Non-Unitary Quantum Walks

• Non-unitary quantum walks are discrete-time systems with non-Hermitian operators that model open-system dynamics using engineered gain/loss and dissipation.
• They exhibit rich symmetry structures, including generalized PT and chiral symmetries, ensuring real quasi-energy spectra even under disorder.
• These walks provide a platform for studying non-Hermitian topological phases and index theorems, with applications in photonics, quantum simulation, and robust state transfer.

Non-unitary quantum walks (QWs) are discrete-time quantum dynamical systems whose time-evolution operators are not unitary, typically due to engineered gain/loss, environmental coupling, or measurement-induced dissipation. Such walks generalize the standard (unitary) quantum walk paradigm, enabling the simulation of open-system dynamics, non-Hermitian band structures, emergent symmetries, and phases without Hermitian analogs. Non-unitary QWs have become a central platform for exploring the interplay between symmetry, topology, decoherence, and disorder in quantum dynamics, particularly their spectral and localization properties, topological indices, and experimental realization in photonic and atomic systems.

1. Definitions and General Structure of Non-Unitary Quantum Walks

A non-unitary quantum walk is characterized by the evolution of a quantum state $|\psi(t)\rangle$ under repeated application of a non-unitary time-step operator $U$, i.e., $|\psi(t+1)\rangle = U|\psi(t)\rangle$. The state space is typically $$\mathcal{H} = \ell^2(\mathbb{Z}^d) \otimes \mathbb{C}^q$$, where $\ell^2$ encodes the positions and $\mathbb{C}^q$ the internal coin structure.

Standard two-step operator (1D, two-level coin):

$$U = S\,G(\gamma_2)\,C(\theta_2)\,S\,G(\gamma_1)\,C(\theta_1)$$

where:

• $$C(\theta) = \sum_n |n\rangle\langle n| \otimes e^{i\theta \sigma_1}$$ is the coin rotation,
• $$S = \sum_n \left( |n-1\rangle\langle n| \otimes |L\rangle\langle L| + |n+1\rangle\langle n| \otimes |R\rangle\langle R|\right )$$ is the conditional shift,
• $$G(\gamma) = \sum_n |n\rangle\langle n| \otimes e^{\gamma \sigma_3}$$ introduces non-Hermiticity via amplification/attenuation.

Non-unitarity enters via $G(\gamma)$ when $\gamma \neq 0$: the resulting $U$ fails $U^\dagger U \neq \mathbb{I}$ and models systems with gain and loss or selective absorption.

2. Symmetry Structures and Spectral Reality

$\mathcal{PT}$ Symmetry and Generalized Anti-unitary Symmetries

The key symmetry paradigm in non-unitary QWs is generalized $\mathcal{PT}$ symmetry, with parity $P$ and time-reversal $T$:

• $P$ acts as site inversion: $P|n\rangle = |-n\rangle$,
• $T$ is anti-unitary (incorporating complex conjugation $K$ and time reversal in the walk operator).
• $\mathcal{PT}$ symmetry is defined via: $(\mathcal{PT})\,U\,(\mathcal{PT})^{-1} = U^{-1}$ If there exists an eigenvector $|\psi\rangle$ such that $$(\mathcal{PT})|\psi\rangle = e^{i\delta}|\psi\rangle$$, then $U|\psi\rangle = \lambda|\psi\rangle$ implies $|\lambda| = 1$, i.e., real quasi-energies.

This is generalized to arbitrary site-local anti-unitary symmetries $A = UK$ with $AUA^{-1} = U^{-1}$ and $A|\psi\rangle = e^{i\delta}|\psi\rangle$ (Mochizuki et al., 2016). These conditions are necessary and sufficient for the entire spectrum to be unimodular ($|\lambda|=1$), ensuring a real spectrum in otherwise non-Hermitian dynamics. Notably, such symmetry protection can persist even with strong spatial disorder that breaks global $P$ (parity).

3. Spectral Topology, Exceptional Points, and Index Theorems

PT Symmetry Breaking and Topological Phases

Spectral phases of non-unitary QWs are governed by the presence or absence of symmetry-protected real spectra. The unbroken $\mathcal{PT}$ regime features all quasi-energies real, while spontaneous symmetry breaking (as a function of gain-loss strength or disorder) leads to complex-conjugate pairs and the appearance of exceptional points (parameter values at which eigenvectors coalesce and the spectrum becomes non-diagonalizable).

For split-step walks: $$M(k) = \cos\theta_1 \cos\theta_2 \cos(2k) - \sin\theta_1 \sin\theta_2 \cosh(2\gamma)$$ The phase boundary (exceptional line) is given by $|M(k)|=1$ for some $k$ (Itable et al., 2022). Topological invariants (e.g., winding numbers or biorthogonal Zak phases) remain quantized in symmetry-unbroken regions and are associated with robust edge states for appropriate boundary conditions.

Chiral Symmetry and Non-Hermitian Index Theory

For chiral-symmetric QWs ($U^* = \Gamma U \Gamma$, $\Gamma^2=1$), the Witten index generalizes topological classification to non-unitary evolution. The index can be defined via the off-diagonal block $Q_0$ of the imaginary part of $U$ in the $\Gamma$-grading basis: $$\mathrm{ind}(\Gamma,U) := \dim\ker Q_0 - \dim\ker Q_0^*$$ Unlike the unitary case, no essential spectral gap is required; the index is well-defined even when the spectrum is gapless or portions of it lie on the real axis or non-unit circle (Asahara et al., 2020). The index predicts topological phase transitions as the asymptotic parameters cross critical values (e.g., $\left|p_\gamma(\pm\infty)\right| = |a(\pm\infty)|$).

4. Effects of Disorder and Robustness of Real Spectra

Spatially random disorder, particularly in coin parameters, can globally break $P$ and hence the $\mathcal{PT}$ symmetry. However, numerical analysis demonstrates that local (anti-unitary) symmetries can persist, such that QWs retain an entirely real spectrum for a wide disorder range (Mochizuki et al., 2016). In some configurations, the introduction of randomness restores spectral realness even when the clean (homogeneous) system had complex quasi-energies.

This phenomenon is tied to the property that time-reversal-like anti-unitary symmetries, acting locally rather than globally, are not affected by site-uncorrelated disorder. This leads to extended parameter regions in which strong disorder does not induce complex quasi-energy bands, and phase diagrams (see figure below) quantify the robust unbroken phases in the presence of randomness.

Case	Coin Disorder	Global $\mathcal{PT}$?	Spectrum Reality
$U_1$ (PT walk)	none (homog.)	Yes	Real for $\gamma<\gamma_c$
$U_1$ (PT walk)	$\theta_1(n)$ only	No	Real over wide parameter ranges
$U_2$ (TR walk)	none	No	Generic complex
$U_2$ (TR walk)	both $\theta_i(n)$	No	Real (restored by disorder)

5. Representative Models and Explicit Constructions

1. Homogeneous Non-Unitary Walks: The archetypal model is the two-step walk with homogeneous coin/gain-loss, as above.
2. Disordered Walks: Models with site-dependent coin angles $\theta_i(n)$ drawn from a uniform box or other random distributions. Enforces breaking of $P$.
3. Exact $\mathcal{PT}$-symmetric Walks: Experimental setups such as the fiber-loop QW of Regensburger et al. with alternating gain/loss and phase engineered to satisfy strict $\mathcal{PT}$ symmetry (Mochizuki et al., 2016).
4. Chiral-Symmetric Non-Unitary Walks: Models such as the Mochizuki–Kim–Obuse (MKO) split-step walk (Asahara et al., 2020), as well as generalizations to two-phase and defect QWs (Endo et al., 6 Nov 2025, Kiumi et al., 2022), supporting bulk–boundary topological indices.
5. Local Anti-unitary-Symmetric Walks: Walks where either coins or gain-loss are site-random, destroying global symmetries but where the spectrum remains real owing to local symmetry.

6. Physical and Experimental Implications

Non-unitary QWs form a flexible platform for the simulation of non-Hermitian physics—including but not limited to, the observation of $\mathcal{PT}$-transition dynamics, edge-state localization, and non-Hermitian topological phases. The existence of exactly real spectra under non-unitary evolution is a feature crucial to robust state transport and topological protection in open quantum systems, with significance for quantum information, photonic circuitry, and dissipation engineering.

Experiments exploiting spatially programmable disorder and tunable gain/loss (via photonics, atomic systems, or engineered dissipation) confirm the numerically predicted phase diagrams and the resilience of real spectra to disorder (Mochizuki et al., 2016). There is ongoing investigation into the microscopic mechanisms responsible for local anti-unitary protection and the explicit forms of such symmetry operators in disordered settings.

7. Outlook and Open Questions

Research on non-unitary quantum walks with generalized $\mathcal{PT}$ symmetry has clarified that real quasi-energy spectra, long thought tied to global $\mathcal{PT}$ symmetry, can survive under significantly weaker, local anti-unitary conditions—even maximal random disorder. Future research directions include:

• Explicit construction of local anti-unitary symmetry operators in general disordered environments;
• Generalization of bulk–boundary correspondences and index theorems for non-unitary (especially locally symmetric) walks;
• Classification of non-Hermitian topological phases in the presence of disorder;
• Applications to robust quantum simulation, photonic state transfer, and the design of open-system quantum devices.

These developments signal a shift in the understanding of symmetry protection and topological robustness in non-Hermitian quantum dynamics, with non-unitary quantum walks providing a fertile framework for foundational and applied advances.