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Cited by 111Quantum Computing Algorithms and ArchitectureQuantum Information and CryptographyQuantum Mechanics and Applications

Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems

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Wang, Xiaoting, Allegra, Michele, Jacobs, Kurt, Lloyd, Seth, Lupo, Cosmo, Mohseni, Masoud·2015·Physical Review Letters
DOI 10.1103/physrevlett.114.170501
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Abstract

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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Cites

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Claims the paper makes

0 of 14 cited sources checked · 0 supported · 0 flagged

  • Findingown data

    For the CNOT example, the geodesic-search method is faster than q jumping.

    “The geodesic-search method is thus faster for this example as well.”
    Quote in the paperNot a substantial claim
  • Findingown data

    For the CNOT example, the minimum protocol time obtained is T=5.75/E.

    “The minimum protocol time we obtain is T ¼ 5.75=E.”
    Quote in the paperNot a substantial claim
  • Interpretationnone given

    The quantum brachistochrone problem can be reformulated as a search for geodesics within the unitary-operator space.

    “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.”
    Quote in the paperSubstantial claim
  • Interpretationreasoning

    Because there is a continuum of geodesics, the q=1 solution can be continuously transformed into the q=∞ solution.

    “(i) Since there is a continuum of geodesics, the solution for q ¼ 1 can be transformed continuously into the solution for q ¼ ∞.”
    Quote in the paperNot a substantial claim
  • Interpretationreasoning

    As q tends to infinity, the geodesic equation becomes the brachistochrone equation, provided that the quantities qβ_k^(q) remain finite in that limit.

    “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.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The quantities qβ_k^(q) remain finite as q tends to infinity.
  • May combine two claims. 23
  • Classified as probably not a substantial claim. 7
  • The quote only partly states this claim. 2

  • May combine two claims.

    Findingown data

    For the CNOT example, the geodesic-search method takes about 15 minutes: about 14 minutes for minimization, 16 seconds for shooting to obtain H100(t), and 29 seconds for shooting to obtain the brachistochrone.

    “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.]”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    For the arbitrary two-qubit example, q jumping with Δq=1 fails for m=1 and m=2, indicating those geodesics cannot be extended to q=100; m=3 succeeds and takes 32 minutes.

    “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.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given

    The shooting method can obtain H_q(0) for a low q, such as q=5, when seeded with a random guess.

    “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.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    For the arbitrary two-qubit example, the geodesic-search method beats q jumping, and both methods arrive at the same minimum-time protocol with T=6.69/E.

    “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.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingown data

    For the arbitrary two-qubit example, the norms of the H_1(0) solutions and the corresponding protocol times increase monotonically with the logarithm branch number m.

    “The norms of these solutions for H 1 ð0Þ, and thus the corresponding protocol times, increase monotonically with m.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Findingown data

    For the arbitrary two-qubit example, using the m=3 geodesic to seed shooting finds the brachistochrone in 19 seconds.

    “We then use this geodesic to seed the shooting method to find the brachistochrone, which takes 19 s.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For the CNOT example, q jumping from m=1 to H5 and then H100 with Δq=1 takes 21.5 minutes, and finding the brachistochrone from q=100 takes 28 seconds, for a total of 22 minutes.

    “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.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    For the CNOT example, q=50 is sufficient to obtain the brachistochrone, reducing the total time to 9 minutes.

    “In fact, for this problem, we find that q ¼ 50 is sufficient to obtain the brachistochrone, reducing the total time to 9 min.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    For the arbitrary two-qubit example, simple shooting with 100 random initial guesses fails to solve the boundary-value QBE, and multiple shooting and finite-difference methods behave similarly.

    “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.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingown data

    For the arbitrary two-qubit example, checking 10 values of m, including nonconvergent ones, takes about 30 minutes each, so the total time to obtain the minimal-time protocol is about 5 hours.

    “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.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    For the arbitrary two-qubit example, the geodesic-search method with 20 segments and 50 searches finds five approximate geodesic solutions with ∥H100∥q<7 in about 14 minutes, then finds five distinct H100(t) geodesics in about 40 seconds each, and finally obtains the brachistochrone in 19 seconds, for a total of about 18 minutes.

    “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.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Interpretationreasoning

    Under a bounded Hamiltonian norm and restriction of the Hamiltonian to an accessible subspace, the minimum-time control problem becomes a shortest-path geodesic problem on a manifold.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation

    The q geodesics satisfy the geodesic equation G_q(Hdot_q) = -i[H_q, G_q(H_q)], which is the Euler-Lagrange equation for length under the constraint ∥H∥_q = E.

    “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”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [41]

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    For q=1, the geodesic equation reduces to Hdot_1=0, so the minimal-time control Hamiltonian is constant.

    “(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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisereasoning

    Conventional numerical methods for this boundary-value problem convert it into nonlinear algebraic equations and degrade rapidly as the ODE dimension grows, failing even at moderate dimensions.

    “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.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [30] source

      Not checked in the free run.

    • cites [31]

      Not checked in the free run.

  • May combine two claims.

    Premisereasoning

    Conventional numerical solvers for the brachistochrone boundary-value problem fail without a sufficiently good initial guess, and even two-qubit systems cannot be solved from a random guess.

    “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].”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [30] source

      Not checked in the free run.

    • cites [31]

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductown data

    For the arbitrary two-qubit example, 50 searches were run to increase the chance of finding the global minimal-time solution.

    “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.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    The methods can also be used when an unalterable drift Hamiltonian H0 lies in A; the total Hamiltonian is H0+H(t) in A, and only the control Hamiltonian is bounded by E.

    “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.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    For q=1, V = exp(-i H_1 T) and H_1 = i log(V)/T.

    “Thus V ¼ expð-iH 1 TÞ and H 1 ¼ i logðVÞ=T.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    A good approximation to H_q(0) can be obtained by optimizing a weighted sum of protocol error d = 1 - ||Tr[V† U(T)]||/N and path length under the q norm; the optimum gives a close-to-minimal path that approximately generates V at time T.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    When a drift Hamiltonian lies in A, the brachistochrone solution can be derived analogously and a similar brachistochrone-geodesic connection can be established.

    “The brachistochrone solution can then be analogously derived for this case, and a similar brachistochrone-geodesic connection established.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationnone given

    Not all q=1 solutions continuously deform into q=∞ solutions; a geodesic can abruptly disappear above some q, so more than one logarithm branch may need to be tried.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The geodesic-search method cannot be applied directly to the brachistochrone because no good approximation to λ(0) is available; the geodesic formulation is essential.

    “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.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

    Interpretationreasoning

    The connection is expected to have broad applications within quantum control.

    “and as such we expect it to have broad applications within quantum control.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    The choice of T only scales the norm of H_1; setting T=1 is convenient, and the minimum-length protocol corresponds to the H_1 solution with the smallest norm, usually the first branch.

    “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).”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisenone given

    Geodesic deformation derives an equation for the derivative of H_q(0) with respect to q and integrates it from q=1; this procedure is slow.

    “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.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [36] source

      Not checked in the free run.

  • May combine two claims.

    Interpretationown data

    Although q jumping is slower than geodesic search, it has the advantage of enumerating solutions and thereby providing confidence that the global time-optimal solution has been found.

    “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.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S4 (based). May combine two claims.

    Interpretationnone given

    The minimal-length geodesic is expected to transform into the minimal-time brachistochrone, though this is not guaranteed.

    “We expect that it is the minimal-length geodesic that will transform into the minimal-time brachistochrone, although this is not guaranteed.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationnone given

    Any geodesic curve is uniquely determined by the initial value H_q(0), whereas the brachistochrone requires both H(0) and λ(0).

    “(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.”
    Quote in the paperNot a substantial claim
29 of 29

  • 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)”
2 of 2

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?

  • The Hilbert-Schmidt norm is widely used in experimental context, not only for quantum optimal control tasks, in order to impose finite energy bandwidth constraints on the control Hamiltonian.

    “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.

  • Supplemental Material is cited under the wrong reference number

    Minor

    The text points readers to reference 42 for the Supplemental Material, but the reference list identifies it as reference 41. Reference 42 is a textbook.

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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)
ORCID
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
DOI
10.1103/physrevlett.114.170501
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
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 › Computer Science › Artificial Intelligence › 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

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