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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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In the paper
“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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“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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In the paper
“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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In the paper
“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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In the paper
“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].”
What the sources say
- Not checked yetcites [28] source
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In the paper
“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 paper
“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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In the paper
“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 the paper
“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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In the paper
“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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In the paper
“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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In the paper
“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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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.”
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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
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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]
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In the paper
“The Fr¨ ohlich Hamiltonian [26] models the interaction of electrons with phonons,”
What the sources say
- Not checked yetcites [26] source
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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”
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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
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