Optimal control theory for unitary transformations

Jose P. Palao; R. Kosloff

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Optimal control theory for unitary transformations

Jose P. Palao, R. Kosloff

Quantum Physics · Submitted 31 Aug 2003

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Light fields are found that are able to implement the Fourier transform within a picosecond time scale.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.76 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.66 · compares with prior results 0.77 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The dynamics of a quantum system driven by an external field is well described by a unitary transformation generated by a time dependent Hamiltonian. The inverse problem of finding the field that generates a specific unitary transformation is the subject of study. The unitary transformation which can represent an algorithm in a quantum computation is imposed on a subset of quantum states embedded in a larger Hilbert space. Optimal control theory (OCT) is used to solve the inversion problem irrespective of the initial input state. A unified formalism, based on the Krotov method is developed leading to a new scheme. The schemes are compared for the inversion of a two-qubit Fourier transform using as registers the vibrational levels of the $X^1\Sigma^+_g$ electronic state of Na$_2$. Raman-like transitions through the $A^1\Sigma^+_u$ electronic state induce the transitions. Light fields are found that are able to implement the Fourier transform within a picosecond time scale. Such fields can be obtained by pulse-shaping techniques of a femtosecond pulse. Out of the schemes studied the square modulus scheme converges fastest. A study of the implementation of the $Q$ qubit Fourier transform in the Na$_2$ molecule was carried out for up to 5 qubits. The classical computation effort required to obtain the algorithm with a given fidelity is estimated to scale exponentially with the number of levels. The observed moderate scaling of the pulse intensity with the number of qubits in the transformation is rationalized.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The dynamics of a quantum system driven by an external field is well described by a unitary transformation generated by a time dependent Hamiltonian.	background 1.00	0.06	0.02	asserted 0.98
s1	The inverse problem of finding the field that generates a specific unitary transformation is the subject of study.	background 0.69	0.18	0.04	asserted 0.99
s2	The unitary transformation which can represent an algorithm in a quantum computation is imposed on a subset of quantum states embedded in a larger Hilbert space.	background 0.71	0.15	0.04	asserted 0.99
s3	Optimal control theory (OCT) is used to solve the inversion problem irrespective of the initial input state.	method 0.94	0.64	0.11	asserted 1.00
s4	A unified formalism, based on the Krotov method is developed leading to a new scheme.	contribution 0.50	0.91	0.08	asserted 1.00
s5	The schemes are compared for the inversion of a two-qubit Fourier transform using as registers the vibrational levels of the $X^1\Sigma^+_g$ electronic state of Na$_2$.	method 0.79	0.88	0.05	asserted 0.97
s6	Raman-like transitions through the $A^1\Sigma^+_u$ electronic state induce the transitions.	method 0.97	0.47	0.03	asserted 1.00
s7	Light fields are found that are able to implement the Fourier transform within a picosecond time scale.	result 1.00	0.89	0.23	asserted 1.00
s8	Such fields can be obtained by pulse-shaping techniques of a femtosecond pulse.	result 0.44	0.52	0.07	asserted 0.80
s9	Out of the schemes studied the square modulus scheme converges fastest.	result 1.00	0.92	0.19	asserted 1.00
s10	A study of the implementation of the $Q$ qubit Fourier transform in the Na$_2$ molecule was carried out for up to 5 qubits.	contribution 0.86	0.93	0.09	asserted 1.00
s11	The classical computation effort required to obtain the algorithm with a given fidelity is estimated to scale exponentially with the number of levels.	result 0.84	0.71	0.17	asserted 0.97
s12	The observed moderate scaling of the pulse intensity with the number of qubits in the transformation is rationalized.	interpretation 0.90	0.83	0.17	asserted 0.97

What kind of paper the classifier reports this is: method or tool 0.92 · evidence from simulation or computation 0.98.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

The dynamics of a quantum system driven by an external field is well described by a unitary transformation generated by a time dependent Hamiltonian. The inverse problem of finding the field that generates a specific unitary transformation is the subject of study. The unitary transformation which can represent an algorithm in a quantum computation is imposed on a subset of quantum states embedded in a larger Hilbert space. Optimal control theory (OCT) is used to solve the inversion problem irrespective of the initial input state. A unified formalism, based on the Krotov method is developed leading to a new scheme. The schemes are compared for the inversion of a two-qubit Fourier transform using as registers the vibrational levels of the $X^1\Sigma^+_g$ electronic state of Na$_2$. Raman-like transitions through the $A^1\Sigma^+_u$ electronic state induce the transitions. Light fields are found that are able to implement the Fourier transform within a picosecond time scale. Such fields can be obtained by pulse-shaping techniques of a femtosecond pulse. Out of the schemes studied the square modulus scheme converges fastest. A study of the implementation of the $Q$ qubit Fourier transform in the Na$_2$ molecule was carried out for up to 5 qubits. The classical computation effort required to obtain the algorithm with a given fidelity is estimated to scale exponentially with the number of levels. The observed moderate scaling of the pulse intensity with the number of qubits in the transformation is rationalized.

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References

How later papers used it

48 later papers cite it

Evidence: substantial. 46 of the citing papers share no author with it, and 16 of those carry a confident label from Pith's citation classifier.

Pith read 22 of the 90 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

9 later papers by other authors are listed below: 9 used its method or tool.

Uses its method or tool 9 of 18 papers with this label

Employed a linear variant of Krotov's method to obtain a monotonically convergent optimization algorithm.

Prospects for fast Rydberg gates on an atom chip (2011) · 1.00 · paraphrased: the licence does not allow quoting
Defined the specific form of its step-size parameter and shape function as given in this paper.

Optimal Control Design of Robust Raman Pulses for High-Fidelity Cold-Atom Interferometry (2026) · 0.88 · paraphrased: the licence does not allow quoting
We take [ref] [equation] where $Ω_k^(t)$ is a reference field for each $Ω_k(t)$, $S(t)$ the shape function from Eq. [ref] and $λ_k$ a numerical parameter that controls the magnitude of update in each iteration.

Engineering Strong Beamsplitter Interaction between Bosonic Modes via Quantum Optimal Control Theory (2021) · Quantum Optimal Control Theory · 0.98
Assumed the function g to take the form given in this paper to minimize the total functional.

Optimally controlled quantum discrimination and estimation (2020) · 0.98 · paraphrased: the licence does not allow quoting
Used a square-modulus overlap functional from this paper to optimize an effective pi/2 pulse.

Robust Optimized Pulse Schemes for Atomic Fountain Interferometry (2022) · 0.87 · paraphrased: the licence does not allow quoting
In order to optimize for quantum gates, we use the error-measure [ref] [equation] and take $|ψ_l^⟩ = O |ψ_l^⟩$ as the desired target states for the target gate $O$.

Comparing planar quantum computing platforms at the quantum speed limit (2023) · Quantum optimal control theory · 0.99
Within 300 iterations, using a square-modulus functional [ref] , we can bring the separation error from 0.648, see Fig. [ref], to $1.26 ×10^-5$.

Rotation Sensing using Tractor Atom Interferometry (2023) · Optimal Control · 0.94
Introduced real and imaginary parts of its coefficient vectors, an arrangement also used in this paper.

Gradient-based reconstruction of molecular Hamiltonians and density matrices from time-dependent quantum observables (2022) · 0.84 · paraphrased: the licence does not allow quoting
Considered Krotov's method from this paper as another quantum optimal control protocol.

Hamiltonian Quantum Generative Adversarial Networks (2022) · 0.89 · paraphrased: the licence does not allow quoting

How the literature uses it

1 arXiv paper cite this in 6 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 2 sentences without a confident label

Uses its method or tool 1 paper

In the following, we use Krotov's method [ref] in its adaption to quantum dynamics [ref] to carry out the optimization.

Fast and accurate circularization of a Rydberg atom · Optimal control theory · 1.00

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