Superadiabatic Controlled Evolutions and Universal Quantum Computation
Open accessAbstract
Adiabatic state engineering is a powerful technique in quantum information and quantum control. However, its performance is limited by the adiabatic theorem of quantum mechanics. In this scenario, shortcuts to adiabaticity, such as provided by the superadiabatic theory, constitute a valuable tool to speed up the adiabatic quantum behavior. Here, we propose a superadiabatic route to implement universal quantum computation. Our method is based on the realization of piecewise controlled superadiabatic evolutions. Remarkably, they can be obtained by simple time-independent counter-diabatic Hamiltonians. In particular, we discuss the implementation of fast rotation gates and arbitrary n-qubit controlled gates, which can be used to design different sets of universal quantum gates. Concerning the energy cost of the superadiabatic implementation, we show that it is dictated by the quantum speed limit, providing an upper bound for the corresponding adiabatic counterparts.
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TL;DR
The paper proposes a superadiabatic shortcut to controlled adiabatic evolutions for universal quantum computation, using simple time-independent counter-diabatic Hamiltonians. It shows that gate times are no longer constrained by the adiabatic theorem and are instead limited by the quantum speed limit and an associated energy cost.
Key findings
- Universal quantum gates can be implemented through piecewise controlled superadiabatic evolutions, including fast rotation gates and arbitrary n-qubit controlled gates, with the counter-diabatic Hamiltonian given by a time-independent operator in Eq. (23).
- The superadiabatic implementation allows the total evolution time τ to be reduced to an arbitrarily small value, with the QSL bound reducing to Eq. (27); saturation is achieved for either τ → 0 or ω → 0, implying τω → 0.
- The energetic cost Σ(τ) of superadiabatic evolutions supersedes the corresponding adiabatic cost Σ0(τ), and for qubit rotation gates it increases inversely proportional to the total evolution time.
- For controlled gates, the sum in the energetic-cost expression runs over 1 to 4 projectors on the subsystem, and the energetic cost ΣCG to implement controlled gates is stated to be [formula omitted in provided text].
Methods
- Theoretical quantum-information design: builds on controlled adiabatic evolutions (CAE) and adiabatic quantum circuits, then applies superadiabatic theory and counter-diabatic driving to implement quantum gates.
- Derives eigenstates and eigenvalues of piecewise adiabatic Hamiltonians, constructs counter-diabatic Hamiltonians, and proves that for arbitrary n-controlled gates the shortcut Hamiltonian is time-independent.
- Analyzes speed and cost using the generalized Margolus-Levitin quantum speed limit from Deffner and Lutz, the Bures metric for pure states, and the Hilbert-Schmidt norm for energetic cost.
Limitations
- The paper states that implications under decoherence and a fault-tolerance analysis of superadiabatic circuits are challenges of immediate interest, and that experimental proposals for superadiabatic circuits are left for future research.
- The paper states that superadiabatic implementations require knowledge of the eigenvalues and eigenstates of the adiabatic Hamiltonian, though for n-controlled gates this reduces to a single-qubit Hamiltonian independent of circuit complexity.
- Evident from the text: the scheme is theoretical and no experimental demonstration is reported; the adiabatic CAE implementation is probabilistic with probability enhanced to one only in the limit θ0 → π, while the superadiabatic shortcut is deterministic only if decoherence effects can be avoided.
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Cites
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Claims the paper makes
0 of 10 cited sources checked · 0 supported · 0 flagged
- Findingreasoning
The addition of a counter-diabatic Hamiltonian implies the QSL bound Eq. (26).
“More specifically, the addition of a counter-diabatic Hamiltonian implies into the QSL bound”
Quote in the paperSubstantial claim- The authors state this as an assumption: The quantum system is closed for the quantum speed limit analysis.
- Findingreasoning
For an arbitrary n-controlled quantum gate, the counter-diabatic Hamiltonians associated with shortcuts to adiabatic evolutions driven by Hξ(s) are time-independent operators given by Eq. (23).
“Hence, the implementation any n-controlled gate is achieved through a time-independent counter-diabatic Hamiltonian.”
Quote in the paperSubstantial claim - Findingown data
The energetic cost Σ(τ) increases inversely proportional to the total time of evolution.
“We illustrate the behavior of Σ(τ) in Fig. 1, where it is apparent that the energetic cost increases inversely proportional to the total time of evolution.”
Quote in the paperSubstantial claim - Findingnone given
The cost of performing superadiabatic evolutions requires knowledge of the eigenvalues and eigenstates of Hl(t).
“Note that the cost of performing superadiabatic evolutions requires the knowledge of the eigenvalues and eigenstates of Hl(t).”
Quote in the paperSubstantial claim - Findingreasoning
Piecewise controlled superadiabatic evolutions can be obtained by simple time-independent counter-diabatic Hamiltonians.
“Remarkably, they can be obtained by simple time-independent counter-diabatic Hamiltonians.”
Quote in the paperSubstantial claim
- Findingreasoning
For a fixed energetic cost, the optimal choice θ0 → π requires a longer evolution.
“In particular, note also that, for a fixed energetic cost, the optimal choice θ0 → π requires a longer evolution.”
Quote in the paperNot a substantial claim - Findingreasoning
The energetic cost Σ(τ) in superadiabatic evolutions supersedes the energetic cost Σ0(τ) for a corresponding adiabatic physical process.
“Then, it follows that the energetic cost Σ(τ) in superadiabatic evolutions supersedes the energetic cost Σ0(τ) for a corresponding adiabatic physical process.”
Quote in the paperSubstantial claim - Findingreasoning
The total time τ that mimics the adiabatic evolution within the superadiabatic approach can be reduced to an arbitrary small value, independently of the boundary states.
“By using the parametrized time s = t/τ, we can show from Eqs. (24) and (25) that the total time τ that mimics the adiabatic evolution within the superadiabatic approach can be reduced to an arbitrary small value.”
- Also stated in: § Quantum speed limit · plain
“This means that the superadiabatic implementation is compatible with an arbitrary reduction of the total time τ, which holds independently of the boundary states and .”
Quote in the paperSubstantial claim- The authors state this as an assumption: The quantum system is closed for the quantum speed limit analysis.
- Also stated in: § Quantum speed limit · plain
- Findingreasoning
The ground state energy for the adiabatic Hamiltonian H(s) in the case of n-controlled gates is E0(s) = −ωħ.
“Since the ground state energy for the adiabatic Hamiltonian H(s) in the case of n-controlled gates is E0(s) = −ωħ [see Eqs. (5) and (9)], we write η1(s) = ωη(s), with .”
Quote in the paperSubstantial claim - Findingreasoning
For the implementation of general n-controlled gates, the Hamiltonian acts over a single qubit, independent of the circuit complexity.
“For the implementation of general n-controlled gates, this is a Hamiltonian acting over a single qubit, which is independent of the circuit complexity.”
Quote in the paperSubstantial claim - Findingreasoning
The spectrum of H(s) for n-controlled gates is (2N)-degenerate, with levels E+ = ωħ and E− = −ωħ.
“By computing the eigenvalues of H(s), we obtain that the spectrum of H(s) is (2N)-degenerate, with and associated with the levels E+ = ωħ and E− = −ωħ, respectively.”
Quote in the paperSubstantial claim - Findingreasoning
The adiabatic implementation of the Toffoli gate can be achieved by a rotation of angle π around the x direction, choosing φ = π and the x axis.
“This can be easily achieved here by a rotation of an angle π around of the direction x, therefore choosing ϕ = π and , with denoting the eigenstates of σx.”
Quote in the paperSubstantial claim - Findingreasoning
The spectra of H0(s) and Hφ(s) are equal.
“By evaluating the eigenvalues of H0(s) and Hϕ(s), we obtain that their spectra are equal, being provided by .”
Quote in the paperSubstantial claim - Interpretationnone given
The time demanded for each gate implies an energy cost, which increases with the speed of the evolution.
“Indeed, the time demanded for each gate will imply in an energy cost, which increases with the speed of the evolution.”
Quote in the paperSubstantial claim - Interpretationreasoning
The counter-diabatic Hamiltonian is universal, holding both for performing single-qubit and n-controlled qubit gates.
“Remarkably, this Hamiltonian is universal, holding both for performing single-qubit and n-controlled qubit gates.”
Quote in the paperSubstantial claim - Interpretationreasoning
Each gate of an adiabatic circuit will be implemented within some fixed probability for a finite evolution time.
“However, since these processes are ruled by the adiabatic approximation, it turns out that each gate of the adiabatic circuit will be implemented within some fixed probability (for a finite evolution time).”
Quote in the paperSubstantial claim - Interpretationcitation
The time for performing each individual gate in an adiabatic circuit is bounded from below by the adiabatic time condition.
“Moreover, the time for performing each individual gate will be bounded from below by the adiabatic time condition45678910.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites 45678910
Not checked in the free run.
- Interpretationnone given
The physical resources spent by the superadiabatic strategy will be governed by the quantum circuit complexity.
“The physical resources spent by this strategy will be governed by the quantum circuit complexity”
Quote in the paperNot a substantial claim - Premisecitation
It is possible to implement universal sets of quantum gates through controlled adiabatic evolutions (CAE).
“In particular, it is possible to implement universal sets of quantum gates through controlled adiabatic evolutions (CAE)24.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [24] Hen, I. Quantum gates with controlled adiabatic evolutions. Phys Rev A 91, 022309 (2015). source
Not checked in the free run.
- Interpretationreasoning
Different sets of universal quantum gates can be designed by using essentially the same counter-diabatic Hamiltonian.
“In particular, different sets of universal quantum gates can be designed by using essentially the same counter-diabatic Hamiltonian.”
Quote in the paperSubstantial claim - Interpretationreasoning
The superadiabatic implementation of universal quantum circuits in the hybrid approach can be promptly achieved because the eigenspectrum of piecewise Hamiltonians acts over a few qubits.
“Here, as we shall see, the superadiabatic implementation of universal quantum circuits in the hybrid approach can be promptly achieved, since we deal with the eigenspectrum of piecewise Hamiltonians, which act over a few qubits.”
Quote in the paperSubstantial claim - Interpretationreasoning
A superadiabatic route can implement universal quantum computation based on piecewise controlled superadiabatic evolutions.
“Here, we propose a superadiabatic route to implement universal quantum computation. Our method is based on the realization of piecewise controlled superadiabatic evolutions.”
Quote in the paperSubstantial claim - Interpretationreasoning
The running time in superadiabatic dynamics is only constrained by the energetic cost of the superadiabatic implementation, within a time-energy complementarity relationship.
“In this context, the running time is only constrained by the energetic cost of the superadiabatic implementation, within a time-energy complementarity relationship.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The quantum system is closed for the quantum speed limit analysis.
- Interpretationreasoning
The energy cost of the superadiabatic implementation is dictated by the quantum speed limit, providing an upper bound for the corresponding adiabatic counterparts.
“Concerning the energy cost of the superadiabatic implementation, we show that it is dictated by the quantum speed limit, providing an upper bound for the corresponding adiabatic counterparts.”
- Also stated in: § Abstract · boosted
“In this context, by analyzing the energy-time complementarity, we will show that the QSL provides an energy cost for superadiabatic evolutions that upper bounds the cost of adiabatic implementations.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The quantum system is closed for the quantum speed limit analysis.
- Also stated in: § Abstract · boosted
- Interpretationnone given
The gates will be deterministically implemented with probability one as long as decoherence effects can be avoided.
“Moreover, the gates will be deterministically implemented with probability one as long as decoherence effects can be avoided.”
Quote in the paperSubstantial claim- Rests on an assumption: Decoherence effects can be avoided so that the superadiabatic gates are deterministically implemented.
- Premisecitation
The adiabatic time is of order τad ∝ 1/ωn.
“Note that this limit is forbidden in the adiabatic regime for finite ω, since the energy gap is proportional to ħω, which implies in an adiabatic time of the order τad ∝ 1/ωn, with 34736.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites 34736
Not checked in the free run.
- Interpretationcitation
The time consumption of individual gates is only dictated by the quantum speed limit.
“This analog approach allows for fast implementation of individual gates, whose time consumption is only dictated by the quantum speed limit (QSL) (for closed systems, see refs 32, 33, 34, 35).”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [32] Mandelstam, L. & Tamm, I. G. The uncertainty relation between energy and time in non-relativistic quantum mechanics. J. Phys. (Moscow) 9, 249 (1945).
Not checked in the free run.
- cites [33] Margolus, N. & Levitin, L. B. The maximum speed of dynamical evolution. Physica D 120, 188 (1998). source
Not checked in the free run.
- cites [34] Giovannetti, V., Lloyd, S. & Maccone, L. Quantum limits to dynamical evolution Phys. Rev. A 67, 052109 (2003). source
Not checked in the free run.
- The authors state this as an assumption: The quantum system is closed for the quantum speed limit analysis.
- Interpretationreasoning
No adiabatic constraint will be required in the individual implementation of the quantum gates.
“but no adiabatic constraint will be required in the individual implementation of the quantum gates.”
Quote in the paperSubstantial claim - Interpretationcitation
The original AQC approach makes superadiabatic implementations seem prohibitive because the whole set of eigenlevels of a many-body Hamiltonian is required.
“For instance, by adopting the original AQC approach14, superadiabatic implementations seem prohibitive, since the whole set of eigenlevels of a many-body Hamiltonian is required.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] Farhi, E. et al. A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292, 472 (2001). source
Not checked in the free run.
- May combine two claims. 5
- Classified as probably not a substantial claim. 2
May combine two claims.
FindingreasoningSaturation of Eq. (27) is achieved for either τ → 0 or ω → 0, with both cases implying τω → 0.
“In particular, saturation of Eq. (27) is achieved for either τ → 0 or ω → 0, with both cases implying in τω → 0.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The quantum system is closed for the quantum speed limit analysis.
May combine two claims.
FindingreasoningH(s) exhibits doubly degenerate levels, with states associated with levels E+ = ωħ and E− = −ωħ.
“Thus, H(s) exhibits doubly degenerate levels, with and associated with levels E+ = ωħ and E− = −ωħ, respectively.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
PremisecitationThe adiabatic theorem ensures transitionless evolution if the system is prepared in an eigenstate of the Hamiltonian at t = 0 and the Hamiltonian is sufficiently slowly varying.
“In particular, if the system is prepared in an eigenstate of the Hamiltonian H at a time t = 0, it will evolve to the corresponding instantaneous eigenstate at later times. This transitionless evolution is ensured by the adiabatic theorem, which is one of the oldest and most explored tools in quantum mechanics123.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [123]
Not checked in the free run.
May combine two claims.
InterpretationreasoningFast rotation gates and arbitrary n-qubit controlled gates can be implemented and used to design different sets of universal quantum gates.
“In particular, we discuss the implementation of fast rotation gates and arbitrary n-qubit controlled gates, which can be used to design different sets of universal quantum gates.”
Quote in the paperSubstantial claimMay combine two claims.
PremisecitationCAE are used to perform one-qubit and two-qubit gates, allowing for universality through the set of one-qubit rotations joint with an entangling two-qubit gate.
“In turn, CAE are used to perform one-qubit and two-qubit gates, allowing for universality through the set of one-qubit rotations joint with an entangling two-qubit gate2526.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [2526]
Not checked in the free run.
Classified as probably not a substantial claim. May combine two claims.
PremisecitationAQC has been proved universal for quantum computing, equivalent to the standard circuit model up to polynomial resource-overhead.
“More generally, AQC has been proved to be universal for quantum computing, being equivalent to the standard circuit model of quantum computation up to polynomial resource-overhead15.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [15] Aharonov, D. et al. Adiabatic quantum computation is equivalent to standard quantum computation. SIAM Journal of Computing 37, 166 (2007). source
Not checked in the free run.
The paper relies on this without support.
Decoherence effects can be avoided so that the superadiabatic gates are deterministically implemented.
“Moreover, the gates will be deterministically implemented with probability one as long as decoherence effects can be avoided.”
The authors state this as an assumption.
The quantum system is closed for the quantum speed limit analysis.
“We will consider a closed quantum system evolving between arbitrary pure states and .”
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?
Santos and Sarandy showed that shortcuts to adiabaticity for quantum gates can perform operations faster, at increasing energetic cost.
“This natural conclusion has recently been shown more explicitly by Santos and Sarandy [97]-wherein by developing shortcuts to adiabaticity that achieve quantum gates, they showed that these operations can be performed faster, however this is accompanied by an increasing energetic cost.”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [97]
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- Authors
- Santos, A. C., & Sarandy, M. S.
- Affiliations
- Universidade Federal Fluminense (BR)
- University of Southern California (US)
- Corresponding author
- Santos, Alan C.
- Sarandy, Marcelo S.
- ORCID
- Sarandy, Marcelo S.
- Venue
- Scientific Reports
- Published
- 2015-10-29
- Volume
- 5
- Issue
- 1
- Pages
- 15775-15775
- ISSN
- 2045-2322 (ISSN-L)
- PMC
- PMC4625175
- OpenAlex
- W2276020317
- Semantic Scholar
- b178b7559d34263a838e8d75308318e5aac5b64d
- Cited by
- 125
- References
- 41
- Citations by year
- 2026: 4 · 2025: 8 · 2024: 4 · 2023: 11 · 2022: 13 · 2021: 11 · 2020: 8 · 2019: 17 · 2018: 15 · 2017: 17 · 2016: 17
- FWCI
- 16.01
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Locations
- Scientific Reports (publishedVersion, cc-by)
- arXiv (Cornell University) (publishedVersion)
- Europe PMC (PubMed Central) (publishedVersion, cc-by)
- PubMed Central (publishedVersion, cc-by)
- PubMed (publishedVersion)
- arXiv (Cornell University) (cc-by)
- Greater South Information System
- Funders
- University of Southern California
- Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
- Conselho Nacional de Desenvolvimento Científico e Tecnológico
- Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro
- Instituto Nacional de Ciência e Tecnologia de Informação Quântica
- 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
- shortcuts to adiabaticity, universal quantum computation, quantum gates, quantum speed limit, quantum control
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