pith. sign in

arxiv: 2606.06919 · v1 · pith:GFJPLB44new · submitted 2026-06-05 · 🪐 quant-ph · cond-mat.str-el· cond-mat.supr-con

Scalable Quantum Algorithms for Gutzwiller Projection

Byungmin Kang , Hyunwoong Kwon , Vito W. Scarola , Kwon Park This is my paper

Pith reviewed 2026-06-27 22:04 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-elcond-mat.supr-con
keywords Gutzwiller projectionamplitude amplificationBCS statesquantum simulationt-J modelinput-state preparationfault-tolerant resources
0
0 comments X

The pith

Amplitude amplification quadratically reduces the number of queries needed to prepare Gutzwiller-projected BCS states.

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

The paper constructs scalable quantum circuits for preparing Gutzwiller-projected BCS states by combining a general circuit for arbitrary BCS states with an amplitude-amplification procedure applied to the projection step. This yields a quadratic reduction in the number of times the projection operator must be queried relative to ordinary measurement-based postselection. The projected weight itself falls exponentially with system size for states optimized on the square-lattice t-J model, yet the quadratic improvement still produces orders-of-magnitude savings at sizes such as 100 sites. The resulting protocol therefore improves fault-tolerant resource estimates enough to make projected BCS states practical input states for quantum simulation of strongly correlated lattice models.

Core claim

Combining a circuit construction for arbitrary BCS states with the amplitude amplification for Gutzwiller projection (AAGP) procedure yields a quadratic reduction in the number of projection queries compared with measurement-based postselection and leads to substantially improved fault-tolerant resource scaling; for projected BCS states optimized for the square-lattice t-J model the projected-state weight decreases exponentially with system size, but the quadratic improvement remains large enough at physically relevant finite sizes that, for a 100-site benchmark, AAGP reduces the required number of projection queries by about seven orders of magnitude.

What carries the argument

Amplitude Amplification for Gutzwiller Projection (AAGP), which uses repeated coherent applications of the projection operator together with a reflection about the initial BCS state to amplify the amplitude of the desired projected component.

If this is right

  • Substantially improved fault-tolerant resource scaling for preparing accurate input states in quantum simulation of strongly correlated models.
  • A decisive practical reduction in query count at system sizes such as 100 sites.
  • Projected BCS states become usable input states for quantum simulation of the t-J model and related lattice Hamiltonians.
  • The quadratic query reduction holds whenever the projection operator can be implemented coherently inside the quantum circuit.

Where Pith is reading between the lines

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

  • The same amplification technique could be applied to other non-unitary projections that appear in variational wavefunctions for quantum many-body systems.
  • Resource estimates for full fault-tolerant algorithms would improve further if AAGP is combined with early fault-tolerant techniques that tolerate some measurement noise.
  • The method may generalize to preparing projected states on lattices with different geometries or with additional variational parameters.

Load-bearing premise

Even though the weight of the projected state decreases exponentially with system size, the quadratic improvement from AAGP is still large enough at physically relevant finite sizes to produce a decisive practical difference in total resource cost.

What would settle it

A direct numerical count, on a fault-tolerant simulator, of the total number of projection-operator applications required to reach a fixed target fidelity for a 100-site projected BCS state when using AAGP versus when using measurement-based postselection.

Figures

Figures reproduced from arXiv: 2606.06919 by Byungmin Kang, Hyunwoong Kwon, Kwon Park, Vito W. Scarola.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows the overall structure of the AAGP al￾gorithm. The essence of the amplitude amplification al￾gorithm (or quantum signal processing in general) can be understood in terms of Grover’s quantum search al￾gorithm, which provides a quadratic speedup over the measurement-based search. Then, the AAGP algorithm can generate the RVB state to within δ using only O(1/ √ W) applications of UBCS and U † BCS, togeth… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
read the original abstract

Quantum simulation requires highly accurate input states. Gutzwiller-projected Bardeen-Cooper-Schrieffer (BCS) states provide physically motivated input states for solving strongly correlated lattice models, but their preparation on a quantum computer is hindered by the non-trivial nature of the Gutzwiller projection. We construct scalable quantum algorithms for this task by combining a circuit construction for arbitrary BCS states with the amplitude amplification for Gutzwiller projection (AAGP) procedure. AAGP yields a quadratic reduction in the number of projection queries compared with measurement-based postselection and leads to substantially improved fault-tolerant resource scaling. For projected BCS states optimized for the square-lattice $t$-$J$ model, we find that the projected-state weight decreases exponentially with system size, but the quadratic improvement is still large enough at physically relevant finite sizes to make a decisive practical difference. In particular, for a 100-site benchmark, AAGP reduces the required number of projection queries by about seven orders of magnitude. These results establish AAGP as an enabling input-state preparation protocol for projected BCS states in quantum simulation.

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

0 major / 0 minor

Summary. The paper presents scalable quantum algorithms for Gutzwiller projection by combining a circuit for arbitrary BCS states with the amplitude amplification for Gutzwiller projection (AAGP) procedure. AAGP achieves a quadratic reduction in the number of projection queries relative to measurement-based postselection. For projected BCS states optimized for the square-lattice t-J model, the projected-state weight is shown to decrease exponentially with system size, yet the quadratic improvement results in a seven-order-of-magnitude reduction in required projection queries for a 100-site benchmark, improving fault-tolerant resource scaling.

Significance. If the claims hold, this work provides an enabling input-state preparation protocol for quantum simulation of strongly correlated lattice models, with substantially improved fault-tolerant scaling. The constructive combination of existing primitives (BCS preparation and amplitude amplification) without free parameters or circularity, together with explicit finite-size numerical benchmarks, is a strength that supports practical relevance at N=100.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept. We appreciate the recognition of the constructive combination of primitives and the practical relevance for N=100 benchmarks.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central claim rests on combining a standard circuit for arbitrary BCS states with amplitude amplification (AAGP) to implement Gutzwiller projection. The quadratic query reduction follows directly from the known scaling of amplitude amplification (O(1/sqrt(w)) versus O(1/w) for post-selection), which is an external quantum primitive and not derived from or fitted to the paper's own data. The reported seven-order reduction at N=100 follows from explicit computation of the projected-state weight w for optimized t-J states; this is a numerical evaluation of an independent quantity, not a parameter fit renamed as a prediction. No self-citation is load-bearing for the scaling argument, and the derivation chain contains no self-definitional steps or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on standard quantum-circuit primitives and the new AAGP construction. No numerical free parameters fitted to data are mentioned; the exponential weight decay is presented as an observed property of the states rather than a fitted constant.

axioms (1)
  • standard math Existence of efficient quantum circuits for arbitrary BCS states and standard amplitude amplification
    The algorithm is built by combining these two established quantum-computing building blocks.
invented entities (1)
  • AAGP procedure no independent evidence
    purpose: Quadratic speedup for Gutzwiller projection on quantum hardware
    New named technique introduced to achieve the reported query reduction.

pith-pipeline@v0.9.1-grok · 5734 in / 1334 out tokens · 30529 ms · 2026-06-27T22:04:30.851092+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

46 extracted references · 1 linked inside Pith

  1. [1]

    R. P. Feynman, Int. J. Theor. Phys.21, 467 (1982)

    1982

  2. [2]

    I. M. Georgescu, S. Ashhab, and F. Nori, Rev. Mod. Phys.86, 153 (2014)

    2014

  3. [3]

    A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Nature607, 667 (2022)

    2022

  4. [4]

    Y. Zhou, K. Kanoda, and T.-K. Ng, Rev. Mod. Phys.89, 025003 (2017)

    2017

  5. [5]

    Auerbach,Interacting Electrons and Quantum Mag- netism(Springer, New York, 1994)

    A. Auerbach,Interacting Electrons and Quantum Mag- netism(Springer, New York, 1994)

    1994

  6. [6]

    D. P. Arovas, E. Berg, S. A. Kivelson, and S. Raghu, Annu. Rev. Condens. Matter Phys.13, 239 (2022)

    2022

  7. [7]

    P. W. Anderson, Science235, 1196 (1987)

    1987

  8. [8]

    Bardeen, L

    J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957). 18

    1957

  9. [9]

    Ortiz, J

    G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Phys. Rev. A64, 022319 (2001)

    2001

  10. [10]

    Wecker, M

    D. Wecker, M. B. Hastings, N. Wiebe, B. K. Clark, C. Nayak, and M. Troyer, Phys. Rev. A92, 062318 (2015)

    2015

  11. [11]

    Peruzzo, J

    A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, Nat. Commun.5, 4213 (2014)

    2014

  12. [12]

    J. R. McClean, J. Romero, R. Babbush, and A. Aspuru- Guzik, New J. Phys.18, 023023 (2016)

    2016

  13. [13]

    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, Nat. Rev. Phys.3, 625 (2021)

    2021

  14. [14]

    A. Y. Kitaev, Quantum measurements and the abelian stabilizer problem (1995), arXiv:quant-ph/9511026

    Pith/arXiv arXiv 1995

  15. [15]

    D. S. Abrams and S. Lloyd, Phys. Rev. Lett.83, 5162 (1999)

    1999

  16. [16]

    Lloyd, Science273, 1073 (1996)

    S. Lloyd, Science273, 1073 (1996)

    1996

  17. [17]

    A. M. Childs and N. Wiebe, Quantum Inf. Comput.12, 901 (2012)

    2012

  18. [18]

    J. M. Martyn, Z. M. Rossi, A. K. Tan, and I. L. Chuang, PRX Quantum2, 040203 (2021)

    2021

  19. [19]

    M. C. Gutzwiller, Phys. Rev. Lett.10, 159 (1963)

    1963

  20. [20]

    M. C. Gutzwiller, Phys. Rev.134, A923 (1964)

    1964

  21. [21]

    M. C. Gutzwiller, Phys. Rev.137, A1726 (1965)

    1965

  22. [22]

    Park, Phys

    K. Park, Phys. Rev. Lett.95, 027001 (2005)

    2005

  23. [23]

    Park, Phys

    K. Park, Phys. Rev. B72, 245116 (2005)

    2005

  24. [24]

    Kwon and K

    H. Kwon and K. Park, Phys. Rev. Research4, 013116 (2022)

    2022

  25. [25]

    Murta and J

    B. Murta and J. Fern´ andez-Rossier, Phys. Rev. B103, L241113 (2021)

    2021

  26. [26]

    K. Seki, Y. Otsuka, and S. Yunoki, Phys. Rev. B105, 155119 (2022)

    2022

  27. [27]

    L. K. Grover, inProceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing (STOC ’96) (ACM, New York, 1996) p. 212

    1996

  28. [28]

    T. J. Yoder, G. H. Low, and I. L. Chuang, Phys. Rev. Lett.113, 210501 (2014)

    2014

  29. [29]

    Gily´ en, Y

    A. Gily´ en, Y. Su, G. H. Low, and N. Wiebe, inProceed- ings of the 51st Annual ACM SIGACT Symposium on Theory of Computing(ACM, New York, 2019) p. 193

    2019

  30. [30]

    Gros, Phys

    C. Gros, Phys. Rev. B38, 931 (1988)

    1988

  31. [31]

    Paramekanti, M

    A. Paramekanti, M. Randeria, and N. Trivedi, Phys. Rev. Lett.87, 217002 (2001)

    2001

  32. [32]

    Wen, Phys

    X.-G. Wen, Phys. Rev. B65, 165113 (2002)

    2002

  33. [33]

    Hermele, T

    M. Hermele, T. Senthil, M. P. A. Fisher, P. A. Lee, N. Na- gaosa, and X.-G. Wen, Phys. Rev. B70, 214437 (2004)

    2004

  34. [34]

    Y. Ran, M. Hermele, P. A. Lee, and X.-G. Wen, Phys. Rev. Lett.98, 117205 (2007)

    2007

  35. [35]

    Hermele, Y

    M. Hermele, Y. Ran, P. A. Lee, and X.-G. Wen, Phys. Rev. B77, 224413 (2008)

    2008

  36. [36]

    Jiang, K

    Z. Jiang, K. J. Sung, K. Kechedzhi, V. N. Smelyanskiy, and S. Boixo, Phys. Rev. Applied9, 044036 (2018)

    2018

  37. [37]

    Bloch and A

    C. Bloch and A. Messiah, Nucl. Phys.39, 95 (1962)

    1962

  38. [38]

    Ring and P

    P. Ring and P. Schuck,The Nuclear Many-Body Problem (Springer-Verlag, Berlin, 1980)

    1980

  39. [39]

    G. Li, S. Feng, and L. Li, Front. Comput. Sci.20, 2010906 (2026)

    2026

  40. [40]

    N. J. Ross and P. Selinger, Quantum Inf. Comput.16, 901 (2016)

    2016

  41. [41]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, 2011)

    2011

  42. [42]

    M. Amy, D. Maslov, M. Mosca, and M. Roetteler, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.32, 818 (2013)

    2013

  43. [43]

    Gosset, V

    D. Gosset, V. Kliuchinikov, M. Mosca, and V. Russo, Quantum Inf. Comput.14, 1261 (2014)

    2014

  44. [44]

    N. M. Myers, R. Scott, K. Park, and V. W. Scarola, Phys. Rev. Research5, 023199 (2023)

    2023

  45. [45]

    C. Gros, R. Joynt, and T. M. Rice, Phys. Rev. B36, 381 (1987)

    1987

  46. [46]

    A. H. MacDonald, S. M. Girvin, and D. Yoshioka, Phys. Rev. B37, 9753 (1988)

    1988