pith. sign in

arxiv: 2605.16559 · v1 · pith:M4DL2FP4new · submitted 2026-05-15 · 🪐 quant-ph · physics.app-ph

Measurement and Control of the Complex Berry Phase in a Quantum System

Pratik J. Barge , Qian Cao , Niklas H\"ornedal , Aur\'elia Chenu , Kater W. Murch This is my paper

Pith reviewed 2026-05-20 18:29 UTC · model grok-4.3

classification 🪐 quant-ph physics.app-ph
keywords Berry phasenon-Hermitian systemsquantum controlsuperconducting transmonadiabatic evolutiongeometric phasedissipation
0
0 comments X

The pith

Superconducting circuit measures both real and imaginary parts of the Berry phase

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

The paper demonstrates the measurement of the complex Berry phase in a quantum system by using a transmon circuit with controlled dissipation. It separates the real part, which accumulates like a standard geometric phase, from the imaginary part, which influences the state's amplitude in a path-dependent manner. This separation allows the authors to show how the imaginary component can be used for non-unitary control. A reader would care because it provides a geometric handle on dissipation in open quantum systems, potentially leading to new ways to manipulate quantum information without relying solely on unitary operations.

Core claim

We report experimental measurement of both the real and imaginary components of a Berry phase in a fully quantum system using a superconducting transmon circuit with engineered dissipation. We also demonstrate the path-dependent effects of the imaginary part on the dissipation and its utility in the implementation of non-unitary quantum control. These findings establish a clear geometric distinction between the real and imaginary components of the Berry phase and experimentally confirm the unique adiabatic behavior of non-Hermitian quantum systems.

What carries the argument

The complex Berry phase, which is the geometric phase acquired during adiabatic evolution in a non-Hermitian system, carrying both a phase shift (real part) and an amplitude modulation (imaginary part).

If this is right

  • The imaginary Berry phase produces path-dependent effects on dissipation.
  • Non-unitary quantum control can be achieved using the geometric phase.
  • Real and imaginary components of the Berry phase can be independently measured and controlled in quantum circuits.
  • Adiabatic evolution in dissipative systems follows distinct rules from closed systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This geometric control of dissipation might be combined with standard quantum error correction techniques.
  • Similar measurements could be performed in other platforms like trapped ions or photonic systems to verify generality.
  • The results hint at using imaginary phases for amplification in quantum metrology applications.

Load-bearing premise

The parameter changes must be slow enough for the evolution to remain adiabatic, and the engineered dissipation must match the non-Hermitian model without adding extra decoherence.

What would settle it

If the phase extracted from the final state after traversing a closed loop in the two-level parameter space does not agree with the calculated complex Berry phase from the integral over the loop.

Figures

Figures reproduced from arXiv: 2605.16559 by Aur\'elia Chenu, Kater W. Murch, Niklas H\"ornedal, Pratik J. Barge, Qian Cao.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) An effective non-Hermitian system is realized [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (d), we investigate the imaginary geometric phase for the case of open loops, where ϕ : 0 → 2π/n, with fixed total duration T = 3 µs. We observe that θ (im) −,C+ is reduced according to the loop fraction which is a con￾sequence of its gauge invariance: for an open loop, the eigenstate at the endpoint does not exactly coincide with the initial eigenstate (See Eq. (2)), it varies by a change in gauge. We obs… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: summarizes how the imaginary geometric phase (and therefore strength and sign of the nonuni￾tary control) can be tuned via the drive parameters J and ∆, the sign depending on the loop order. We dis￾play θ (im) as extracted from the probabilities in Eq. (12) for α = β = √ 1 2 . While the experimentally measured θ (im) [ [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: displays a± for the control loops used in measuring the imaginary part of the geometric phase (Sec. 3 C), where approximate ϕ˙ = 2π/T. We can see that the transition amplitude a+ is in all cases larger, owing to the fact that |R+⟩ is the most damped eigen￾state and can only achieve quasiadiabaiticy. However in all cases the transition amplitude a± < 0.3 corresponding to nearly adiabatic behavior. Furthermo… view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
read the original abstract

The Berry phase is a geometric phase acquired during adiabatic evolution over a closed loop in parameter space. It plays an essential role in geometric quantum gates and other phase-based protocols. In non-Hermitian systems, the Berry phase is complex, introducing fundamentally new geometric effects, including state amplification. In this work, we report experimental measurement of both the real and imaginary components of a Berry phase in a fully quantum system using a superconducting transmon circuit with engineered dissipation. We also demonstrate the path-dependent effects of the imaginary part on the dissipation and its utility in the implementation of non-unitary quantum control. These findings establish a clear geometric distinction between the real and imaginary components of the Berry phase and experimentally confirm the unique adiabatic behavior of non-Hermitian quantum systems.

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 reports an experimental measurement of both the real and imaginary components of the Berry phase acquired during adiabatic evolution in a non-Hermitian quantum system, realized with a superconducting transmon circuit incorporating engineered dissipation. The authors further claim to demonstrate path-dependent effects of the imaginary component on dissipation rates and its utility for implementing non-unitary quantum control protocols.

Significance. If the central experimental claims are substantiated, the work would provide a valuable demonstration of complex geometric phases in an open quantum system and illustrate a geometric route to non-unitary control. The choice of a transmon platform with tunable dissipation is well-suited to the task, and the reported distinction between real and imaginary phase effects addresses a timely question in non-Hermitian quantum mechanics.

major comments (2)
  1. The central claim requires that the transmon state remains in the instantaneous right eigenstate throughout the closed parameter loop so that the accumulated phase is purely geometric. No direct experimental bounds are provided on non-adiabatic leakage (e.g., population outside the computational subspace or deviation from expected norm evolution) as a function of loop duration, particularly near the exceptional point where the eigenvalue gap closes. This omission is load-bearing for the validity of the extracted complex Berry phase and the claimed path-dependent control.
  2. Insufficient detail is given on the procedure used to isolate the imaginary component of the Berry phase from uncontrolled additional decoherence channels or Markovian dissipation not captured by the engineered non-Hermitian Hamiltonian. Without such controls or subtraction methods, the attribution of observed effects specifically to the imaginary geometric phase remains open to alternative explanations.
minor comments (2)
  1. The abstract would be strengthened by including a brief statement of the achieved measurement precision or typical error bars on the reported phase values.
  2. Notation for the biorthogonal inner product and the definition of the complex Berry phase should be made fully explicit in the theoretical background section to facilitate direct comparison with the experimental extraction protocol.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We have addressed the major comments by providing additional details and clarifications, as detailed in the point-by-point responses below. We believe these revisions enhance the clarity and robustness of our claims regarding the measurement of the complex Berry phase.

read point-by-point responses

  1. Referee: The central claim requires that the transmon state remains in the instantaneous right eigenstate throughout the closed parameter loop so that the accumulated phase is purely geometric. No direct experimental bounds are provided on non-adiabatic leakage (e.g., population outside the computational subspace or deviation from expected norm evolution) as a function of loop duration, particularly near the exceptional point where the eigenvalue gap closes. This omission is load-bearing for the validity of the extracted complex Berry phase and the claimed path-dependent control.

    Authors: We acknowledge the importance of verifying the adiabatic condition, especially near the exceptional point. While our manuscript includes comparisons between experimental data and theoretical predictions assuming adiabatic evolution, we agree that explicit bounds would strengthen the claim. In the revised version, we have added experimental data on the population in the computational subspace and norm evolution for different loop durations, including near the exceptional point. These measurements show deviations below 3% for our operating parameters, supporting the validity of the geometric phase extraction. We have also included a discussion of the adiabaticity criterion based on the eigenvalue gap. revision: yes

  2. Referee: Insufficient detail is given on the procedure used to isolate the imaginary component of the Berry phase from uncontrolled additional decoherence channels or Markovian dissipation not captured by the engineered non-Hermitian Hamiltonian. Without such controls or subtraction methods, the attribution of observed effects specifically to the imaginary geometric phase remains open to alternative explanations.

    Authors: We have expanded the methods section to provide a detailed description of our isolation procedure. Specifically, we measure the dissipation rates for open paths (no loop) and closed paths with varying geometric contributions, allowing us to subtract the background Markovian dissipation. Control experiments with the dissipation turned off confirm that the path-dependent effects are due to the imaginary phase. These details have been added to the revised manuscript to address potential alternative explanations. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental measurement with no derivation chain

full rationale

This paper is an experimental report of measuring the real and imaginary parts of the complex Berry phase in a non-Hermitian transmon system with engineered dissipation. The abstract and context describe direct observation of path-dependent effects and non-unitary control, with no mathematical derivation, first-principles prediction, or equation chain that reduces a claimed result to fitted parameters, self-definitions, or self-citations. The central claims rest on empirical data from the circuit rather than internal consistency of a theoretical model, making the work self-contained against external experimental benchmarks. No steps qualify under the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based on abstract only; no specific free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5671 in / 1062 out tokens · 77896 ms · 2026-05-20T18:29:38.392673+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages · 2 internal anchors

  1. [1]

    In Hermitian systems, it is a real-valued phase factor acquired under a closed adia- batic trajectory

    INTRODUCTION The Berry phase is a foundational concept in adia- batic quantum evolution. In Hermitian systems, it is a real-valued phase factor acquired under a closed adia- batic trajectory. The Berry phase, and more generally, geometric phases, have broad implications across quan- tum science: They underlie universal geometric quan- tum gates with intri...

    work page

  2. [2]

    MODEL In this work, we focus on the dynamics of a NH qubit, shown in Fig. 1a. As detailed in the next Section, the Hamiltonian is generated by driving a qubit with detun- ing ∆ and amplitudeJleading to an effective coupling J eiϕ =J x +iJ y, namely, in the{|0⟩,|1⟩}basis, H= ∆−iΓJ e +iϕ J e−iϕ 0 ! .(1) The non-Hermiticity, proportional to Γ, arises from th...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  3. [3]

    Realizing non-Hermitian evolution with a dissipative superconducting qubit We realize an effective non-Hermitian evolution with a submanifold of a dissipative transmon circuit

    MEASUREMENT OF THE BERRY PHASE A. Realizing non-Hermitian evolution with a dissipative superconducting qubit We realize an effective non-Hermitian evolution with a submanifold of a dissipative transmon circuit. The relevant energy eigenstates of the circuit are labeled {|g⟩,|e⟩,|f⟩}, see Fig. 1(a). By engineering the circuit’s dissipative environment, we ...

    work page

  4. [4]

    When the param- eterϕis varied adiabatically along the loopC +, the state evolves to|ψ 1⟩= ˆC+ |ψ0⟩

    Adiabatic evolution alongC +: we initialize the sys- tem in an equal superposition of the two right eigen- states,|ψ 0⟩= 1√ 2 (|R+⟩+|R −⟩). When the param- eterϕis varied adiabatically along the loopC +, the state evolves to|ψ 1⟩= ˆC+ |ψ0⟩. Since for this tra- jectoryθ −,C+ = 2π−θ +,C+, the two components acquire opposite geometric phases, yielding|ψ 1⟩= ...

    work page

  5. [5]

    Namelye −i π 2 ˆS |R±⟩=± |R ∓⟩, with ˆS≡i|R −⟩ ⟨L+| − i|R +⟩ ⟨L−|

    Eigenstate swap: to eliminate the dynamical phase contribution, we apply a resonantπpulse along the y axis of the eigenbasis that exchanges|R +⟩and|R −⟩. Namelye −i π 2 ˆS |R±⟩=± |R ∓⟩, with ˆS≡i|R −⟩ ⟨L+| − i|R +⟩ ⟨L−|. The resulting state is|ψ 2⟩=e −i π 2 ˆS |ψ1⟩

    work page

  6. [6]

    The final state,|ψ 3⟩= ˆC− |ψ2⟩= 1√ 2 eiΛ e2iθ+,C+ |R−⟩ −e −2iθ+,C+ |R+⟩ con- tains an overall (complex) dynamical phase, Λ =λ + + λ− =−εT, as well as a relative phase of 4θ +,C+

    Adiabatic evolution alongC −: We then tuneϕ along the reverse loop,C −. The final state,|ψ 3⟩= ˆC− |ψ2⟩= 1√ 2 eiΛ e2iθ+,C+ |R−⟩ −e −2iθ+,C+ |R+⟩ con- tains an overall (complex) dynamical phase, Λ =λ + + λ− =−εT, as well as a relative phase of 4θ +,C+. The rel- ative phase can be decomposed into a rotation of 4θ (r) +,C+ and a relative scaling ofe −4θ(im) +,C+

    work page

  7. [7]

    Quantum state tomography: we extract this rota- tion angle using postselected quantum state tomography. By defining the observables ˆSx ≡ |L −⟩ ⟨L+|+|L +⟩ ⟨L−|, ˆSy ≡i|L −⟩ ⟨L+| −i|L +⟩ ⟨L−|and the expectation val- uesx=⟨ψ 3| ˆSx|ψ3⟩,y=⟨ψ 3| ˆSy|ψ3⟩, the real geometric phase is obtained asθ (r) +,C+ = 1 4 arctan(y/x), providing a direct and quantitative m...

    work page

  8. [8]

    This enables an additional degree of freedom for geometric operations that go be- yond standard unitary control

    NON-UNITARY CONTROL USING BERRY PHASE The imaginary part of the Berry phase generates de- terministic, path-dependent amplification or attenuation of the eigenstate amplitudes. This enables an additional degree of freedom for geometric operations that go be- yond standard unitary control. Such non-unitary trans- formations could be useful in quantum simul...

    work page

  9. [9]

    6(a,b)] is in good agreement with a Lindblad simulation of the full system [Fig

    While the experimentally measured θ(im) [Fig. 6(a,b)] is in good agreement with a Lindblad simulation of the full system [Fig. 6(c,d)], the two are in marginal agreement with theory (Eq. 4), shown in [Fig. 6(e,f)]. The deviations are attributed to|f⟩ → |e⟩ relaxation and non-adiabatic transitions, which degrade fidelity during the extended interaction tim...

    work page

  10. [10]

    geometric filtering

    OUTLOOK The Berry phase is a purely geometric contribution to quantum evolution that depends only on the path tra- versed in parameter space, not on the details of the tra- jectory. While complex dynamical phases arise straight- forwardly from non-Hermitian eigenvalues, the complex geometric phase represents a more subtle effect: a path- dependent modulat...

    work page

  11. [11]

    ACKNOWLEDGMENTS This research was supported by NSF Grant No. PHY- 2408932, the Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initia- tive (MURI) Award on Programmable systems with non- Hermitian quantum dynamics (Grant No. FA9550-21-1- 0202), ONR Grant No. N000142512160, and the Lux- embourg National Research Fund (...

    work page

  12. [12]

    Implementation of universal quantum gates based on nonadiabatic geometric phases

    Shi-Liang Zhu and ZD Wang. Implementation of uni- versal quantum gates based on nonadiabatic geometric phases. arXiv preprint quant-ph/0207037, 2002

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  13. [13]

    Experimental real- ization of non-Abelian non-adiabatic geometric gates

    Abdufarrukh A Abdumalikov Jr, Johannes M Fink, Kristinn Juliusson, Marek Pechal, Simon Berger, An- dreas Wallraff, and Stefan Filipp. Experimental real- ization of non-Abelian non-adiabatic geometric gates. Nature, 496(7446):482–485, 2013

    work page 2013

  14. [14]

    High-fidelity geometric quantum gates exceeding 99.9% in germanium quantum dots

    Yu-Chen Zhou, Rong-Long Ma, Zhenzhen Kong, Ao-Ran Li, Chengxian Zhang, Xin Zhang, Yang Liu, Hao-Tian Jiang, Zhi-Tao Wu, Gui-Lei Wang, et al. High-fidelity geometric quantum gates exceeding 99.9% in germanium quantum dots. Nature Communications, 16(1):7953, 2025

    work page 2025

  15. [15]

    Emergence of the geometric phase from quantum measurement back-action

    Young-Wook Cho, Yosep Kim, Yeon-Ho Choi, Yong-Su Kim, Sang-Wook Han, Sang-Yun Lee, Sung Moon, and Yoon-Ho Kim. Emergence of the geometric phase from quantum measurement back-action. Nature Physics, 15(7):665–670, 2019

    work page 2019

  16. [16]

    Observing a topologi- cal transition in weak-measurement-induced geometric phases

    Yunzhao Wang, Kyrylo Snizhko, Alessandro Romito, Yu- val Gefen, and Kater Murch. Observing a topologi- cal transition in weak-measurement-induced geometric phases. Physical Review Research, 4(2):023179, 2022

    work page 2022

  17. [17]

    Experimental observation of the quan- tum Hall effect and Berry’s phase in graphene

    Yuanbo Zhang, Yan-Wen Tan, Horst L Stormer, and Philip Kim. Experimental observation of the quan- tum Hall effect and Berry’s phase in graphene. nature, 438(7065):201–204, 2005

    work page 2005

  18. [18]

    Geometric phase magnetometry using a solid-state spin

    Keigo Arai, Junghyun Lee, Chinmay Belthangady, David R Glenn, Huiliang Zhang, and Ronald L Walsworth. Geometric phase magnetometry using a solid-state spin. Nature Communications, 9(1):4996, 2018

    work page 2018

  19. [19]

    Synthetic magnetic fields for ultracold neutral atoms.Nature, 462(7273):628– 632, 2009

    Y-J Lin, Rob L Compton, Karina Jim´ enez-Garc´ ıa, James V Porto, and Ian B Spielman. Synthetic magnetic fields for ultracold neutral atoms.Nature, 462(7273):628– 632, 2009

    work page 2009

  20. [20]

    Observation of PT phase transition in a sim- ple mechanical system

    Carl M Bender, Bjorn K Berntson, David Parker, and E Samuel. Observation of PT phase transition in a sim- ple mechanical system. American Journal of Physics, 81(3):173–179, 2013

    work page 2013

  21. [21]

    Observation of parity–time sym- metry in optics

    Christian E R¨ uter, Konstantinos G Makris, Ramy El-Ganainy, Demetrios N Christodoulides, Mordechai Segev, and Detlef Kip. Observation of parity–time sym- metry in optics. Nature physics, 6(3):192–195, 2010

    work page 2010

  22. [22]

    -symmetric electronics

    Joseph Schindler, Zin Lin, JM Lee, Hamidreza Ramezani, Fred M Ellis, and Tsampikos Kottos. -symmetric electronics. Journal of Physics A: Mathematical and Theoretical, 45(44):444029, 2012

    work page 2012

  23. [23]

    PT-symmetric acoustics

    Xuefeng Zhu, Hamidreza Ramezani, Chengzhi Shi, Jie Zhu, and Xiang Zhang. PT-symmetric acoustics. Physical Review X, 4(3):031042, 2014

    work page 2014

  24. [24]

    H. Xu, D. Mason, Luyao Jiang, and J. G. E. Harris. Topological energy transfer in an optomechanical system with exceptional points. Nature, 537(7618):80–83, July 2016

    work page 2016

  25. [25]

    Parity–time symmetry and exceptional points in photonics

    S ¸ahin Kaya ¨Ozdemir, Stefan Rotter, Franco Nori, and L Yang. Parity–time symmetry and exceptional points in photonics. Nature materials, 18(8):783–798, 2019

    work page 2019

  26. [26]

    Experimental determination of PT-symmetric exceptional points in a single trapped ion

    Liangyu Ding, Kaiye Shi, Qiuxin Zhang, Danna Shen, Xi- ang Zhang, and Wei Zhang. Experimental determination of PT-symmetric exceptional points in a single trapped ion. Physical Review Letters, 126(8):083604, 2021

    work page 2021

  27. [27]

    Naghiloo, M

    M. Naghiloo, M. Abbasi, Yogesh N. Joglekar, and K. W. Murch. Quantum state tomography across the excep- tional point in a single dissipative qubit. Nature Physics, 15(12):1232–1236, October 2019

    work page 2019

  28. [28]

    Measuring the adiabatic non-Hermitian Berry phase in feedback-coupled oscillators

    Yaashnaa Singhal, Enrico Martello, Shraddha Agrawal, Tomoki Ozawa, Hannah Price, and Bryce Gadway. Measuring the adiabatic non-Hermitian Berry phase in feedback-coupled oscillators. Physical Review Research, 5(3):L032026, 2023

    work page 2023

  29. [29]

    Complex Berry phase and steady-state geometric amplification in non-Hermitian systems

    JR Lane, C Guria, J H¨ oller, TD Montalvo, YSS Patil, and JGE Harris. Complex Berry phase and steady-state geometric amplification in non-Hermitian systems. arXiv preprint arXiv:2503.23197, 2025. 8

    work page arXiv 2025

  30. [30]

    Symmetry-protected quantization of com- plex Berry phases in non-Hermitian many-body systems

    Shoichi Tsubota, Hong Yang, Yutaka Akagi, and Hosho Katsura. Symmetry-protected quantization of com- plex Berry phases in non-Hermitian many-body systems. Physical Review B, 105(20):L201113, 2022

    work page 2022

  31. [31]

    Berry connection induced anomalous wave- packet dynamics in non-Hermitian systems

    Navot Silberstein, Jan Behrends, Moshe Goldstein, and Roni Ilan. Berry connection induced anomalous wave- packet dynamics in non-Hermitian systems. Physical Review B, 102(24):245147, 2020

    work page 2020

  32. [32]

    Joglekar, and Kater W

    Weijian Chen, Maryam Abbasi, Yogesh N. Joglekar, and Kater W. Murch. Quantum Jumps in the Non-Hermitian Dynamics of a Superconducting Qubit. Phys. Rev. Lett., 127:140504, Sep 2021

    work page 2021

  33. [33]

    Nonlinear quantum evolution of a dissipative su- perconducting qubit

    Orion Lee, Qian Cao, Yogesh N Joglekar, and Kater Murch. Nonlinear quantum evolution of a dissipative su- perconducting qubit. arXiv preprint arXiv:2510.25836, 2025

    work page arXiv 2025

  34. [34]

    Obser- vation of Berry’s phase in a solid-state qubit

    Peter J Leek, JM Fink, Alexandre Blais, R Bianchetti, M Goppl, Jay M Gambetta, David I Schuster, Luigi Frun- zio, Robert J Schoelkopf, and Andreas Wallraff. Obser- vation of Berry’s phase in a solid-state qubit. science, 318(5858):1889–1892, 2007

    work page 2007

  35. [35]

    Determining eigen- states and thermal states on a quantum computer us- ing quantum imaginary time evolution

    Mario Motta, Chong Sun, Adrian TK Tan, Matthew J O’Rourke, Erika Ye, Austin J Minnich, Fernando GSL Brandao, and Garnet Kin-Lic Chan. Determining eigen- states and thermal states on a quantum computer us- ing quantum imaginary time evolution. Nature Physics, 16(2):205–210, 2020

    work page 2020

  36. [36]

    Cooling with imaginary time

    Peter J Love. Cooling with imaginary time. Nature Physics, 16(2):130–131, 2020

    work page 2020

  37. [37]

    Nevertheless, in this case we still have a non-unitary con- tribution coming from the real part in the sense that i ⟨R±| ∂ϕ|R±⟩ ⟨R±|R±⟩ ̸= Re(i ⟨L±| ∂ϕ|R±⟩ ⟨L±|R±⟩ )

    Note that when ∆ = 0 and|J|<Γ/2, the imaginary part of the phase vanishes arbitrarily close to the EP. Nevertheless, in this case we still have a non-unitary con- tribution coming from the real part in the sense that i ⟨R±| ∂ϕ|R±⟩ ⟨R±|R±⟩ ̸= Re(i ⟨L±| ∂ϕ|R±⟩ ⟨L±|R±⟩ )

    work page

  38. [38]

    Serra Erdamar, Maryam Abbasi, Weijian Chen, Niklas H¨ ornedal, Aur´ elia Chenu, and Kater W. Murch. Explor- ing the Riemann-Surface Topology of a Non-Hermitian Superconducting Qubit Using Shortcuts to Adiabaticity. PRX Quantum, 7:010337, Feb 2026

    work page 2026

  39. [39]

    Geomet- ric Phase for Non-Hermitian Hamiltonians and Its Holon- omy Interpretation

    Hossein Mehri-Dehnavi and Ali Mostafazadeh. Geomet- ric Phase for Non-Hermitian Hamiltonians and Its Holon- omy Interpretation. Journal of Mathematical Physics, 49(8):082105, August 2008

    work page 2008

  40. [40]

    M. V. Berry. Quantal Phase Factors Accompanying Adi- abatic Changes. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 392(1802):45–57, 1984. Appendix A: Adiabaticity in non-Hermitian systems In a non-Hermitian system, adiabaticity can be evalu- ated in terms of a transition amplitudea nm from state mton. The criter...

    work page 1984