Pith. sign in

REVIEW 3 major objections 5 minor 55 references

Large time-step discretisation of adiabatic quantum dynamics

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read Discretizing adiabatic dynamics tolerates time steps independent of the tolerated error and evolution time, cutting the step count from O(1/ε²) to O(1/ε).

desk verdict Proven core: uniform O(1) step and O(1/eps) total steps via the discrete adiabatic walk viewpoint. The advertised exponential convergence rests on an explicitly flagged unproven conjecture, so read that part as strong evidence, not theorem. read the letter →

arxiv 2509.00171 v1 pith:VBUFRCPB submitted 2025-08-29 quant-ph cs.NAmath.NA

Dong An , Pedro C. S. Costa , Dominic W. Berry This is my paper · ORCID
classification quant-phcs.NAmath.NA
keywords adiabaticquantumcomputingdiscretetheoremtimediscretizationTrotterproductformulaeigenstatepreparationunstructuredsearchGroverspeedupboundarycancellation
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

Adiabatic quantum computing prepares a target eigenstate by slowly evolving a simple Hamiltonian into the problem Hamiltonian; running it on a digital device means chopping that evolution into discrete time steps. The paper's central claim: the standard prescription—shrink the step until the local truncation error is below the target precision ε—is far too pessimistic. Seen through the discrete adiabatic theorem, each numerical propagator is itself a slowly varying unitary walk operator with its own spectral gap, so a uniform step h = O(min{1/α, Δ_*^{1/p} C_p^{-1/p}}) keeps the discretization errors subdominant for any ε and T, where α is the Hamiltonian norm scale, Δ_* the minimum gap, and C_p the scale of nested commutators of order p. The total step count then drops from O(α³Δ_*⁻³ε⁻²) to O(α³Δ_*⁻³ε⁻¹). Under the boundary cancellation condition, the paper offers evidence—explicitly conditional on an unproven conjecture whose only literature proof is missing a step—that even first-order methods with O(1) steps converge faster than any polynomial in 1/T. For unstructured search, the resulting Trotterized adiabatic algorithm matches the Grover lower bound without knowing the number of marked states, and ties the QAOA performance up to a constant factor.

What carries the argument

The discrete adiabatic theorem: the first-order version (Lemma 2) with explicit gap dependence, plus the hypothetical higher-order version (Conjecture 10). The paper's move is to identify the numerical integrator with the walk operator W(s) of a discrete adiabatic evolution—W(s) = e^{−ihH(s)} for the exponential integrator, the (simplified) product formula of order p otherwise. Slow variation is quantified by finite-difference bounds c₁(s), c₂(s) = O(hα + h²α²); the Hamiltonian's spectral gap is transferred to the walk operator through eigenvalue-perturbation and Trotter-error estimates (Lemmas 7 and 27), which pins the admissible step size to norms and nested commutators rather than to ε an

What would settle it

Reproduce the paper's boundary-leakage experiment (Appendix F.2, Fig. 5) at higher dimension—say 8 to 10 walk-operator levels with the same glue-function scheduling—and check whether ∥Q₀Ω(1)P₀∥ decays faster than every polynomial in 1/T; if it stalls at O(1/T), Conjecture 10 fails and the exponential-convergence theorems 11, 12, and 16 lose their basis. Separately, to test the proved core, run the first-order exponential integrator at h = 1/(‖H₀‖+‖H₁‖) on a gapped interpolation and verify the final state-preparation error falls as O(1/T) as T grows; an error that grows with T would refute the

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that time discretization of adiabatic dynamics need not be analyzed as a perturbation of the continuous Schrödinger evolution. Each local numerical propagator—e^{−ihH(s)} for the first-order exponential integrator, the simplified order-p product formula otherwise—is itself a walk operator in a discrete adiabatic evolution, changing slowly because the Hamiltonian changes on the rescaled time s = t/T. The discrete adiabatic theorem then bounds the leakage of the sequence of walk operators directly, and the walk operator inherits a spectral gap from H(s) once h is below a threshold set by Hamiltonian norms, nested commutators, and the minimum gap—never

Load-bearing premise

The exponential-convergence claims rest on Conjecture 10, a high-order discrete adiabatic theorem whose only published proof, the paper reports, has a missing step that 'seemingly cannot be fixed in a simple way' and which is backed only by small 4-level numerical tests; without it, the rigorous results are linear in 1/T with an O(1/ε) step count.

Editorial extensions

If this is right

  • A first-order exponential integrator can run at h = 1/(‖H₀‖+‖H₁‖): total steps to reach error ε are O(α³Δ_*⁻³ε⁻¹), a factor 1/ε better than the standard O(α³Δ_*⁻³ε⁻²) estimate (Corollary 4).
  • Simplified product formulae of any order p reach the same O(1/ε) step count with step size Θ(min{α⁻¹, Δ_*^{1/p} eα_p^{−1/p}}); higher order buys a larger gap-matching step rather than better ε-dependence (Corollaries 8, 9).
  • If Conjecture 10 holds, boundary-cancelled schedules make even first-order Trotter and exponential-integration steps O(1)-sized with error O(1/T^k) for any k, so T = Td = O(1/ε^{1/k})—super-polynomial precision at first-order cost (Theorems 11, 12).
  • For unstructured search with an unknown number M of marked states, step-1 Trotterized AQC with a gap-adapted schedule achieves Õ(√(N/M)) steps in dimension, matching the Grover lower bound, and the glue-function schedule adds super-polynomial precision convergence conditional on Conjecture 10 (Theorems 15, 16).
  • Because only the walk operator's gap matters, Trotterized evolution can remain gapped where the continuous Hamiltonian is gapless, so discretized AQC can solve state-preparation problems continuous AQC provably cannot (Section V).

Reading between the lines

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

  • If Conjecture 10 is proved with a constant C_k growing only polynomially in k, the conditional O(1/ε^{1/k}) bounds would become a rigorous exponential-in-precision speedup for first-order circuits, not merely a linear one—the paper notes the missing gap-dependence and k-dependence in C_k as open problems.
  • The gap-matching analysis suggests a design heuristic the paper does not spell out: choose the step size to maximize the walk operator's gap rather than to minimize local truncation error; systematic scans over random gapped/gapless Hamiltonian pairs could map when Trotterization rescues a continuous problem that fails adiabatically.
  • The gapless-example observation opens a testable route to 'discrete-only' adiabatic algorithms: an eigenstate preparation blocked in the continuous formulation by a level crossing might be reachable by a large-step Trotter path whose walk-operator spectrum is gapped, with the paper's Section V toy model as the template.
  • The QAOA-angle construction (βⱼ = 1 − f(j/T), γⱼ = f(j/T)) yields a concrete hypothesis: on Grover-type instances, any QAOA schedule is matched up to constants by the discrete-adiabatic schedule whenever the schedule satisfies the gap-adaptation condition.
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 proposes a discrete-adiabatic viewpoint for analyzing time-discretized adiabatic quantum evolution. Rather than combining a continuous adiabatic theorem with Trotter error bounds, the authors treat the local numerical propagators (exponential integrator or simplified product formula) as slowly varying unitary walk operators and apply the discrete adiabatic theorem of Costa et al. [22]. This yields uniform time-step thresholds independent of ε and T: Corollary 4 gives h=1/(∥H0∥+∥H1∥) and Td=O(α^3 Δ_*^{-3} ε^{-1}) for the first-order exponential integrator; Corollaries 8 and 9 give analogous thresholds for simplified product formulae, with h=O(min{α^{-1}, Δ_*^{1/p} eα_p^{-1/p}}). Under boundary cancellation, Theorems 11 and 12 claim super-polynomial convergence O(1/T^k) for any k even for first-order methods, conditional on Conjecture 10 (the high-order discrete adiabatic theorem of Dranov et al.), whose literature proof is incomplete. The paper applies these results to adiabatic Grover search: Theorem 15 gives O(√(N/M) log N / ε) steps for p=1 schedule, Theorem 16 gives eO(√(N/M)) plus super-polynomial precision under Conjecture 10, and links the resulting angles to QAOA.

Significance. The unconditional core is a genuine improvement: for gapped bounded Hamiltonians, the step size and hence total step count O(α^3 Δ_*^{-3} ε^{-1}) replace standard O(1/ε^2) first-order bounds, and the derivation is self-contained and parameter-free, with explicit constants depending only on norms, commutators, and gaps. The discrete-path interpretation also yields the interesting observation that Trotterization with O(1) step can have a gap when the Hamiltonian is gapless. The Grover application with unknown M and no prior estimate is a nice feature. The paper is unusually transparent: it explicitly labels Conjecture 10, documents the missing step in [21] in Appendix F.2, and provides numerical evidence. However, the advertised exponential/super-polynomial convergence is not a theorem: it rests entirely on Conjecture 10, and the numerical evidence is limited to one 4-level exponential-integrator example. If the conjecture is false or C_k has hidden exponential gap dependence, those claims collapse. The proven O(1/ε) complexity remains valuable regardless.

major comments (3)
  1. [Section IV.A, Conjecture 10; Theorems 11, 12, 16; Corollary 13] The super-polynomial claims are conditional on an unproved conjecture, and Appendix F.2 shows the only proof in the literature has an invalid induction step (Eq. 40 in [21] is stronger than the claim and is numerically false in Fig. 5a). The numerical validation of Conjecture 10 is for W(s)=e^{-iH(s)} on a 4-level system, not for a Trotter or product-formula walk operator. Since these theorems are the source of the "exponential convergence" headline, they should be presented as conditional results and the abstract/conclusions should not imply a proof.
  2. [Appendix G.2, Eq. (G8)] Lemma 7 is stated only for h≤1/(∥H0∥+∥H1∥). In the Grover search application, ∥H0∥=∥H1∥=1, so h=1 is outside the lemma's hypothesis. The bound Δ_W ≥ (2/3)Δ_H in Lemma 14 is load-bearing for Theorems 15 and 16. It can likely be repaired because ∥H(s)∥≤1, so e^{-iH(s)} has no phase wrap, but that extension is not supplied; as written the proof of Lemma 14 is incomplete.
  3. [Section II.B.2 and Theorems 11/12] The text claims the boundary-cancellation case gives total steps O(1/ε^{o(1)}). The theorem statements only give T=O(1/ε^{1/k}) for each fixed k, with constants C_k depending on k in Eq. (59). Without control of C_k^{1/k}, this does not imply O(1/ε^{o(1)}), a point the authors acknowledge later but should be reflected in the theorem statements and summary. Please either prove a sub-polynomial C_k dependence or state the weaker super-polynomial-in-T result.
minor comments (5)
  1. [Lemma 7 proof, page 12] Typo: "according to to Lemma 6" should be "according to Lemma 6".
  2. [Table II] The reported gap of H(s) at ε=10^{-2} (3.0×10^{-2}) breaks monotonicity with neighboring rows and is likely a typo (3.0×10^{-3}?).
  3. [Section II.B.2] The notation O(1/ϵo(1)) is nonstandard; use O(ε^{-o(1)}) or write "super-polynomial" explicitly.
  4. [Appendix F.2] The numerical test for Ω_1 uses only four levels and one fixed schedule; a brief statement of what would need to be checked for Trotter walk operators would help.
  5. [Theorems 3 and 5] The phrases "actual and ideal evolution" refer to U and UA from Lemma 2; consider defining these in the theorem statements for self-containment.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the complexity claims are parameter-free consequences of discrete adiabatic theorems; the exponential-convergence claims are explicitly conditional on an unproven conjecture, which is a correctness caveat, not a circularity.

full rationale

The paper's derivation chain does not reduce to its own inputs by construction. The central results (Corollaries 4, 8, 9) follow from the discrete adiabatic theorem (Lemma 2, cited from the authors' prior work [22]) by computing explicit, parameter-free bounds on the finite-difference coefficients c1(s), c2(s) and by choosing step sizes h and evolution times T that depend only on norms, commutators, and gaps — never on the target error or on fitted data. The Grover schedules are explicit formulas (Eqs. (76) and (82)), and the error analyses in Theorems 15 and 16 are analytic. The paper transparently flags the one major unproven premise: the high-order discrete adiabatic theorem with boundary cancellation is stated only as Conjecture 10, and Appendix F.2 documents that the proof in [21] has a missing step ('the induction assumption ... is much stronger than the original claim', 'seemingly cannot be fixed in a simple way'). Theorems 11, 12, and 16 are explicitly conditional on Conjecture 10. This is a correctness limitation, not circularity: the conjecture is not secretly assumed as the conclusion, and the paper does not present the conditional statements as unconditional. Likewise, the use of [22] is a citation to a published theorem with stated assumptions that do not include the present target results; it is load-bearing but independently stated, so it does not constitute circular self-citation under the given rules. There are correctness risks (e.g., Lemma 14 invokes Lemma 7 with h=1 outside Lemma 7's stated h <= 1/(||H0||+||H1||) regime, and Conjecture 10 is unproved), but none of these amount to a derivation that is equivalent to its inputs by definition.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities, forces, or dimensions; the 'discrete adiabatic path' is an analysis viewpoint, not an entity. The gapless-Hamiltonian observation concerns the existing Trotter operator's gap, not a new degree of freedom. The only nontrivial input is the unproven Conjecture 10, which is explicitly flagged. No free parameters are fitted to data: all constants are universal and all problem-dependent quantities (gaps, norms, commutators) are inputs from the problem instance.

assumptions (5)
  • ad hoc to paper Conjecture 10: high-order discrete adiabatic theorem with boundary cancellation
    The paper's exponential-convergence theorems (11, 12, 16, Corollary 13) all assume this conjecture. Appendix F 2 shows the proof in [21] has a missing step that the authors cannot fix, and only numerical evidence is provided. This is the central unproven input.
  • domain assumption Discrete adiabatic theorem with explicit gap dependence (Lemma 2 from [22])
    The general-case results (Corollaries 4, 8, 9) rely on this published theorem from Costa, An, Sanders, Su, Babbush, Berry, PRX Quantum 2022. It is a prior result, not machine-checked, and the present paper builds on it as a black box.
  • domain assumption Eigenvalue perturbation bounds relating gaps of Trotter walk operators to gaps of H(s) (Lemmas 7 and 27)
    Used to transfer the Hamiltonian gap condition to the product-formula walk operator for h = O(1/alpha). The derivation is given in the appendix but relies on known eigenvalue perturbation results [35, 36].
  • domain assumption Smoothness and boundary cancellation of the scheduling function f(s)
    The main theorems assume |f'| and |f''| uniformly bounded, and the exponential results assume f^{(k)}(0)=f^{(k)}(1)=0 for all k. Stated in Section II B and used throughout.
  • domain assumption Spectral gap and norm bounds on H0 and H1
    Complexity estimates assume H(s) has minimum gap Delta_* and alpha = ||H0||+||H1||; Section IV additionally bounds norms by 1 and asserts a constant-level gap. These are domain assumptions about the problem instance.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Large time-step discretisation of adiabatic quantum dynamics." pith.science (2026). https://pith.science/paper/VBUFRCPB

@misc{pith2026250900171,
  author       = {Pith},
  title        = {Pith review of: Large time-step discretisation of adiabatic quantum dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBUFRCPB}},
  note         = {Machine review of arXiv:2509.00171}
}
read the original abstract

Adiabatic quantum computing is a general framework for preparing eigenstates of Hamiltonians on quantum devices. However, its digital implementation requires an efficient Hamiltonian simulation subroutine, which may introduce extra computational overhead or complicated quantum control logic. In this work, we show that the time step sizes in time discretization can be much larger than expected, and the overall complexity is greatly reduced. Remarkably, regardless of the general convergence order of the numerical method, we can choose a uniform time step size independent of tolerated error and evolution time for sufficiently accurate simulation. Furthermore, with the boundary cancellation condition where the continuous diabatic errors are exponentially suppressed, we provide strong evidence on an exponential convergence of even first-order Trotter with uniform time step size. We apply our analysis to the example of adiabatic unstructured search and show several preferable features of the Trotterized adiabatic approach: it can match the Grover lower bound, it does not require a priori knowledge on the number of marked states, and its performance can be asymptotically comparable with that of the quantum approximate optimization algorithm.

Figures

Figures reproduced from arXiv: 2509.00171 by the authors.

Figure 1
Figure 1. FIG. 1. Left: spectrum of [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. shows the spectrums of H(s) and W(s), and confirms that H(s) is gapless and W(s) is gapped (though the gap is indeed very small). We choose time T = 10k for k = 3, 4, 5, 6, 7, and implement first-order Trotter method with time step sizes h = 2−k for k = 0, 1, 2, 3, 4, 5. In [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Fidelity with the ground state (left) and the first excited state (right) of the numerical solutions from first-order Trotter [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The scheduling function in robust Grover search with boundary cancellation. [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Numerical tests of the high-order discrete adiabatic theorem and its intermediate step. Left: numerical scaling of [PITH_FULL_IMAGE:figures/full_fig_p037_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

55 extracted references · 46 canonical work pages

  1. [22]

    L. K. Kovalsky, F. A. Calderon-Vargas, M. D. Grace, A. B. Magann, J. B. Larsen, A. D. Baczewski, and M. Sarovar, Physical Review Letters 131, 060602 (2023)

    work page 2023

  2. [21]

    Yi, Physical Review A 104, 052603 (2021)

    C. Yi, Physical Review A 104, 052603 (2021)

    work page 2021

  3. [1]

    the overall error between the actual and the ideal evolution can be bounded by O (∥H0∥ + ∥H1∥)2 T ∆3∗ . (55)

    work page

  4. [2]

    in order to bound the overall error by ϵ, it suffices to choose T = O (∥H0∥ + ∥H1∥)2 ∆3∗ϵ , (56) and h = Θ  min    1 ∥H0∥ + ∥H1∥ , ∆1/p ∗ P γ0,··· ,γp∈{0,1} ∥[Hγp , · · ·, [Hγ1 , Hγ0 ]]∥ 1/p      , (57) and the overall number of steps becomes Td = O   (∥H0∥ + ∥H1∥)3 ∆3∗ϵ max    1, P γ0,··· ,γp∈{0,1} ∥[Hγp , · · ·, [Hγ1 , Hγ0 ]]∥ 1/...

    work page

  5. [3]

    For k ≥ 1, there exist real-valued functions ck(s) such that ∥D(k)W (s)∥ ≤ck(s)/T k for 0 ≤ s ≤ 1 − k/T , where D(k)W (s) represents the k-th order finite difference of W (s) with step size 1 /T

    work page

  6. [4]

    The eigenvalues of W (s) can be separated into two groups σP (s) and σQ(s) such that each continuous eigenpath λ(s) of W (s) consistently belongs to one of the two groups and the distance betweenσP (s) and σQ(s) is bounded from below by ∆( s) > 0

    work page

  7. [5]

    Under these conditions, Ref

    The input state |ψ⟩ is within the eigenspace corresponding to σP (0). Under these conditions, Ref. [21] claims an upper bound of the discrete diabatic error, achieving O(1/T k) for any positive integer k. Unfortunately, after a careful examination, we find a missing step in the proof provided in Ref. [21], so the claimed high-order discrete adiabatic theo...

    work page

  8. [6]

    (76) with p = 1

    Suppose that T ≥ O(log(N )) and the scheduling function is defined in Eq. (76) with p = 1. Then the overall error is bounded by O log(N ) T r N M ! . (78) 22 FIG. 4. The scheduling function in robust Grover search with boundary cancellation

    work page

Show all 55 references
  1. [7]

    (76) with 1 < p <2

    Suppose that T ≥ O(N p−1 2 ) and the scheduling function is defined in Eq. (76) with 1 < p <2. Then the overall error is bounded by O 1 T √ N M 1− p 2 + N p−1 T 2 max 1, N 3/2−p M 3/2−p ! . (79) Theorem 15 implies that, to achieve an ϵ error, the smallest number of Trotter ste...

  2. [8]

    Albash and D

    T. Albash and D. A. Lidar, Rev. Mod. Phys. 90, 015002 (2018)

    2018

  3. [9]

    Hochbruck and A

    M. Hochbruck and A. Ostermann, Acta Numerica 19, 209–286 (2010)

    2010

  4. [10]

    B. F. Schiffer, J. Tura, and J. I. Cirac, PRX Quantum 3, 020347 (2022)

    2022

  5. [11]

    Farhi, J

    E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, arXiv:quant-ph/0001106 (2000)

    2000 arXiv

  6. [12]

    van Dam, M

    W. van Dam, M. Mosca, and U. Vazirani, in Proceedings 42nd IEEE Symposium on Foundations of Computer Science (2001) pp. 279–287

    2001

  7. [13]

    A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Physical Review X 11, 011020 (2021)

    2021

  8. [14]

    D. An, D. Fang, and L. Lin, Quantum 5, 459 (2021)

    2021

  9. [15]

    Suzuki, Proceedings of the Japan Academy, Series B 69, 161–166 (1993)

    M. Suzuki, Proceedings of the Japan Academy, Series B 69, 161–166 (1993)

    1993

  10. [16]

    Kieferov´ a, A

    M. Kieferov´ a, A. Scherer, and D. W. Berry, Physical Review A 99, 042314 (2019)

    2019

  11. [17]

    G. H. Low and N. Wiebe, arXiv:1805.00675 (2018)

    2018 arXiv

  12. [18]

    Y. Su, D. W. Berry, N. Wiebe, N. Rubin, and R. Babbush, PRX Quantum 2, 040332 (2021)

    2021

  13. [19]

    Rajput, A

    A. Rajput, A. Roggero, and N. Wiebe, Quantum 6, 780 (2022)

    2022

  14. [20]

    Farhi, J

    E. Farhi, J. Goldstone, and S. Gutmann, arXiv:1411.4028 (2014)

    2014 arXiv

  15. [23]

    Wiebe, D

    N. Wiebe, D. Berry, P. Høyer, and B. C. Sanders, Journal of Physics A: Mathematical and Theoretical 43, 065203 (2010)

    2010

  16. [24]

    Yoshida, Physics Letters A 150, 262 (1990)

    H. Yoshida, Physics Letters A 150, 262 (1990)

    1990

  17. [25]

    M. E. S. Morales, P. C. S. Costa, G. Pantaleoni, D. K. Burgarth, Y. R. Sanders, and D. W. Berry, Quantum Information & Computation 25, 1 (2025)

    2025

  18. [26]

    Blanes, F

    S. Blanes, F. Casas, and A. Murua, Acta Numerica 33, 1–161 (2024)

    2024

  19. [27]

    Jansen, M.-B

    S. Jansen, M.-B. Ruskai, and R. Seiler, Journal of Mathematical Physics 48, 102111 (2007)

    2007

  20. [28]

    Dranov, J

    A. Dranov, J. Kellendonk, and R. Seiler, Journal of Mathematical Physics 39, 1340 (1998)

    1998

  21. [29]

    P. C. Costa, D. An, Y. R. Sanders, Y. Su, R. Babbush, and D. W. Berry, PRX Quantum 3, 040303 (2022)

    2022

  22. [30]

    Nenciu, Communications in Mathematical Physics 152, 479–496 (1993)

    G. Nenciu, Communications in Mathematical Physics 152, 479–496 (1993)

    1993

  23. [31]

    Y. Ge, A. Moln´ ar, and J. I. Cirac, Physical Review Letters 116, 080503 (2016)

    2016

  24. [32]

    J. E. Avron and A. Elgart, Communications in Mathematical Physics 203, 445 (1999)

    1999

  25. [33]

    Roland and N

    J. Roland and N. J. Cerf, Physical Review A 65, 042308 (2002)

    2002

  26. [34]

    An and L

    D. An and L. Lin, ACM Transactions on Quantum Computing 3, 5 (2022)

    2022

  27. [35]

    A. M. Dalzell, T. J. Yoder, and I. L. Chuang, Physical Review A 95, 012311 (2017)

    2017

  28. [36]

    T. J. Yoder, G. H. Low, and I. L. Chuang, Physical Review Letters 113, 210501 (2014). 25

    2014

  29. [37]

    L. K. Grover, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing , STOC ’96 (Associ- ation for Computing Machinery, New York, NY, USA, 1996) pp. 212–219

    1996

  30. [38]

    Jiang, E

    Z. Jiang, E. G. Rieffel, and Z. Wang, Physical Review A 95, 062317 (2017)

    2017

  31. [39]

    Gily´ en, Y

    A. Gily´ en, Y. Su, G. H. Low, and N. Wiebe, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (2019) pp. 193–204

    2019

  32. [40]

    G. H. Low and I. L. Chuang, Quantum 3, 163 (2019)

    2019

  33. [41]

    Quantum singular value transformation and beyond: exponential improve- ments for quantum matrix arithmetics,

    A. Gily´ en, Y. Su, G. H. Low, and N. Wiebe, “Quantum singular value transformation and beyond: exponential improve- ments for quantum matrix arithmetics,” (2019), arXiv:1806.01838

    2019 arXiv

  34. [42]

    Bhatia and C

    R. Bhatia and C. Davis, Linear and Multilinear Algebra 15, 71 (1984)

    1984

  35. [43]

    Elsner and C

    L. Elsner and C. He, Linear Algebra and its Applications 188-189, 207 (1993)

    1993

  36. [44]

    An, UC Berkeley Electronic Theses and Dissertations (2021)

    D. An, UC Berkeley Electronic Theses and Dissertations (2021)

    2021

  37. [45]

    Milne-Thomson, The Calculus of Finite Differences (Macmillan and Co, 1933)

    L. Milne-Thomson, The Calculus of Finite Differences (Macmillan and Co, 1933). 26 Appendix A: Continuous adiabatic theorems Here we briefly review the previous results about continuous adiabatic theorems, which aim to bound the distance between the dynamics in Eq. (1) and the ...

    1933

  38. [46]

    Lemma 25

    Standard time discretization error bounds For the first-order exponential integrator, its standard error bound is given as follows. Lemma 25. Let Uexp(t + h, t) be the first-order exponential integrator defined in Eq. (5). Suppose that |f ′(s)| is uniformly bounded over [0, 1]...

  39. [47]

    Standard complexity estimates Now we show the standard approach to analyze the complexity of using discretized AQC to prepare an eigenstate. Specifically, let |ϕ⟩ denote the exact target eigenstate and | eϕ⟩ = Td−1Y j=0 Unum((j + 1)h, jh)|ψ(0)⟩ (D13) where Unum is a p-th order...

  40. [48]

    We need to introduce some operators before proceeding

    A slight variant We show how to derive Conjecture 10 from the original result in [21]. We need to introduce some operators before proceeding. Let P (s) and Q(s) denote the spectral projections onto σP (s) and σQ(s), respectively. Define S(s) = P (s + 1/Td)P (s) + Q(s + 1/Td)Q(...

  41. [49]

    Let P (s) and Q(s) be the spectral projections as before, and V (s) is defined through Eq

    Missing steps in the proof of the high-order discrete adiabatic theorem and a numerical validation We first follow the notations in [21] to define several operators. Let P (s) and Q(s) be the spectral projections as before, and V (s) is defined through Eq. (F3). Let Ω(s) = U †...

  42. [50]

    First we diagonalise H0 using Eq

    Direct computations Here we determine the spectral gap of the walk operator W (s) for a step of Trotter as applied to a search problem in Section VI. First we diagonalise H0 using Eq. (71) as H0 = p M/N p (N − M )/Np (N − M )/N − p M/N ! 0 0 0 1 ! p M/N p (N − M )/Np (N − M )/...

  43. [51]

    Proof of Lemma 14 This Lemma lower bounds the gap of the walk operator W (s) for a step of Trotter in terms of the spectral gap of the Hamiltonian. To bound this spectral gap, we may use Lemma 7 with h = 1 to give ∆W (s; N, M, f) ≥ ∆H (s; N, M, f) − π 144 √ 3 (2∥[H1, [H1, H0]]...

  44. [52]

    The error bound Eq

    p = 1 We first consider the limit case when p = 1. The error bound Eq. (H15) becomes Error ≤ O   dN,1 T + d2 N,1 T T −1X j=0 1 T ∆(j/T ; 1) ∆(j/T ; M )2   . (H17) For dN,1, using Eq. (H16) we have dN,1 = r N N − 1 Z arctan √N −1 0 1 cos θ dθ = r N N − 1 log 1 + sinθ cos θ ...

  45. [53]

    Continuing with Eq

    1 < p <2 Now we consider 1 < p <2. Continuing with Eq. (H16) and using integration by parts, we may obtain Z arctan √N −1 0 cosp−2 θ dθ= − 1 p − 1 cosp−1 θ sin θ arctan √N −1 0 + p p − 1 Z arctan √N −1 0 cosp θ dθ. (H28) 43 Both terms on the right hand side of Eq. (H28) are bo...

  46. [54]

    f (0) = 0, f (1) = 1, and f (k)(0) = f (k)(1) = 0 for all k ≥ 1,

  47. [55]

    To prove Theorem 16, we will frequently use the following lemma

    for all s ∈ [0, 1/2] and k ≥ 1, f (s) + f (1 − s) = 1 and f (k)(s) = f (k)(1 − s). To prove Theorem 16, we will frequently use the following lemma. Lemma 30. Let ∆H (s) be the gap defined in Eq. (74). Then for any s ∈ [0, 1/2] and any integer l ≥ 0, we have exp − 1 2s(1−2s) ∆H...

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.