EFT-Based Pipeline for Quantum Simulations

• The EFT-based pipeline is a modular framework that integrates operator construction, matching, simulation, and measurement to efficiently target low-energy degrees of freedom.
• It employs systematic factorization, qubit mapping, and digitization to reduce resource demands and achieve controlled approximations in complex quantum systems.
• The pipeline’s modular design enables first-principles predictions for collider observables and jet substructure, significantly cutting computational costs in high-energy physics.

An Effective Field Theory (EFT)-based pipeline is a modular computational or analytical framework that implements the full lifecycle of EFT methodology for a specific physical system or a class of problems. It orchestrates the construction, matching, simulation, and analysis of EFTs while optimally exploiting the scale separation, operator reduction, and computational tractability that the EFT paradigm enables. In collider physics and quantum simulation contexts, such a pipeline can dramatically reduce the required resources and facilitate first-principles predictions by narrowing the focus to the relevant low-energy degrees of freedom and systematically integrating out heavy/fast dynamics (Bauer et al., 2021). The pipeline is composed of well-defined stages ranging from operator basis construction and matching to simulation and measurement.

1. Factorization, Matching, and Hamiltonian Construction

A central element in the EFT-based pipeline is the factorization of relevant observables into hierarchically separated contributions. For example, in collider observables, the cross section $\sigma$ is decomposed as

$$\sigma = H(\mu) \otimes J_1(\mu) \otimes ... \otimes J_n(\mu) \otimes S(\mu)$$

where $H(\mu)$ (hard) is evaluated at high scales via perturbation theory, $J_i(\mu)$ (jet) encode collinear sector dynamics, and $S(\mu)$ (soft) represents matrix elements of Wilson line operators (Bauer et al., 2021).

Matching involves projecting observables of the full theory onto the EFT at an appropriate scale, introducing Wilson coefficients $C_H(\alpha_s, \mu_H)$ that satisfy perturbative matching conditions and encode all high-energy corrections up to power-suppressed terms:

$$L_\mathrm{full} \rightarrow L_\mathrm{EFT} + C_H(\alpha_s, \mu_H) O_H + ...$$

In a scalar toy model, the low-energy Lagrangian is

$$L_s = \frac{1}{2} (\partial_t \phi)^2 - \frac{1}{2} (\partial_x \phi)^2$$

with a corresponding Hamiltonian density

$$H_s = \frac{1}{2} \pi^2 + \frac{1}{2} (\partial_x \phi)^2$$

after discretization, yielding a tractable lattice Hamiltonian (Bauer et al., 2021).

2. Qubit Mapping, Digitization, and Operator Realization

The pipeline digitizes field configurations for quantum simulation. The field value $\phi_i$ at each site is mapped to $n_\phi=2^{n_Q}$ discrete levels, encoded in $n_Q$ qubits per site. The field operator acts diagonally:

$$\hat{\phi}_i = \sum_{j=0}^{n_Q-1} 2^j \sigma_z^{(j)}(i)$$

and conjugate momentum $\pi_i$ is diagonal in the Fourier basis, applying quantum Fourier transforms to implement kinetic terms:

$$e^{-i \Delta t \pi^2} = QFT^\dagger e^{-i \Delta t \phi^2} QFT$$

The insertion of Wilson line operators becomes sequences of exponentials of lattice $\phi_i$, realized as single-qubit rotations and interleaved with time-evolution via Trotter–Suzuki steps (Bauer et al., 2021).

3. State Preparation and Simulation Protocols

Preparation of the initial quantum state, typically the free-theory vacuum, employs the Kitaev–Webb algorithm for multivariate Gaussian states. The protocol involves:

1. LDL$^T$ decomposition of the covariance matrix,
2. Preparation of uncorrelated Gaussians,
3. Application of shear unitaries to correlate sites.

Each stage has polynomial resource cost in system size. For three-site, two-qubit lattices, state preparation requires at most six CNOT gates, with circuit depth scaling polynomially for larger systems (Bauer et al., 2021).

4. Time Evolution, Measurement, and Resource Accounting

Time evolution of states is performed via Lie–Trotter splitting of the Hamiltonian:

$$[e^{-i H t}]_n \approx [e^{-i H_\phi \delta t} e^{-i H_\pi \delta t}]^n$$

Each Trotter step involves CNOT–R$_z$–CNOT gadgets for nearest-neighbor interactions and QFT operations for kinetic terms—about 60 CNOTs per step for the $N=3$, $n_Q=2$ implementation. The scaling is $O(N n_Q^2)$ per step.

Measurements of transition amplitudes such as $$\mathcal{Y}_X = |\langle X| T[Y_n Y_{\bar{n}}^\dagger] |0\rangle|^2$$ are achieved by overlap circuits combining state preparation, operator insertion, and projection onto computational basis states (Bauer et al., 2021).

Resource summary:

• Sites: $N=3$
• Qubits per site: $n_Q=2$ ($6$ total qubits)
• Gates: $<6$ CNOTs for state prep, $60$ per Trotter step, single-qubit rotations for operator insertions

Accuracy benchmarks:

• Digitization error $\sim10\%$ for $n_Q=2$, $<2\%$ for $n_Q=3$
• Quantum device result (IBMQ Manhattan): within $5\%$ of noiseless simulation after error mitigation

5. Generalization, Scaling, and Theoretical Justification

The pipeline generalizes to full SCET, where soft gluon fields $A_s^\mu$ and static Wilson lines in $\mathrm{SU}(3)$ replace scalar analogues. Collinear fields $\chi_n$ inhabit independent lattice Hilbert spaces. The entire protocol—matching, Hamiltonian discretization, qubit mapping, state preparation, evolution, and measurement—carries over (Bauer et al., 2021).

The critical scaling advantage lies in restricting simulation energy ranges (e.g., soft jet masses $m_J \lesssim 50$ GeV), reducing necessary lattice dimensions by $O(10^2)$ per spatial direction and total qubits by $O(10^6)$ in 3D relative to brute-force QFT approaches at LHC scales. This scaling establishes the EFT quantum simulation pipeline as the only viable prospect for first-principles studies of jet substructure and soft-radiation collider observables (Bauer et al., 2021).

6. Impact, Limitations, and Outlook

The effective field theory-based pipeline enables resource-efficient quantum simulations of collider observables and systematically incorporates perturbative and nonperturbative physics. It allows rigorously controlled approximations via matching, discretization, and operator reduction, with demonstrable accuracy and scalability. Limitations stem from digitization errors, quantum gate fidelity, and the need for further scaling to higher-dimensional and more complex field content (gauge fields, spinors).

Its adoption justifies large-scale quantum simulation initiatives in high energy physics, especially for jet physics, and is directly extensible to other domains where EFT constructions are tractable and quantum computational speedups are anticipated (Bauer et al., 2021).