Observation of Geometric Phase in a Molecular Aharonov-Bohm System Using IBM Quantum Computer

Bikash K. Behera; Gaurav Rudra Malik; Prasanta K. Panigrahi; Shubham Kumar; Sushree Swateeprajnya Behera

REVIEW 4 major objections 5 minor 19 references

Observation of Geometric Phase in a Molecular Aharonov-Bohm System Using IBM Quantum Computer

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper reports exact estimation of the molecular Aharonov-Bohm geometric phase on a quantum computer: a full 2π parameter sweep produces the expected π phase.

desk verdict A clean but circular educational demonstration: the measured phase is the gate angle programmed into the circuit, not an independently observed molecular Berry phase. read the letter →

arxiv 1909.00298 v1 pith:IBCZ2GUK submitted 2019-09-01 quant-ph

Gaurav Rudra Malik , Sushree Swateeprajnya Behera , Shubham Kumar , Bikash K. Behera , Prasanta K. Panigrahi This is my paper · ORCID

classification quant-ph

keywords geometricphaseBerrymolecularAharonov-BohmeffectconicalintersectionBorn-Oppenheimerapproximationtwo-qubitcircuitJahn-Tellerquantumsimulation

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 reports an exact estimation of the molecular Aharonov-Bohm geometric phase on a quantum computer. In the molecular version of the Aharonov-Bohm effect, a closed loop of nuclear coordinates around a conical intersection changes the sign of the electronic wavefunction, a global π phase that acts like a vector potential on the nuclear motion. The authors implement a two-qubit interferometric protocol: a system qubit carries the molecular eigenstate, a probe qubit controls the application of the rotation matrix built from the instantaneous eigenstates, and the accumulated overlap phase is recovered from the probe's σx and σy expectation values. The measured phase grows linearly from 0 to π as the angle parameter runs from 0 to 2π, which the paper takes as confirmation that the protocol estimates the molecular geometric phase exactly.

What carries the argument

The load-bearing object is the 2×2 operation matrix M(t), built from the instantaneous eigenstates of the molecular Hamiltonian and applied as a controlled gate on the system qubit; the paper writes this matrix as $e^{{∫ h dt'}}$ without identifying the generator h. Starting from the φ=0 eigenstates |0⟩ and |1⟩, the rotation by φ/2 maps each basis state to the corresponding eigenstate at angle φ, so the inner product between the initial state and the rotated state carries the phase accumulated over the parameter interval. The probe qubit turns that inner product into an observable: tracing out the system leaves the probe in a state whose σx and σy expectation values are the real and imaginary parts of the overlap, and the geometric phase is their argument. The circuit therefore works as a direct phase meter requiring only single-qubit measurements on the probe, not full tomography of the system.

What would settle it

Replace the hand-built matrix M(t) with a time-ordered evolution generated by the adiabatic Hamiltonian H(t) = K(I + cos φ(t) σz + sin φ(t) σx) integrated over the same closed loop, and compare the measured overlap phase with the rotation-gate result; if the two disagree, the reported phase is the gate's rotation angle rather than a dynamically accumulated geometric phase.

Watch

Extended reading notes

Core claim

The central claim is that the π Berry phase of the molecular Aharonov-Bohm effect can be measured directly and exactly with a state-independent two-qubit circuit. For the two-level conical-intersection Hamiltonian whose eigenstates are [cos(φ/2), sin(φ/2)]^T and [-sin(φ/2), cos(φ/2)]^T, the operation matrix M(t) formed from these eigenstates is the rotation matrix [[cos(φ/2), -sin(φ/2)], [sin(φ/2), cos(φ/2)]]. The circuit prepares the φ=0 eigenstate, applies M(t) conditionally on the probe qubit, and reads the geometric phase as arg(⟨σx⟩ + i⟨σy⟩) from the probe's reduced state. At φ=2π the rotation becomes -I, the overlap is -1, and the phase is π; intermediate sweeps yield the reported linear phase accumulation, plotted against the rotation angle. The authors present this agreement as exact estimation of the geometric phase and propose the protocol as a first-principles route for more complex molecular Hamiltonians.

Load-bearing premise

The protocol rests on identifying the eigenvector matrix M(t) of the instantaneous molecular Hamiltonian with the actual adiabatic evolution operator, so the phase read from the circuit is assumed to be the geometric phase accumulated by a nuclear traversal rather than merely the phase of the rotation that was programmed into the gate.

Editorial extensions

If this is right

A complete 2π cycle of the molecular parameter returns the eigenstates with a sign change, so the circuit confirms the Born-Oppenheimer wavefunction's multivaluedness that the molecular Aharonov-Bohm vector potential is designed to remove.
Because the phase is read from Pauli expectation values on a single probe qubit, the protocol does not require state tomography and can be run on two-qubit processors as they exist today.
For Hamiltonians with m-fold degeneracy, the same construction generalizes to an m×m secular problem, and the paper argues that quantum eigensolvers can supply the eigenstates when analytic ones are unavailable.
The reported linear phase-versus-angle dependence provides a calibration curve: any two-level adiabatic loop whose eigenstates wind by half the loop angle should show the same linear readout.

Reading between the lines

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

A direct dynamical test would replace the hand-built matrix M(t) with a time-ordered evolution generated by the time-dependent molecular Hamiltonian and check that the readout still equals the rotation-matrix phase; that would separate a programmed rotation phase from a genuinely accumulated geometric phase.
The probe-qubit overlap readout is a pure-state version of the mixed-state Uhlmann phase protocol, so the same circuit could be adapted to measure molecular geometric phases at finite temperature or under decoherence.
Because the measured phase is the argument of an overlap, the protocol is effectively a universal geometric-phase meter for two-level systems; any parameter loop, not only a conical intersection, could be characterized with the same basic circuit.
The linear phase plot implies the circuit could be used as a calibration device for quantum gates: deviations from linearity in the phase-versus-angle curve would expose rotation-angle errors in the applied single-qubit gate.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know two things about this paper. First, the circuit is simple and the data show a clean linear phase accumulation ending at pi for a 2pi rotation. Second, the central claim that this observes the molecular Aharonov-Bohm geometric phase does not survive reading the methods. The measured quantity is essentially the angle programmed into the rotation gate, so the experiment cannot distinguish a Berry phase from a preprogrammed unitary.

What is genuinely here: the authors take the Longuet-Higgins two-level Hamiltonian, map its eigenstates onto qubit rotations, and run a textbook interferometric phase measurement on IBM hardware. The one-step circuit in Fig. 2 is transparent, and reproducing the expected pi phase with a simulator and real hardware is a reasonable pedagogical exercise. If reframed as a demonstration of how a Berry phase would appear in a qubit rotation, the paper has some value for students.

The load-bearing problem is in Eqs. (13)-(18). The operator M(t) is introduced as an exponential of an integral involving h(t'), but h is never defined. Then M is explicitly constructed as the sum of |output> is the phase written into the gate by construction. The adiabatic theorem, the Berry connection, and the dynamical phase are never used to connect the Hamiltonian H(phi) to M(t). This is circular as an observation of a geometric phase, and it is the central unsupported step. This is not a minor gap; it is the claim.

There are smaller issues. Equation (3) appears dimensionally off: the off-diagonal term should be squared under the square root, likely a typo. There are no error bars, no statistical analysis, and no code or raw data. The reference list has some garbled entries, though not enough to matter.

Who gets value from this: instructors looking for a simple IBM Q exercise, and students learning why one must be careful about what a circuit actually simulates. It is not a research result on molecular Aharonov-Bohm systems. I would not cite it as an observation, but I might point students to it as an example of how easy it is to program a phase and then measure it back.

If I were the editor, I would send it to a referee rather than desk-reject, because the flaw is instructive and the paper makes a concrete, checkable claim. I would expect the referee to reject the central claim but to allow that the protocol could be salvaged if reframed as a simulation exercise.

Referee Report

4 major / 5 minor

Summary. The paper reports an experiment on IBM quantum hardware and simulators aimed at observing the geometric (Berry) phase of the Longuet-Higgins molecular Aharonov-Bohm model. The authors define a two-qubit circuit in which a probe qubit controls the application of an operator M(t) to a system qubit, and the geometric phase is extracted from the expectation values of σx and σy on the probe. They find that the phase accumulates linearly with the rotation angle and reaches π for a 2π rotation, which they interpret as the molecular Aharonov-Bohm phase. The abstract and discussion claim an exact estimation of the geometric phase for this molecular system.

Significance. If the claim were correct, it would provide a simple quantum-computer demonstration of the molecular Aharonov-Bohm phase and a scalable protocol for evaluating geometric phases in more complex molecular Hamiltonians. The paper has some positive features: it presents a concrete ancilla-based interferometric scheme for measuring an overlap phase, it implements the circuit on real IBM hardware, and it connects the problem to the Longuet-Higgins conical-intersection literature. However, the central physical claim is not supported by the presented evidence. The operator M(t) is defined by the eigenvector matrix of the Hamiltonian, and the circuit applies that matrix as a gate, so the measured phase is essentially the phase programmed into the gate. The connection between M(t) and the adiabatic evolution under the molecular Hamiltonian is never derived, and the Berry connection and adiabatic theorem play no role in the analysis. The paper therefore does not demonstrate an observation of a geometric phase.

major comments (4)

[II, Eqs. (13)-(18)] The operator M(t) applied in the circuit is defined in Eqs. (16)-(18) as the matrix mapping the φ=0 eigenstates of the Hamiltonian in Eq. (5) to the eigenstates at arbitrary φ, namely a rotation by φ/2. No derivation is given that this matrix is the unitary evolution generated by the molecular Hamiltonian H(φ(t)) during an adiabatic traversal; Eq. (13) invokes an exponent of an integral of h(t') but h(t') is never identified. Consequently, the measured quantity arg⟨ψ(0)|M(tf)|ψ(0)⟩ is the argument of the overlap with the programmed rotation matrix, not a geometric phase accumulated through the adiabatic theorem, and the central claim in the Discussion is unsupported.
[Eq. (3)] The secular equation in Eq. (3) reads ϵ = ±[(αx+βy)^2 + by]^{1/2}; this is dimensionally inconsistent because the first term in the parentheses has dimensions of energy squared while the second has dimensions of energy. The standard double-cone expression requires (by)^2, so the algebra preceding the conical-intersection model contains an error. While this error does not directly enter the quantum circuit, it undermines the presentation of the model.
[III, Figs. 4-7] The results report a linear accumulation of phase with the programmed rotation angle, but no error bars, shot counts, or statistical analysis are given, and there is no comparison between the simulator results and the two hardware backends shown in Figs. 5-7. Because the angle θ of the U3 gate is set to the same value as φ/2 used to compute the expected phase, the agreement between the measured phase and the theoretical curve is a check of the gate calibration rather than an independent observation of a geometric phase.
[II, protocol and circuit] The protocol places the system in an eigenstate at φ=0 and then directly applies M(t) as a controlled gate, rather than evolving the system under H(φ(t)) through a closed path with a time-dependent parameter. The Berry connection A(R) in Eqs. (7)-(9) and the parallel-transport condition are never used to relate H(φ(t)) to M(t). Thus the experiment cannot discriminate a geometric phase from a preprogrammed unitary rotation, and the connection to the molecular Aharonov-Bohm effect is not established.

minor comments (5)

[Abstract and Introduction] The name 'Lounguet-Higgins' appears in the abstract and Introduction; it should be 'Longuet-Higgins'.
[References] Reference 12 is incomplete and garbled, listing multiple author sequences without a coherent citation.
[Eq. (10)] The notation |ψθ(0)⟩ and |ψθ(tf)⟩ is used in Eq. (10) before the state |ψθ(t)⟩ is defined; the subscripts and arguments should be specified.
[Fig. 4] The horizontal axis is labeled 'angle(theta)' and the vertical axis 'phase(theta)', while the text calls the quantity 'berry phase'; these labels should be made consistent.
[Results and Discussion] The paper does not clarify which results were obtained on the quantum simulator versus the real devices IBMqx4 and IBMqx2; the Discussion refers to the 'nominal circuit on the quantum simulator' without specifying the hardware provenance of each figure.

Circularity Check

2 steps flagged · score 8.0 of 10

The measured 'geometric phase' is the U3 gate angle by construction: M is defined as the eigenvector map, not derived from Hamiltonian dynamics.

self definitional [Section II (Methods), Eqs. (13)-(18); protocol steps 2-3]
"Thus if the evolution is represented by the operator M(t) then it’s form is: M(t) = e^{∫t 0 h(t′)dt′} ... We assume that at time “t” the angle changed is φ. ... For an arbitrary angle φ we define the operation matrix M(t) as: M(t) = ∑ |output⟩⟨input| ... and hence we obtain the matrix, M(t) = [ [cos(φ/2), − sin(φ/2)], [sin(φ/2), cos(φ/2)] ]"

Equation (13) is only a formal placeholder: h(t′) is never identified, so M is never derived from H(φ) or from the adiabatic theorem. Equation (16) defines M purely as the unitary sending the φ=0 eigenstates (15) to the φ-dependent eigenstates (5); the resulting Eq. (18) is exactly the programmed controlled-U3 rotation. Equation (14) then defines the geometric phase as Φ = arg⟨ψ(0)|M(tf)|ψ(0)⟩, so the measured phase is the gate angle by construction. The experiment never tests the Longuet-Higgins Hamiltonian, the Berry connection, or the cyclic adiabatic condition: it reads back the rotation that was inserted.
other [Section III (Results), Fig. 4 and accompanying text]
"For the intermediate values of φ, the circuit remains unchanged with the only varying parameter ‘theta’ of the U3 gate. ... The included graph shows the linear variation of the acquired geometric phase in steps of π/6 radian."

This passage identifies the U3-gate angle theta as the sole tunable in the experiment and the plotted 'acquired geometric phase' as a linear function of that angle. Thus the data are a relabeling of the input rotation angle, not an independent observable. The claimed observation of the molecular Aharonov-Bohm π phase after 2π is already fixed by M(2π) = -I, which is baked into the same eigenvector-map definition.

full rationale

The central claim in the Discussion—'Exact estimation of the geometric phase in context of the molecular Aharonov-Bohm system undergoing cyclic adiabatic evolution within the described protocol is reported'—reduces to the circuit construction. The paper defines the evolution operator M(t) first as a formal exponential (Eq. 13) with an unspecified h(t′), then as the eigenvector matrix Eq. (16)-(18) that maps φ=0 eigenstates onto φ-eigenstates. Because that matrix, a rotation by φ/2, is what is implemented by the controlled U3 gate, the measured overlap phase Φ=arg⟨ψ(0)|M(tf)|ψ(0)⟩ is exactly the prescribed gate angle. No step links the Berry connection A(R), the parallel-transport condition, or the Hamiltonian H(φ) to the gate; the adiabatic theorem is never used to obtain M. The 2π result reduces to M(2π)=-I, which is already contained in the definition of M. There is no self-citation circularity here; the circularity is definitional and is located in the paper's own equations. The result is therefore a calibrated rotation experiment relabeled as a geometric-phase observation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central result rests on the ad hoc identification of the eigenvector matrix with the evolution operator and on the unstated assumption that no dynamical phase appears; no free parameters are fitted, but the measured phase is determined by the chosen gate angles.

assumptions (4)

domain assumption The 2x2 Hamiltonian in Eq. (4) with parameters K and φ is a valid description of the Longuet-Higgins molecular Aharonov-Bohm system near a conical intersection.
Used throughout; no justification beyond citing ref. 6.
ad hoc to paper The operator M(t) defined in Eqs. (16)-(18) represents the unitary evolution of the system during the adiabatic traversal.
This is the load-bearing assumption; M is the matrix of instantaneous eigenstates, not a time evolution under H.
domain assumption The measured phase from the probe qubit is purely geometric and contains no dynamical phase contribution.
The paper never subtracts or cancels the dynamical phase; it is assumed absent.
standard math The Born-Oppenheimer approximation and the adiabatic theorem apply to the nuclear coordinate evolution.
Background; standard but not actually used in the circuit.

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Pith. "Pith review of Observation of Geometric Phase in a Molecular Aharonov-Bohm System Using IBM Quantum Computer." pith.science (2026). https://pith.science/paper/IBCZ2GUK

@misc{pith2026190900298,
  author       = {Pith},
  title        = {Pith review of: Observation of Geometric Phase in a Molecular Aharonov-Bohm System Using IBM Quantum Computer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IBCZ2GUK}},
  note         = {Machine review of arXiv:1909.00298}
}

abstract

The evolution of a quantum system is governed by the associated Hamiltonian. A system defined by a parameter-dependent Hamiltonian acquires a geometric phase when adiabatically evolved. Such an adiabatic evolution of a system having non-degenerate quantum states gives the well-studied Berry phase. Lounguet-Higgins and co-workers discovered a geometric phase when considering the Jahn-Teller distortion described by the nuclear coordinates traversing a closed path about the point of intersection of the electronic potential energy surfaces. Under such a condition, the Born-Oppenheimer wave function undergoes a sign change corresponding to an introduced global phase of $\pi$ radian. This change further introduces a multiple valuedness in the wavefunction which may be removed by adding a vector potential like term in the Hamiltonian for the nuclear motion giving the Molecular Aharonov Bohm effect. Here, we demonstrate a scheme to evaluate the introduced global phase for the molecular system considered by Longuet-Higgins and propose methods as the first principle to do the same in more complex examples for the molecular Hamiltonian on a quantum computer.

Figures

Figures reproduced from arXiv: 1909.00298 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Aharonov, D

author Y. Aharonov, D. Bohm title Significance of Electromagnetic Potentials in the Quantum Theory . journal The Physical Review volume 115 , pages 485 ( year 1959 )

work page 1959

[2] [2]

author M.V Berry title Quantal phase factors accompanying adiabatic changes journal Proceedings of the Royal Society A volume 392 , pages 1802 ( year 1984 )

work page 1984

[3] [3]

Sutcliffe, R

author Brian T. Sutcliffe, R. Guy Woolley title Quantum Theory of Molecules arxiv preprint arXiv:1206.4239 [quant-ph]

arXiv

[4] [4]

Allen Mead title The geometric phase in Molecular Systems journal Review Modern Physics volume 64 , pages 51 ( year 1992 )

author C. Allen Mead title The geometric phase in Molecular Systems journal Review Modern Physics volume 64 , pages 51 ( year 1992 )

work page 1992

[5] [5]

author Eric Sj o qvist title journal arxiv pre print arXiv:1503.04847v2 [quant-ph] ( year 2015 )

work page Pith review arXiv 2015

[6] [6]

Hertzburg and H.C

author G. Hertzburg and H.C. Longuet-Higgins title Intersection of potential energy surfaces in polyatomic molecules journal Discussions of the Faraday Society volume 35 ,( year 1963 )

work page 1963

[7] [7]

Teller title Crossing of potential energy surfaces journal American Chemical Society ,( year 1936 )

author E. Teller title Crossing of potential energy surfaces journal American Chemical Society ,( year 1936 )

work page 1936

[8] [8]

Anandan and Y.Aharonov title Geometric Quantum Phase and Angles journal Physics Review D , volume 38 , pages 1863 ( year 1988 )

author J. Anandan and Y.Aharonov title Geometric Quantum Phase and Angles journal Physics Review D , volume 38 , pages 1863 ( year 1988 )

work page 1988