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REVIEW 2 major objections 3 minor 107 references

Optimal Distributed Similarity Estimation of Quantum Channels

T0 review · 2 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper proves that distributed comparison of two unknown quantum channels has optimal query complexity Θ(max{√d/ε, 1/ε²}), matching the state-comparison rate and achievable with non-adaptive, ancilla-free measurements plus shared randomn

desk verdict The lower-bound and unitary-channel pieces are solid, but the general-channel upper bound has an unstated unitality assumption, so the main Θ theorem is not established. read the letter →

arxiv 2512.10465 v4 pith:ZCQXENRO submitted 2025-12-11 quant-ph

Congcong Zheng , Kun Wang , Xutao Yu , Ping Xu , Zaichen Zhang This is my paper · ORCID
classification quant-ph MSC 81P4581P50 PACS 03.67.-a
keywords distributedsimilarityestimationquantumchannelsChoistatequerycomplexityrandomizedmeasurementssharedrandomnesscross-platformverificationLOCC
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 aims to pin down the exact query cost of distributed similarity estimation of quantum channels (DSEC): two parties with black-box access to unknown channels E and F must estimate the normalized inner product of their Choi states, Tr[J_EJ_F]/d², to additive error ε. It claims the optimal query complexity is Θ(max{√d/ε, 1/ε²}), proving a lower bound that holds even for adaptive, coherent, LOCC strategies with ancillas, and an upper bound achieved by a non-adaptive, ancilla-free, incoherent protocol that uses shared randomness. If correct, this tight bound means cross-platform verification of quantum processes costs essentially the same as comparing quantum states, and shared randomness rather than quantum memory is the resource that buys the √d speedup over independent classical shadows.

What carries the argument

The central object is the Choi-state inner product Tr[J_EJ_F]/d², a measure of channel similarity via max fidelity. The lower bound argument uses Le Cam's two-point method with an intermediate depolarizing channel for incoherent access and an approximate T-design channel for coherent access, together with permutation (Weingarten) calculus and quantum cloning bounds. The upper bound relies on shared-randomness randomized measurements: both devices sample the same state from a 4-design and the same basis from a unitary 4-design, converting the two channel outputs into correlated classical bit-strings; the collision frequency g̃ and the affine estimator ω̃ extract the Choi inner product. The id

What would settle it

Run Algorithm 2 with E=F being the non-unital channel that measures in the computational basis and prepares |0⟩. The paper's estimator ω̃ has expectation 1 by the algorithm's own computation, whereas the claimed unbiasedness would require 1/d. A simulation or analytic calculation showing E[ω̃] = 1 ≠ Tr[J_EJ_F]/d² for this pair of channels exposes the false identity and falsifies Theorem 5 as stated.

Watch

Extended reading notes

Core claim

The paper proves that DSEC has query complexity Θ(max{√d/ε, 1/ε²}). The lower bound is obtained by reducing DSEC to a hypothesis test between two independent Haar-random unitaries and the same Haar-random unitary, bounded via Le Cam's two-point method; it holds even in the strongest access model. The matching upper bound is given by two randomized-measurement algorithms: a coherent one for unitary channels and an incoherent one for general channels. Algorithm 2 returns the estimator ω̃ = ((d+1)²/d) g̃ − (d+2)/d, with g̃ the empirical collision probability of the two devices' measurement outcomes under a shared random state and basis, and the paper claims E[ω̃] = Tr[J_EJ_F]/d² with variance O

Load-bearing premise

The proof that Algorithm 2's estimator is unbiased for general channels assumes the identity Σ_{i,j} Tr[E_i†F_jF_j†E_i] = d for the two channels' Kraus operators, which holds only when at least one channel is unital; this unstated premise is load-bearing, and if it fails (as for identical measure-and-prepare channels) the claimed Θ upper bound for general channels collapses.

Editorial extensions

If this is right

  • If the bound is tight as claimed, DSEC has the same optimal query complexity as distributed inner product estimation for states, up to constants, so process comparison is no harder asymptotically than state comparison.
  • The incoherent algorithm can serve as a practical, near-term tool for cross-platform verification, unitarity estimation, and circuit equivalence checking without needing entangled states or coherent access.
  • Shared randomness in both state preparation and measurement is the resource that buys the quadratic improvement over independent classical-shadows, not coherent access or collective measurements.
  • The lower bound applies to any DSEC algorithm, including the strongest imaginable settings (multi-round LOCC, adaptivity, ancillas, coherent access), so the Θ form is robust to the access model.
  • A quadratic reduction in the dimension-dependent term (from O(d/ε) to O(√d/ε)) means the protocol can certify device similarity with far fewer queries for large Hilbert space dimension.

Reading between the lines

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

  • The paper's unbiasedness proof for Algorithm 2 uses the identity Σ_{i,j} Tr[E_i†F_jF_j†E_i] = d, which is stated without qualification; a direct calculation shows it fails for non-unital channels, e.g., E=F=measure-and-prepare-|0⟩, where the true value is d². The general-channel claim therefore appears to rest on an unstated unitality assumption.
  • If the gap is confirmed, the matching upper bound for general channels may still be salvageable by adding a small correction term that depends on the non-unital part of the channels, or by restricting the claim to unital channels; a reader should not rely on Theorem 5 for arbitrary amplitude-damping-like channels without a revised proof.
  • A testable extension: numerically evaluate Algorithm 2 on a pair of identical amplitude-damping channels of varying strength; the estimator's bias should depart from zero exactly when the identity above fails, which would make the theoretical gap directly observable.
  • The lower bound's ε-dependence comes from a reduction to a distinguishing problem with random phases; it is plausible the same reduction technique can be adapted to give tight bounds for related distributed estimation tasks, such as unitarity or process fidelity estimation, where the state-DIPE analogy breaks down.
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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

2 major / 3 minor

Summary. The paper introduces distributed similarity estimation of quantum channels (DSEC): given access to two unknown d-dimensional channels E and F, estimate the normalized Choi inner product Tr[J_E J_F]/d^2 up to additive error ε. It claims the exact query complexity Θ(max{√d/ε, 1/ε^2}), proved by (i) a lower bound that holds even for adaptive, multi-round LOCC with coherent access, and (ii) a matching upper bound by an ancilla-free, non-adaptive, incoherent randomized-measurement algorithm (Algorithm 2) that is claimed to work for general channels. A second upper-bound algorithm for unitary channels uses symmetric collective measurements, and a classical-shadow baseline is analyzed. The lower-bound half is technically substantial and appears coherent; however, the general-channel upper-bound theorem is invalid as stated because a key algebraic identity in the proof of unbiasedness fails for non-unital channels.

Significance. If the main result were correct, it would settle the query complexity of a natural distributed channel-comparison primitive, identify shared randomness as the key resource, and give a practical near-term protocol. The lower-bound theorems and the coherent unitary protocol are valuable and may survive independently: the lower-bound argument uses learning trees, approximate designs, and cloning, and the unitary protocol's unbiasedness proof is genuine. However, the central general-channel claim is unsupported. Since Algorithm 2 is the only protocol that matches the lower bound for general channels, the paper's headline optimality statement collapses. The failure is not a presentation issue but a false equality in the proof of Theorem 5.

major comments (2)
  1. [Appendix C.2, Eq. (C24)] The equality at Eq. (C24) is false for general channels. For Kraus decompositions E={E_i}, F={F_j}, the second term in the two-design identity is S = Σ_{i,j} Tr[E_i† F_j F_j† E_i] = Σ_j Tr[F_j F_j† ρ_E], with ρ_E = Σ_i E_i E_i†. Trace preservation gives only Σ_i E_i†E_i=I and Σ_j F_j†F_j=I; S=d requires ρ_E=I or Σ_j F_jF_j†=I, i.e. E or F unital. Counterexample: E=F(ρ)=|0⟩⟨0| Trρ, with Kraus K_a=|0⟩⟨a|. Then ρ_E=d|0⟩⟨0| and S=d², whereas the proof assumes S=d. For this pair A=E_ψ Tr[E(|ψ⟩⟨ψ|)F(|ψ⟩⟨ψ|)]=1, while the claimed formula gives 2/(d+1). Hence E[g]=2/(d+1) and the estimator in Eq. (C15) has expectation 1, but Tr[J_EJ_F]/d²=1/d. The bias is constant and does not vanish as m,T grow. This invalidates the unbiasedness assertion of Theorem 5 for non-unital channels, which are explicitly in the theorem's stated scope.
  2. [Theorem 5 / main text, 'Incoherent Access'] The failure of Eq. (C24) is load-bearing. Algorithm 2 is the only protocol in the paper that claims to match the lower bound for general channels; the abstract, Theorem 5, and the conclusion assert Θ(max{√d/ε, 1/ε²}) for DSEC with general channels. Without an unbiased general-channel estimator, this optimality statement is unsupported. The lower-bound Theorems 1–3 and the coherent unitary protocol (Theorem 4) appear unaffected by this specific error, but they do not establish the paper's headline result. A corrected version would need either a genuinely unbiased estimator for non-unital channels or an explicit restriction to unital (or at least one unital) channels, which would change the claimed scope.
minor comments (3)
  1. [Appendix D, Proposition 6] The final sentence 'For s=1, the result extends to general channels' is not proved in Appendix D; Algorithm 3 and Lemmas 19–21 are unitary-specific. Please provide a proof or a reference for the claimed general-channel extension.
  2. [Table I] Table I omits the ε-dependence and lists only O(d) or O(√d). Including the full ε factors would be more consistent with Theorems 4–5 and Proposition 6 and avoid potential misreading.
  3. [Appendix B.2, Eq. (B56)] The displayed equation contains a stray 'H' and the binomial-coefficient formatting is garbled. Please clean up the notation.

Circularity Check

0 steps flagged · score 1.0 of 10

No meaningful circularity; Theorem 5 has a non-circular unitality gap that invalidates the general-channel claim.

full rationale

The derivation does not fit parameters to data or rename an input as a prediction. The lower bound relies on external techniques (Le Cam, learning trees [44], approximate designs [50], cloning [51], DIPE [15]); the upper-bound estimator constants in Eqs. (5), (8), (C1), (C15) are obtained from Haar/design moment identities, not from the target value. Self-citations [10,11,12,17,24,32] are contextual and not load-bearing. The one serious issue is a missing assumption in Appendix C.2: Eq. (C24) implicitly uses the sum over i,j of Tr[E_i^dagger F_j F_j^dagger E_i] = d, which holds only if a channel is unital; for non-unital channels such as K_a = |0><a|, the estimator is biased and Theorem 5 is unsupported. This is a correctness gap, not circularity: the estimator is not calibrated by the target or by a self-citation.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The result rests on standard imported machinery (Weingarten calculus, approximate unitary designs, learning trees, cloning bounds) plus one unstated and false identity in Eq. (C24) that effectively assumes unitality. No free parameters are fitted and no new physical entities are introduced.

assumptions (7)
  • standard math Weingarten calculus and Haar twirl identities (Lemma 7-8, App. A.2)
    Imported from [50,56]; used in both lower- and upper-bound computations. External and standard.
  • standard math Exact state/unitary 4-design ensembles and symmetric collective measurement M_s (Lemma 9, from [52])
    The algorithms sample |ψ⟩ from a state 4-design and Q from a unitary 4-design, and use POVM M_s; standard randomized-measurement tools.
  • standard math Approximate T-design channel J_a with relative error T²/(2d−T²) for T² ≤ 2d (Eqs. A17-A19, [50])
    Used in the coherent lower bound; validity restricted to T=O(√d), which is the intended regime.
  • domain assumption Learning-tree representation of incoherent algorithms (Def. 12, App. B.1.a, from [44])
    Models arbitrary adaptive incoherent strategies as trees with rank-1 POVMs; cited framework, not re-derived.
  • domain assumption Coherent strategies reduce to Choi operators of U^⊗T with interleaving absorbed (Eqs. B29-B33, from [50])
    Reformulates coherent access for the lower bound; imported from [50].
  • ad hoc to paper Eq. (C24): Σ_{i,j}Tr[E_i†F_jF_j†E_i] = d for the two unknown channels
    Unstated premise of Theorem 5; false in general. Holds only when at least one channel is unital. For E=F=Σ_a|0⟩⟨a|(·)|a⟩⟨0| the sum is d², making E[ω̃]=1 versus the true 1/d.
  • domain assumption Problem 2 ≡ Problem 3 with binomial t~B(T,ε) simulation (App. B.3)
    ε-dependent lower-bound reduction, mirroring [15, Lemma 10]; asymptotically plausible, with constants left implicit.

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Pith. "Pith review of Optimal Distributed Similarity Estimation of Quantum Channels." pith.science (2026). https://pith.science/paper/ZCQXENRO

@misc{pith2026251210465,
  author       = {Pith},
  title        = {Pith review of: Optimal Distributed Similarity Estimation of Quantum Channels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZCQXENRO}},
  note         = {Machine review of arXiv:2512.10465}
}
abstract

As quantum processors are deployed across different hardware platforms and remote cloud laboratories, a basic physical question is whether two black-box devices realize the same quantum process, without relying on a trusted classical description. We formulate the core primitive for this comparison task as \emph{distributed similarity estimation of quantum channels} (DSEC): given local access to two unknown channels, estimate the normalized inner product of their Choi states. We prove that the optimal query complexity of DSEC is $\Theta(\max\{\sqrt{d}/\varepsilon,1/\varepsilon^2\})$, where $d$ is the channel dimension and $\varepsilon$ is the additive error. This matching query complexity is nontrivial: channel learning permits input choices and interleaving known operations, which makes channel learning strictly harder than state learning. We first prove an information-theoretic lower bound with this scaling, which holds even in the \emph{strongest setting}, allowing adaptive strategies, multiple rounds of classical communication, and coherent access with arbitrary ancillas. We then give a matching upper bound in the \emph{weakest setting}, namely non-adaptive and ancilla-free incoherent access, via a randomized measurement algorithm achieving this bound. Finally, we show that our algorithm achieves a quadratic improvement over classical shadow baselines. Our results provide theoretically optimal and practical algorithms for quantum device benchmarking and distributed quantum learning.

Figures

Figures reproduced from arXiv: 2512.10465 by the authors.

Figure 1
Figure 1. (a). (i) Incoherent access, also known as learning with￾out quantum memory: In each query, the learner prepares an arbitrary quantum state, applies the unknown channel, and im￾mediately measures the output state. The key restriction is that the unknown channel is queried exactly once before each mea￾surement. (ii) Coherent access, also known as learning with quantum memory: In each experimental round, the learner pr… view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

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Reference graph

Works this paper leans on

107 extracted references · 8 linked inside Pith

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    Apply the unitary operators 𝑈 ⊗𝑇 and𝑉 ⊗𝑇 on |𝜓⟩⊗𝑇

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    Record the measurement results on two devices as |𝜙𝐴⟩ and |𝜙𝐵⟩, and compute ˜𝑓 = |⟨𝜙𝐴|𝜙𝐵⟩| 2

    Measure (𝑈|𝜓⟩) ⊗𝑇 and (𝑉 |𝜓⟩) ⊗𝑇 with the POVM M𝑇 . Record the measurement results on two devices as |𝜙𝐴⟩ and |𝜙𝐵⟩, and compute ˜𝑓 = |⟨𝜙𝐴|𝜙𝐵⟩| 2

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    (5) This algorithm requires shared randomness only for state preparation and is described in pseudocode in Algorithm 1

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    Record the measurement results as {𝑎𝑖}𝑚 𝑖=1 and {𝑏𝑖}𝑚 𝑖=1, respectively. Compute ˜𝑔 = 1 𝑚2 𝑚Õ 𝑖,𝑗 =1 𝟙(𝑎𝑖,𝑏 𝑗), (7) where 𝟙(𝑎,𝑏 ) = 1 if𝑎 =𝑏; otherwise 0

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    Technical Lemmas 25 Appendix A: Preliminaries In this section, we first set some notation used in this work. Then, we introduce two central tools utilized in this work: the Haar measure and approximate unitary designs. Both play fundamental roles in quantum information theory ...

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    The set of Hermitian operators on H is denoted by B (H ), and the set of density matrices on H is denoted by D (H )

    Notation Let H be the Hilbert space of a 𝑑-dimensional system. The set of Hermitian operators on H is denoted by B (H ), and the set of density matrices on H is denoted by D (H ). For a quantum channel E, we define its Choi operator as [ 39] 𝐽E := (E ⊗ I) (| Φ⟩⟨Φ|), (A1) where...

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    The Haar random channel is defined as E (𝑘) 𝐻 (𝐴) := 𝔼𝑈∼𝜇𝐻𝑈 ⊗𝑘𝐴𝑈†⊗𝑘, 𝐴 ∈ B (H ⊗𝑘)

    Haar Measure and Permutation Operators Let𝜇𝐻 be the Haar measure over the unitary group. The Haar random channel is defined as E (𝑘) 𝐻 (𝐴) := 𝔼𝑈∼𝜇𝐻𝑈 ⊗𝑘𝐴𝑈†⊗𝑘, 𝐴 ∈ B (H ⊗𝑘). (A6) To describe the Haar random channel, we introduce permutation operators. Let S𝑘 be the symmetric gro...

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    Here E1 ⪯ E 2 denotes that E2 − E 1 is a completely-positive map

    Approximate Unitary Designs A unitary ensemble A is an 𝜀-approximate unitary 𝑘-design if [ 50] (1 −𝜀)E (𝑘) A ⪯ E (𝑘) 𝐻 ⪯ ( 1 +𝜀)E (𝑘) A , (A17) where 𝜀 is called relative error. Here E1 ⪯ E 2 denotes that E2 − E 1 is a completely-positive map. A standard approximate 𝑘-design c...

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    Two quantum devices perform the same unitary𝑈, which is a Haar random unitary

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    Define the Choi operators of𝑈 and𝑉 as𝐽𝑈 and𝐽𝑉 , respectively

    Two quantum devices independently perform two unitaries𝑈 and𝑉, which are two independent Haar random unitaries. Define the Choi operators of𝑈 and𝑉 as𝐽𝑈 and𝐽𝑉 , respectively. If an estimation algorithm can accurately estimateTr[𝐽𝑈𝐽𝑉 ]/𝑑2, then it can be leveraged to solve the d...

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    First, we introduce the tree representation of channel learning with incoherent access, which is a powerful tool in proving lower bounds for learning tasks [ 41, 44, 47–49]

    Incoherent Access Here, we prove the lower bound for completing DSEC with incoherent access. First, we introduce the tree representation of channel learning with incoherent access, which is a powerful tool in proving lower bounds for learning tasks [ 41, 44, 47–49]. 11 a. Proo...

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    𝔼𝑈∼𝜇𝐻𝑝𝑈,𝑈 − 𝑝D TV = 1 2 Õ ℓ 𝔼𝑈∼𝜇𝐻𝑝𝑈,𝑈 (ℓ) − 𝑝D (ℓ) (B16) = Õ ℓ:𝑝D (ℓ) ≥𝔼𝑈∼𝜇𝐻𝑝𝑈,𝑈 (ℓ) 𝑝D (ℓ) 1 − 𝔼𝑈∼𝜇𝐻𝑝𝑈,𝑈 (ℓ) 𝑝D (ℓ)

    For the first term, we can also write it in the following form. 𝔼𝑈∼𝜇𝐻𝑝𝑈,𝑈 − 𝑝D TV = 1 2 Õ ℓ 𝔼𝑈∼𝜇𝐻𝑝𝑈,𝑈 (ℓ) − 𝑝D (ℓ) (B16) = Õ ℓ:𝑝D (ℓ) ≥𝔼𝑈∼𝜇𝐻𝑝𝑈,𝑈 (ℓ) 𝑝D (ℓ) 1 − 𝔼𝑈∼𝜇𝐻𝑝𝑈,𝑈 (ℓ) 𝑝D (ℓ) . (B17) Therefore, we can focus on the lower bound for 𝔼𝑈∼𝜇𝐻𝑝𝑈,𝑈 (ℓ)/𝑝D (ℓ) and have 𝔼𝑈∼𝜇𝐻𝑝𝑈,𝑈 (...

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    (B26) Therefore, we have 𝑝D − 𝔼𝑈,𝑉 ∼𝜇𝐻𝑝𝑈,𝑉 TV ≤ 1 − 1 − 𝑇 2 𝑑 2 ≤ 2𝑇 2 𝑑

    Likewise, we have 𝔼𝑈,𝑉 ∼𝜇𝐻𝑝𝑈,𝑉 (ℓ) = (𝑑𝑑 ′)2𝑇𝑊ℓ 2Ö 𝑖=1 ©­ « Õ 𝒋,𝒌 ⟨Ψℓ,𝑖, 𝒋 ⊗ Φ∗ ℓ,𝑖, 𝒋|𝐽 (𝑇 ) 𝐻 |Ψℓ,𝑖,𝒌 ⊗ Φ∗ ℓ,𝑖,𝒌⟩ª® ¬ (B23) ≥ (𝑑𝑑 ′)𝑇 𝑑(𝑑 + 1) · · · (𝑑 +𝑇 − 1) 2 𝑊ℓ 2Ö 𝑖=1 ©­ « Õ 𝒋,𝒌 ⟨Ψℓ,𝑖, 𝒋|Ψℓ,𝑖,𝒌⟩⟨ Φℓ,𝑖,𝒌|Φℓ,𝑖, 𝒋⟩ª® ¬ (B24) = (𝑑𝑑 ′)𝑇 𝑑(𝑑 + 1) · · · (𝑑 +𝑇 − 1) 2 𝑊ℓ Õ 𝒋,𝒌 ⟨...

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    measure-and-prepare

    Coherent Access We now consider the lower bound for completing DSEC with coherent access. a. Proof of Theorem2 We first establish some notations. Recall that any learning algorithms with coherent access can be described as follows. Consider two unknown unitary channels U and V...

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    The proof is similar to [ 15]

    Lower bound for DSEC Furthermore, we consider the lower bound for DSEC. The proof is similar to [ 15]. We first define a distinguishing problem for estimation, then analyze the lower bound for solving this problem with the result in Appendix B 2. 17 a. Distinguishing Problem f...

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    Twoquantumdevicesimplementtwooperators √ 1 −𝜀𝑒𝑖𝜃1𝐼 + √𝜀𝑈 and √ 1 −𝜀𝑒𝑖𝜃2𝐼 + √𝜀𝑈,respectively,where 𝑈 isaHaar random unitary inHmain and𝜃1,𝜃 2 are independent random phases in[0, 2𝜋]

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    Twoquantumdevicesindependentlyimplementtwooperators √ 1 −𝜀𝑒𝑖𝜃1𝐼 +√𝜀𝑈and √ 1 −𝜀𝑒𝑖𝜃2𝐼 +√𝜀𝑉,respectively,where 𝑈,𝑉 are two independent Haar random unitaries inHmain and𝜃1,𝜃 2 are independent random phases in[0, 2𝜋]. When𝑑 is large enough, case 1 gives 𝑓1 := 𝔼𝑈∼𝜇𝐻 Tr[𝐽E𝐽F] 𝑑2 ≈ (1...

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    Obtain the estimation result 𝑓 with the estimation algorithm 𝔸

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    The performance of the estimation algorithm 𝔸 guarantees that if the result 𝑓 is in case 𝑖(𝑖 = 1, 2), then |𝑓 − 𝑓𝑖| ≤ 𝜀/100 holds with probability at least 0.99

    Output case 2 if 𝑓 ∈ [( 1 −𝜀)2 −𝜀/50, (1 −𝜀)2 +𝜀/50], and output case 1 otherwise. The performance of the estimation algorithm 𝔸 guarantees that if the result 𝑓 is in case 𝑖(𝑖 = 1, 2), then |𝑓 − 𝑓𝑖| ≤ 𝜀/100 holds with probability at least 0.99. Then, if 𝑓 is from case 1, we ha...

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    Two quantum devices perform𝑇 queries and the corresponding Choi operators are|Φ𝑈,𝑡 ⟩⟨Φ𝑈,𝑡 | and |Φ𝑈,𝑡 ′ ⟩⟨Φ𝑈,𝑡 ′ |, where 𝑡,𝑡 ′ ∼ 𝐵(𝑇,𝜀 ) and𝑈 is a Haar random unitary inHmain

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    It should be noted that Problem 2 is equivalent to Problem 3, as these two problems have the same average inputs

    Two quantum devices perform𝑇 queries and the corresponding Choi operators are|Φ𝑈,𝑡 ⟩⟨Φ𝑈,𝑡 | and |Φ𝑉,𝑡 ′ ⟩⟨Φ𝑉,𝑡 ′ |, where 𝑡,𝑡 ′ ∼ 𝐵(𝑇,𝜀 ) and𝑈,𝑉 are two independent Haar random unitaries inHmain. It should be noted that Problem 2 is equivalent to Problem 3, as these two proble...

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    For device 1, randomly sample 𝑡 ∼ 𝐵(𝑇,𝜀 ). (a) If 𝑡 ≤ 𝐾, we construct the queries corresponding to |Φ𝑈,𝑡 ⟩⟨Φ𝑈,𝑡 | with 𝑡 queries of unknown operator 𝑈 based on Lemma 17; (b) Otherwise, all 𝑇 queries are identity channel queries

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    For device 2, we perform the same procedure as device 1

  87. [95]

    It should be noted that we can simulate the input of Problem 3 with high probability, since 𝐾 = 100𝑇𝜀 + 100 ⇒𝑡 ≤ 𝐾 with high probability

    Run the algorithm 𝔸′ and decide accordingly. It should be noted that we can simulate the input of Problem 3 with high probability, since 𝐾 = 100𝑇𝜀 + 100 ⇒𝑡 ≤ 𝐾 with high probability. Lastly, as shown in Appendix B 2, to complete Problem 1, we require 100𝑇𝜀 + 100 = Ω( √ 𝑑) ⇒ 𝑇 ...

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    Let 𝐽𝑈 and 𝐽𝑉 be the Choi operators of channels U and V

    Coherent Access: Proof of Theorem 4 Suppose two quantum devices perform two unitary channels U (·) =𝑈 (·)𝑈† and V (·) =𝑉 (·)𝑉 †, respectively. Let 𝐽𝑈 and 𝐽𝑉 be the Choi operators of channels U and V. The algorithm solving DSEC with coherent access is summarized in Algorithm 1....

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    The algorithm solving DSEC with incoherent access is summarized in Algorithm 2

    Incoherent Access: Proof of Theorem 5 Suppose two quantum devices perform two channels E and F , respectively. The algorithm solving DSEC with incoherent access is summarized in Algorithm 2. In the following, we will prove Theorem 5. For each SPAM setting {𝑄,𝜓 }, we define the...

  90. [98]

    (C31) As shown in Eq

    For the first term, with Lemma 14 in [ 15], we have 𝔼𝑄,𝜓𝕍 [˜𝑔(𝑄,𝜓,𝑆 |𝑄,𝜓 )] ≤ 𝔼𝑄,𝜓 " 𝑔(𝑄,𝜓 ) 𝑚2 + 1 𝑚 Õ 𝑎 𝑝2 𝑄,𝜓 (𝑎)𝑞𝑄,𝜓 (𝑎) + 𝑝𝑄,𝜓 (𝑎)𝑞2 𝑄,𝜓 (𝑎) # . (C31) As shown in Eq. ( C25), we have 𝔼𝑄,𝜓 𝑔(𝑄,𝜓 ) 𝑚2 = 1 𝑚2 (𝑑 + 1) + 1 𝑚2𝑑 Tr[𝐽E𝐽F] + 𝑑 𝑑(𝑑 + 1) = O 1 𝑚2𝑑 . (C32) Additional...

  91. [99]

    For the second term, we have 𝕍𝑄,𝜓𝔼 [˜𝑔(𝑄,𝜓,𝑆 |𝑄,𝜓 )] = 𝕍𝑄,𝜓 𝑔(𝑄,𝜓 ) = 𝔼𝑄,𝜓 𝑔2 (𝑄,𝜓 ) − 𝔼𝑄,𝜓 𝑔(𝑄,𝜓 ) 2 (C37) = 𝔼𝑄,𝜓 𝑔2 (𝑄,𝜓 ) − 𝔼𝜓 1 + Tr[E (|𝜓⟩⟨𝜓|)F (|𝜓⟩⟨𝜓|)] 𝑑 + 1 2 . (C38) We have 𝔼𝑄 𝑔2 (𝑄,𝜓 ) = 𝔼𝑄 Õ 𝑎 ⟨𝑎|𝑄E (|𝜓⟩⟨𝜓|)𝑄†|𝑎⟩⟨𝑎|𝑄F (|𝜓⟩⟨𝜓|)𝑄†|𝑎⟩ ! 2 (C39) =𝑑 𝔼𝜙⟨𝜙|E (|𝜓⟩⟨𝜓|)|𝜙⟩2⟨...

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    Given an unknown unitary channel U (·) = 𝑈 (·)𝑈†, we can obtain the following classical snapshot by inputting the state |𝜓⟩⊗𝑠 and performing the POVM M𝑠 (defined in Eq

    Algorithm & Proof of Proposition 6 We first briefly introduce CSEU. Given an unknown unitary channel U (·) = 𝑈 (·)𝑈†, we can obtain the following classical snapshot by inputting the state |𝜓⟩⊗𝑠 and performing the POVM M𝑠 (defined in Eq. ( 4)), ˜𝑋 = Snap(𝜓,𝜙,𝑠 ) := 𝑑(𝑑 + 1) (𝑑 ...

  93. [101]

    For𝑖 =𝑘 and 𝑗 =𝑙, there are 𝑇 2 terms, and using Lemma 20, we have 𝔼 Õ 𝑖,𝑗 Tr2 ˜𝑋𝑖 ˜𝑌𝑗 ! =𝑇 2𝔼 Tr2 [𝑋𝑌 ] =𝑇 2O 𝑑4 (𝑑 +𝑠)2 𝑠2 . (D8)

  94. [102]

    For𝑖 ≠ 𝑘 and 𝑗 ≠ 𝑙, there are 𝑇 2 (𝑇 − 1)2 terms, and we have 𝔼©­ « Õ 𝑖≠𝑘,𝑗 ≠𝑙 Tr ˜𝑋𝑖 ˜𝑌𝑗 Tr ˜𝑋𝑘 ˜𝑌𝑙 ª® ¬ =𝑇 2 (𝑇 − 1)2 Tr4 [𝑈†𝑉]. (D9)

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    For𝑖 =𝑘 and 𝑗 ≠ 𝑙, there are 𝑇 2 (𝑇 − 1) terms, and using Lemma 21 we have 𝔼©­ « Õ 𝑖=𝑘,𝑗 ≠𝑙 Tr ˜𝑋𝑖 ˜𝑌𝑗 Tr ˜𝑋𝑘 ˜𝑌𝑙 ª® ¬ =𝑇 2 (𝑇 − 1)𝔼 Tr 𝑋 ˜𝑌1 Tr 𝑋 ˜𝑌2 ≤ O 𝑇 3𝑑4 . (D10)

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    (D11) Therefore, we have 𝕍 (˜𝛾) = 1 𝑇 2𝑑4 O 𝑑4 (𝑑 +𝑠)2 𝑠2 + (𝑇 − 1)2 𝑇 2𝑑4 Tr4 [𝑈†𝑉] + O 1 𝑇 − 1 𝑑4 Tr4 [𝑈†𝑉] (D12) ≤ O (𝑑 +𝑠)2 𝑇 2𝑠2 + 1 𝑇 = O 𝑑2 𝑇 2𝑠2 + 1 𝑇

    For𝑖 ≠ 𝑘 and 𝑗 =𝑙, there are 𝑇 2 (𝑇 − 1) terms, and using Lemma 21 we have 𝔼©­ « Õ 𝑖≠𝑘,𝑗 =𝑙 Tr ˜𝑋𝑖 ˜𝑌𝑗 Tr ˜𝑋𝑘 ˜𝑌𝑙 ª® ¬ =𝑇 2 (𝑇 − 1)𝔼 Tr ˜𝑋1𝑌 Tr ˜𝑋2𝑌 ≤ O 𝑇 3𝑑4 . (D11) Therefore, we have 𝕍 (˜𝛾) = 1 𝑇 2𝑑4 O 𝑑4 (𝑑 +𝑠)2 𝑠2 + (𝑇 − 1)2 𝑇 2𝑑4 Tr4 [𝑈†𝑉] + O 1 𝑇 − 1 𝑑4 Tr4 [𝑈†𝑉] (D12) ...

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    Technical Lemmas Lemma 19 (Lemma D1 in [ 37]). Suppose that𝜙 and𝜓 are the random input states and measurement outcomes of learning a unitary channelU, then we have 𝔼𝜙⊗2 ⊗𝜓⊗2 = 2 (𝑑 +𝑠) (𝑑 +𝑠 + 1) 4Õ 𝑖=1 Δ𝑈,𝑖, (D15) where Δ𝑈,1 := 1 𝜅2 Π (𝑑,2) sym ⊗ Π (𝑑,2) sym , (D16) Δ𝑈,2 := 𝑠...

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    For the first term, with Lemma 19, there are 16 terms: 𝔼 Tr2 [𝜙1𝜙2 ⊗𝜓2𝜓2] = 𝔼 Tr h 𝜙⊗2 1 ⊗𝜓⊗2 1 𝜙⊗2 2 ⊗𝜓⊗2 2 i = 4 (𝑑 +𝑠)2 (𝑑 +𝑠 + 1)2 4Õ 𝑘,𝑙=1 Tr Δ𝑈,𝑘 Δ𝑉,𝑙 (D23) We analyze these terms as follows: Tr[Δ𝑈,1Δ𝑉,1] = Tr2 h Π (𝑑,2) sym i 𝜅2 2 = 1, (D24) Tr[Δ𝑈,1Δ𝑉,2] = Tr[Δ𝑈,2Δ𝑉,1] ...

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    (D34) Likewise, 𝔼 Tr2 𝜙2 ⊗𝜓𝑇 2 = O (1)

    For the second term, with Lemma 19, we have 𝔼 Tr2 𝜙1 ⊗𝜓𝑇 1 = 𝔼 Tr 𝜙⊗2 1 ⊗𝜓⊗2 1 = 2 (𝑑 +𝑠) (𝑑 +𝑠 + 1) 4Õ 𝑖=1 Tr Δ𝑈,𝑖 (D33) = 2 (𝑑 +𝑠) (𝑑 +𝑠 + 1) 𝜅2 + 2 ·𝑑(𝑑 + 1)2 (𝑑 + 2) 12 ·𝜅3 + 𝑠(𝑠 − 1) 2 = O (1). (D34) Likewise, 𝔼 Tr2 𝜙2 ⊗𝜓𝑇 2 = O (1). Therefore, we have 𝔼 Tr2 [𝑋𝑌 ] ≤ O 𝑑4 ...

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.