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Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions
Open accessAbstract
We propose a method of simulating efficiently many-body interacting fermion lattice models in trapped ions, including highly nonlinear interactions in arbitrary spatial dimensions and for arbitrarily distant couplings. We map products of fermionic operators onto nonlocal spin operators and decompose the resulting dynamics in efficient steps with Trotter methods, yielding an overall protocol that employs only polynomial resources. The proposed scheme can be relevant in a variety of fields such as condensed-matter or high-energy physics, where quantum simulations may solve problems intractable for classical computers.
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TL;DR
The paper proposes a trapped-ion protocol for quantum simulating many-body interacting fermion lattice models by mapping fermionic operators to nonlocal spin operators via Jordan-Wigner, decomposing the evolution with Trotter methods, and implementing the resulting nonlocal terms with Mølmer-Sørensen gates. It argues the protocol uses polynomial resources and can handle long-range, highly nonlinear interactions in arbitrary dimensions, with numerical Fermi-Hubbard simulations showing convergence of the Trotter approximation.
Key findings
- The protocol simulates general fermionic lattice Hamiltonians with arbitrary spatial dimension and arbitrarily distant couplings, avoiding auxiliary virtual Majorana fermions and the usual restriction to few neighbors.
- For a Fermi-Hubbard example with three lattice sites and six modes (two spins per site), the number of Hamiltonian terms scales as \(5N/2 - 4\), giving 11 terms for \(N=6\); the total number of gates per Trotter step is 33, on average 3 gates per Hamiltonian term.
- Trotter fidelity approaches 1 with increasing Trotter steps \(n_T\); for \(n_T=10\), fidelity is 0.99 in one parameter case and 0.97 in another. Supplementary fits show the error scales as \(1 - C/n_T^2\).
- The authors state that for more than \(\sim 30\) particles, foreseeable in the near future, one could simulate fermionic systems intractable for classical computers.
Methods
- Theoretical proposal in three steps: Jordan-Wigner mapping of \(N\) fermionic modes to \(N\) nonlocal spin operators; Trotter decomposition of the total evolution into exponentials of each nonlocal spin term; implementation of each exponential on \(N\) two-level trapped ions using Mølmer-Sørensen gates, local gates, and local rotations.
- Numerical validation compares exact diagonalization with Trotter-evolved dynamics for the Fermi-Hubbard Hamiltonian on three lattice sites with six modes, simulated by six two-level trapped ions, using initial state \(b^\dagger_{1\uparrow} b^\dagger_{1\downarrow}|0\rangle\).
- Supplementary analysis argues the number of elementary gates scales polynomially in the Trotter error \(\epsilon\), total evolution time \(t\), and number of fermionic modes \(N\), for bounded Hamiltonians.
Limitations
- The paper does not include a dedicated limitations section; stated caveats include that the Trotter expansion is approximate and its error decreases with more Trotter steps, and that the error analysis focuses on time-independent Hamiltonians.
- The numerical demonstration is limited to a small Fermi-Hubbard system of three lattice sites and six modes; no experimental implementation is reported.
- The text notes that current technology has realized more than 100 gates in a single experiment and that without error correction one would expect more than a thousand gates in the near future, indicating gate counts and scalability depend on future experimental capabilities.
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26 Not checked yet · 20 Own interpretation · 3 No basis given · 2 Own result
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“Thus, the error goes to zero polynomially in n T , as expected.”
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“Notice that the number of terms to be implemented scales linearly with the number of modes, 5 N/ 2 -4 (11 in this case, for N = 6).”
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“This includes the calculation of time evolutions and ground state computations, e.g. through adiabatic protocols [34].”
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- Not checked yetcites [34] source
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“Each of these exponentials can be implemented, for arbitrary k and up to local rotations, with a Mølmer-Sørensen gate upon k ions, one local gate upon one of the ions, and the inverse Mølmer-Sørensen gate [31].”
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- Not checked yetcites [31] source
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“We can simulate this kind of Hamiltonians in order to recover the polaron physics, a critical open issue for the deep understanding of correlated electrons in solids [38].”
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- Not checked yetcites [38]
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“ii) each H j is always of the form of a product of arbitrary number of Pauli matrices times a coupling h j , such that its norm || H j || = h j , given that the 2-norm of a product of arbitrary number of Pauli matrices is always 1.”
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“One of the main appeals of our method is that the efficient encoding of fermionic models in a lattice with arbitrarily long-range couplings is feasible.”
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“This technique establishes a correspondence between a set of fermionic operators and a set of spin operators, transforming a local Hamiltonian of fermions onto a nonlocal Hamiltonian of spins.”
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- Not checked yetcites [28] source
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“Finally, we implement each of these exponentials in polynomial time on a set of N two-level trapped ions with a reduced number of laser pulses [31, 32].”
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“Only in one spatial dimension, and for couplings extended to a reduced number of neighbors, the correspondence is from a local model to a similar one [28].”
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- Not checked yetcites [28] source
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“Quantum simulations appear as a tool that will allow us to compute the time evolution of free and interacting fermion lattice theories with minimal experimental resources.”
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“The last term contains the product b † p + q b p ( a q + a † -q ), whose dynamics can be implemented with our technique above according to Eq. (6).”
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“The last term contains the product of four fermionic operators, allowing for efficient implementation.”
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“Our protocol consists of three steps, gathering techniques that have not been previously considered for quantum simulators of fermionic lattice models [35]:”
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- Not checked yetcites [35] source
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“We propose a method of simulating efficiently many-body interacting fermion lattice models in trapped ions, including highly nonlinear interactions in arbitrary spatial dimensions and for arbitrarily distant couplings.”
- Also stated in: § Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions · p. 1 · plain
“Our method can simulate highly nonlinear and long-range interactions in arbitrary spatial dimensions, applying the Jordan-Wigner transformation without the usual restriction of a reduced number of neighbors, and without the need of auxiliary virtual Majorana fermions [33].”
- Also stated in: § Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions · p. 4 · plain
“In conclusion, we have introduced a method for simulating efficiently, i.e. with polynomial resources, manybody fermionic lattice models in an ion string.”
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“In this Section we show that the resources needed in our protocol, including number of elementary gates and time scaling, are polynomial on the Trotter error, the total time simulated, and the total size of the system, in terms of the number of fermionic modes.”
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“Notice that, e.g., the operators b † p ↑ b p ′ ↓ σ -j can be now mapped to a sum of products of Pauli matrices, leading to an efficient implementation.”
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“Here, the total number of gates needed, N e , scales in the error ϵ , the total evolution time t , and the norm of the Hamiltonian H , || H || , where || . || is the 2-norm, in the form [5]”
What the sources say
- Not checked yetcites [5] D. W. Berry, G. Ahokas, R. Cleve and Barry C. Sanders, Commun. Math. Phys. 270 , 359 (2007). source
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“Furthermore, quantum simulations will allow us to reproduce the complete dynamics of these systems, avoiding mean field approximations as Hartree-Fock to simplify nonlinear interactions [27].”
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- Not checked yetcites [27]
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“We map products of fermionic operators onto nonlocal spin operators and decompose the resulting dynamics in efficient steps with Trotter methods, yielding an overall protocol that employs only polynomial resources.”
- Also stated in: § Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions · p. 1 · plain
“Firstly, we map the set of N fermionic modes, via the Jordan-Wigner transformation [28], to a set of N nonlocal spin operators.”
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- Not checked yetcites [28] source
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- Also stated in: § Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions · p. 1 · plain
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“The proposed protocol opens the possibility to simulate a wide range of interesting condensed-matter and highenergy physics fermionic systems for a large number of particles.”
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“The spin degrees of freedom can be encoded in long-lived electronic states of the ions [5, 6, 12, 14, 19, 32, 39].”
What the sources say
- Not checked yetcites [5] D. W. Berry, G. Ahokas, R. Cleve and Barry C. Sanders, Commun. Math. Phys. 270 , 359 (2007). source
- Not checked yetcites [6] M. Suzuki, Proc. Japan Acad. B 69 , 161 (1993). source
- Not checked yetcites [12] source
- Not checked yetcites [14] source
- Not checked yetcites [19] source
- Not checked yetcites [32] source
- Not checked yetcites [39]
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“The operators b i k , b † i k can always be mapped to products of Pauli matrices, i.e., nonlocal spin operators, using the Jordan-Wigner transformation”
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- Not checked yetcites [28] source
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“The latter can be implemented by using a red and a blue sideband interactions for each of the a j modes in the context of standard trapped-ion technology [37].”
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- Not checked yetcites [37]
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“For a number of particles above ∼ 30, which is foreseeable in the near future, one could already simulate fermionic systems that are intractable for classical computers.”
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“The exponentials of each nonlocal spin operator are efficiently implementable, given that their exponents consist of tensor products of k Pauli matrices.”
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“With the proposed method, we can simulate Kondo Hamiltonians [24], modelling the interaction of a Fermi sea of electrons with several magnetic impurities at positions R j ,”
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- Not checked yetcites [24] source
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“Notice that in all fermionic Hamiltonians we are considering, we have i) a polynomial number of nonlocal spin operators, i.e., m is polynomial in N , the total number of fermionic modes.”
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“Indeed, without error correction, one would expect the realization of more than thousand gates in the near future [40].”
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- Not checked yetcites [40]
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“This also means that we can apply the Jordan-Wigner transformation for two and three spatial dimensions, not just for one, without employing additional virtual Majorana fermions [33].”
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- Not checked yetcites [33] source
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“The numerical simulation of fermionic systems is, in general, a hard problem due to the huge increase of the Hilbert space dimension with the number of modes [1, 21].”
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“With current technology, more than 100 gates have been realized in a single experiment [39].”
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- Not checked yetcites [39]
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“The bosonic operators a j can be implemented with the vibrational modes of the ion chain.”
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“Using customized numerical methods as quantum Monte Carlo is not always possible due to the well-known sign problem [22, 23].”
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- Not checked yetcites [22]
- Not checked yetcites [23] source
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“We could as well implement the coupling of arbitrary products of fermionic operators [similarly to Eq. (1)] to linear sum of bosonic operators,”
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“This will allow, for n T = 10 Trotter steps, the implementation of hundreds of gates per step, giving us the possibility to simulate a wide variety of fermionic models.”
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“The three steps of our protocol amounts to an efficient method for simulating fermionic models with long-range couplings in arbitrary dimensions with trapped ions.”
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“Note that for bounded Hamiltonians, the Trotter expansion associated with the exponential of the polynomial sum of efficiently implementable nonlocal terms is also efficient [2, 29, 30], i.e., it only requires polynomial resources [35].”
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“At the same time, the nonlocal gates upon several ions are efficiently implementable with few lasers, such that the number of gates in each term of the Hamiltonian is, in the worst case, linear in the number of modes, and in many cases just constant.”
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“In the general case, it would require a number of elementary gates of 2 20 ≃ 10 6 [2]. With our method, the number of gates can be as small as 10 per Trotter step, consisting of Mølmer-Sørensen gates (dark blue and green), local exp( iφ ′ σ y 2 ) gates (red), exp[ ± i ( π/ 4) ∑ i σ y i ] (yellow) and exp[ ± i ( π/ 4) ∑ i σ x i ] (cyan) gates (see Fig. 1b) [35].”
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“Notice that the fidelity of the Trotter expansion goes to 1 with the number of Trotter steps n T , and for n T = 10 it is 0.99 (b) or 0.97 (d).”
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“In this particular example the total number of gates per Trotter step is of 33, i.e., on average 3 gates per Hamiltonian term, which is very efficient.”
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“Notice that our method is general and extends the hopping terms to arbitrarily distant pairs of electrons.”
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“Our method can simulate highly nonlinear and long-range interactions in arbitrary spatial dimensions, applying the Jordan-Wigner transformation without the usual restriction of a reduced number of neighbors, and without the need of auxiliary virtual Majorana fermions [33].”
What the sources say
- Not checked yetcites [33] source
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“Thus, the mapped spin Hamiltonians are highly nonlocal but their evolution is efficiently realizable.”
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In the paper
“We map products of fermionic operators onto nonlocal spin operators and decompose the resulting dynamics in efficient steps with Trotter methods, yielding an overall protocol that employs only polynomial resources.”
- Also stated in: § Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions · p. 1 · plain
“These three steps yield an efficient protocol that employs only polynomial resources.”
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- Also stated in: § Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions · p. 1 · plain
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“iii) the total norm of H is bounded by || H || ≤ ∑ m j =1 || H j || = ∑ m j =1 h j ≤ mh max j , where h max j is the maximum among all h ′ j s . Thus, || H || is polynomial in m , and in consecuence, also in N .”
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Accordingly, we have shown that the scaling of the number of elementary gates needed in our expansion, is polynomial (more specifically, a power law) in ϵ , t , and N , such that our method for implementing arbitrary fermionic Hamiltonians that occur in nature is efficient.”
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“The Fermi-Hubbard Hamiltonian [25] takes into account a range of effects in condensed-matter physics, as the Mott transition, and is also believed to be relevant in highT c superconductivity.”
What the sources say
- Not checked yetcites [25] source
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
In the paper
“The long debated Kondo Hamiltonian provides a variety of interesting features in different systems, as the minimum in the resistivity at low temperatures [36].”
What the sources say
- Not checked yetcites [36]
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“The Fr¨ ohlich Hamiltonian [26] models the interaction of electrons with phonons,”
What the sources say
- Not checked yetcites [26] source
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
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
“This kind of second quantized fermionic Hamiltonians are efficiently simulatable in trapped ions23.”
From From transistor to trapped-ion computers for quantum chemistry (2014) · cites this paper as [23]
In the citing paper
“The phonon degrees of freedom in trapped-ion systems provide a natural platform for addressing spin-boson or fermion-boson-type problems through quantum simulation233637383940.”
From From transistor to trapped-ion computers for quantum chemistry (2014) · cites this paper as [23]
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
§ IMPLEMENTATION OF FERMION LATTICE MODELS IN TRAPPED IONS: EXAMPLES · p. 5
“H 2 = g ( σ z j + I j ) ⊗ ( σ z j +1 + I j +1 ) / 4”
§ IMPLEMENTATION OF FERMION LATTICE MODELS IN TRAPPED IONS: EXAMPLES · p. 5
“= gσ z j ⊗ σ z j +1 / 4 + gσ z j / 4 + gσ z j +1 / 4.”
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 Simulation of Interacting Fermion Lattice Models in Trapped Ions · p. 3
“In (b) and (c) we include the fidelity |〈 ψ ( t F ) | ψ ( t F ) T 〉| 2 as a function of the number of Trotter steps n T”
§ Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions · p. 4
“(c) 〈 b † 2 ↓ b 2 ↓ 〉 ( t ) (dashed, blue), and 〈 b † 3 ↑ b 3 ↑ 〉 ( t ) (solid, red), for n T = 15, and (d) fidelity |〈 ψ ( t F ) | ψ ( t F ) T 〉| 2”
Citations
Cites
Quantum Computation and Quantum Information
Nielsen, M. A., Chuang, I., & Grover, L. K. (2002) · American Journal of Physics · doi:10.1119/1.1463744 · Cited by 21,895
Quantum computation and quantum information
Law, J. (2001) · ACM SIGSOFT Software Engineering Notes · doi:10.1145/505482.505499 · Cited by 20,450
Mahan, G. D. (1990) · Springer eBooks · doi:10.1007/978-1-4613-1469-1 · Cited by 8,358
Mahan, G. D. (2000) · doi:10.1007/978-1-4757-5714-9 · Cited by 8,249
Simulating physics with computers
Feynman, R. P. (1982) · International Journal of Theoretical Physics · doi:10.1007/bf02650179 · Cited by 7,691
Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms
Greiner, M., Mandel, O., Esslinger, T., Hänsch, T. W., & Bloch, I. (2002) · Nature · doi:10.1038/415039a · Cited by 5,935
Resistance Minimum in Dilute Magnetic Alloys
Kondo, J. (1964) · Progress of Theoretical Physics · doi:10.1143/ptp.32.37 · Cited by 3,658
Lloyd, S. (1996) · Science · doi:10.1126/science.273.5278.1073 · Cited by 3,088
Quantum dynamics of single trapped ions
Leibfried, D., Blatt, R., Monroe, C., & Wineland, D. (2003) · Reviews of Modern Physics · doi:10.1103/revmodphys.75.281 · Cited by 2,739
�ber das Paulische �quivalenzverbot
Jordan, P., & Wigner, E. (1928) · Zeitschrift f�r Physik · doi:10.1007/bf01331938 · Cited by 2,597
Cited by
Sorted by citations
Georgescu, I. M., Ashhab, S., & Nori, F. (2014) · Reviews of Modern Physics · doi:10.1103/revmodphys.86.153 · Cited by 2,932
“There have also been related proposals using trapped ions (Casanova et al., 2012, 2011), cavity QED systems (Barrett et al., 2013) and superconducting circuits (Marcos et al., 2013; Pedernales et al., 2013).”
BackgroundStrategies for quantum computing molecular energies using the unitary coupled cluster ansatz
Romero, J., Babbush, R., McClean, J. R., Hempel, C., Love, P. J., & Aspuru-Guzik, A. (2019) · Quantum Science and Technology · doi:10.1088/2058-9565/aad3e4 · Cited by 726
Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator
Hempel, C., Maier, C., Romero, J., McClean, J., Monz, T., Shen, H., Jurcevic, P., Lanyon, B. P., Love, P., Babbush, R., et al. (2018) · Physical Review X · doi:10.1103/physrevx.8.031022 · Cited by 629
Quantum Algorithms for Quantum Field Theories
Jordan, S. P., Lee, K. S. M., & Preskill, J. P. (2012) · Science · doi:10.1126/science.1217069 · Cited by 511
Digital quantum simulation of fermionic models with a superconducting circuit
Barends, R., Lamata, L., Kelly, J., García-Álvarez, L., Fowler, A. G., Megrant, A., Jeffrey, E. R., White, T., Sank, D., Mutus, J., et al. (2015) · Nature Communications · doi:10.1038/ncomms8654 · Cited by 347
Self-citation
Metadata
- Authors
- Casanova, J., Mezzacapo, A., Lamata, L., & Solano, E.
- Affiliations
- University of the Basque Country (ES)
- Ikerbasque (ES)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2012-05-10
- Volume
- 108
- Issue
- 19
- Pages
- 190502-190502
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2088422359
- Cited by
- 118
- References
- 42
- Citations by year
- 2026: 1 · 2025: 2 · 2024: 7 · 2023: 2 · 2022: 11 · 2021: 6 · 2020: 6 · 2019: 8 · 2018: 14 · 2017: 8 · 2016: 5 · 2015: 16 · 2014: 19 · 2013: 7 · 2012: 6
- FWCI
- 15.11
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Cold Atom Physics and Bose-Einstein Condensates
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- quantum simulation, trapped ions, Trotter decomposition, long-range coupling, quantum many-body systems
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 2026-10-07, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-07 (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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