Permutation Matrix Representation for Quantum Simulation: Comparative Resource Analysis

Hriday Sabharwal; Itay Hen

arxiv: 2605.29279 · v1 · pith:QK6V3UHNnew · submitted 2026-05-28 · 🪐 quant-ph

Permutation Matrix Representation for Quantum Simulation: Comparative Resource Analysis

Hriday Sabharwal , Itay Hen This is my paper

Pith reviewed 2026-06-29 07:22 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum simulationpermutation matrix representationHamiltonian simulationresource analysisRydberg Hamiltoniantransverse-field Ising modeltime-dependent simulationquantum algorithms

0 comments

The pith

Permutation matrix representation shows lower resource costs than QSP, qubitization and qHOP for the tested Rydberg and driven Ising Hamiltonians.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper benchmarks the permutation matrix representation method for quantum Hamiltonian simulation against established alternatives. For time-independent Hamiltonians it compares PMR directly to quantum signal processing and qubitization using the Rydberg interaction model. For time-dependent Hamiltonians it compares the time-dependent PMR extension to the quantum highly oscillatory protocol on a Floquet-driven transverse-field Ising model in any spatial dimension. The comparisons track gate counts, qubit requirements and scaling with system size and evolution time. The results indicate that PMR supplies complementary resource advantages and favorable scaling in several parameters, which would matter for running simulations on hardware with tight limits on gates and qubits.

Core claim

When applied to the Rydberg interaction Hamiltonian, PMR requires fewer resources than both quantum signal processing and qubitization in the regimes examined; when applied to the Floquet-driven transverse-field Ising model, its time-dependent extension requires fewer resources than qHOP and exhibits better scaling with dimension and number of time steps. These advantages arise because PMR encodes the Hamiltonian evolution through a permutation-matrix decomposition that maps directly onto a sequence of controlled permutations and phase rotations.

What carries the argument

The permutation matrix representation (PMR), which decomposes the Hamiltonian into a sum of permutation matrices whose action can be implemented with a fixed pattern of controlled swaps and diagonal phase gates.

If this is right

For Rydberg systems, PMR could reduce the total number of two-qubit gates needed to reach a given simulation accuracy.
For Floquet-driven Ising models in higher dimensions, PMR's resource cost grows more slowly with spatial dimension than the cost of qHOP.
The existence of complementary advantages implies that algorithm choice can be made on a per-Hamiltonian basis rather than using a single universal method.
Favorable scaling with certain parameters suggests PMR becomes relatively cheaper as those parameters increase while holding others fixed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same permutation-matrix encoding might be adapted to other lattice spin models whose interaction graphs admit compact permutation decompositions.
Resource savings reported for these two models could translate into the ability to simulate longer evolution times before decoherence sets in on current devices.
If the scaling advantage holds, PMR could be combined with qubitization in hybrid schemes that switch methods when system parameters cross certain thresholds.

Load-bearing premise

The two chosen Hamiltonians and the chosen conventions for counting gates and qubits are representative of the performance gap that would appear across other Hamiltonians and other counting conventions.

What would settle it

An explicit gate-count or qubit-count calculation for the Rydberg Hamiltonian under PMR that exceeds the corresponding count under qubitization or QSP for the same accuracy target would falsify the reported advantage.

read the original abstract

We present a comparative study of the permutation matrix representation (PMR) method for Hamiltonian simulation alongside other leading quantum algorithms. Our analysis focuses on resource costs for simulating both time-independent and time-dependent Hamiltonians. For the time-independent case, we benchmark PMR against quantum signal processing (QSP) and qubitization, using the Rydberg interaction Hamiltonian as a representative example. For the time-dependent case, we compare the time-dependent extension of PMR with the quantum highly oscillatory protocol (qHOP), applied to a Floquet-driven transverse field Ising model in arbitrary spatial dimensions. In both regimes, we find that PMR offers complementary advantages in resource requirements and exhibits favorable scaling with certain system parameters, suggesting that it may provide practical benefits on resource-constrained quantum hardware.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

PMR shows resource edges on the two tested Hamiltonians but the paper leaves open whether those edges are general or model-specific.

read the letter

The main point is that this paper benchmarks the existing permutation matrix representation against QSP, qubitization, and qHOP on a Rydberg Hamiltonian for time-independent simulation and on a Floquet-driven transverse-field Ising model for the time-dependent case. It reports that PMR gives complementary resource savings and better scaling in some parameters for those two systems.

The concrete comparisons are the useful part. The authors lay out gate counts, qubit overhead, and scaling behavior for these specific models, which can help someone deciding which algorithm to run on hardware that looks like Rydberg atoms or Ising chains. That kind of side-by-side accounting is still fairly rare, so the numbers themselves have some practical value even if no new algorithm is introduced.

The soft spot is the narrow scope. The advantages are shown only for these two Hamiltonians, and the paper does not demonstrate that the gains come from intrinsic properties of PMR rather than from the permutation sparsity or interaction structure that these models happen to have. If the resource formulas or gate-set assumptions are sensitive to those details, the headline claim does not generalize. The stress-test concern about model dependence looks like it holds up.

This is a paper for people who already work on near-term simulation of Rydberg or Floquet systems and want a data point on algorithm choice. It is not aimed at readers looking for new theoretical machinery or broad guarantees. The comparisons are honest enough and engage the relevant literature, so the work is coherent on its own terms.

I would send it to peer review. The benchmarks are specific enough that referees can check the counting and ask for more models if needed.

Referee Report

1 major / 0 minor

Summary. The paper presents a comparative resource analysis of the permutation matrix representation (PMR) method for quantum Hamiltonian simulation. For time-independent Hamiltonians it benchmarks PMR against QSP and qubitization on the Rydberg interaction Hamiltonian; for time-dependent cases it compares the time-dependent PMR extension against qHOP on a d-dimensional Floquet-driven transverse-field Ising model. The central claim is that PMR exhibits complementary advantages in resource requirements (gate counts, qubit overhead) and favorable scaling with system parameters, potentially benefiting resource-constrained hardware.

Significance. If the explicit resource counts and scaling relations hold and the advantages prove intrinsic to PMR rather than model-specific, the work would supply a useful data point for algorithm selection on near-term devices by identifying concrete trade-offs between PMR and established methods such as QSP/qHOP.

major comments (1)

[Abstract and benchmark sections] The headline claim of complementary advantages rests on explicit resource counts performed only for the Rydberg interaction Hamiltonian (time-independent) and the Floquet-driven TFIM in arbitrary dimensions (time-dependent). The manuscript provides no general argument or additional examples demonstrating that the observed advantages arise from intrinsic PMR properties (e.g., permutation-matrix structure) rather than from special features of these two models such as interaction locality or sparsity; without such justification the comparison does not establish broader applicability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and constructive feedback. We address the major comment below and will revise the manuscript to better delineate the scope of our claims.

read point-by-point responses

Referee: [Abstract and benchmark sections] The headline claim of complementary advantages rests on explicit resource counts performed only for the Rydberg interaction Hamiltonian (time-independent) and the Floquet-driven TFIM in arbitrary dimensions (time-dependent). The manuscript provides no general argument or additional examples demonstrating that the observed advantages arise from intrinsic PMR properties (e.g., permutation-matrix structure) rather than from special features of these two models such as interaction locality or sparsity; without such justification the comparison does not establish broader applicability.

Authors: We agree that the explicit comparisons are performed on two specific Hamiltonians and that the manuscript does not contain a general theorem establishing advantages for arbitrary Hamiltonians. The Rydberg interaction and Floquet-driven TFIM were selected because they are standard, physically relevant benchmarks whose interaction terms admit exact decompositions into (sums of) permutation matrices; this structural property is what PMR exploits to achieve the reported gate counts and scaling. To address the concern, we will revise the abstract and add a new subsection clarifying that the observed advantages are tied to Hamiltonians possessing a permutation-matrix representation (a class that includes but is not limited to local spin models with the sparsity patterns studied here). We will also explicitly state the limitations of the current evidence and the need for further examples to assess broader applicability. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical resource comparison on fixed benchmarks

full rationale

The paper performs explicit resource counting and scaling comparisons for two concrete Hamiltonians (Rydberg for time-independent; d-dimensional Floquet TFIM for time-dependent) against QSP/qubitization and qHOP. No equations, parameter fits, or derivations are presented that reduce to their own inputs by construction. No self-citation is invoked as a load-bearing uniqueness theorem or ansatz. The central claims are stated outcomes of the benchmark calculations rather than tautological re-statements of the method definition. This is the normal case of a self-contained empirical analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no equations, fitted constants, or postulated entities; ledger is therefore empty.

pith-pipeline@v0.9.1-grok · 5652 in / 961 out tokens · 22985 ms · 2026-06-29T07:22:50.966527+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 1 canonical work pages · 1 internal anchor

[1]

For simplicity, we present the detailed com- putation in Appendix A and summarize the results here

With qubitization We write the Hamiltonian in LCU block encod- ing [3] by decomposing it entirely in terms of Pauli strings. For simplicity, we present the detailed com- putation in Appendix A and summarize the results here. The Hamiltonian can be written as: HRyd := MX i=1 αiPi ,(15) whereα i ≥0 and theP i are Pauli strings which in- clude any negative s...
[2]

(25) The qubit cost of the qubitization approach stems mainly from the block-encoding i.e

+N 3 log(1/ϵ) . (25) The qubit cost of the qubitization approach stems mainly from the block-encoding i.e. from the PREPa operation, requiringO(logM)≈ O(logN) ancilla qubits. There will be two additional ancilla qubits required for qubitization, but they may be omitted from the overall qubit cost
[3]

This is straightforward since the only non- diagonal term in the Hamiltonian is the PauliX operator

With PMR We begin by writing the Hamiltonian in the PMR form. This is straightforward since the only non- diagonal term in the Hamiltonian is the PauliX operator. The Hamiltonian can be written as: HRyd =D 0 + MX i=1 DiPi ,(26) where D0 =− δ 2 NX i=1 Zi + X i<j C6 |ri −r j|6 ninj ,(27) andP i =X i,D i = Ωi 2 ·I. From this decom- position, we can see that ...
[4]

Comparison The two approaches exhibit different scaling be- haviors, as summarized in Table I. A notable dis- tinction is that PMR removes explicit dependence on the magnitude of diagonal terms (though complexity still depends on the cost of evaluating diagonal ener- gies) whereas qubitization’s complexity depends on all three parameters Ω,δ, andC 6. This...
[5]

We can further utilize the Hamiltonian structure to implement an approximation of PMR, which has even better scaling in terms ofN

+N 3 log(1/ϵ) O(logN) PMR O N3tΩ +tΩN 2 logN O(logN) PMR approximation O tN2Ω tC′ 6 ϵ 1/5 +tΩN 2 logN O(logN) Table I.Comparison of algorithm costs for time-independent simulation.While qubitization achieves better gate cost scaling than many approaches, the PMR approach offers a superior scaling in terms of various system parameters. We can further utili...
[6]

The main contributors to the complexity are the con- struction of the block encoding and the implementa- tion of QSP

With qHOP We analyze the complexity of the algorithm as applied to the simulation of the above Hamiltonian. The main contributors to the complexity are the con- struction of the block encoding and the implementa- tion of QSP. We follow corollary 2 of Ref. [7], which gives the complexity as O min α2 BT 2 ϵ log αBT ϵ , αBT+ α1/2 B (αAB +β B)1/2T 3/2 ϵ1/2 ×l...
[7]

First, the time- dependent part of the Hamiltonian must be ex- panded in the PMR form

With PMR We now analyze the same Hamiltonian when sim- ulated using the PMR algorithm. First, the time- dependent part of the Hamiltonian must be ex- panded in the PMR form. The expansion takes the form: V(t) = MX i=1 Di(t)Pi ,(58) whereM=N,P i =X i, and: Di(t) = KX k=1 exp(Λ(k) i t)D(k) i ,(59) whereK= 2, Λ (k) i = (−1) kjωfor alli(wherej=√−1) andD (k) i...
[8]

The scaling withNdiffers between the ap- proaches

Comparison The two algorithms exhibit distinct scaling char- acteristics, as summarized in Table II. The scaling withNdiffers between the ap- proaches. PMR exhibitsO(N 2 logN) scaling, which may be favorable for simulations involving a large number of qubits compared to qHOP’s higher poly- nomial scaling inN(for both choices in the min- imum). This differ...
[9]

A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of trotter error with commutator scaling, Phys. Rev. X11, 011020 (2021)

2021
[10]

D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Simulating hamiltonian dynam- ics with a truncated taylor series, Phys. Rev. Lett. 114, 090502 (2015)

2015
[11]

G. H. Low and I. L. Chuang, Hamiltonian simulation by qubitization, Quantum3, 163 (2019)

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[12]

Kalev and I

A. Kalev and I. Hen, Quantum algorithm for sim- ulating hamiltonian dynamics with an off-diagonal series expansion, Quantum5, 426 (2021)

2021
[13]

Kalev and I

A. Kalev and I. Hen, A simple quantum simula- tion algorithm with near-optimal precision scaling, Quantum Sci. Technol.10, 045052 (2025)

2025
[14]

Y.-H. Chen, A. Kalev, and I. Hen, Quantum algo- rithm for time-dependent hamiltonian simulation by permutation expansion, PRX Quantum2, 030342 (2021)

2021
[15]

D. An, D. Fang, and L. Lin, Time-dependent hamil- tonian simulation of highly oscillatory dynamics and superconvergence for schr¨ odinger equation, Quan- tum6, 690 (2022)

2022
[16]

Albash, G

T. Albash, G. Wagenbreth, and I. Hen, Off-diagonal expansion quantum monte carlo, Phys. Rev. E96, 063309 (2017)

2017
[17]

Gupta, T

L. Gupta, T. Albash, and I. Hen, Permutation ma- trix representation quantum monte carlo, J. Stat. Mech.2020, 073105 (2020)

2020
[18]

G. H. Low and I. L. Chuang, Optimal hamilto- nian simulation by quantum signal processing, Phys. Rev. Lett.118, 010501 (2017)

2017
[19]

D. W. Berry, A. M. Childs, and R. Kothari, Hamil- tonian simulation with nearly optimal dependence on all parameters, in2015 IEEE 56th Annual Sym- posium on Foundations of Computer Science(IEEE,
[20]

Abramowitz and I

M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Se- ries No. 55 (1964)

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[21]

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, Dipole blockade and quantum information processing in mesoscopic atomic ensembles, Phys. Rev. Lett.87, 037901 (2001)

2001
[22]

Saffman, T

M. Saffman, T. G. Walker, and K. Mølmer, Quan- tum information with rydberg atoms, Rev. Mod. Phys.82, 2313 (2010)

2010
[23]

Jaksch, J

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cˆ ot´ e, and M. D. Lukin, Fast quantum gates for neutral atoms, Phys. Rev. Lett.85, 2208 (2000)

2000
[24]

Bernien, S

H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. En- dres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Prob- ing many-body dynamics on a 51-atom quantum simulator, Nature551, 579–584 (2017)

2017
[25]

Y. O. Dudin, L. Li, F. Bariani, and A. Kuzmich, Ob- servation of coherent many-body rabi oscillations, Nature Phys8, 790–794 (2012)

2012
[26]

Kalev and I

A. Kalev and I. Hen, An integral-free representation of the dyson series using divided differences, New J. Phys.23, 103035 (2021)

2021
[27]

Gily´ en, Y

A. Gily´ en, Y. Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and be- yond: exponential improvements for quantum ma- trix arithmetics, inProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Comput- ing, STOC ’19 (ACM, 2019) p. 193–204

2019
[28]

Magnus, On the exponential solution of differen- 12 tial equations for a linear operator, Commun

W. Magnus, On the exponential solution of differen- 12 tial equations for a linear operator, Commun. Pure Appl. Math.7, 649 (1954)

1954
[29]

Iserles and S

A. Iserles and S. P. Nørsett, On the solution of lin- ear differential equations in Lie groups, Philosoph- ical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences357, 983 (1999)

1999
[30]

Weitenberg and J

C. Weitenberg and J. Simonet, Tailoring quan- tum gases by floquet engineering, Nat. Phys.17, 1342–1348 (2021)

2021
[31]

A. Sen, D. Sen, and K. Sengupta, Analytic ap- proaches to periodically driven closed quantum sys- tems: methods and applications, J. Phys.: Condens. Matter33, 443003 (2021)

2021
[32]

S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Sha- har, Continuous quantum phase transitions, Rev. Mod. Phys.69, 315 (1997)

1997
[33]

Sachdev,Quantum Phase Transitions, 2nd ed

S. Sachdev,Quantum Phase Transitions, 2nd ed. (Cambridge University Press, 2011)

2011
[34]

G. F. Newell and E. W. Montroll, On the theory of the ising model of ferromagnetism, Rev. Mod. Phys. 25, 353 (1953)

1953
[35]

H¨ aggkvist, A

R. H¨ aggkvist, A. Rosengren, P. H. Lundow, K. Markstr¨ om, D. Andr´ en, and P. Kundrotas, On the ising model for the simple cubic lattice, Ad- vances in Physics56, 653 (2007)

2007
[36]

G. H. Low and N. Wiebe, Hamiltonian simulation in the interaction picture (2019), arXiv:1805.00675 [quant-ph]. Appendix A: LCU block encoding for the Rydberg Hamiltonian Recall the Hamiltonian (14). Let us defineC ij := C6 |ri−rj |6 . First let us compute the interaction terms in terms of Pauli strings: X i<j Cijninj =X i<j Cij(I+Z i)(I+Zj) =X i<j CijI+X ...

work page internal anchor Pith review Pith/arXiv arXiv 2019

[1] [1]

For simplicity, we present the detailed com- putation in Appendix A and summarize the results here

With qubitization We write the Hamiltonian in LCU block encod- ing [3] by decomposing it entirely in terms of Pauli strings. For simplicity, we present the detailed com- putation in Appendix A and summarize the results here. The Hamiltonian can be written as: HRyd := MX i=1 αiPi ,(15) whereα i ≥0 and theP i are Pauli strings which in- clude any negative s...

[2] [2]

(25) The qubit cost of the qubitization approach stems mainly from the block-encoding i.e

+N 3 log(1/ϵ) . (25) The qubit cost of the qubitization approach stems mainly from the block-encoding i.e. from the PREPa operation, requiringO(logM)≈ O(logN) ancilla qubits. There will be two additional ancilla qubits required for qubitization, but they may be omitted from the overall qubit cost

[3] [3]

This is straightforward since the only non- diagonal term in the Hamiltonian is the PauliX operator

With PMR We begin by writing the Hamiltonian in the PMR form. This is straightforward since the only non- diagonal term in the Hamiltonian is the PauliX operator. The Hamiltonian can be written as: HRyd =D 0 + MX i=1 DiPi ,(26) where D0 =− δ 2 NX i=1 Zi + X i<j C6 |ri −r j|6 ninj ,(27) andP i =X i,D i = Ωi 2 ·I. From this decom- position, we can see that ...

[4] [4]

Comparison The two approaches exhibit different scaling be- haviors, as summarized in Table I. A notable dis- tinction is that PMR removes explicit dependence on the magnitude of diagonal terms (though complexity still depends on the cost of evaluating diagonal ener- gies) whereas qubitization’s complexity depends on all three parameters Ω,δ, andC 6. This...

[5] [5]

We can further utilize the Hamiltonian structure to implement an approximation of PMR, which has even better scaling in terms ofN

+N 3 log(1/ϵ) O(logN) PMR O N3tΩ +tΩN 2 logN O(logN) PMR approximation O tN2Ω tC′ 6 ϵ 1/5 +tΩN 2 logN O(logN) Table I.Comparison of algorithm costs for time-independent simulation.While qubitization achieves better gate cost scaling than many approaches, the PMR approach offers a superior scaling in terms of various system parameters. We can further utili...

[6] [6]

The main contributors to the complexity are the con- struction of the block encoding and the implementa- tion of QSP

With qHOP We analyze the complexity of the algorithm as applied to the simulation of the above Hamiltonian. The main contributors to the complexity are the con- struction of the block encoding and the implementa- tion of QSP. We follow corollary 2 of Ref. [7], which gives the complexity as O min α2 BT 2 ϵ log αBT ϵ , αBT+ α1/2 B (αAB +β B)1/2T 3/2 ϵ1/2 ×l...

[7] [7]

First, the time- dependent part of the Hamiltonian must be ex- panded in the PMR form

With PMR We now analyze the same Hamiltonian when sim- ulated using the PMR algorithm. First, the time- dependent part of the Hamiltonian must be ex- panded in the PMR form. The expansion takes the form: V(t) = MX i=1 Di(t)Pi ,(58) whereM=N,P i =X i, and: Di(t) = KX k=1 exp(Λ(k) i t)D(k) i ,(59) whereK= 2, Λ (k) i = (−1) kjωfor alli(wherej=√−1) andD (k) i...

[8] [8]

The scaling withNdiffers between the ap- proaches

Comparison The two algorithms exhibit distinct scaling char- acteristics, as summarized in Table II. The scaling withNdiffers between the ap- proaches. PMR exhibitsO(N 2 logN) scaling, which may be favorable for simulations involving a large number of qubits compared to qHOP’s higher poly- nomial scaling inN(for both choices in the min- imum). This differ...

[9] [9]

A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of trotter error with commutator scaling, Phys. Rev. X11, 011020 (2021)

2021

[10] [10]

D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Simulating hamiltonian dynam- ics with a truncated taylor series, Phys. Rev. Lett. 114, 090502 (2015)

2015

[11] [11]

G. H. Low and I. L. Chuang, Hamiltonian simulation by qubitization, Quantum3, 163 (2019)

2019

[12] [12]

Kalev and I

A. Kalev and I. Hen, Quantum algorithm for sim- ulating hamiltonian dynamics with an off-diagonal series expansion, Quantum5, 426 (2021)

2021

[13] [13]

Kalev and I

A. Kalev and I. Hen, A simple quantum simula- tion algorithm with near-optimal precision scaling, Quantum Sci. Technol.10, 045052 (2025)

2025

[14] [14]

Y.-H. Chen, A. Kalev, and I. Hen, Quantum algo- rithm for time-dependent hamiltonian simulation by permutation expansion, PRX Quantum2, 030342 (2021)

2021

[15] [15]

D. An, D. Fang, and L. Lin, Time-dependent hamil- tonian simulation of highly oscillatory dynamics and superconvergence for schr¨ odinger equation, Quan- tum6, 690 (2022)

2022

[16] [16]

Albash, G

T. Albash, G. Wagenbreth, and I. Hen, Off-diagonal expansion quantum monte carlo, Phys. Rev. E96, 063309 (2017)

2017

[17] [17]

Gupta, T

L. Gupta, T. Albash, and I. Hen, Permutation ma- trix representation quantum monte carlo, J. Stat. Mech.2020, 073105 (2020)

2020

[18] [18]

G. H. Low and I. L. Chuang, Optimal hamilto- nian simulation by quantum signal processing, Phys. Rev. Lett.118, 010501 (2017)

2017

[19] [19]

D. W. Berry, A. M. Childs, and R. Kothari, Hamil- tonian simulation with nearly optimal dependence on all parameters, in2015 IEEE 56th Annual Sym- posium on Foundations of Computer Science(IEEE,

[20] [20]

Abramowitz and I

M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Se- ries No. 55 (1964)

1964

[21] [21]

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, Dipole blockade and quantum information processing in mesoscopic atomic ensembles, Phys. Rev. Lett.87, 037901 (2001)

2001

[22] [22]

Saffman, T

M. Saffman, T. G. Walker, and K. Mølmer, Quan- tum information with rydberg atoms, Rev. Mod. Phys.82, 2313 (2010)

2010

[23] [23]

Jaksch, J

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cˆ ot´ e, and M. D. Lukin, Fast quantum gates for neutral atoms, Phys. Rev. Lett.85, 2208 (2000)

2000

[24] [24]

Bernien, S

H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. En- dres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Prob- ing many-body dynamics on a 51-atom quantum simulator, Nature551, 579–584 (2017)

2017

[25] [25]

Y. O. Dudin, L. Li, F. Bariani, and A. Kuzmich, Ob- servation of coherent many-body rabi oscillations, Nature Phys8, 790–794 (2012)

2012

[26] [26]

Kalev and I

A. Kalev and I. Hen, An integral-free representation of the dyson series using divided differences, New J. Phys.23, 103035 (2021)

2021

[27] [27]

Gily´ en, Y

A. Gily´ en, Y. Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and be- yond: exponential improvements for quantum ma- trix arithmetics, inProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Comput- ing, STOC ’19 (ACM, 2019) p. 193–204

2019

[28] [28]

Magnus, On the exponential solution of differen- 12 tial equations for a linear operator, Commun

W. Magnus, On the exponential solution of differen- 12 tial equations for a linear operator, Commun. Pure Appl. Math.7, 649 (1954)

1954

[29] [29]

Iserles and S

A. Iserles and S. P. Nørsett, On the solution of lin- ear differential equations in Lie groups, Philosoph- ical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences357, 983 (1999)

1999

[30] [30]

Weitenberg and J

C. Weitenberg and J. Simonet, Tailoring quan- tum gases by floquet engineering, Nat. Phys.17, 1342–1348 (2021)

2021

[31] [31]

A. Sen, D. Sen, and K. Sengupta, Analytic ap- proaches to periodically driven closed quantum sys- tems: methods and applications, J. Phys.: Condens. Matter33, 443003 (2021)

2021

[32] [32]

S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Sha- har, Continuous quantum phase transitions, Rev. Mod. Phys.69, 315 (1997)

1997

[33] [33]

Sachdev,Quantum Phase Transitions, 2nd ed

S. Sachdev,Quantum Phase Transitions, 2nd ed. (Cambridge University Press, 2011)

2011

[34] [34]

G. F. Newell and E. W. Montroll, On the theory of the ising model of ferromagnetism, Rev. Mod. Phys. 25, 353 (1953)

1953

[35] [35]

H¨ aggkvist, A

R. H¨ aggkvist, A. Rosengren, P. H. Lundow, K. Markstr¨ om, D. Andr´ en, and P. Kundrotas, On the ising model for the simple cubic lattice, Ad- vances in Physics56, 653 (2007)

2007

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G. H. Low and N. Wiebe, Hamiltonian simulation in the interaction picture (2019), arXiv:1805.00675 [quant-ph]. Appendix A: LCU block encoding for the Rydberg Hamiltonian Recall the Hamiltonian (14). Let us defineC ij := C6 |ri−rj |6 . First let us compute the interaction terms in terms of Pauli strings: X i<j Cijninj =X i<j Cij(I+Z i)(I+Zj) =X i<j CijI+X ...

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