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Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems
Open accessAbstract
Most methods of optimal control cannot obtain accurate time-optimal protocols. The quantum brachistochrone equation is an exception, and has the potential to provide accurate time-optimal protocols for a wide range of quantum control problems. So far, this potential has not been realized, however, due to the inadequacy of conventional numerical methods to solve it. Here we show that the quantum brachistochrone problem can be recast as that of finding geodesic paths in the space of unitary operators. We expect this brachistochrone-geodesic connection to have broad applications, as it opens up minimal-time control to the tools of geometry. As one such application, we use it to obtain a fast numerical method to solve the brachistochrone problem, and apply this method to two examples demonstrating its power.
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“The geodesic-search method is thus faster for this example as well.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“The minimum protocol time we obtain is T ¼ 5.75=E.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“Here we show that the quantum brachistochrone problem can be recast as that of finding geodesic paths in the space of unitary operators.”
- Also stated in: § Geodesic interpretation. · boosted
“We have revealed a fundamental connection between time-optimal control and geodesics in the space of unitary operators.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Abstract
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“(i) Since there is a continuum of geodesics, the solution for q ¼ 1 can be transformed continuously into the solution for q ¼ ∞.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“It is now straightforward to show that when q → ∞ the geodesic equation becomes the brachistochrone equation, under the assumption that the values qβ ðqÞ k remain finite in this limit.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“Thus the minimum-time curve must also be the minimum-distance curve connecting I and V.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > R
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“Byvirtue of the analysis above, constraint (i) can be replaced by the equality Tr½H 2 ðtÞ ¼ E 2 .”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > R
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“It has the potential to provide accurate time-optimal protocols under two physically relevant constraints: (i) the system has a finite energy bandwidth (the norm of the Hamiltonian is bounded); and (ii) the Hamiltonian is restricted to a subspace of Hermitian operators.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Abstract
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“the QBE has been solved only for special cases that possess analytic solutions [32][33][34].”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Abstract
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- Not checked yetcites [32][33][34]
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“Minimum-time quantum control protocols can therefore be obtained as the limit of a continuum of geodesics, which makes timeoptimal control amenable to the use of geometric tools.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“Discretizing the path into 20 segments as above, the geodesic-search method takes about 15 min [about 14 min for the minimization to find the approximation to H 100 ðtÞ, 16 s for the shooting method to obtain H 100 ðtÞ, and 29 s for the shooting method to obtain the brachistochrone.]”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“Starting with m ¼ 1, we attempt to use q jumping to obtain H 100 ðtÞ from H 1 using steps of size Δq ¼ 1.This fails for both m ¼ 1 and m ¼ 2, indicating that these geodesics cannot be extended to q ¼ 100.The solution for m ¼ 3 succeeds, and takes 32 min.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“Fortunately, the shooting method is able to obtain H q ð0Þ for a low value of q (e.g., q ¼ 5) when seeded with a random guess, and having obtained that we proceed as before.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“The geodesic-search method therefore takes a total time of about 18 min, and thus beats the q jumping method hands-down.Both methods arrive at the same minimum-time protocol, which has a time of T ¼ 6.69=E.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“The norms of these solutions for H 1 ð0Þ, and thus the corresponding protocol times, increase monotonically with m.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“We then use this geodesic to seed the shooting method to find the brachistochrone, which takes 19 s.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“Using the q jumping method for m ¼ 1 to obtain first H 5 ðtÞ and then H 100 ðtÞ, using a step size Δq ¼ 1, takes a total time of 21.5 min, and finding the brachistochrone from q ¼ 100 takes 28 s for a total time of 22 min.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“In fact, for this problem, we find that q ¼ 50 is sufficient to obtain the brachistochrone, reducing the total time to 9 min.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“Running the simple shooting method 100 times with randomly chosen initial guesses fails to obtain a solution, and the multipleshooting and finite-difference methods have similar behavior.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“We have checked 10 values of m, including those that do not converge, and each takes approximately 30 min, so the total time to obtain the minimal-time protocol is about 5 h.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“Using these approximate solutions to seed the shooting method, we find five distinct geodesics H 100 ðtÞ, each taking about 40 s.The final step is to obtain the brachistochrone starting from the shortest geodesic, just as we did in the method above, and so takes 19 s.The geodesic-search method therefore takes a total time of about 18 min, and thus beats the q jumping method hands-down.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“Here we show that, under constraints (i) and (ii) above, the minimum-time control problem can be transformed into that of finding a shortest path-a geodesic-on a manifold.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Abstract
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“The q geodesics obey the following geodesic equation, which is the Euler-Lagrange equation for L under the constraint ∥H∥ q ¼ E [41]: G q ð _ H q Þ ¼ -i½H q ; G q ðH q Þ;ð4Þ”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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- Not checked yetcites [41]
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“(ii) For q ¼ 1, the solution is trivial: the geodesic equation is merely _ H 1 ¼ 0, so the minimal-time control Hamiltonian is constant.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“As discussed above, all conventional numerical methods for this BVP convert it into that of solving a set of nonlinear equations.As a result, all these methods suffer a rapidly decreasing performance as the dimension of the ODE increases, failing even for moderately large dimensions.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > R
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- Not checked yetcites [30] source
- Not checked yetcites [31]
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“These search methods fail unless provided with a sufficiently good initial guess.Even for systems as small as two qubits, a random guess is insufficient, with the result that the QBE has been solved only for special cases that possess analytic solutions [32][33][34].”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Abstract
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- Not checked yetcites [30] source
- Not checked yetcites [31]
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“To increase the chance of finding the global minimal-time solution we run 50 searches, which altogether provides 5 different approximate geodesic solutions with ∥H 100 ∥ q < 7.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“Themethods we have described above can also be used when there is an additional drift Hamiltonian H 0 that cannot be altered by the controller, and which is contained in A (H 0 ¼ P j ν j A j ).The total Hamiltonian is H tot ðtÞ ¼ H 0 þ HðtÞ ∈ A, and it is the control Hamiltonian that is bounded: ∥HðtÞ∥ ≤ E.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“Thus V ¼ expð-iH 1 TÞ and H 1 ¼ i logðVÞ=T.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“This can be done by optimizing a weighted sum of two quantities.The first is a measure of the error of the protocol which we quantify using d ¼ 1 -∥Tr½V † UðTÞ∥=N, with N being the dimension of the system.The second quantity is the length of the path under the q norm.The optimal solution for this weighted sum gives a close-to-minimal path that approximately generates V at time T.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“The brachistochrone solution can then be analogously derived for this case, and a similar brachistochrone-geodesic connection established.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“The first is that not all solutions for q ¼ 1 continuously deform into solutions for q ¼ ∞; a given geodesic can abruptly disappear above some value of q.Thus, we may have to try more than one branch of the logarithm [more than one of the solutions H 1 ðtÞ] to find the brachistochrone.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“We cannot apply this method to the brachistochrone directly, because there is no way to obtain a good approximation to λð0Þ: the geodesic formulation is essential.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“and as such we expect it to have broad applications within quantum control.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“Further, the choice of T merely scales the norm of H 1 .Thus, it is convenient to set T ¼ 1, and the minimumlength protocol is given by the solution for H 1 with the smallest norm (usually the first branch).”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“This can be done by deriving an equation for the derivative of H q ð0Þ with respect to q, and then integrating this equation starting at q ¼ 1, a method known as “geodesic deformation” [36].This procedure is slow, however, and a much faster one, that we will refer to as “q jumping,” is as follows.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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- Not checked yetcites [36] source
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“Although q jumping is slower than the geodesic-search method, it has the advantage of enumerating a set of solutions, providing confidence that we have obtained the global time-optimal solution.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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“We expect that it is the minimal-length geodesic that will transform into the minimal-time brachistochrone, although this is not guaranteed.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
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In the paper
“(iii) Any geodesic curve will be uniquely determined by the initial value of H q ð0Þ, unlike the brachistochrone, which requires both Hð0Þ and λð0Þ.”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
It may state two separate claims as one.
The authors state this as an assumption.
The quantities qβ_k^(q) remain finite as q tends to infinity.
“under the assumption that the values qβ ðqÞ k remain finite in this limit.”
The authors state this as an assumption.
The experimenter can vary the six parameters {ω_m^(l)} and the interqubit coupling rate κ.
“We assume that the experimenter has the ability to vary the six parameters fω We choose the target unitary V to be a randomly selected two-qubit operator in SU(4)”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“The Hilbert-Schmidt norm is also widely used in experimental context, not only for quantum optimal control tasks, in order to impose finite energy bandwidth constraints on the control Hamiltonian [70,71].”
From Tight, robust, and feasible quantum speed limits for open dynamics (2019) · cites this paper as [70]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > Geodesic interpretation.
“(see the Supplemental Material [42])”
§ Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems > References
“41. See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.114.170501”
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Metadata
- Authors
- Wang, X., Allegra, M., Jacobs, K., Lloyd, S., Lupo, C., & Mohseni, M.
- Affiliations
- Louisiana State University (US)
- Institute for Scientific Interchange (IT)
- Massachusetts Institute of Technology (US)
- Google (United States) (US)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2015-04-28
- Volume
- 114
- Issue
- 17
- Pages
- 170501-170501
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2180439073
- Semantic Scholar
- 074654491d3acabd4488171b9d456ab7b5e6eb69
- Cited by
- 111
- References
- 51
- Citations by year
- 2026: 7 · 2025: 8 · 2024: 10 · 2023: 15 · 2022: 16 · 2021: 12 · 2020: 10 · 2019: 9 · 2018: 9 · 2017: 5 · 2016: 5 · 2015: 5
- FWCI
- 7.85
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- DSpace@MIT (Massachusetts Institute of Technology) (submittedVersion, cc-by-nc)
- Padua Research Archive (University of Padova) (submittedVersion, other-oa)
- arXiv (Cornell University)
- Funders
- National Science Foundation (PHY-1005571, PHY-1212413)
- Army Research Office (W911NF-11-1-0268)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- quantum brachistochrone, geodesics, time-optimal control, quantum control, unitary operators, Riemannian geometry
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-06, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-06 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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