Developing techniques for Simulation of SU(3) Quantum Field Theories on State-of-the-Art Quantum Devices

Ivan Chernyshev

REVIEW 3 major objections 4 minor 300 references

Developing techniques for Simulation of SU(3) Quantum Field Theories on State-of-the-Art Quantum Devices

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Dense three-flavor neutrino systems that start in all three flavors develop the highest, most persistent non-stabilizer 'magic', making initial flavor composition a control knob for quantum advantage.

desk verdict Solid SU(3) simulation toolbox; headline neutrino-magic claim needs qualification on basis-dependence and scan scope. read the letter →

arxiv 2502.02502 v1 pith:ALUD3LYI submitted 2025-02-04 quant-ph hep-latnucl-th

Ivan Chernyshev This is my paper · ORCID

classification quant-phhep-latnucl-th

keywords quantumsimulationSU(3)latticegaugetheorycollectiveneutrinooscillationsnon-stabilizernessmagicvariationaleigensolverTrotterizationNISQdevices

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This thesis develops circuit-level techniques for simulating SU(3) quantum field theories—lattice QCD and dense three-flavor neutrino systems—on noisy near-term quantum devices, and it reports a physics result that emerges from those simulations. The result is that, under collective neutrino oscillations with all three physical flavors, initial states containing all three flavors develop the highest and most persistent 'magic', a measure of how far a quantum state is from being classically simulable. Because magic is considered necessary for quantum advantage, the finding suggests that initial flavor composition is a practical control knob for choosing which neutrino simulations are most likely to outperform classical computers. The thesis also contributes a reusable toolbox: VQE vacuum preparation with gradient descent and Lanczos preconditioning, Trotterized time-evolution circuits for QCD and weak decays, and qutrit and qubit circuits for neutrino oscillations that run on trapped-ion and superconducting hardware.

What carries the argument

The load-bearing object for the neutrino result is the collective-oscillation Hamiltonian $H_{\nu\nu} = \sum_{i<j} J_{ij}\,\boldsymbol{\lambda}^{(i)}\cdot\boldsymbol{\lambda}^{(j)}$, written in terms of SU(3) Gell-Mann matrices, together with the M2 magic measure, a stabilizer Rényi entropy that quantifies deviation from the classically simulable stabilizer states. Trotterized time evolution is implemented either on qutrits through the natural $\boldsymbol{\lambda}\cdot\boldsymbol{\lambda}$ interaction or on qubits via a swap network; the swap network converts the all-to-all interaction into nearest-neighbor form at no extra circuit cost. For the lattice-gauge-theory half of the thesis, the analogous machinery is the axial-gauge Kogut-Susskind Hamiltonian, in which Gauss's law eliminates the gauge links and turns the chromo-electric energy into a non-local sum over color charges, plus VQE circuits that prepare the vacuum and hadronic states.

What would settle it

Prepare $N_\nu=8$ systems in single-flavor and all-three-flavor tensor-product states, evolve them under the collective Hamiltonian, and tomographically measure M2 per neutrino over time; if any state with fewer than three flavors has higher or more persistent asymptotic magic, the paper's ordering claim is refuted. Adding a matter potential to the Hamiltonian and repeating the scan is a second decisive test.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the asymptotic magic per neutrino—quantified by the M2 stabilizer Rényi entropy—is larger, and stays larger over time, for collective-neutrino-oscillation systems whose initial tensor-product state contains all three flavors than for systems starting in one flavor alone. This ordering is established numerically for systems of up to eight neutrinos under the all-to-all two-body SU(3) flavor Hamiltonian, using neutrino mixing parameters taken from experiment. The thesis presents this as a result with implications for the Standard Model: dense three-flavor neutrino environments, such as the interiors of core-collapse supernovae or compact-object mergers, are natural places to look for quantum advantage in simulation. The same work develops the circuit toolbox that makes the claim testable, including swap-network Trotterizations for nearest-neighbor hardware and a two-neutrino qutrit circuit with only four two-qutrit entangling gates.

Load-bearing premise

The magic-ordering result is computed for tensor-product initial states with up to eight neutrinos under the collective-oscillation Hamiltonian; if a wider class of initial states, a different magic measure, or extra physical terms such as matter or collisions changes which initial flavor composition wins, the headline claim fails.

Editorial extensions

If this is right

Preparing initial neutrino states that contain all three flavors should be prioritized in quantum simulations of dense neutrino systems, since the numerics show these states carry the most magic.
The M2 measure can serve as a state-selection benchmark: experiments aiming at quantum advantage should evolve until magic is high and persistent rather than stopping at early times.
The qutrit encoding reduces the two-neutrino circuit to four two-qutrit entangling gates, and the qubit swap network makes the all-to-all neutrino interaction implementable on nearest-neighbor superconducting devices with no extra SWAP overhead.
The VQE and domain-decomposition results indicate that gradient-descent optimization with Lanczos-preconditioned starts, rather than Bayesian optimization, is the more scalable route for SU(3) lattice-gauge-theory state preparation.
The Trotterized circuits for beta decay and neutrinoless double beta decay provide a reusable starting point for simulating weak-interaction processes in 1+1D lattice QCD on near-term hardware.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the magic ordering holds for larger $N_\nu$, initial flavor composition is not just a physics detail but a resource-allocation decision: the hardest circuits should be spent on all-three-flavor initial states.
The same SU(3) all-to-all structure appears in other many-body settings, such as color systems in quark matter; it is a natural extension to test whether maximally symmetric initial states generically maximize non-stabilizerness there, though the thesis does not make that claim.
Persistence may matter more than peak magic under hardware noise; an implied, testable extension is to compare the magic decay time against device decoherence time to see whether all-three-flavor states are also the most noise-resilient.
A scaling study beyond $N_\nu=8$, using tensor networks or sampling, could reveal whether the per-neutrino magic gap between flavor compositions saturates or grows, sharpening the experimental target.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this is a solid thesis: the SU(3) lattice-gauge-theory circuit constructions in Chapters 3–6 are careful, honestly reported, and genuinely tested on hardware. Second, the headline magic claim in Chapter 7 – that three-flavor neutrino systems with all three flavors have the highest, most persistent magic – is a numerical observation over a finite scan, not a proven theorem, and the abstract states it without qualification.

What is new: the thesis presents the first quantum-device simulation of 1+1D SU(3) lattice gauge theory with quarks (Ch. 3), new qutrit and qubit circuits for three-flavor collective neutrino oscillations (Ch. 6), and a numerical magic characterization of ultradense neutrino systems (Ch. 7). The device work is real: beta-decay on Quantinuum H1-1, neutrino oscillations on both H1-1 and IBM torino, with explicit circuits, Trotter steps, resource counts and error-mitigation details. The authors are also honest about what did not work – VQE on hardware failed to converge in some cases, and post-selection was the only mitigation in places.

The soft spots are mostly around Chapter 7. The abstract's 'highest, most-persistent magic' is an empirical result for tensor-product initial states up to N=8, evaluated with the M2 measure in a chosen computational basis. The thesis does not prove the ordering for arbitrary initial states, larger N, or other measures. The stress-test concern about basis-dependence is worth taking seriously: M2 is a stabilizer Renyi entropy, so the ordering could shift if you use a different logical basis (e.g., mass vs flavor) or a different qubit encoding. I don't think that invalidates the numerical result – it just means the claim should be stated as 'in the flavor basis, for the states we scanned' rather than as a Standard Model statement. No code is shipped, which makes exact reproduction harder.

Who it's for: people working on quantum simulation of gauge theories or collective neutrino oscillations. The circuit toolbox is the lasting value. The magic chapter is more of a pointer for future work than a definitive result.

Recommendation: it deserves a serious referee. A referee should ask the author to qualify the abstract, provide the data/code, and address the basis-dependence question. With those changes, the technical core is publishable.

Referee Report

3 major / 4 minor

Summary. This dissertation develops circuit-level techniques for simulating SU(3) lattice gauge theories and three-flavor neutrino oscillations on NISQ hardware, and it studies "magic" (non-stabilizerness) in collective neutrino oscillations. Chapters 2–5 cover VQE preparation of the SU(3) Yang–Mills vacuum, an axial-gauge Hamiltonian for 1+1D QCD, beta-decay and neutrinoless double-beta decay circuits, and optimizations for nearest-neighbor connectivity. Chapter 6 presents qutrit and qubit Trotter circuits for three-flavor collective neutrino oscillations and reports hardware results on Quantinuum H1-1 and IBM ibm torino. Chapter 7 defines the M2 magic measure and reports numerical integrations showing that, among the tensor-product initial states scanned up to Nnu=8, states containing all three flavors achieve higher asymptotic magic per neutrino than all-electron-type initial states (Figs. 7.4–7.6, Tables 7.1–7.2). The abstract generalizes this to "the 3 flavor ultradense neutrino systems with the highest, most-persistent magic" and to "implications for the Standard Model in general." The manuscript is a mixed compilation of prior collaborative work, but it is self-contained enough for review.

Significance. The thesis has concrete strengths: Trotter circuit counts are tabulated (Table 6.2 and Tables 3.4, 4.3), ODE solver tolerances are stated in Tables 7.1–7.2, device parameters are collected in Tables 6.6–6.7, and several chapters report genuine hardware runs. Section 5.4 includes a proof of the color-singlet space construction, and Appendix 7.D gives analytic expressions for the one-body magic power. The central magic claim, if it survives scrutiny, is interesting: initial flavor composition would be a control knob for non-stabilizerness, one candidate resource for quantum advantage in neutrino simulations. However, the claim is currently an empirical result over a finite scan, and the M2 measure is stabilizer-basis dependent. The physical significance is therefore real but narrower than the abstract states.

major comments (3)

[§7, Eq. (7.4) and App. 7.C] The M2 measure is defined with respect to a fixed Weyl–Heisenberg group and computational basis, yet the abstract presents the ordering as a property of the neutrino system. A unitary change of basis (e.g., flavor basis vs. mass basis) or a change of encoding (qutrit basis in App. 7.C vs. the qubit mappings used in Chapter 6) can change M2 values. The thesis does not show that the ordering "all three flavors > other tensor products" is invariant under such choices. I request a numerical check of the ordering under at least one alternative basis/encoding, or an explicit argument for stability, before the claim is stated without qualification.
[§7, Figs. 7.4–7.6 and Tables 7.1–7.2] The supporting evidence covers tensor-product initial states with Nnu up to 8. The abstract generalizes to "the 3 flavor ultradense neutrino systems with the highest, most-persistent magic" without restricting to this scanned class. No proof or scaling argument is given for arbitrary Nnu or for non-product or entangled initial states. The abstract and the Chapter 7 conclusions should either be restricted to the scanned class or supplemented with an argument that the ordering persists for larger systems and more general initial states.
[§7, Fig. 7.1 and Tables 7.1–7.2] Fig. 7.1 shows that the one-body magic power M2(U1) varies appreciably when Δm²32 and δm²21 are sampled over their 68% confidence intervals, but Tables 7.1–7.2 report asymptotic per-neutrino magic as point values. Since the abstract makes a universal claim, the multi-neutrino ordering should be checked against parameter variation as well; otherwise the conclusion is tied to a single point in the neutrino parameter space and may not be robust to updated measurements.

minor comments (4)

[Abstract and §§1.1, 1.2.1, 2.5] There are several typographical errors, including "out-of-equilbrium" in the abstract, "aformentioned" in §1.2.1, "of-nonperturbative QCD" in §1.1, and "vaccum" in §2.5; these should be corrected.
[App. 7.E, Tables 7.1–7.2] The table captions refer to "Fig. 4 of the main text," but the corresponding figures in the thesis are numbered 7.4–7.6; the cross-references should be updated.
[§7] The term "persistent" is used informally. The authors should define whether it refers to the late-time asymptotic value of M2 per neutrino and, if a different time horizon is intended, specify the window over which persistence is evaluated.
[§6 and §7] The connection between the qutrit-based magic calculation and the qubit encodings used for hardware in Chapter 6 is not discussed; a short remark in Chapter 7 explaining how the encoding choice affects the reported magic values would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the thesis's results are computed from external Hamiltonian parameters, exact diagonalization benchmarks, and standard magic measures, with no target result built into the inputs.

full rationale

The derivation chain is self-contained and non-circular. The central claim—that three-flavor ultradense neutrino systems starting with all three flavors have the highest, most-persistent magic—is obtained by numerically evolving the collective-neutrino-oscillation Hamiltonian with externally fixed parameters from Refs. [230, 5] and evaluating the standard M2 stabilizer-Rényi measure against tensor-product initial states. No fitted parameter, ansatz, or cited result is used to define the magic ordering; the ordering is the output of the computation. The hardware and circuit chapters are similarly benchmarked against exact diagonalization or exact classical simulation (e.g., β-decay probabilities in Chapter 4 compared with exact results, and collective neutrino flavor evolution in Chapter 6 compared with exact evolution). The thesis does cite prior work from the same research group for circuit building blocks (e.g., Ref. [238] for e^{iθ/2(XY±YX)} building blocks), but these citations are implementation details, not load-bearing justifications of the paper's physical conclusions. The basis-dependence of the magic measure is a legitimate correctness or interpretation concern, but it is not circular: the paper does not define the measure in terms of the claimed ordering, nor does it input the ordering into the calculation. No step in the paper reduces by construction to its own inputs, and no self-citation chain is invoked to forbid alternatives or to force the stated result. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 6 free parameters · 8 assumptions · 0 invented entities

The central technical results rely on standard quantum simulation machinery (Jordan-Wigner, Suzuki-Trotter, VQE, Lanczos) and on domain assumptions about gauge fixing and noise models. No new physical entities are introduced. Free parameters are dominated by variational ansatz angles, regularization choices (h, lambda, truncation), and state-preparation circuit angles.

free parameters (6)

h (color edge-state penalty coefficient) = h -> infinity (large limit)
Added in Eq. (3.7) to push color non-singlet edge states out of the low-lying spectrum; a regularization choice rather than a fit.
lepton state-preparation angles (theta, phi, chi, omega, psi) = theta=-0.83015, phi=-0.83015, chi=0.24090, omega=-1.02630, psi=-1.28030 (Fig. 5.5)
Variational parameters optimized to initialize electron and neutrino ground states on nearest-neighbor PBC circuits for |mM|<=1.
single-plaquette VQE ansatz angles = optimized on IBM Manila; not tabulated as constants
Gradient-descent optimized angles in CP-symmetric and unconstrained ansatze (Ch. 2, Figs. 2.7-2.8).
domain-decomposition stitching angles (R1-R7) = optimized via simulated VQE; values in Ch. 2/C
Givens rotation angles in the domain-stitching ansatz optimized to reduce vacuum energy.
Bayesian optimization regularization lambda = lambda = 0.0036, 0.0009, 1e-6, 1e-9, 1e-12 (Fig. 2.4)
Tikhonov regularization added to covariance matrix; convergence depends on this hand-chosen value.
field truncation cutoff (p,q max) = 3, 6, 8, 31 depending on coupling and accuracy
Truncation of the chromo-electric multiplet basis used throughout Ch. 2; chosen so truncation error in observables is <=1% for p,q<=31.

assumptions (8)

standard math Jordan-Wigner transformation maps fermionic operators to Pauli operators while preserving anticommutation relations.
Used in Chs. 3-5 to map Kogut-Susskind fermions to qubits (Eq. 1.9).
standard math Suzuki-Trotter decomposition approximates time evolution by products of exponentials of individual Hamiltonian terms.
Used for all time-evolution circuits (Eqs. 1.10-1.12).
domain assumption Deviation from stabilizer states is necessary for quantum advantage, motivated by the Gottesman-Knill theorem.
The thesis treats non-stabilizerness as necessary for quantum advantage (Ch. 1.2.1); this is an interpretation of Gottesman-Knill, not a theorem.
domain assumption Axial gauge A_x=0 with Gauss's law determines chromo-electric fields non-locally and allows removing gauge links.
Used in Chs. 3-5 to eliminate gauge fields; valid for 1+1D lattice with OBC and zero background field.
domain assumption Collective neutrino oscillations are described by all-to-all two-body SU(3) flavor interactions with parameters from Refs. [230,5].
Basis for Chs. 6-7 Hamiltonian; if this Hamiltonian is incomplete, the magic results change.
domain assumption ODR error mitigation assumes all errors are depolarizing.
ODR in Eq. (1.6) assumes depolarizing noise; non-depolarizing errors break the ratio relation.
domain assumption The vacuum state of the SU(3) plaquette system respects CP symmetry under link reversal.
Used in Ch. 2 to reduce VQE ansatz parameters; explicitly tested on hardware.
ad hoc to paper Color edge-state penalty term H1 with large h projects onto the color-singlet sector.
Introduced in Eq. (3.7) to lift non-singlet color states; not part of the original Kogut-Susskind Hamiltonian.

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Cite this review

Pith. "Pith review of Developing techniques for Simulation of SU(3) Quantum Field Theories on State-of-the-Art Quantum Devices." pith.science (2026). https://pith.science/paper/ALUD3LYI

@misc{pith2026250202502,
  author       = {Pith},
  title        = {Pith review of: Developing techniques for Simulation of SU(3) Quantum Field Theories on State-of-the-Art Quantum Devices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ALUD3LYI}},
  note         = {Machine review of arXiv:2502.02502}
}

read the original abstract

Quantum computing has long been an experimental technology with the potential to simulate, at scale, phenomena which on classical devices would be too expensive to simulate at any but the smallest scales. Over the last several years, however, it has entered the NISQ era, where the number of qubits are sufficient for quantum advantage but substantial noise on hardware stands in the way of this achievement. This thesis details NISQ device-centered improvements to techniques of quantum simulation of the out-of-equilbrium real-time dynamics of lattice quantum chromodynamics (LQCD) and of dense 3-flavor neutrino systems on digital quantum devices. The first project concerning LQCD is a comparison of methods for implementing the variational quantum eigensolver (VQE) that initializes the ground state of an SU(3) plaquette-chain. The thesis then pivots to a 1+1D lattice of quarks interacting with an SU(3) gauge-field. A VQE-based state-preparation for the vacua and a Trotterized time-evolution circuit is designed and applied to the problems of simulating beta and neutrinoless double beta decay. Finally, these circuits are adapted to a version useable on quantum devices with nearest-neighbor connectivity with minimal overhead, with an eye towards utilizing the higher qubit count of such devices for hadron dynamics and scattering. This thesis covers two projects that concern dense 3-flavor neutrino systems. The first details design and testing of Trotterized time-evolution circuits on state-of-the-art quantum devices. The second, motivated by the Gottesman-Knill theorem's result that deviation from stabilizer states ("magic") is necessary for a problem to exhibit quantum advantage, details results with implications for the Standard Model in general that the 3 flavor ultradense neutrino systems with the highest, most-persistent magic are those that start with neutrinos in all 3 flavors.

Figures

Figures reproduced from arXiv: 2502.02502 by the authors.

**Figure 1.1.** An example of a quantum circuit diagram, with each component labeled. The [PITH_FULL_IMAGE:figures/full_fig_p039_1_1.png] view at source ↗

**Figure 1.2.** Top: a graphic of a quantum circuit whose form is specified by Eq. [PITH_FULL_IMAGE:figures/full_fig_p048_1_2.png] view at source ↗

**Figure 2.1.** An SU(3) plaquette in a 1D chain of plaquettes. The electric multiplet basis [PITH_FULL_IMAGE:figures/full_fig_p057_2_1.png] view at source ↗

Figures from the paper (103 more)

**Figure 2.2.** Figure 2.2: A single SU(3) plaquette. p and q label the chromo-electric flux on each link. [PITH_FULL_IMAGE:figures/full_fig_p058_2_2.png]

**Figure 2.3.** Figure 2.3: This figure shows the dimension of the Krylov subspace required for the overlap [PITH_FULL_IMAGE:figures/full_fig_p060_2_3.png]

**Figure 2.4.** Figure 2.4: The relative error in the estimation of the vacuum energy obtained by perform [PITH_FULL_IMAGE:figures/full_fig_p063_2_4.png]

**Figure 2.5.** Figure 2.5: The relative error in the estimation of the vacuum energy obtained by performing [PITH_FULL_IMAGE:figures/full_fig_p064_2_5.png]

**Figure 2.6.** Figure 2.6: The left panel shows the number of steps needed for VQE using a backtracking [PITH_FULL_IMAGE:figures/full_fig_p066_2_6.png]

**Figure 2.7.** Figure 2.7: Variational state preparation of the vacuum state for a single plaquette truncated [PITH_FULL_IMAGE:figures/full_fig_p068_2_7.png]

**Figure 2.8.** Figure 2.8: Variational state preparation of the vacuum state for a single plaquette truncated [PITH_FULL_IMAGE:figures/full_fig_p069_2_8.png]

**Figure 2.9.** Figure 2.9: A lattice composed of a chain of plaquettes. [PITH_FULL_IMAGE:figures/full_fig_p071_2_9.png]

**Figure 2.10.** Figure 2.10: The left panel shows the overlap of different domain decompositions with the [PITH_FULL_IMAGE:figures/full_fig_p076_2_10.png]

**Figure 2.11.** Figure 2.11: The top panel shows the expectation of the electric energy for a five plaquette [PITH_FULL_IMAGE:figures/full_fig_p077_2_11.png]

**Figure 2.12.** Figure 2.12: The left panel shows the expectation of a plaquette operator at the center of a [PITH_FULL_IMAGE:figures/full_fig_p078_2_12.png]

**Figure 2.13.** Figure 2.13: Variational state preparation of the vacuum state for a two plaquette system [PITH_FULL_IMAGE:figures/full_fig_p080_2_13.png]

**Figure 2.14.** Figure 2.14: The top circuit is used to compute the expectation of [PITH_FULL_IMAGE:figures/full_fig_p084_2_14.png]

**Figure 2.15.** Figure 2.15: An infinite chain of SU(3) plaquettes can be mapped onto a 1D quantum system [PITH_FULL_IMAGE:figures/full_fig_p087_2_15.png]

**Figure 2.16.** Figure 2.16: This figure shows the required sequence of SVDs that must be performed to [PITH_FULL_IMAGE:figures/full_fig_p088_2_16.png]

**Figure 3.1.** Figure 3.1: The encoding of Nf = 2 QCD onto a lattice of spins describing L = 2 spatial sites. Staggering is used to discretize the quark fields, which doubles the number of lattice sites, with (anti)quarks on (odd) even sites. The chromo-electric field resides on the links betw…

**Figure 3.2.** Figure 3.2: The spectrum of the Hamiltonian as the couplings [PITH_FULL_IMAGE:figures/full_fig_p103_3_2.png]

**Figure 3.3.** Figure 3.3: The spectrum of the Hamiltonian as g increases for h = 0. When g = h = 0 there is an exact SU(12) symmetry and the σ- and π-mesons are a part of the antisymmetric 66 irrep. When g > 0 but h = 0 the spectrum splits into irreps of global SU(3)c ⊗ SU(2)f , and non-singl…

**Figure 3.4.** Figure 3.4: The mass splitting between the σ- and π-mesons for L = 1 (left panel) and L = 2 (right panel). makes this system valuable from the standpoint of quantum simulations of the formation of nuclei in a model of reduced complexity. The mass of the ∆, M∆, and the binding en…

**Figure 3.5.** Figure 3.5: The decomposition of vacuum energy (EΩ) and the masses of the lightest hadrons (Mσ, Mπ and M∆) into contributions from the mass, the kinetic and the chromo-electric field terms in the Hamiltonian, defined in axial gauge, for 1 + 1D QCD with Nf = L = 2 and m = g = 1. …

**Figure 3.6.** Figure 3.6: The left panel shows the deuteron binding energy, [PITH_FULL_IMAGE:figures/full_fig_p108_3_6.png]

**Figure 3.7.** Figure 3.7: Iterative convergence of the energy, masses and wavefunctions for the three [PITH_FULL_IMAGE:figures/full_fig_p111_3_7.png]

**Figure 3.8.** Figure 3.8: The linear entropy between quarks and antiquarks in [PITH_FULL_IMAGE:figures/full_fig_p112_3_8.png]

**Figure 3.9.** Figure 3.9: The expectation value of quark occupation in the [PITH_FULL_IMAGE:figures/full_fig_p113_3_9.png]

**Figure 3.10.** Figure 3.10: The quantum circuit that implements time evolution by the mass term, [PITH_FULL_IMAGE:figures/full_fig_p114_3_10.png]

**Figure 3.11.** Figure 3.11: A circuit that implements the time evolution from two sequential hopping [PITH_FULL_IMAGE:figures/full_fig_p115_3_11.png]

**Figure 3.12.** Figure 3.12: Two GHZ state-preparation circuits. become G † (σ +σ −σ −σ + + h.c.) G = 1 8 (IIZI − ZIZZ − ZZZZ + ZIZI +IZZI − IIZZ − IZZZ + ZZZI) , G˜† (σ +σ −σ −σ + + h.c.) G˜ = 1 8 (IIIZ − IZZZ − IIZZ + ZIIZ +IZIZ − ZZZZ − ZIZZ + ZZIZ) . (3.12) Another simplification comes from…

**Figure 3.13.** Figure 3.13: The circuits that implement the time evolution of exp( [PITH_FULL_IMAGE:figures/full_fig_p118_3_13.png]

**Figure 3.14.** Figure 3.14: The trivial vacuum-to-vacuum and trivial vacuum-to- [PITH_FULL_IMAGE:figures/full_fig_p126_3_14.png]

**Figure 3.15.** Figure 3.15: The number of Trotter steps, NTrott, required to achieve a systematic fractional error of ϵTrott ≤ 0.1 at time t in the trivial vacuum-to-vacuum probability (left panel) and the trivial vacuum-to-drdr probability (right panel) for QCD with Nf = 2 and m = g = L = 1. …

**Figure 3.16.** Figure 3.16: Building upon the trivial vacuum, this circuit initializes the most general [PITH_FULL_IMAGE:figures/full_fig_p128_3_16.png]

**Figure 3.17.** Figure 3.17: A circuit that initializes the most general [PITH_FULL_IMAGE:figures/full_fig_p130_3_17.png]

**Figure 3.18.** Figure 3.18: A circuit that implements Um(t) = exp(−iHmt) for Nf = 1 and L = 1 [PITH_FULL_IMAGE:figures/full_fig_p131_3_18.png]

**Figure 3.19.** Figure 3.19: A circuit that implements exp −i t 2 (σ +ZZσ− + h.c.) [PITH_FULL_IMAGE:figures/full_fig_p131_3_19.png]

**Figure 3.20.** Figure 3.20: Two potential quantum device topologies for the implementation of Trotterized [PITH_FULL_IMAGE:figures/full_fig_p132_3_20.png]

**Figure 3.21.** Figure 3.21: The trivial vacuum-to-vacuum (left panel) and trivial vacuum-to- [PITH_FULL_IMAGE:figures/full_fig_p135_3_21.png]

**Figure 3.22.** Figure 3.22: Histograms of the post-processed vacuum-to-vacuum results obtained using [PITH_FULL_IMAGE:figures/full_fig_p136_3_22.png]

**Figure 3.23.** Figure 3.23: Iterative convergence of the energy, masses and wavefunctions for the three [PITH_FULL_IMAGE:figures/full_fig_p149_3_23.png]

**Figure 3.24.** Figure 3.24: The X and Z circuit identities. X circuit identity to move all Xs past the CNOTs. The third equality moves the Zs past the controls of the CNOTs and uses the Z circuit identity. The other Pauli strings are diagonalized in a similar manner. It is also straightforward…

**Figure 3.25.** Figure 3.25: The diagonalization of XXY Y via a GHZ state-preparation circuit [PITH_FULL_IMAGE:figures/full_fig_p150_3_25.png]

**Figure 3.26.** Figure 3.26: The complete circuit that implements a single Trotter step for [PITH_FULL_IMAGE:figures/full_fig_p151_3_26.png]

**Figure 3.27.** Figure 3.27: The complete circuit that implements a single Trotter step for [PITH_FULL_IMAGE:figures/full_fig_p152_3_27.png]

**Figure 3.28.** Figure 3.28: The time evolution of the decomposition of the energy starting from the trivial [PITH_FULL_IMAGE:figures/full_fig_p153_3_28.png]

**Figure 3.29.** Figure 3.29: The trivial vacuum-to-BB probability for 1 + 1D QCD with m = g = L = 1. Shown are the results obtained from exact exponentiation of the Hamiltonian (dashed red curve) and from the Trotterized implementation with 1, 2 and 3 Trotter steps. to ∆∆∆∆. 3.H Supplementary D…

**Figure 4.1.** Figure 4.1: The qubit layout of a L = 2 lattice, where fermions and anti-fermions are grouped together (which will be preferred if electromagnetism is included). This layout extends straightforwardly to L > 2. is given by Hquarks → 1 2 X L−1 l=0 X f=u,d X 2 c=0 mf [PITH_FULL_IM…

**Figure 4.2.** Figure 4.2: The L = 1 lattice qubit layout of one generation of the SM that is used in this paper for quantum simulation. Fermion (anti-fermion) sites are occupied when the spin is up (down), and the spins at the lepton sites represent occupation in the tilde basis. Specifically…

**Figure 4.3.** Figure 4.3: A quantum circuit for preparing the ∆−-baryon on L = 1 spatial site. i.e., the quark configuration associated with the “bare” baryon in the d-sector and the trivial vacuum in the u-sector. This implies that the dominant contribution to the β-decay is from the ϕ (u)† …

**Figure 4.4.** Figure 4.4: The probability of β-decay, ∆− → ∆0 + e + ν, with mu = 0.9, md = 2.1, me,ν = 0, g = 2 and G = 0.5 computed via exact diagonalization (dotted black line) and on the qiskit quantum simulator [640] using 1, 2, 5, 20 Trotter steps. Entanglement in quantum simulations of …

**Figure 4.5.** Figure 4.5: The linear entanglement entropy, SL, between quarks and antiquarks plus leptons during the β-decay of an initial ∆−-baryon. somewhat uninteresting, it does demonstrate that when multiple final states are accessible, the time-dependence of the entanglement structure m…

**Figure 4.6.** Figure 4.6: The probability of β-decay, ∆− → ∆0+e+ν, with mu = 0.9, md = 2.1, me,ν = 0, g = 2 and G = 0.5, using one (left panel) and two (right panel) Trotter steps (requiring 59 and 212 ZZ gates, respectively), as given in [PITH_FULL_IMAGE:figures/full_fig_p182_4_6.png]

**Figure 4.7.** Figure 4.7: Ensemble averages (over 2000 random samples) of the persistence probability [PITH_FULL_IMAGE:figures/full_fig_p192_4_7.png]

**Figure 4.8.** Figure 4.8: Two GHZ state preparation circuits. G † (XXXX + Y Y XX − Y XY X + Y XXY + XY Y X − XY XY + XXY Y + Y Y Y Y )G = IIIZ − ZIIZ + ZZIZ − ZZZZ − IZIZ + IZZZ − IIZZ + ZIZZ , (4.31) [PITH_FULL_IMAGE:figures/full_fig_p193_4_8.png]

**Figure 4.9.** Figure 4.9: A quantum circuit that provides the time evolution associated with the [PITH_FULL_IMAGE:figures/full_fig_p194_4_9.png]

**Figure 4.10.** Figure 4.10: A qubit layout that is efficient for the simulation of [PITH_FULL_IMAGE:figures/full_fig_p195_4_10.png]

**Figure 4.11.** Figure 4.11: The probability of β-decay using both the approximate β-decay operator which only acts on valence quarks (blue) and the full operator (orange) [PITH_FULL_IMAGE:figures/full_fig_p198_4_11.png]

**Figure 5.1.** Figure 5.1: The decomposition of the lepton-qubit register state preparation circuits into [PITH_FULL_IMAGE:figures/full_fig_p205_5_1.png]

**Figure 5.2.** Figure 5.2: The circuit for insertion into the electron ground-state initializer subcomponent [PITH_FULL_IMAGE:figures/full_fig_p206_5_2.png]

**Figure 5.3.** Figure 5.3: The circuit for insertion into the neutrino ground-state initializer subcomponent [PITH_FULL_IMAGE:figures/full_fig_p206_5_3.png]

**Figure 5.4.** Figure 5.4: The circuit for insertion into the electron ground-state initializer subcomponent [PITH_FULL_IMAGE:figures/full_fig_p207_5_4.png]

**Figure 5.5.** Figure 5.5: The circuit for insertion into the electron ground-state initializer subcomponent of [PITH_FULL_IMAGE:figures/full_fig_p207_5_5.png]

**Figure 5.6.** Figure 5.6: Left: The circuit for insertion into the neutrino ground-state initializer subcom [PITH_FULL_IMAGE:figures/full_fig_p208_5_6.png]

**Figure 5.7.** Figure 5.7: The actions of the SU(3) annihilation operators on the [PITH_FULL_IMAGE:figures/full_fig_p213_5_7.png]

**Figure 5.8.** Figure 5.8: The result of acting the Q (a) n,fQ (a) m,f′ on two sites from two different meson excitations but of the same irrep. The 3 and ¯3 denote the irreps on the sites; the red connections denote which sites are part of the same meson excitation [PITH_FULL_IMAGE:figures/…

**Figure 5.9.** Figure 5.9: The result of acting the Q (a) n,fQ (a) m,f′ on two sites from two different meson excitations and of different irreps. The 3 and ¯3 denote the irreps on the sites; the red connections denote which sites are part of the same meson excitation [PITH_FULL_IMAGE:figure…

**Figure 5.10.** Figure 5.10: The two remaining overlapping two-meson-excitation states. [PITH_FULL_IMAGE:figures/full_fig_p219_5_10.png]

**Figure 5.11.** Figure 5.11: The result of acting (Q (a) n,fQ (a) m,f′)trun1 or (Q (a) n,fQ (a) m,f′)trun2 on two sites from two different meson excitations but of the same irrep. The 3 and ¯3 denote the irreps on the sites; the red connections denote which sites are part of the same meson exci…

**Figure 5.12.** Figure 5.12: The result of acting (Q (a) n,fQ (a) m,f′)trun1 or (Q (a) n,fQ (a) m,f′)trun2 on two sites from two different meson excitations and of different irreps. The 3 and ¯3 denote the irreps on the sites; the red connections denote which sites are part of the same meson ex…

**Figure 5.13.** Figure 5.13: A Trotter decomposition of the kinetic and chromoelectric terms from the PBC [PITH_FULL_IMAGE:figures/full_fig_p225_5_13.png]

**Figure 5.14.** Figure 5.14: The implementation of the FSWAP gate, which simulataneously executes a [PITH_FULL_IMAGE:figures/full_fig_p226_5_14.png]

**Figure 5.15.** Figure 5.15: A Trotter decomposition of the kinetic and chromoelectric terms from the PBC [PITH_FULL_IMAGE:figures/full_fig_p227_5_15.png]

**Figure 5.16.** Figure 5.16: The SWAP-network needed to execute the portion of the chromoelectric term in [PITH_FULL_IMAGE:figures/full_fig_p228_5_16.png]

**Figure 5.17.** Figure 5.17: Top: the Trotterization of the kinetic term and the [PITH_FULL_IMAGE:figures/full_fig_p229_5_17.png]

**Figure 5.18.** Figure 5.18: The explicit implmenentations of the kinetic terms used in Figs. [PITH_FULL_IMAGE:figures/full_fig_p229_5_18.png]

**Figure 5.19.** Figure 5.19: A naive combination of two implementations of an [PITH_FULL_IMAGE:figures/full_fig_p230_5_19.png]

**Figure 5.20.** Figure 5.20: The definitions of four circuits based on the mappings in Tab. [PITH_FULL_IMAGE:figures/full_fig_p232_5_20.png]

**Figure 5.21.** Figure 5.21: Constructions of the 4-qubit chromoelectric term defined in Fig. [PITH_FULL_IMAGE:figures/full_fig_p234_5_21.png]

**Figure 5.22.** Figure 5.22: The implementation of the Trotterization of the kinetic portion of the quark [PITH_FULL_IMAGE:figures/full_fig_p236_5_22.png]

**Figure 5.23.** Figure 5.23: The results of time evolution under the Hamiltonian in Eq. [PITH_FULL_IMAGE:figures/full_fig_p237_5_23.png]

**Figure 6.1.** Figure 6.1: Quantum circuit implementing the term e −itJijλ(i) ·λ(j) from the two-neutrino part Hνν. Definitions of the gates can be found in App. 6.B. trino system. Our proposed qutrit circuit follows the native qutrit gate set and the notation of the transmon qudits in Ref. [2…

**Figure 6.2.** Figure 6.2: (a) Quantum circuit implementing a single LO Trotterized time evolution step via [PITH_FULL_IMAGE:figures/full_fig_p248_6_2.png]

**Figure 6.3.** Figure 6.3: Circuit A implementing e −iαλ(i) ·λ(j) in the physical subspace, using 24 CNOTs. The gates R± z represents the short-hand version of Rz(± π 2 ). • Ry(π 4 ) Ry(− π 4 ) • • • • • • Ry(π 4 ) Ry(− π 4 ) • • Ry(π 4 ) Ry(− π 4 ) • Rz(−2 α) • • • Ry(π 4 ) Ry(− π 4 ) • • Rz(…

**Figure 6.4.** Figure 6.4: Circuit B implementing e −iαλ(i) ·λ(j) in the physical subspace, using 18 CNOTs. The two-neutrino term Hνν is more delicate in this case, compared to the qutrit implementation. As mentioned, while the physical subspace is fixed, we have the freedom on the unphysical…

**Figure 6.5.** Figure 6.5: Different post-selecting procedures for computing the single-neutrino flavor prob [PITH_FULL_IMAGE:figures/full_fig_p262_6_5.png]

**Figure 6.6.** Figure 6.6: Flavor evolution of a two-neutrino system as a function of time. Panels (a) and [PITH_FULL_IMAGE:figures/full_fig_p263_6_6.png]

**Figure 6.7.** Figure 6.7: Flavor evolution of a four-neutrino system as a function of time, using the [PITH_FULL_IMAGE:figures/full_fig_p264_6_7.png]

**Figure 6.8.** Figure 6.8: Flavor evolution for an eight-neutrino system as a function of time, using the [PITH_FULL_IMAGE:figures/full_fig_p265_6_8.png]

**Figure 6.9.** Figure 6.9: Flavor evolution for two neutrinos as a function of time obtained from [PITH_FULL_IMAGE:figures/full_fig_p265_6_9.png]

**Figure 6.10.** Figure 6.10: Flavor evolution for four neutrinos as a function of time obtained from the [PITH_FULL_IMAGE:figures/full_fig_p266_6_10.png]

**Figure 6.11.** Figure 6.11: Flavor evolution for eight neutrinos as a function of time obtained from the [PITH_FULL_IMAGE:figures/full_fig_p266_6_11.png]

**Figure 6.12.** Figure 6.12: Flavor evolution for an eight-neutrino system as a function of time obtained [PITH_FULL_IMAGE:figures/full_fig_p267_6_12.png]

**Figure 6.13.** Figure 6.13: (a) Fidelity and (b) single-neutrino entropy for different ∆ [PITH_FULL_IMAGE:figures/full_fig_p267_6_13.png]

**Figure 6.14.** Figure 6.14: Flavor evolution for four neutrinos as a function of time from the [PITH_FULL_IMAGE:figures/full_fig_p270_6_14.png]

**Figure 6.15.** Figure 6.15: Flavor evolution for eight neutrinos as a function of time obtained from the [PITH_FULL_IMAGE:figures/full_fig_p271_6_15.png]

**Figure 7.1.** Figure 7.1: The magic power, M2(Uˆ 1), of the free-space one-body evolution operator for three flavors of neutrinos given in Eq. (7.7). The solid blue line shows the central value of the magic power, while the khaki region corresponds to the values of magic power from a sampling…

**Figure 7.2.** Figure 7.2: The normalized magic in the two-flavor (lighter, cream) and three-flavor (darker, [PITH_FULL_IMAGE:figures/full_fig_p281_7_2.png]

**Figure 7.3.** Figure 7.3: The normalized magic in the two-flavor (lighter, cream) and three-flavor (darker, [PITH_FULL_IMAGE:figures/full_fig_p282_7_3.png]

**Figure 7.4.** Figure 7.4: M2 per neutrino in systems initially in tensor-product states of |νe⟩ ⊗Nν only values are decreasing with increasing Nν. In contrast, wavefunctions from initial states containing all three flavors support magic that exceeds the maximum value in tensor-product states…

**Figure 7.5.** Figure 7.5: M2 per neutrino in systems initially in tensor-products of all three |νe⟩, |νµ⟩, |ντ ⟩ (lower curves), as a function of time. Initial states with the maximum asymptotic values of M2 from the possible flavor combinations for a given Nν are shown, i.e., |νeνµντ ⟩, |νeν…

**Figure 7.6.** Figure 7.6: The asymptotic values of M2 per neutrino in systems initially in a tensor-product state of |νe⟩ ⊗Nν (brown points and dashed curve) and in systems initially in tensor-products of all three |νe⟩, |νµ⟩, |ντ ⟩ (blue points and dashed curve). The maximum value of M2 from…

**Figure 7.7.** Figure 7.7: The left panel shows the probabilities of neutrinos initially in the [PITH_FULL_IMAGE:figures/full_fig_p296_7_7.png]

**Figure 7.8.** Figure 7.8: The left panel shows the sum of the concurrence (C) and generalized-concurrence [PITH_FULL_IMAGE:figures/full_fig_p296_7_8.png]

**Figure 7.9.** Figure 7.9: The left panel shows the probabilities of neutrinos initially in the [PITH_FULL_IMAGE:figures/full_fig_p297_7_9.png]

**Figure 7.10.** Figure 7.10: The left panel shows the sum of the concurrence (C) and generalized [PITH_FULL_IMAGE:figures/full_fig_p298_7_10.png]

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