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REVIEW 4 major objections 4 minor 42 references

Complexity and multi-functional variants of the Quantum-to-Quantum Bernoulli Factories

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper proves that quantum-to-quantum Bernoulli factories simulate exactly the rational functions, and that simulating f(z) needs deg(f) input qubits—with an explicit circuit and closed-form maximal success probability.

desk verdict A useful resource-counting paper for QQBFs: the qubit lower bound and matching construction are real additions, but the proof has two repairable gaps and the success-probability claim is stronger than what is shown. read the letter →

arxiv 2512.10810 v1 pith:CD5I426C submitted 2025-12-11 quant-ph

Francesco Hoch , Taira Giordani , Gonzalo Carvacho , Nicolò Spagnolo , Fabio Sciarrino This is my paper · ORCID
classification quant-ph PACS 03.67.-a03.67.Lx
keywords quantum-to-quantumBernoullifactoryrationalfunctionsimulationqubitcomplexitylowerboundsuccessprobabilityoptimizationmultivariatemultifunctionalquantumrandomnessmanipulationcircuitsynthesis
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

Quantum-to-quantum Bernoulli factories turn quantum coins—qubits whose amplitudes encode an unknown bias z—into new quantum coins with bias f(z). This paper characterises how costly that transformation is. It proves that the class of simulable functions is exactly the complex rational functions, that any implementation of f(z) must use at least deg(f) copies of the input state, and that the bound is tight: every rational f can be implemented with exactly deg(f) copies, no ancillas (except degree-one functions, which may need at most one), via an explicit unitary circuit. It also supplies a closed-form expression for the maximum success probability and shows the construction reaches it. Knowing the exact qubit count and maximal success probability is what makes these factories viable subroutines for quantum algorithms that rely on Bayesian inference, Monte Carlo methods, and blind quantum computation.

What carries the argument

The key object is the quantum coin state |z> = (z|0>+|1>)/sqrt(1+|z|²), a parametrisation of qubit states by a complex number. The argument is carried by the canonical-form reduction: every QQBF becomes one fixed unitary U acting on n copies of |z> plus m ancillas, followed by all-zero postselection. In that form the output amplitude pair (P(z), Q(z)) is a linear combination of monomials z^j with binomial weights, which directly ties the degree of the rational function to the number n of input qubits. The paper constructs the optimal U from two orthonormal vectors |v0>, |v1> whose overlaps with the symmetric states |s^n_j> store the coefficients of P and Q; a parameter w parametrises the unu

What would settle it

A numerical or experimental search for a circuit with only one copy of |z> that outputs |z²> with nonzero probability for a dense set of z values, using any postselection or adaptive strategy, would falsify the lower bound n ≥ deg(f). On the optimality side, computing the success probability of the paper's degree-one circuit for f(z)=z and finding any input z for which a different circuit exceeds Eq. (26) would falsify the maximal-success-probability claim.

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Extended reading notes

Core claim

Any valid QQBF can be reduced to a fixed unitary U acting on n copies of |z> plus m ancillas, followed by computational-basis measurements and acceptance only when all non-output qubits read 0. In that canonical form every output amplitude is a polynomial in z of degree at most n, so the output is |P(z)/Q(z)> and simulable functions are rational. The paper's constructive proof makes the bound tight: given P,Q of degree at most n, two orthonormal rows of U can be chosen with overlaps KP(z) and KQ(z), which fixes all but one free parameter |w|². Since |w|² only lowers the success probability, the optimal choice is w=0, giving a circuit with n=deg(f) qubits, no ancillas for degree at least 2, a

Load-bearing premise

The load-bearing premise is that every valid factory can be rearranged into a single fixed unitary followed by measuring all non-output qubits and accepting only the all-zero pattern; if adaptive mid-circuit measurements, feed-forward, or postselection on other outcome patterns could do the job with fewer than deg(f) copies of |z>, the claimed optimality would collapse.

Editorial extensions

If this is right

  • Every rational function f with deg(f) ≥ 2 can be implemented with exactly deg(f) quantum coins and no auxiliary qubits; degree-one functions consume at most one auxiliary qubit.
  • The success probability formula (Eq. 26) gives the exact maximum for a fixed qubit budget, so no circuit using the same number of qubits can exceed it.
  • Using more than deg(f) copies of |z> can raise the success probability for particular inputs or state ensembles, and the optimal number of qubits depends on the function and on the input distribution.
  • Multivariate rational functions of k biases are simulable with per-variable qubit counts equal to their partial degrees, and one auxiliary qubit is needed only when every partial degree is 1.
  • Any pair of simulable functions can be combined in a single multifunctional circuit; if the pair satisfies the compatibility conditions (29)–(30), the primary function's success probability is preserved, as when the product operation is added to the sum operation.

Reading between the lines

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

  • The degree-equals-qubit-count result suggests that the minimal circuit complexity is essentially the degree of a rational map on the Riemann sphere; one might therefore compare quantum factory complexity directly with classical Bernoulli factories, which are restricted to rational functions with no zeros or poles on [0,1].
  • The compatibility conditions define a partial ordering on rational functions: g1 is compatible with g0 when extra unitary rows can be made orthogonal to g0's rows without touching g0's success probability. A useful next step would be to characterise maximal sets of pairwise compatible functions and schedule many factories in a single circuit.
  • The lower bound depends on accepting only the all-zero outcome pattern. A testable extension is to search systematically for adaptive or multi-outcome-postselection factories that simulate degree-d functions with fewer than d copies of |z>; finding one would sharpen the boundary of the canonical-form argument.
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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

4 major / 4 minor

Summary. This paper studies the Quantum-to-Quantum Bernoulli Factory (QQBF). It claims that any QQBF can be represented by a canonical circuit with n copies of the input state |z⟩, a fixed unitary, computational-basis measurements, and acceptance only when all non-output qubits read 0; from this it derives a lower bound n ≥ deg(f) on the number of required |z⟩ copies. The paper then constructs, for every rational function f = P/Q, a circuit using n = deg(f) copies and no ancillas for n ≥ 2 (with at most one ancilla for degree 1), and gives a closed-form expression for the maximal success probability (Eq. 26). It also formalizes two extensions: multivariate QQBFs and multifunctional QQBFs, with applications to sum and product operations.

Significance. If the main claims hold, the paper resolves the qubit-resource complexity of QQBFs, provides an explicit circuit construction that saturates the degree lower bound, and gives a closed-form success probability. The constructive proof is self-contained and does not merely import the rational-function characterization from prior work; it also provides a unitary-completion algorithm. The multivariate and multifunctional extensions are natural and potentially useful for quantum algorithms and experimental implementations. These are substantial contributions. The main caveat is that the lower-bound proof relies on an incompletely justified canonical-form reduction, and some technical details in the explicit formulas and multifunctional characterization need clarification.

major comments (4)
  1. [Sec. III.A, Fig. 1, Eq. (8)] The reduction to a single accepted outcome (all non-output qubits read 0) is asserted via the deferred measurement principle. Deferred measurement preserves the number n of |z⟩ copies and moves measurements to the end, but it does not imply that a protocol with several accepted outcome strings can be rewritten as one accepting only the all-zero string. Because a unitary is injective, distinct post-measurement ancilla states generally cannot be mapped to a single |0...0⟩ while keeping the output factorized. Since the lower bound n ≥ deg(f) is derived only for the all-zero branch, the proof of the lower bound is incomplete as written. A repair is available: for a fixed accepted measurement record, the unnormalized output amplitudes are degree-≤n polynomials in z; any branch with nonzero success probability succeeds on an open dense set, so its output ratio is a rational P/Q with deg ≤ n, a
  2. [Sec. III.C, Eqs. (24)-(26)] The text claims to maximize the success probability for a fixed number of qubits n and treats |w|^2 as the only free parameter. But for n > deg(f), the polynomials P and Q representing f are not unique: multiplying both by any polynomial R of degree at most n−deg(f) leaves f unchanged but changes a, b, c, and the numerator in Eq. (25). Thus Eq. (26) is not the maximum over all implementations with n qubits unless one explicitly restricts to the canonical choice deg(P), deg(Q) ≤ n with no common factor. The central tightness claim for n = deg(f) is unaffected, but the stated generality of the optimization is not proven.
  3. [Sec. III.B, Eqs. (21)-(23)] The closed-form solution for y contains the factor c/|c|, so it is undefined when c = 0. The paper then uses c = 0 examples (Sec. VI, product and sum operations) and claims the construction is general. A separate treatment of the c = 0 case is needed, e.g., choosing y = e^{iφ} sqrt((l − |w|^2 − a + b)/2) with φ fixed by Eq. (17). Without this, the explicit algorithm does not cover all rational functions as stated.
  4. [Sec. V, Eqs. (29)-(30); Supplementary V] The proof that Eqs. (29)-(30) are necessary and sufficient for compatibility is sketched rather than completed. The text says the conditions allow deriving a1 and a3, and that orthonormality allows always deriving a2, a4, H, but it does not show explicitly that the remaining orthonormality equations are satisfied, nor that setting a5 = 0 yields the maximum probability. In addition, Supplementary V's unitary-dilation construction appears to give a multifunctional circuit for any two functions, which is a different existential claim; the relationship between that construction and the compatibility conditions should be clarified.
minor comments (4)
  1. [Supplementary V, Eq. (47)] The definition of M contains a duplicated term |10⟩⟨r| and |10⟩⟨s|; the third term should presumably be |01⟩⟨r| (with the four basis states |00⟩, |10⟩, |01⟩, |11⟩ corresponding to the two output cases and the herald q2). Please correct this typo, as it affects the subsequent dilation argument.
  2. [Sec. II, Eq. (4)] The notation |s_n_j⟩ is used for symmetric states, but it is not explicitly defined whether j counts zeros or ones in the computational basis. Please define it precisely, as the binomial coefficients in Eq. (4) depend on this convention.
  3. [Throughout] There are several grammatical and typographical errors: 'In this contest' should be 'In this context' (Sec. II), 'desidered' should be 'desired' (Sec. VII), and 'simulatable' and 'simulable' are used inconsistently. A proofreading pass is recommended.
  4. [Sec. III.C, Eqs. (27)-(28)] The derivation of the averaged success probabilities over the uniform Bloch-sphere and covariant-state ensembles is not shown. Since these formulas are used to draw conclusions about optimal n, please provide the integration details or a reference where the measures are defined.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity — rational-function characterization and qubit-count bounds are re-derived from unitary linearity; self-citations are not load-bearing.

full rationale

The derivation chain is self-contained. In Sec. III.A the input state is expanded in monomials (Eq. 4); linearity of U forces every computational-basis amplitude to be a polynomial of degree at most n (Eqs. 5–6); postselecting on all-zero ancillas gives an output ratio P/Q with degree at most n (Eq. 7), yielding the lower bound n ≥ deg(f) (Eq. 8). This reconstructs the rational-function characterization rather than importing it from Ref. [22]. The converse is constructive: for any P,Q the paper defines coefficients K,x,y,w satisfying the orthonormality conditions for two rows of a unitary (Eqs. 10–23), and the dimension count 2^n−n−1>0 for n≥2 supplies the needed orthogonal vector |θ0⟩, giving m=0. The success probability (Eqs. 24–26) follows from those normalizations and the maximization over the single free parameter w. No parameter is fitted to a data subset, and no "prediction" is an input renamed. The only self-citations, Refs. [25,26], are used for the |z⟩ parametrization and experimental context; the central theorem is re-proved in the paper. One genuine gap exists, but it is not circular: the "without loss of generality" reduction in Sec. III.A to a single all-zero acceptance branch is asserted via deferred measurement and is not separately proven for protocols that could merge multiple heralded branches. That is a completeness or correctness concern, not an equivalence of outputs to inputs, so it does not raise the circularity score.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The derivation relies on standard quantum-information background (deferred measurement, unitary dilation, Householder completion) and on the stereographic qubit parameterization. No data are fitted; one internal construction parameter w is optimized to zero. No new physical entities are postulated.

free parameters (1)
  • w (coefficient of |θ1> in |v1>) = 0 (optimal)
    A degree of freedom in the vector construction; the paper argues the success probability decreases with |w|^2 and sets w=0 for the claimed optimal circuit.
assumptions (5)
  • domain assumption Deferred measurement principle: all measurements can be postponed to the end and all non-output outcomes relabeled as 0.
    Used in Sec. III.A to reduce arbitrary QQBF circuits to Fig. 1; this is the key structural premise behind the polynomial-degree bound.
  • standard math Stereographic parameterization |z> = (z|0>+|1>)/sqrt(1+|z|^2) describes all pure qubit states up to global phase.
    Definition used throughout; it lets the input state expansion in Eq. (4) have coefficients z^j sqrt(C(n,j)).
  • standard math The n-qubit symmetric states |s^n_j> form an orthonormal set.
    Needed in Eqs. (4), (12)-(14), and in the construction of |v0>, |v1>.
  • standard math Finite-dimensional unitary dilation theorem: any contraction can be embedded as a block of a unitary.
    Proved in Supplementary V and used to show that any pair of functions admits a multifunctional factory.
  • standard math Householder reflection algorithm completes two orthonormal vectors to a unitary basis.
    Supplementary II constructs the explicit U; the text cites standard numerical linear algebra and does not discuss degenerate cases where Householder vectors vanish.

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Pith. "Pith review of Complexity and multi-functional variants of the Quantum-to-Quantum Bernoulli Factories." pith.science (2026). https://pith.science/paper/CD5I426C

@misc{pith2026251210810,
  author       = {Pith},
  title        = {Pith review of: Complexity and multi-functional variants of the Quantum-to-Quantum Bernoulli Factories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CD5I426C}},
  note         = {Machine review of arXiv:2512.10810}
}
read the original abstract

A Bernoulli factory is a model for randomness manipulation that transforms an initial Bernoulli random variable into another Bernoulli variable by applying a predetermined function relating the output bias to the input one. In literature, quantum-to-quantum Bernoulli factory schemes have been proposed, which encode both the input and output variables using qubit amplitudes. This fundamental concept can serve as a subroutine for quantum algorithms that involve Bayesian inference and Monte Carlo methods, or that require data encryption, like in blind quantum computation. In this work, we present a characterisation of the complexity of the quantum-to-quantum Bernoulli factory by providing a lower bound on the required number of qubits needed to implement the protocol, an upper bound on the success probability and the quantum circuit that saturates the bounds. We also formalise and analyse two different variants of the original problem that address the possibility of increasing the number of input biases or the number of functions implemented by the quantum-to-quantum Bernoulli factory. The obtained results can be used as a framework for randomness manipulation via such an approach.

Figures

Figures reproduced from arXiv: 2512.10810 by the authors.

Figure 1
Figure 1. Schematic representation of the general circuit for the Quantum Bernoulli. The circuit takes as input n qubit in the state |z⟩ and m ancillary qubits in the state |0c⟩. The resulting state evolves through a unitary evolution U. All qubits are then measured in the computational basis, except the first one, which carries the out￾put state of the Bernoulli factory. The output state |ψO⟩ is accepted if all the measureme… view at source ↗
Figure 2
Figure 2. shows the trends of those mean probabilities for the function g(z) = ηz2 as a function of the parameter η and for different numbers of qubits n employed by the quantum circuit of the QQBF. We note that, as the parameter η changes, the number of optimal qubits changes in a non-trivial manner, and the regions are different depending on the set of states considered. With these last considerations on success probabiliti… view at source ↗
Figure 3
Figure 3. Multivariate Quantum to Quantum Bernoulli factory general circuit. The circuit takes in input nk qubits in the state |zk⟩ for a total of nt and m ancillary qubits in the state |0⟩. The rest of the circuit is similar to that of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Multifunctional Quantum Bernoulli factory general circuit. This circuit implementing a Multifunctional Quantum Bernoulli factory is similar to the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 1
Figure 1. Figure 1: Quantum Bernoulli factory general circuit with traced qubits. The considered circuit is the one presented in [PITH_FULL_IMAGE:figures/full_fig_p013_1.png]

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Pith tools

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