Yang-Lee Criticality as a Dissipative Dynamical Phase Transition: Quantum Simulation of non-Hermitian Physics without Post-selection

Technical Summary: Yang-Lee Criticality as a Dissipative Dynamical Phase Transition

Problem Statement
The Yang-Lee theory describes classical Ising spins in an imaginary magnetic field, a model historically significant for understanding first-order phase transitions and non-unitary conformal field theories (specifically the minimal model $M_{2,5}$). While analytically tractable, direct physical realization of Yang-Lee criticality is obstructed by the requirement of an imaginary field. Previous proposals to simulate this physics have relied on indirect analytic continuation, non-local probe couplings, or non-Hermitian time-evolution achieved through post-selection. Post-selection is particularly problematic as it incurs an exponential sampling overhead, rendering the approach infeasible for near-term quantum devices. The central problem addressed is how to realize the non-Hermitian generator of the Yang-Lee transfer matrix—and more generally, arbitrary non-Hermitian Hamiltonians—within an open quantum system evolving under unconditional (post-selection free) dynamics.

Methodology
The author proposes a framework where the dynamics of specific linear observables in an open quantum system are governed by a non-Hermitian effective Hamiltonian. The core methodology relies on the following mechanisms:

1. Doubled Hilbert Space and Lindbladian Dynamics: The system is modeled as an open quantum system of qubits evolving under a Lindbladian superoperator $\mathcal{L}$. Using the Choi-Jamiołkowski isomorphism, the density matrix $\rho$ is mapped to a pure state $|\rho\rangle\!\rangle$ in a doubled Hilbert space $\mathcal{H} \otimes \mathcal{H}^*$. The Lindbladian acts as a linear operator on this space.
2. Weak-Symmetry and Non-Hermitian Subspace: The construction utilizes an extensive number of "weak-symmetries" of the form $Z_i \otimes Z_i$ (where $Z$ is the Pauli operator). These symmetries lead to a fragmentation of the Hilbert space, isolating a specific subspace $\Lambda$ spanned by Pauli strings consisting of $X$ and $Y$ operators.
3. Projection via Observables: By choosing an observable $O$ that lies within the non-Hermitian subspace $\Lambda$, the expectation value $\text{Tr}(O\rho(T))$ is determined entirely by the projection of the Lindbladian onto $\Lambda$. Within this subspace, the dynamics are isomorphic to the time-evolution generated by a non-Hermitian Hamiltonian $\hat{H}_{\text{eff}}$.
4. Engineered Dissipation: The specific terms of the target non-Hermitian Hamiltonian (e.g., the Yang-Lee transfer matrix generator) are realized through engineered dissipative channels (jump operators) and local unitaries.
   • Unitary terms (imaginary field) are generated by single-site $Z$-rotations.
   • Dissipative terms (Ising coupling and transverse field) are generated by $X$-dephasing channels and nearest-neighbor channels involving weak measurements and feedback (or ancilla coupling).
5. Discrete-Time Implementation: For digital quantum simulators, the continuous-time Lindbladian evolution is approximated via Suzuki-Trotter decomposition into a sequence of local unitaries and quantum channels. This avoids the need for continuous analog control.

Key Contributions and Results

• Realization of Yang-Lee Theory without Post-Selection: The paper demonstrates that the $d+0$-dimensional Yang-Lee theory can be realized as the effective dynamics of a $(d-1)+1$-dimensional open quantum system. The partition function of the classical Yang-Lee model is mapped exactly to the expectation value of a specific observable ($O = \prod X_i$) in the open system.
• Dynamical Phase Transition: The Yang-Lee critical point is identified as a dynamical phase transition in the time-dependence of the observable.
  • Overdamped Phase: Corresponds to the $PT$-symmetric phase of the non-Hermitian Hamiltonian (real eigenvalues), resulting in purely exponential decay of the observable.
  • Underdamped Phase: Corresponds to the $PT$-symmetry broken phase (complex conjugate eigenvalues), resulting in damped oscillatory decay.
  • Critical Point: The transition between these regimes coincides with the exceptional point of the non-Hermitian Hamiltonian, where the $PT$-symmetry is broken.
• Measurement Protocols: The author details protocols to measure spin correlation functions and "Loschmidt Echo" correlators directly.
  • Spin Correlators: By locally modifying the dynamics (inserting unitary defects $V = e^{i\pi Z/4}$) at specific spacetime points, the author shows that spin-spin correlation functions can be extracted from the change in the observable's expectation value. Numerical simulations confirm the unbounded growth of correlations consistent with the negative scaling dimension of the Yang-Lee CFT.
  • Loschmidt Echo: A related correlator, $LE^{(n)}$, is defined to measure sensitivity to defects under imperfect time-reversal, showing bounded decay consistent with positive scaling dimensions.
• Generalization to Arbitrary Non-Hermitian Hamiltonians: The construction is generalized to embed any non-Hermitian Hamiltonian $\hat{H}_{\text{target}}$ into an open quantum system. This is achieved by engineering jump operators (proportional to Clifford unitaries) that act on an extended system (including ancillary qudits) to reproduce the matrix elements of $\hat{H}_{\text{target}}$ within a weak-symmetry sector. This proves that any non-Hermitian Hamiltonian can be realized via Lindbladian evolution without post-selection.

Significance and Claims
The paper claims that Yang-Lee theory is a natural description of the spacetime dynamics of an open quantum system when probed by specific observables. The primary significance lies in providing an exact, post-selection-free method to simulate non-Hermitian physics and exceptional point phenomena.

• Experimental Feasibility: The proposed scheme is tailored for near-term quantum simulators (e.g., Rydberg tweezer arrays). It requires only local 2-qubit gates (CNOT, CZ), single-site unitaries, and engineered dissipation via ancilla coupling. The measurement of the non-local string observable $\prod X_i$ can be reconstructed from site-resolved readouts.
• Theoretical Connection: The work establishes a rigorous link between unconditional open quantum dynamics, exceptional point physics, and non-unitary statistical mechanics. It clarifies that the "symmetry-broken" states associated with the non-Hermitian exceptional point are unphysical (zero trace) but control the relaxation dynamics of physical observables.
• Scalability: While the measurement of the partition function itself (the observable value) may require an exponential number of runs to resolve, the protocol allows for the direct measurement of correlation functions and the exploration of criticality without the exponential overhead associated with post-selection.

The author concludes that this framework opens a pathway to experimentally explore non-unitary critical points and general non-Hermitian Hamiltonians in dimensions and regimes previously inaccessible due to the constraints of post-selection.