pith. sign in

arxiv: 2606.07376 · v1 · pith:PKD3S5PZnew · submitted 2026-06-05 · 🪐 quant-ph

Measurement circuit ansatz: Naimark versus quantum neural-network measurements

Sung Won Yun , Thi Ha Kyaw , Joonwoo Bae This is my paper

Pith reviewed 2026-06-27 21:41 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum measurementsNaimark extensionquantum neural networksstate discriminationparameterized quantum circuitsPOVMminimum-error measurementmaximum-confidence measurement
0
0 comments X

The pith

Quantum neural-network circuits achieve near-optimal measurements for state discrimination with fewer training iterations than Naimark-based circuits.

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

This paper constructs three types of quantum circuit ansatzes for implementing general quantum measurements on hardware: Naimark-based circuits with parameterized single-qubit gates, hybrid circuits combining Naimark and QNN elements, and fully QNN circuits with shallow parameterized layers. It evaluates these on quantum state discrimination tasks, specifically minimum-error and maximum-confidence strategies. The comparison reveals that fully QNN measurements reach performance close to the theoretical optimum using substantially fewer optimization steps. A sympathetic reader would care because efficient circuit implementations of measurements are essential for practical quantum information processing, where general measurements beyond projective ones enable better protocols but are hard to realize directly.

Core claim

The authors construct Naimark quantum measurement circuits by extending the system with an ancilla and using parameterized gates optimized classically. They then relax this to hybrid Naimark-QNN and fully QNN circuits. When tested on minimum-error and maximum-confidence state discrimination, the QNN-based approaches achieve near-optimal discrimination probabilities with fewer training iterations.

What carries the argument

Parameterized quantum circuit ansatz optimized to approximate a target positive operator-valued measure (POVM) for state discrimination.

Load-bearing premise

The parameterized circuit forms used are sufficiently expressive to approximate the target general measurements closely enough for the performance advantages to appear.

What would settle it

Running the optimization for a two-qubit state discrimination problem and finding that the QNN circuit's achieved success probability remains below the Naimark circuit's or the theoretical bound after convergence.

read the original abstract

In this work, we present constructions of quantum circuits to implement general measurements on quantum hardware. Firstly, we investigate a quantum circuit ansatz by following the Naimark extension with a universal set of gates, such as controlled-NOT and single-qubit gates; we call it a Naimark quantum measurement. We present a circuit ansatz framed by the Naimark extension, leaving single-qubit gates with parameters, and apply a classical optimizer to determine their parameters to approximate a desired quantum measurement. Secondly, we relax the Naimark measurement with quantum neural-network (QNN) circuits, employing parameterized quantum circuits. We present hybrid Naimark-QNN measurements by incorporating QNN circuits into Naimark measurements. Thirdly, we also consider fully QNN measurements with shallow parameterized circuits. Then, we compare the constructed measurement circuits, Naimark, hybrid Naimark-QNN, and fully QNN measurements, for strategies of state discrimination, such as minimum-error and maximum-confidence measurements. We demonstrate that QNN circuits can efficiently and effectively achieve near-optimal quantum measurements with fewer training iterations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs three families of parameterized quantum circuits for implementing general measurements: a Naimark-extension ansatz with tunable single-qubit gates, hybrid Naimark-QNN circuits, and fully QNN circuits using shallow parameterized gates. These are applied to minimum-error and maximum-confidence state discrimination tasks on low-dimensional ensembles, with numerical optimization showing that the QNN variants reach near-optimal performance using fewer training iterations than the pure Naimark ansatz.

Significance. If the reported iteration counts and performance levels prove robust, the constructions offer practical routes to hardware-native implementations of optimal POVMs, potentially lowering the classical optimization cost in variational quantum algorithms for discrimination. The explicit side-by-side comparison of the three ansatz families supplies a useful empirical benchmark, even in the absence of theoretical expressivity guarantees.

major comments (2)
  1. [§4] §4 (numerical demonstrations for minimum-error and maximum-confidence discrimination): the central claim that QNN circuits achieve the reported performance 'with fewer training iterations' is unsupported by any description of the optimizer (e.g., Adam vs. gradient descent), number of random initializations, convergence criteria, or error bars on iteration counts; without these, it is impossible to determine whether the efficiency advantage is reproducible or an artifact of the chosen hyper-parameters and small system sizes.
  2. [§3] §3 (QNN and hybrid constructions): no approximation-error bounds, expressivity analysis, or landscape characterization is supplied for the shallow parameterized circuits; the claim that they 'efficiently and effectively achieve near-optimal quantum measurements' therefore rests entirely on the specific low-dimensional numerical instances, leaving open the possibility that the ansatz is insufficiently expressive for the general case asserted in the abstract.
minor comments (2)
  1. The abstract states that 'demonstrations were performed' yet the main text supplies no explicit table or figure caption listing the exact state ensembles, Hilbert-space dimensions, or target POVM elements used in the comparisons.
  2. Notation for the variational parameters in the Naimark and QNN blocks is introduced inline without a consolidated parameter table or diagram legend, complicating reproduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help improve the clarity and reproducibility of our work. We address each major comment below.

read point-by-point responses

  1. Referee: [§4] §4 (numerical demonstrations for minimum-error and maximum-confidence discrimination): the central claim that QNN circuits achieve the reported performance 'with fewer training iterations' is unsupported by any description of the optimizer (e.g., Adam vs. gradient descent), number of random initializations, convergence criteria, or error bars on iteration counts; without these, it is impossible to determine whether the efficiency advantage is reproducible or an artifact of the chosen hyper-parameters and small system sizes.

    Authors: We agree that the optimization details are essential for supporting the reported iteration counts. In the revised manuscript we will explicitly state the optimizer (Adam), learning rate, number of random initializations, convergence criterion, and include error bars on the iteration counts obtained from multiple independent runs. revision: yes

  2. Referee: [§3] §3 (QNN and hybrid constructions): no approximation-error bounds, expressivity analysis, or landscape characterization is supplied for the shallow parameterized circuits; the claim that they 'efficiently and effectively achieve near-optimal quantum measurements' therefore rests entirely on the specific low-dimensional numerical instances, leaving open the possibility that the ansatz is insufficiently expressive for the general case asserted in the abstract.

    Authors: The manuscript centers on explicit circuit constructions and their empirical performance on the low-dimensional discrimination tasks considered; the abstract uses the verb 'demonstrate' rather than claiming a general proof. We will add a short paragraph in the revised version acknowledging the lack of theoretical bounds and stating that a full expressivity analysis lies outside the present scope. revision: partial

Circularity Check

0 steps flagged

No circularity; empirical circuit comparisons rest on independent numerical optimization

full rationale

The paper constructs Naimark-based and QNN circuit families, optimizes parameters via classical methods to approximate target POVMs, and reports numerical performance on small discrimination tasks. No derivation reduces a claimed result to its inputs by construction, no self-citation supplies a load-bearing uniqueness theorem, and no fitted quantity is relabeled as a prediction. All performance statements are direct outputs of the reported simulations rather than algebraic identities or self-referential fits.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.1-grok · 5723 in / 833 out tokens · 19271 ms · 2026-06-27T21:41:51.569255+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

45 extracted references · 3 canonical work pages

  1. [1]

    Popescu, Bell’s inequalities and density matrices: Re- vealing “hidden” nonlocality, Phys

    S. Popescu, Bell’s inequalities and density matrices: Re- vealing “hidden” nonlocality, Phys. Rev. Lett.74,2619 (1995)

    1995

  2. [2]

    Hirsch, M

    F. Hirsch, M. T. Quintino, J. Bowles, and N. Brunner, Gen- uine hidden quantum nonlocality, Phys. Rev. Lett.111, 160402(2013)

    2013

  3. [3]

    Heinosaari, T

    T. Heinosaari, T. Miyadera, and M. Ziman, An invitation to quantum incompatibility, Journal of Physics A: Math- ematical and Theoretical49,123001(2016)

    2016

  4. [4]

    H. M. Wiseman, S. J. Jones, and A. C. Doherty, Steering, entanglement, nonlocality, and the einstein-podolsky- rosen paradox, Phys. Rev. Lett.98,140402(2007)

    2007

  5. [5]

    R. Uola, A. C. S. Costa, H. C. Nguyen, and O. G ¨uhne, Quantum steering, Rev. Mod. Phys.92,015001(2020)

    2020

  6. [6]

    Rudinger, G

    K. Rudinger, G. J. Ribeill, L. C. Govia, M. Ware, E. Nielsen, K. Young, T. A. Ohki, R. Blume-Kohout, and T. Proctor, Characterizing midcircuit measurements on a superconducting qubit using gate set tomography, Phys. Rev. Appl.17,014014(2022)

    2022

  7. [7]

    Bharti, A

    K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quan- tum algorithms, Rev. Mod. Phys.94,015004(2022)

    2022

  8. [8]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P . J. Coles, Variational quantum algo- rithms, Nature Reviews Physics3,625(2021)

    2021

  9. [9]

    Blekos, D

    K. Blekos, D. Brand, A. Ceschini, C.-H. Chou, R.-H. Li, K. Pandya, and A. Summer, A review on quan- tum approximate optimization algorithm and its vari- ants, Physics Reports1068,1(2024), a review on Quan- tum Approximate Optimization Algorithm and its vari- ants

    2024

  10. [10]

    Weber, A

    M. Weber, A. Anand, A. Cervera-Lierta, J. S. Kottmann, T. H. Kyaw, B. Li, A. Aspuru-Guzik, C. Zhang, and Z. Zhao, Toward reliability in the nisq era: Robust in- terval guarantee for quantum measurements on approx- imate states, Phys. Rev. Res.4(2022)

    2022

  11. [11]

    T. H. Kyaw, M. B. Soley, B. Allen, P . Bergold, C. Sun, V . S. Batista, and A. Aspuru-Guzik, Boosting quantum ampli- tude exponentially in variational quantum algorithms, Quantum Sci. Technol.9,01LT01(2023)

    2023

  12. [12]

    M. A. Naimark, On a representation of additive operator set functions, Comptes Rendus (Doklady) de l’Acad´emie des Sciences de l’URSS41,359(1943)

    1943

  13. [13]

    Y. S. Yordanov and C. H. W. Barnes, Implementation of a general single-qubit positive operator-valued measure on a circuit-based quantum computer, Physical Review A100,10.1103/physreva.100.062317(2019)

    work page doi:10.1103/physreva.100.062317(2019 2019

  14. [14]

    Barenco, C

    A. Barenco, C. H. Bennett, R. Cleve, D. P . DiVincenzo, N. Margolus, P . Shor, T. Sleator, J. A. Smolin, and H. We- infurter, Elementary gates for quantum computation, Phys. Rev. A52,3457(1995)

    1995

  15. [15]

    Preskill, Quantum Computing in the NISQ era and be- yond, Quantum2,79(2018)

    J. Preskill, Quantum Computing in the NISQ era and be- yond, Quantum2,79(2018)

    2018

  16. [16]

    Camps and R

    D. Camps and R. Van Beeumen, Fable: Fast approximate quantum circuits for block-encodings, in2022IEEE Inter- national Conference on Quantum Computing and Engineering (QCE)(IEEE,2022) pp.104–113

    2022

  17. [17]

    G. H. Low and I. L. Chuang, Optimal hamiltonian simu- lation by quantum signal processing, Phys. Rev. Lett.118, 10.1103/physrevlett.118.010501(2017)

    work page doi:10.1103/physrevlett.118.010501(2017 2017

  18. [18]

    Andersson and D

    E. Andersson and D. K. L. Oi, Binary search trees for gen- eralized measurements, Phys. Rev. A77,052104(2008)

    2008

  19. [19]

    Oszmaniec, L

    M. Oszmaniec, L. Guerini, P . Wittek, and A. Ac ´ın, Simu- lating positive-operator-valued measures with projective measurements, Phys. Rev. Lett.119,190501(2017)

    2017

  20. [20]

    Oszmaniec, F

    M. Oszmaniec, F. B. Maciejewski, and Z. Puchała, Sim- ulating all quantum measurements using only projec- tive measurements and postselection, Phys. Rev. A100, 012351(2019)

    2019

  21. [21]

    Singal, F

    T. Singal, F. B. Maciejewski, and M. Oszmaniec, Imple- mentation of quantum measurements using classical re- sources and only a single ancillary qubit, npj Quantum Information8,82(2022)

    2022

  22. [22]

    Reitzner and J

    D. Reitzner and J. Bouda, General measurements with limited resources and their application to quantum un- ambiguous state discrimination, Quantum Information Processing23,268(2024)

    2024

  23. [23]

    Peres, Neumark’s theorem and quantum inseparabil- ity, Foundations of Physics20,1441(1990)

    A. Peres, Neumark’s theorem and quantum inseparabil- ity, Foundations of Physics20,1441(1990)

    1990

  24. [24]

    Vatan and C

    F. Vatan and C. Williams, Optimal quantum circuits for general two-qubit gates, Phys. Rev. A69,032315(2004)

    2004

  25. [25]

    S. Sim, P . D. Johnson, and A. Aspuru-Guzik, Expressibil- ity and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms, Adv. Quantum Technologies2(2019)

    2019

  26. [26]

    C. W. Helstrom, Quantum detection and estimation the- ory, Journal of Statistical Physics1,231(1969). 13

    1969

  27. [27]

    J. A. Bergou, Discrimination of quantum states, Journal of Modern Optics57,160(2010)

    2010

  28. [28]

    S. M. Barnett and S. Croke, Quantum state discrimina- tion, Advances in Optics and Photonics1,238(2009)

    2009

  29. [29]

    Bae and L.-C

    J. Bae and L.-C. Kwek, Quantum state discrimination and its applications, Journal of Physics A: Mathematical and Theoretical48,083001(2015)

    2015

  30. [30]

    Soro and A

    S. Croke, E. Andersson, S. M. Barnett, C. R. Gilson, and J. Jeffers, Maximum confidence quantum mea- surements, Physical Review Letters96,10.1103/phys- revlett.96.070401(2006)

    work page doi:10.1103/phys- 2006

  31. [31]

    Dieks, Overlap and distinguishability of quantum states, Physics Letters A126,303(1988)

    D. Dieks, Overlap and distinguishability of quantum states, Physics Letters A126,303(1988)

    1988

  32. [32]

    H. Lee, K. Flatt, and J. Bae, Sequential quantum maximum-confidence discrimination, Phys. Rev. A112, 052206(2025)

    2025

  33. [33]

    Bae, Structure of minimum-error quantum state dis- crimination, New Journal of Physics15,073037(2013)

    J. Bae, Structure of minimum-error quantum state dis- crimination, New Journal of Physics15,073037(2013)

    2013

  34. [34]

    W. K. Wootters, Distinguishing unentangled states with an unentangled measurement, International Journal of Quantum Information4,219(2006)

    2006

  35. [35]

    D. Lee, K. Baek, J. Huh, and D. K. Park, Variational quan- tum state discriminator for supervised machine learning (2023), arXiv:2303.03588[quant-ph]

    arXiv 2023

  36. [36]

    Davies, Information and quantum measurement, IEEE Transactions on Information Theory24,596(1978)

    E. Davies, Information and quantum measurement, IEEE Transactions on Information Theory24,596(1978)

    1978

  37. [37]

    H. Lee, K. Flatt, C. R. i Carceller, J. B. Brask, and J. Bae, Maximum-confidence measurement for qubit states, Physical Review A106(2022)

    2022

  38. [38]

    Flatt, H

    K. Flatt, H. Lee, C. R. I. Carceller, J. B. Brask, and J. Bae, Contextual advantages and certification for maximum- confidence discrimination, PRX Quantum3,030337 (2022)

    2022

  39. [39]

    N. P . D. Sawaya, T. Menke, T. H. Kyaw, S. Johri, A. Aspuru-Guzik, and G. G. Guerreschi, Resource- efficient digital quantum simulation of d-level systems for photonic, vibrational, and spin-s hamiltonians, npj Quantum Inf.6(2020)

    2020

  40. [40]

    Camps and R

    D. Camps and R. Van Beeumen, Fable: Fast approximate quantum circuits for block-encodings, in2022IEEE Inter- national Conference on Quantum Computing and Engineering (QCE)(2022) pp.104–113

    2022

  41. [41]

    V . V . Shende and I. L. Markov, On the cnot-cost of toffoli gates, Quantum Info. Comput.9,461(2009)

    2009

  42. [42]

    M ¨ott¨onen, J

    M. M ¨ott¨onen, J. J. Vartiainen, V . Bergholm, and M. M. Salomaa, Quantum circuits for general multiqubit gates, Phys. Rev. Lett.93,130502(2004). Appendix A: Types of theCX (x) In this section, we describe the types ofCX (x) gates required for Naimark measurement circuit acting onm qubits. We first define the gate MCX l k, which denotes a multi-controlle...

    2004

  43. [43]

    Both thei-thB SA gate and sub- sequentCX (x) gate act on⌊log 2 i⌋+1 ancillary qubits

    CNOT Count in thei-thCX (x) We analyze the CNOT count required to implement thei-thCX (x) gate. Both thei-thB SA gate and sub- sequentCX (x) gate act on⌊log 2 i⌋+1 ancillary qubits. Here, thei-thCX (x) gate denotes the one applied im- mediately after thei-thB SA gate. Assume that the bitstringxcontainskbits equal to 0, corresponding tok Xgates applied to ...

  44. [44]

    As shown in Ref

    CNOT Count in the Uniformly Controlled Rotation Gate at thei-thB SA Gate The uniformly controlled rotation gate in thei-thB SA gate has two types of controls:nsystem qubits that act as uniformly controlling qubits, and⌊log 2 i⌋ancillary qubits that act as standard control qubits. As shown in Ref. [42], a uniformly controlled rota- tion gate withmuniformly...

  45. [45]

    Each Vgate is controlled by⌊log 2 i⌋+1 qubits

    CNOT Count in the V gate at thei-thB SA Gate AB SA gate contains twoVgates, each composed ofn single-qubit rotation gates andn−1 CNOT gates. Each Vgate is controlled by⌊log 2 i⌋+1 qubits. We first con- sider the rotation gates, more precisely, multi-controlled rotation gates. Since a multi-controlled rotation gate withmcontrol qubit is decomposed into 2 m...