer, 2017)

    2017

Show all 73 references
  1. [6]

    W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned, Nature299, 802 (1982)

    1982

  2. [7]

    Buˇ zek and M

    V. Buˇ zek and M. Hillery, Quantum copying: Beyond the no-cloning theorem, Physical Review A54, 1844 (1996), arXiv:quant-ph/9607018

    1996 arXiv

  3. [8]

    D. Bruß, D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchiavello, and J. A. Smolin, Optimal universal and state-dependent quantum cloning, Physical Review A57, 2368 (1998), arXiv:quant-ph/9705038

    1998 arXiv

  4. [9]

    Gisin and S

    N. Gisin and S. Massar, Optimal quantum cloning machines, Physical review letters79, 2153 (1997), arXiv:quant-ph/9705046

    1997 arXiv

  5. [10]

    R. F. Werner, Optimal cloning of pure states, Physical Review A58, 1827 (1998), arXiv:quant-ph/9804001

    1998 arXiv

  6. [11]

    Keyl and R

    M. Keyl and R. F. Werner, Optimal cloning of pure states, testing single clones, Journal of Mathematical Physics40, 3283 (1999), arXiv:quant-ph/9807010

    1999 arXiv

  7. [12]

    Scarani, S

    V. Scarani, S. Iblisdir, N. Gisin, and A. Ac ´ ın, Quan- tum cloning, Reviews of Modern Physics77, 1225 (2005), arXiv:quant-ph/0511088

    2005 arXiv

  8. [13]

    Marvian and R

    I. Marvian and R. W. Spekkens, Extending Noether’s theorem by quantifying the asymmetry of quan- tum states, Nature communications5, 3821 (2014), arXiv:1404.3236

    2014 arXiv

  9. [14]

    C ˆ ırstoiu, K

    C. C ˆ ırstoiu, K. Korzekwa, and D. Jennings, Robustness of Noether’s principle: Maximal disconnects between conservation laws and symmetries in quantum theory, Physical Review X10, 041035 (2020), arXiv:1908.04254

    2020 arXiv

  10. [15]

    R. F. Werner and A. S. Holevo, Counterexample to an additivity conjecture for output purity of quantum chan- nels, Journal of Mathematical Physics43, 4353 (2002), arXiv:quant-ph/0203003

    2002 arXiv

  11. [16]

    King, The capacity of the quantum depolarizing chan- nel, IEEE Transactions on Information Theory49, 221 (2003), arXiv:quant-ph/0204172

    C. King, The capacity of the quantum depolarizing chan- nel, IEEE Transactions on Information Theory49, 221 (2003), arXiv:quant-ph/0204172

    2003 arXiv

  12. [17]

    Holevo, Additivity conjecture and covariant chan- nels, International Journal of Quantum Information3, 41 (2005)

    A. Holevo, Additivity conjecture and covariant chan- nels, International Journal of Quantum Information3, 41 (2005)

    2005

  13. [18]

    Datta, A

    N. Datta, A. S. Holevo, and Y. Suhov, Additivity for transpose depolarizing channels, International Journal of Quantum Information4, 85 (2006), arXiv:quant- ph/0412034

    2006

  14. [19]

    Koenig and S

    R. Koenig and S. Wehner, A strong converse for classical channel coding using entangled inputs, Physical Review Letters103, 070504 (2009), arXiv:0903.2838. 6

    2009 arXiv

  15. [20]

    Datta, M

    N. Datta, M. Tomamichel, and M. M. Wilde, On the second-order asymptotics for entanglement-assisted com- munication, Quantum Information Processing15, 2569 (2016), arXiv:1405.1797

    2016 arXiv

  16. [21]

    M. M. Wilde, M. Tomamichel, and M. Berta, Converse bounds for private communication over quantum chan- nels, IEEE Transactions on Information Theory63, 1792 (2017), arXiv:1602.08898

    2017 arXiv

  17. [22]

    Brannan, B

    M. Brannan, B. Collins, H. H. Lee, and S.-G. Youn, Temperley–Lieb quantum channels, Communi- cations in Mathematical Physics376, 795 (2020), arXiv:1810.08001

    2020 arXiv

  18. [23]

    Kong and Z.-W

    L. Kong and Z.-W. Liu, Near-optimal covariant quan- tum error-correcting codes from random unitaries with symmetries, PRX Quantum3, 020314 (2022), arXiv:2112.01498

    2022 arXiv

  19. [24]

    Lin, Z.-W

    C.-J. Lin, Z.-W. Liu, V. V. Albert, and A. V. Gorshkov, Covariant Quantum Error-Correcting Codes with Metro- logical Entanglement Advantage, Physical Review Let- ters135, 110801 (2025), arXiv:2409.20561

    2025

  20. [25]

    J. R. Glick, T. P. Gujarati, A. D. Corcoles, Y. Kim, A. Kandala, J. M. Gambetta, and K. Temme, Covariant quantum kernels for data with group structure, Nature Physics20, 479 (2024), arXiv:2105.03406

    2024 arXiv

  21. [26]

    Grinko and M

    D. Grinko and M. Ozols, Linear programming with unitary-equivariant constraints, Communications in Mathematical Physics405, 278 (2024)

    2024

  22. [27]

    Grinko, A

    D. Grinko, A. Burchardt, and M. Ozols, Gelfand-Tsetlin basis for partially transposed permutations, with applica- tions to quantum information, arXiv:2310.02252 (2023)

    2023 arXiv

  23. [28]

    Chiribella and D

    G. Chiribella and D. Ebler, Optimal quantum networks and one-shot entropies, New Journal of Physics18, 093053 (2016), arXiv:1606.02394

    2016 arXiv

  24. [29]

    Tavakoli, A

    A. Tavakoli, A. Pozas-Kerstjens, P. Brown, and M. Ara´ ujo, Semidefinite programming relaxations for quantum correlations, Reviews of Modern Physics96, 045006 (2024), arXiv:2307.02551

    2024 arXiv

  25. [30]

    Bacon, I

    D. Bacon, I. L. Chuang, and A. W. Harrow, Effi- cient quantum circuits for schur and clebsch-gordan transforms, Physical review letters97, 170502 (2006), arXiv:quant-ph/0407082

    2006 arXiv

  26. [31]

    Bacon, I

    D. Bacon, I. L. Chuang, and A. W. Harrow, The quantum schur transform: I. efficient qudit circuits, inProceed- ings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms(2007) pp. 1235–1244, arXiv:quant- ph/0601001

    2007

  27. [32]

    Q. T. Nguyen, The mixed schur transform: efficient quan- tum circuit and applications, arXiv preprint (2023), arXiv:2310.01613

    2023 arXiv

  28. [33]

    J. Fei, S. Timmerman, and P. Hayden, Efficient quantum algorithm for port-based teleportation, arXiv preprint (2023), arXiv:2310.01637

    2023 arXiv

  29. [34]

    Buhrman, N

    H. Buhrman, N. Linden, L. Manˇ cinska, A. Montanaro, and M. Ozols, Quantum majority vote, arXiv:2211.11729 (2022)

    2022 arXiv

  30. [35]

    Cervero and L

    E. Cervero and L. Manˇ cinska, Weak schur sampling with logarithmic quantum memory, arXiv:2309.11947 (2023)

    2023 arXiv

  31. [36]

    Cervero-Mart ´ ın, L

    E. Cervero-Mart ´ ın, L. Manˇ cinska, and E. Theil, A mem- ory and gate efficient algorithm for unitary mixed schur sampling, arXiv:2410.15793 (2024)

    2024 arXiv

  32. [37]

    Chitambar and G

    E. Chitambar and G. Gour, Quantum resource the- ories, Reviews of modern physics91, 025001 (2019), arXiv:1806.06107

    2019 arXiv

  33. [38]

    Taranto, S

    P. Taranto, S. Milz, M. Murao, M. T. Quintino, and K. Modi, Higher-order quantum operations, arXiv:2503.09693 (2025)

    2025 arXiv

  34. [39]

    Chiribella, G

    G. Chiribella, G. M. D’Ariano, and P. Perinotti, Quan- tum circuit architecture, Physical Review Letters101, 060401 (2008), arXiv:0712.1325

    2008 arXiv

  35. [40]

    F. A. Pollock, C. Rodr ´ ıguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Non-markovian quantum processes: Complete framework and efficient characteri- zation, Phys. Rev. A97, 012127 (2018)

    2018

  36. [41]

    Oreshkov, F

    O. Oreshkov, F. Costa, and ˇC. Brukner, Quantum cor- relations with no causal order, Nat. Commun.3, 1092 (2012)

    2012

  37. [42]

    Y. Mo, L. Zhang, Y.-A. Chen, Y. Liu, T. Lin, and X. Wang, Parameterized quantum comb and simpler cir- cuits for reversing unknown qubit-unitary operations, npj Quantum Information11, 32 (2025), arXiv:2403.03761

    2025 arXiv

  38. [43]

    I. S. Sardharwalla, T. S. Cubitt, A. W. Harrow, and N. Linden, Universal refocusing of systematic quantum noise, arXiv:1602.07963 (2016)

    2016 arXiv

  39. [44]

    M. T. Quintino, Q. Dong, A. Shimbo, A. Soeda, and M. Murao, Probabilistic exact universal quantum circuits for transforming unitary operations, Physical Review A 100, 062339 (2019), arXiv:1909.01366

    2019 arXiv

  40. [45]

    M. T. Quintino, Q. Dong, A. Shimbo, A. Soeda, and M. Murao, Reversing unknown quantum transforma- tions: Universal quantum circuit for inverting general unitary operations, Phys. Rev. Lett.123, 210502 (2019), arXiv:1810.06944

    2019 arXiv

  41. [46]

    Q. Dong, M. T. Quintino, A. Soeda, and M. Mu- rao, Success-or-draw: A strategy allowing repeat-until- success in quantum computation, Phys. Rev. Lett.126, 150504 (2021)

    2021

  42. [47]

    M. T. Quintino and D. Ebler, Deterministic transforma- tions between unitary operations: Exponential advantage with adaptive quantum circuits and the power of indefi- nite causality, Quantum6, 679 (2022)

    2022

  43. [48]

    Yoshida, A

    S. Yoshida, A. Soeda, and M. Murao, Reversing unknown qubit-unitary operation, deterministically and exactly, Phys. Rev. Lett.131, 120602 (2023)

    2023

  44. [49]

    Navascu´ es, Resetting Uncontrolled Quantum Sys- tems, Phys

    M. Navascu´ es, Resetting Uncontrolled Quantum Sys- tems, Phys. Rev. X8, 031008 (2018), arXiv:1710.02470

    2018 arXiv

  45. [50]

    Trillo, B

    D. Trillo, B. Dive, and M. Navascu´ es, Translating uncon- trolled systems in time, Quantum4, 374 (2020)

    2020

  46. [51]

    Trillo, B

    D. Trillo, B. Dive, and M. Navascu´ es, Universal quantum rewinding protocol with an arbitrarily high probability of success, Phys. Rev. Lett.130, 110201 (2023)

    2023

  47. [53]

    Eggeling, D

    T. Eggeling, D. Schlingemann, and R. F. Werner, Sem- icausal operations are semilocalizable, Europhysics Let- ters57, 782 (2002), arXiv:quant-ph/0104027

    2002 arXiv

  48. [54]

    See Supplementary Material for the details of the proofs

  49. [55]

    Yoshida, Y

    S. Yoshida, Y. Koizumi, M. Studzi´ nski, M. T. Quintino, and M. Murao, One-to-one Correspondence between De- terministic Port-Based Teleportation and Unitary Esti- mation, arXiv:2408.11902 (2024)

    2024

  50. [56]

    K. Chen, N. Yu, and Z. Zhang, Tight Bound for Quantum Unitary Time-Reversal, arXiv:2507.05736 (2025)

    2025

  51. [59]

    Yoshida, A

    S. Yoshida, A. Soeda, and M. Murao, Universal construc- tion of decoders from encoding black boxes, Quantum7, 957 (2023), arXiv:2110.00258

    2023 arXiv

  52. [60]

    Ebler, M

    D. Ebler, M. Horodecki, M. Marciniak, T. M lynik, M. T. Quintino, and M. Studzi´ nski, Optimal universal quan- tum circuits for unitary complex conjugation, IEEE Transactions on Information Theory69, 5069 (2023), arXiv:2206.00107

    2023 arXiv

  53. [61]

    Klimyk and N

    A. Klimyk and N. Y. Vilenkin, Representations of lie groups and special functions, inRepresentation The- ory and Noncommutative Harmonic Analysis II: Homo- geneous Spaces, Representations and Special Functions (Springer, 1995) pp. 137–259. 8 CONTENTS A. SDP of deterministic qu...

    1995

  54. [62]

    Commutant of the tensor representation of compact groups 13

  55. [63]

    Definition of generalized Clebsch–Gordan transforms 15

  56. [64]

    Comb condition in the commutant of the tensor representation 18

  57. [65]

    1: Circuit construction of the (G×H)-invariant quantum comb based on the generalized CG transforms 20

    Proof of Thm. 1: Circuit construction of the (G×H)-invariant quantum comb based on the generalized CG transforms 20

  58. [66]

    Circuit construction of theG-covariant quantum comb based on the generalized CG transforms 22

  59. [67]

    Proof of Lem. 3: Partial trace and tensor product in the commutant algebra 24 Appendix A: SDP of deterministic quantum combs for unitary transposition and inversion Following the work of [47] and others [44, 45, 48, 59, 60], we consider a task of universal transformation a bla...

  60. [68]

    Commutant of the tensor representation of compact groups We consider unitary representationsρ i :G→End(V ρi ) of a compact groupGfori∈[N], whereV ρi is a repre- sentation space and End(V ρi ) represents the group of invertible operators onV ρi . In this section, we investigate...

  61. [69]

    As shown in Eq

    Definition of generalized Clebsch–Gordan transforms We define generalized Clebsch–Gordan (CG) transforms corresponding to a unitary representationρ:G→End(V ρ) for a compact groupG. As shown in Eq. (C14), the tensor product of an irrepµ∈ bGandρis decomposed into irreps as Vµ ⊗V...

  62. [70]

    Comb condition in the commutant of the tensor representation As shown in the main text, the quantum comb is characterized as a Choir matrixC∈End(I n ⊗ On) satisfying the following comb condition [39]: C⪰0, TrIi Ci =C i−1 ⊗1 Oi−1 ∀i∈[n+ 1], C0 = 1, (C54) whereI n andO n are joi...

  63. [71]

    1: Circuit construction of the(G×H)-invariant quantum comb based on the generalized CG transforms Proof.Using the operatorsC λiµi i defined in Lem

    Proof of Thm. 1: Circuit construction of the(G×H)-invariant quantum comb based on the generalized CG transforms Proof.Using the operatorsC λiµi i defined in Lem. 4, we define an isometry operator V λiµi−1 i : M λi−1∈ bG(i−1) Supp(C λi−1µi−1 i−1 )T ⊗C c λi λi−1 ρi → M µi∈ bH (i...

  64. [72]

    Circuit construction of theG-covariant quantum comb based on the generalized CG transforms This section shows the following theorem that is aG-covariant version of Thm. 1. Using the operatorsC λ2i i defined in Lem. 5, we define an isometry operator V λ2i−1 i : M λ2i−2∈ bG(2i−2...

  65. [73]

    3: Partial trace and tensor product in the commutant algebra Proof.The first property of Lem

    Proof of Lem. 3: Partial trace and tensor product in the commutant algebra Proof.The first property of Lem. 3 is shown as follows: Eµ rs ⊗1 dn ∼= (1Vµ ⊗ |r⟩⟨s|)⊗1dn (C108) = (1Vµ ⊗ |r⟩⟨s|)⊗ M ν∈ bGn 1Vν ⊗1 mνρn (C109) = M ν∈ bGn (1Vµ ⊗1 Vν ⊗1 mνρn )⊗ |r⟩⟨s|(C110) = M λ∈ bG(n) ...

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