[论文解读] Quantum simulation of general spin-1/2 Hamiltonians with parity-violating fermionic Gaussian states

Michael Kaicher, Joseph Vovrosh+2 more|arXiv (Cornell University)|Jan 20, 2026

Quantum many-body systems被引用 0

一句话总结

This paper develops parity-violating fermionic mean-field theory (PV-FMFT) based on parity-violating fermionic Gaussian states (PV-FGS) to efficiently simulate real- and imaginary-time dynamics of general spin-1/2 Hamiltonians, including those with parity-violating terms, via a Colpa mapping and an extended Hilbert space.

ABSTRACT

We introduce equations of motion for a parity-violating fermionic mean-field theory (PV-FMFT): a numerically efficient fermionic mean-field theory based on parity-violating fermionic Gaussian states (PV-FGS). This work provides explicit equations of motion for studying the real- and imaginary-time evolution of spin-1/2 Hamiltonians with arbitrary geometries and interactions. We extend previous formulations of parity-preserving fermionic mean-field theory (PP-FMFT) by including fermionic displacement operators in the variational Ansatz. Unlike PP-FMFT, PV-FMFT can be applied to general spin-1/2 Hamiltonians, describe quenches from arbitrary initial spin-1/2 product states, and compute local and non-local observables in a straight-forward manner at the same modest computational cost as PP-FMFT -- scaling as $O(N^3)$ in the worst case for a system of $N$ spins or fermionic modes. We demonstrate that PV-FMFT can exactly capture the imaginary- and real-time dynamics of non-interacting spin-1/2 Hamiltonians. We then study the post quench-dynamics of the one- and two-dimensional Ising model in presence of longitudinal and transversal fields with PV-FMFT and compute the single site magnetization and correlation functions, and compare them against results from other state-of-the-art numerical approaches. In two-dimensional spin systems, we show that the employed spin-to-fermion mapping can break rotational symmetry within the PV-FMFT description, and we discuss the resulting consequences for the calculated correlation functions. Our work introduces PV-FMFT as a benchmark for other numerical techniques and quantum simulators, and it outlines both its capabilities and its limitations.

研究动机与目标

• Motivate and formulate a numerically efficient mean-field framework for parity-violating fermionic systems that arise when mapping spin-1/2 Hamiltonians to fermions.
• Extend existing parity-preserving FMFT to handle arbitrary spin-1/2 Hamiltonians via parity-violating Gaussian states.
• Provide explicit, stable equations of motion for both imaginary-time (ground-state) and real-time (dynamics) evolution within PV-FMFT.

提出的方法

• Introduce PV-FGS as a variational Ansatz that includes fermionic displacement terms to break parity.
• Derive closed-form equations of motion for the covariance matrix Γ in the extended PV-FGS space (imaginary time: dΓ/dτ = -Hm - Γ Hm Γ; real time: dΓ/dt = [Hm, Γ]).
• Implement the Colpa mapping to convert PV problems into a PP framework in an extended Hilbert space (N+1 modes) enabling linear canonical transformations.
• Provide explicit expressions for energy expectations and variational matrices (Hm, M, etc.) and discuss numerical stability and Pfaffian-based evaluations.
• Demonstrate exactness for non-interacting PV spin Hamiltonians and study quenches in one- and two-dimensional Ising models with longitudinal/transverse fields.

Figure 1: Schematic relation between spin mean-field theory (SMFT) and fermionic mean-field theory (FMFT). Each circle represents a family of Hamiltonians and states which are exactly described by a mean-field theory based on spin-1/2 product states (purple), parity-preserving fermionic Gaussian sta

实验结果

研究问题

• RQ1 Can PV-FMFT with PV-FGS exactly capture imaginary- and real-time dynamics of non-interacting spin-1/2 Hamiltonians?
• RQ2 How can PV terms arising from spin-to-fermion mappings be efficiently handled within a Gaussian mean-field framework?
• RQ3 Does the Colpa mapping allow PV problems to be treated with PP-FMFT methods without introducing gauge constraints or mode doubling beyond one auxiliary mode?
• RQ4 What are the capabilities and limitations of PV-FMFT when applied to quenches in 1D and 2D Ising models with longitudinal fields?

主要发现

• PV-FMFT with PV-FGS can exactly describe imaginary- and real-time dynamics of non-interacting spin-1/2 Hamiltonians.
• The framework remains computationally tractable, scaling as O(N^3) in the worst case for N spins or fermionic modes.
• PV-FMFT can simulate quenches in the Ising model with longitudinal and transverse fields and compute local magnetization and correlation functions.
• Colpa mapping enables mapping PV-FGS to a linear combination of two PP-FGS in an extended N+1 mode space, preserving similar computational cost to PP-FMFT.
• In 2D, the spin-to-fermion mapping can break rotational symmetry within PV-FMFT, affecting computed correlation functions and highlighting mapping-induced artifacts.
• The PV-FMFT framework serves as a benchmark for other numerical techniques and quantum simulators, outlining capabilities and limitations.

Figure 2: Workflow diagram describing the steps to represent an arbitrary spin-1/2 Hamiltonian (here a two-dimensional square lattice) as a PP fermionic Hamiltonian. We consider the TFIM model where the spin-1/2 Hamiltonian is mapped to its fermionic description via the Jordan-Wigner transformation.