Pith. sign in

REVIEW 3 major objections 5 minor 34 references

Classical analog circuit emulation of quantum Grover search algorithm

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read This paper claims that a classical analog circuit built from operational amplifiers, resistors, and capacitors can emulate universal quantum gates and run a 3-qubit Grover search, reaching measured success probabilities of 99.0% for one…

desk verdict A real op-amp Grover emulator with one-iteration data close to theory, but two-iteration data contradict the paper's central claim. read the letter →

arxiv 2506.06502 v1 pith:2DT2WXME submitted 2025-06-06 quant-ph cond-mat.dis-nn

Samuel Feldman , Hassam Ghazali , Andrey Rogachev This is my paper
classification quant-phcond-mat.dis-nn
keywords analogquantumemulationGroversearchalgorithmoperationalamplifiercircuitsuniversalgatesetcomputationalbasisstateencodingcircuitdecoherence-free
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

This paper reports a hardware design that emulates quantum computation using analog electronics: operational amplifiers, resistors, and capacitors, with one electrical line per computational basis state and AC amplitude and phase encoding each probability amplitude. The authors build and test the universal gate set of phase-shift, Hadamard, and controlled-NOT gates, then assemble a 3-qubit Grover search emulator. Measured success probabilities are 79.4% after one iteration and 99.0% after two for a single marked state, and 96.4% after one iteration for two marked states. The paper positions these results as fulfilling the quantitative predictions of Grover's algorithm and argues that the emulator can serve as a sub-component of physical quantum computers or for tasks that need only a few qubits.

What carries the argument

The central mechanism is the computational-basis-state (CBS) line encoding: every possible 2^n basis state of an n-qubit register is assigned its own electrical line, and an AC signal on that line carries both the magnitude and phase of the corresponding probability amplitude. Quantum gates are then implemented as op-amp circuits: a single-op-amp phase shifter, a three-op-amp Hadamard gate, and a wired swap for controlled-NOT. The Grover diffusion operator is implemented as one combined matrix circuit that scales each line to one quarter and sums contributions through op-amp summers, rather than decomposing the operation into elementary gates. The encoding makes multi-qubit gates simple when applied only to selected CBS lines, and it allows output signals to be renormalized after each gate.

What would settle it

Feed the measured output voltages of the first Grover iteration directly back into the circuit as the second iteration's input, without manual reconfiguration, and compare the resulting two-iteration success probabilities to the theoretical values of 94.5% for one marked state and 25% for two marked states; a disagreement beyond component tolerance would show that the emulator does not reproduce the algorithm's iteration dynamics.

Watch

Extended reading notes

Core claim

The paper's central claim is that a completely analog circuit framework can emulate a universal quantum computer. In this framework, 2^n lines represent the computational basis states of n qubits, and the complex amplitude of each state is carried by an AC signal's amplitude and relative phase. Because the emulator is classical, it is free from decoherence, measurements do not collapse states, and gates need not be implemented unitarily as long as the output is renormalized. The authors demonstrate the framework by building and testing the phase-shift, Hadamard, and controlled-NOT gates, and by constructing a 3-qubit Grover search emulator in which the oracle is simply an inverter on the marked line and the diffusion operator is a single op-amp matrix. They report that their emulator fulfills all quantitative predictions of the Grover search algorithm, and they claim greater accuracy with less computational complexity than current physical quantum computers for this 3-qubit demonstration.

Load-bearing premise

The multi-iteration results depend on the assumption that manually adjusting the voltage dividers between runs genuinely performs the second Grover iteration on the first iteration's outputs, rather than merely re-running the circuit with new input amplitudes.

Editorial extensions

If this is right

  • Because gates do not have to be unitary, entire algorithms can be collapsed into single analog matrix circuits, reducing the number of components needed for a fixed gate sequence.
  • The emulator is free of decoherence and measurement collapse, so repeated Grover iterations can be run continuously without the state being destroyed by readout.
  • The oracle in this emulator is only an inverter on the marked state line, making test oracles trivial to insert, whereas physical quantum computers require ancilla qubits and multi-qubit controlled gates.
  • The hardware is compatible with standard semiconductor technology and, as the paper notes, could be built on a single integrated circuit; one cited CMOS-based simulation estimates a 400x speed-up over corresponding digital simulation.
  • The main drawback is the exponential growth of lines with qubit count, which confines the approach to few-qubit operations or algorithms that can be split into weakly entangled independent blocks.

Reading between the lines

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

  • Beyond the paper: the measured two-iteration probabilities (99.0% for one marked state and 72.3% for two) deviate from the theoretical values (94.5% and 25%), suggesting that the manual reconfiguration procedure may not implement the second Grover iteration with the same fidelity as the first.
  • Beyond the paper: a natural next test is to build a fully connected 4-qubit version of the emulator; if line count becomes prohibitive, the approach could serve as an analog coprocessor for weakly entangled sub-blocks inside a hybrid setup with a physical quantum computer.
  • Beyond the paper: the diffusion-operator analogy of a negative population attracting signal away from unmarked states hints that the same analog mean-inversion circuit could be adapted to classical optimization heuristics, though the paper does not pursue that direction.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper reports an analog electronic circuit framework in which each computational basis state of an n-qubit register is represented by an AC signal on a separate line, and quantum gates are implemented with operational amplifiers, resistors, and capacitors. The authors describe and test emulators for the phase-shift, Hadamard, and controlled-NOT gates, and then construct a 3-qubit circuit emulating Grover's search algorithm. For one Grover iteration they report 79.4% success probability for one marked state and 96.4% for two marked states, close to the ideal values of 78.1% and 100%. For two iterations they report 99.0% for one marked state and 72.3% for two marked states. The paper concludes that the emulator fulfills all quantitative predictions of the Grover search algorithm and that it demonstrates greater accuracy with less computational complexity than current physical quantum computers.

Significance. The basic idea of representing complex amplitudes by AC signals and implementing gate matrices with op-amp circuits is interesting and potentially useful for small-scale, decoherence-free demonstrations of quantum algorithms. A strength of the paper is that the hardware results are compared with independent analytical predictions of Grover's algorithm, and no parameter fitting is evident, so the comparison is not circular. However, the central accuracy claim is directly contradicted by the two-iteration data: the measured values are inconsistent with the unitary Grover dynamics that the paper claims to emulate. The reported one-iteration results alone would be a modest proof of concept, but the manuscript as written does not support the stronger claims made in the abstract and summary.

major comments (3)
  1. [Section III, Fig. 5 and following text] The two-iteration measured success probabilities contradict the paper's central claim. For a 3-qubit Grover search with k marked states, the ideal success probability after r iterations is ASP = sin^2((2r+1)theta) with theta = arcsin(sqrt(k/8)). For k=1 and r=2 this gives 121/128 = 94.5%, while Fig. 5(a) reports 99.0%; for k=2 and r=2 it gives 25%, while Fig. 5(b) reports 72.3%. The 99.0% value exceeds the ideal two-iteration Grover maximum for one marked state in N=8, and the 72.3% value is 47 percentage points above the prediction. These deviations cannot be attributed to the invoked 'tolerance of components.' The sentence 'our emulator fulfills all quantitative predictions of the Grover search algorithm' is therefore not supported by the data. In addition, no error bars, repeated trials, or uncertainty analysis are provided, so even the one-iteration deviations cannot be properly assessed.
  2. [Section III, second-iteration procedure] The description of the second Grover iteration is physically incomplete. The paper states that 'we ran a second iteration of the algorithm by adjusting the voltage dividers so that the initial state matched the output of the first iteration.' A Grover iteration requires the full complex amplitude vector, including relative phase signs, as its input; reproducing only the magnitudes does not implement the unitary Grover operator. No transfer matrix, phase verification, or circuit configuration for the second pass is reported. This explains why the two-iteration results deviate so strongly from theory. The general statement in Section II that gates 'need not be implemented unitarily' and that outputs can be renormalized is especially problematic for Grover amplification, which relies on destructive interference between positive and negative amplitudes.
  3. [Section III, ASP extraction] The manuscript does not define how the Algorithm Success Probability is computed from the measured output voltages, how normalization is performed, or what measurement uncertainties follow from the component tolerances. Since the conclusion of 'greater accuracy' than physical quantum computers rests on precise ASP values, the absence of a defined error budget and of repeated measurements is a load-bearing omission for any accuracy claim.
minor comments (5)
  1. [Abstract and Section III] The abstract mentions a 'matrix-based emulator' of the 3-qubit Grover algorithm, but the body of the paper describes only the analog circuit implementation. Please clarify whether this is a separate software matrix emulator or the hardware implementation of the diffusion operator as a matrix, and report its results separately if it is a distinct object.
  2. [Fig. 5] The bar chart does not explain how the plotted voltages are converted into the reported probabilities, and the legend for the one- and two-iteration bars is not self-contained. A short description in the caption or text would improve reproducibility.
  3. [Section II, gate circuits] The paper gives circuit diagrams but not the resistor and capacitor values, input amplitudes, or operating frequency for the gate tests. For an experimental paper, this level of detail is needed for reproducibility.
  4. [Section IV] There is a typo, 'semicoductor technology,' and the closing paragraph contains 'Correspondance' for 'Correspondence.'
  5. [Section III, fourth paragraph] The sentence reporting the two-iteration two-state result, 'the ASP dropped to 72.3% as indeed expected when the iterations go beyond the optimal number,' is misleading: the ideal value after two iterations is 25%, not a value near 72.3%. The qualitative statement that the ASP drops is correct, but the quantitative expectation is not met.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: hardware results are compared against independent Grover theory; no parameter fitting or self-citation chain is load-bearing.

full rationale

The paper's derivation chain is an experimental implementation: circuits for Hadamard, phase-shift, and CNOT gates are built, and the Grover diffusion operator is implemented as the matrix in Eq. (3). The measured success probabilities are compared to textbook Grover predictions (e.g., 78.1% and 94.5% for one marked state after one and two iterations). No parameter is fitted to the data to force agreement; the initial equal superposition is set directly by 50/50 voltage dividers, and the oracle is implemented as inverters. The two-iteration results (99.0% and 72.3%) deviate substantially from the theoretical values (94.5% and 25%), which is a correctness/accuracy problem, not a circularity problem: the outputs are not equal to the inputs by construction, nor are the predictions derived from fitted parameters. The citations to Ezawa are background; no load-bearing self-citation or imported uniqueness theorem appears. The 'matrix-based emulator' mentioned in the abstract is at most an independent simulation of the same algorithm, and the paper's central quantitative claim is checked against the known Grover formula rather than against the emulator itself. Hence no circular step can be exhibited.

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

The framework rests on the mapping of quantum amplitudes to AC signal amplitudes and phases, and on the assumption that op-amp circuits can perform linear operations (summation, inversion, scaling) faithfully. The Grover implementation further assumes that the manual reconfiguration between iterations correctly applies the second iteration; the data suggests this assumption fails.

assumptions (3)
  • domain assumption Quantum amplitudes can be represented by the amplitude and phase of AC signals on separate lines, and linear operations by op-amp circuits.
    This is the foundational mapping of the emulator, introduced in Section II.
  • domain assumption The diffusion operator can be implemented as the matrix in Eq. (3) using gain -3 and summing amplifiers.
    The circuit in Fig. 4 realizes this matrix; the correctness relies on the circuit performing linear superposition and scaling as intended.
  • ad hoc to paper The initial equal superposition is created by feeding equal-amplitude AC signals through 50/50 voltage dividers.
    The paper skips the Hadamard layer and manually sets the initial state, which is a simplification specific to this demonstration.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Classical analog circuit emulation of quantum Grover search algorithm." pith.science (2026). https://pith.science/paper/2DT2WXME

@misc{pith2026250606502,
  author       = {Pith},
  title        = {Pith review of: Classical analog circuit emulation of quantum Grover search algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2DT2WXME}},
  note         = {Machine review of arXiv:2506.06502}
}
abstract

We construct a completely analog framework that emulates universal quantum gates and quantum algorithms. It is based on electronic circuits made of operational amplifiers, resistors and capacitors. In these circuits, input and output lines represent the computational basis states (CBSs) and thus $2^n$ lines are required to represent $n$ qubits. An operation of the circuits is based on classical evolution and interference of complex amplitudes associated with each CBS. The framework can emulate entangled states and is free from decoherence, measurements are classical and do not collapse states. Similar to physical quantum computers, emulated quantum algorithms can be constructed as a sequence of the gates belonging to a universal set (phase shift, Hadamard, controlled-NOT), as a unitary matrix and as combinations of the two. Circuits representing the universal gates have been made and tested. We also have made a matrix-based emulator of a 3-qubit Grover search algorithm. We tested it by searching for one and two particular states with one and two iterations and found that its outputs accurately match predicted values. We anticipate that the emulators can work as sub-components of physical quantum computers. On their own, the emulators can be used for operations that require a few qubits or operations that can be split into independent (and perhaps weakly entangled) blocks of qubits.

Figures

Figures reproduced from arXiv: 2506.06502 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Shor, SIAM J

    P. Shor, SIAM J. Sci. Statist. Comput. 26 1484 (1997)

    work page 1997

  2. [2]

    particles

    On the circuit diagram of Fig. 1(b), this means that if the input signal is applied only to |0⟩in, the outputs, marked as |0⟩out and|1⟩out, assume the same am- plitude and phase. The basis state |1⟩ evolves into a state (|0⟩−|1⟩)/ √ 2, meaning that the output|1⟩out isπ-shifted with respect to|0⟩out. The outputs of the Hadamard gate are the eigenvectors of...

    work page

  3. [3]

    Santagati et al., Nat

    R. Santagati et al., Nat. Phys. 20, 549 (2024)

    work page 2024

  4. [4]

    Aspuru-Guzik, A.D

    A. Aspuru-Guzik, A.D. Dutoi, P.J. Love, and M. Head-Gordon, Science 309, 5741 (2005)

    work page 2005

  5. [5]

    Clinton et al., Nature Communications, 15, 211 (2024)

    L. Clinton et al., Nature Communications, 15, 211 (2024)

    work page 2024

  6. [6]

    H. Ma, M. Govoni, and G. Galli, npj Computational Materials 6, 85 (2020)

    work page 2020

  7. [7]

    Grover, PRL 79, 325 (1997)

    L. Grover, PRL 79, 325 (1997)

    work page 1997

  8. [8]

    Brassard, P

    G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, Quantum Com- putation and Quantum Information 305:53-74 (2002)

    work page 2002

Show all 34 references
  1. [9]

    Arute, et al

    F. Arute, et al. Nature 574, 505 (2019)

    2019

  2. [10]

    Ebadi et al., Nature 595, 227 (2021)

    S. Ebadi et al., Nature 595, 227 (2021)

    2021

  3. [11]

    Wang, et al., Nature Communications 12, 233 (2021)

    P. Wang, et al., Nature Communications 12, 233 (2021)

    2021

  4. [12]

    King, et al

    A. King, et al. Beyond-classical Computation in Quantum Sim- ulation, Science 388, 199 (2025)

    2025

  5. [13]

    Ningyuan, C

    J. Ningyuan, C. Owens, A. Sommer, D. Schuster, and J. Simon, Phys. Rev. X 5, 021031 (2015)

    2015

  6. [14]

    Rosenthal, N.K

    E.I. Rosenthal, N.K. Ehrlich, M.S. Rudner, A.P. Higginbotham, and K.W. Lehnert, Phys. Rev. B 97, 220301 (2018)

    2018

  7. [15]

    Serra-Garcia, R

    M. Serra-Garcia, R. S ¨usstrunk, and S.D. Huber, Phys. Rev. B 99, 020304(R) (2019)

    2019

  8. [16]

    Helbig, et al., Nature Physics 16, 747 (2020)

    T. Helbig, et al., Nature Physics 16, 747 (2020)

    2020

  9. [17]

    Ezawa, Phys

    M. Ezawa, Phys. Rev. B 102, 075424 (2020)

    2020

  10. [18]

    Lu, et al

    Y . Lu, et al. Phys. Rev. B 99, 020302 (2019)

    2019

  11. [19]

    Wang et al., Science Advances 5, 7854 (2019)

    P. Wang et al., Science Advances 5, 7854 (2019)

    2019

  12. [20]

    Deng, Journal of Integrative Neuroscience 8, 255 (2009)

    B. Deng, Journal of Integrative Neuroscience 8, 255 (2009)

    2009

  13. [21]

    Brink et al., IEEE Transactions on Biomedical Circuits and Systems 7, 71 (2013)

    S. Brink et al., IEEE Transactions on Biomedical Circuits and Systems 7, 71 (2013)

    2013

  14. [22]

    Ezawa, Phys

    M. Ezawa, Phys. Rev. Research 3, 023051 (2021)

    2021

  15. [23]

    Mcrae, B.G

    B.H. Mcrae, B.G. Dickson, T.H. Keitt, V .B. Shah, Ecology 89, 2712 (2008)

    2008

  16. [24]

    Keller, Electronic Notes in Discrete Mathematics 28, 469 (2007)

    A.A. Keller, Electronic Notes in Discrete Mathematics 28, 469 (2007)

    2007

  17. [25]

    Ezawa, Phys

    M. Ezawa, Phys. Rev. B 100, 165419 (2019)

    2019

  18. [26]

    Ezawa, Phys

    M. Ezawa, Phys. Rev. Research 2, 023278 (2020)

    2020

  19. [27]

    Ezawa, Journal of the Physical Society of Japan 89, 124712 6 (2020)

    M. Ezawa, Journal of the Physical Society of Japan 89, 124712 6 (2020)

    2020

  20. [28]

    Mourya, B

    S. Mourya, B. R. L. Cour and B. D. Sahoo, IEEE Transactions on Quantum Engineering, 4, 1, (2023)

    2023

  21. [29]

    Negovetic, M

    G. Negovetic, M. Perkowski, M. Lukac, A. Buller, 5th In- ternational Workshop Boolean Problems, Freiburg, Germany, (September 2002)

    2002

  22. [30]

    Khalid, Z

    A. Khalid, Z. Zilic and K. Radecka, Proc. IEEE Int. Conf. Com- put. Des.: VLSI Comput. Processors, 310-315, (2004)

    2004

  23. [31]

    Pilch, and J

    J. Pilch, and J. Dlugopolski, Journal of Computational Elec- tronics, 18, 329-342, (2019)

    2019

  24. [32]

    Hundt, Quantum computing for programmers, (Cambridge University Press, Cambridge, UK, 2022)

    R. Hundt, Quantum computing for programmers, (Cambridge University Press, Cambridge, UK, 2022)

    2022

  25. [33]

    Zhang et al., Quant

    K. Zhang et al., Quant. Inf. Proc. 20, 1 (2021)

    2021

  26. [34]

    Zhang, K

    K. Zhang, K. Yu and V . Korepin, Europhys. Lett.140, 18002 (2022)

    2022

Pith tools

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