Non-unitary Quantum Walks Research Articles

Spectral mapping theorem of an abstract non-unitary quantum walk

This paper continues the previous work (Quantum Inf. Process 11 (2019)) by two authors of the present paper about a spectral mapping property of chiral symmetric unitary operators. In physics, they treat non-unitary time-evolution operators to consider quantum walks in open systems. In this paper, we generalize the above result to include a chiral symmetric non-unitary operator whose coin operator only has two eigenvalues. As a result, the spectra of such non-unitary operators are included in the (possibly non-unit) circle and the real axis in the complex plane. We also give some examples of our abstract results, such as non-unitary quantum walks defined by Mochizuki et al. Furthermore, the main theorem applies not only to quantum walks but also to the Ihara zeta function and correlated random walks on regular graphs.

Anomalous non-Hermitian dynamical phenomenon on the quantum circuit

The anomalous non-Hermitian dynamical phenomenon with the non-Hermitian skin effect (NHSE) attracts wide attention due to its novel physics and promising applications. Here, we propose a new type of non-unitary discrete-time quantum walk system demonstrating the NHSE and anomalous non-Hermitian dynamical phenomena, including the dynamical chiral phenomenon, the funneling phenomenon on the domain wall, and the anomalous reflection on the phase impurity. Furthermore, we design the quantum circuit experiments of these quantum walk systems and numerically simulate them with quantum noises to verify the robustness of the non-Hermitian dynamical phenomenon on the noisy intermediate-scale quantum (NISQ) devices. Our work paves the way for implementing the non-Hermitian dynamical phenomenon on the quantum circuit.

Entanglement entropy distinguishes PT-symmetry and topological phases in a class of non-unitary quantum walks

We calculate the hybrid entanglement entropy between coin and walker degrees of freedom in a non-unitary quantum walk. The model possesses a joint parity and time-reversal symmetry or PT-symmetry and supports topological phases when this symmetry is unbroken by its eigenstates. An asymptotic analysis at long times reveals that the quantum walk can indefinitely sustain hybrid entanglement in the unbroken symmetry phase even when gain and loss mechanisms are present. However, when the gain-loss strength is too large, the PT-symmetry of the model is spontaneously broken and entanglement vanishes. The entanglement entropy is therefore an effective and robust parameter for constructing PT-symmetry and topological phase diagrams in this non-unitary dynamical system.

Topological invariants of nonunitary quantum walk with chiral symmetry

We propose an one-dimensional model of nonunitary quantum walk with chiral symmetry. The nonunitarity is caused by gain and loss that they stagger in time and act on different walking directions. The degenerate point of the quasi-energy on the dispersion-relation diagram becomes a line segment and the corresponding topological defect is characterized by the vorticity. The winding number is still well defined due to chiral symmetry. It is found that the vorticity, as a topological invariant, equals the difference of winding numbers within both sides of topological defect. Meanwhile, a pair of vorticities can be derived in two time symmetric frames and they count the number of topological defects with zero- and π- quasi-energy. At boundary in inhomogeneous system, appearing or disappearing of edge state is not affected by nonunitary action. The existing of the edge states can be measured by the average displacement calculated by the escape probability, and the bulk-boundary corresponding is shown to be invalid.

An index theorem for one-dimensional gapless non-unitary quantum walks

Recent developments in the index theory of discrete-time quantum walks allow us to assign a certain well-defined supersymmetric index to a pair of a unitary time-evolution $U$ and a $\mathbb{Z}_2$-grading operator $\varGamma$ satisfying the chiral symmetry condition $U^* = \varGamma U \varGamma.$ In this paper, this index theory will be extended to encompass non-unitary $U$. The existing literature for unitary $U$ makes use of the indispensable assumption that $U$ is essentially gapped; that is, we require that the essential spectrum of $U$ contains neither $-1$ nor $+1$ to define the associated index. It turns out that this assumption is no longer necessary, if the given time-evolution $U$ is non-unitary. As a concrete example, we shall consider a well-known non-unitary quantum walk model on the one-dimensional integer lattice, introduced by Mochizuki-Kim-Obuse.

Observation of Non-Bloch Parity-Time Symmetry and Exceptional Points.

Parity-time (PT)-symmetric Hamiltonians have widespread significance in non-Hermitian physics. A PT-symmetric Hamiltonian can exhibit distinct phases with either real or complex eigenspectrum, while the transition points in between, the so-called exceptional points, give rise to a host of critical behaviors that holds great promise for applications. For spatially periodic non-Hermitian systems, PT symmetries are commonly characterized and observed in line with the Bloch band theory, with exceptional points dwelling in the Brillouin zone. Here, in nonunitary quantum walks of single photons, we uncover a novel family of exceptional points beyond this common wisdom. These "non-Bloch exceptional points" originate from the accumulation of bulk eigenstates near boundaries, known as the non-Hermitian skin effect, and inhabit a generalized Brillouin zone. Our finding opens the avenue toward a generalized PT-symmetry framework, and reveals the intriguing interplay between PT symmetry and non-Hermitian skin effect.

High winding number of topological phase in non-unitary periodic quantum walk* *Project supported by the National Natural Science Foundation of China (Grant No. 12004231).

Topological phases and their associated multiple edge states are studied by constructing a one-dimensional non-unitary multi-period quantum walk with parity-time symmetry. It is shown that large topological numbers can be obtained when choosing an appropriate time frame. The maximum value of the winding number can reach the number of periods in the one-step evolution operator. The validity of the bulk–edge correspondence is confirmed, while for an odd-period quantum walk and an even-period quantum walk, they have different configurations of the 0-energy edge state and π-energy edge state. On the boundary, two kinds of edge states always coexist in equal amount for the odd-period quantum walk, however three cases including equal amount, unequal amount or even only one type may occur for the even-period quantum walk.

Non-Bloch quench dynamics

We study the quench dynamics of non-Hermitian topological models with non-Hermitian skin effects. Adopting the non-Bloch band theory and projecting quench dynamics onto the generalized Brillouin zone, we find that emergent topological structures, in the form of dynamic skyrmions, exist in the generalized momentum-time domain, and are correlated with the non-Bloch topological invariants of the static Hamiltonians. The skyrmion structures anchor on the fixed points of dynamics whose existence are conditional on the coincidence of generalized Brillouin zones of the pre- and post-quench Hamiltonians. Global signatures of dynamic skyrmions, however, persist well beyond such a condition, thus offering a general dynamic detection scheme for non-Bloch topology in the presence of non-Hermitian skin effects. Applying our theory to an experimentally relevant, non-unitary quantum walk, we explicitly demonstrate how the non-Bloch topological invariants can be revealed through the non-Bloch quench dynamics.

Bulk–edge correspondence and stability of multiple edge states of a $\mathcal{PT}$-symmetric non-Hermitian system by using non-unitary quantum walks

Abstract Topological phases and the associated multiple edge states are studied for parity and time-reversal ($\mathcal{PT}$)-symmetric non-Hermitian open quantum systems by constructing a non-unitary three-step quantum walk retaining $\mathcal{PT}$ symmetry in one dimension. We show that the non-unitary quantum walk has large topological numbers of the $\mathbb{Z}$ topological phase and numerically confirm that multiple edge states appear as expected from the bulk–edge correspondence. Therefore, the bulk–edge correspondence is valid in this case. Moreover, we study the stability of the multiple edge states against a symmetry-breaking perturbation so that the topological phase is reduced to $\mathbb{Z}_2$ from $\mathbb{Z}$. In this case, we find that the number of edge states does not become one unless a pair of edge states coalesce at an exceptional point. Thereby, this is a new kind of breakdown of the bulk–edge correspondence in non-Hermitian systems. The mechanism of the prolongation of edge states against the symmetry-breaking perturbation is unique to non-Hermitian systems with multiple edge states and anti-linear symmetry. Toward experimental verifications, we propose a procedure to determine the number of multiple edge states from the time evolution of the probability distribution.

Non-Hermitian physics

A review is given on the foundations and applications of non-Hermitian classical and quantum physics. First, key theorems and central concepts in non-Hermitian linear algebra, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity and parity-time symmetry, are delineated in a pedagogical and mathematically coherent manner. Building on these, we provide an overview of how diverse classical systems, ranging from photonics, mechanics, electrical circuits, acoustics to active matter, can be used to simulate non-Hermitian wave physics. In particular, we discuss rich and unique phenomena found therein, such as unidirectional invisibility, enhanced sensitivity, topological energy transfer, coherent perfect absorption, single-mode lasing, and robust biological transport. We then explain in detail how non-Hermitian operators emerge as an effective description of open quantum systems on the basis of the Feshbach projection approach and the quantum trajectory approach. We discuss their applications to physical systems relevant to a variety of fields, including atomic, molecular and optical physics, mesoscopic physics, and nuclear physics with emphasis on prominent phenomena/subjects in quantum regimes, such as quantum resonances, superradiance, continuous quantum Zeno effect, quantum critical phenomena, Dirac spectra in quantum chromodynamics, and nonunitary conformal field theories. Finally, we introduce the notion of band topology in complex spectra of non-Hermitian systems and present their classifications by providing the proof, firstly given by this review in a complete manner, as well as a number of instructive examples. Other topics related to non-Hermitian physics, including nonreciprocal transport, speed limits, nonunitary quantum walk, are also reviewed.

Non-Bloch ${\cal P}{\cal T}$PT symmetry breaking in non-Hermitian photonic quantum walks.

A hallmark of topological band theory in periodic media is that bulk properties are not affected by boundary conditions. Remarkably, in certain non-Hermitian lattices, the bulk properties are affected largely by boundaries, leading to such major effects as the non-Hermitian skin effect and violation of the bulk-boundary correspondence. Here, we unveil that non-unitary discrete-time quantum walks of photons in systems involving gain and loss show rather generally non-Bloch parity-time symmetry-breaking phase transitions and suggest a bulk probing method to detect such boundary-driven phase transitions.

Higher winding number in a nonunitary photonic quantum walk

Topological matter exhibits exotic properties yet phases characterized by large topological invariants are difficult to implement, despite rapid experimental progress. A promising route toward higher topological invariants is via engineered Floquet systems, particularly in photonics, where flexible control holds the potential of extending the study of conventional topological matter to novel regimes. Here we implement a one-dimensional photonic quantum walk to explore large winding numbers. By introducing partial measurements and hence loss into the system, we detect winding numbers of three and four in multi-step non-unitary quantum walks, which agree well with theoretical predictions. Moreover, by probing statistical moments of the walker, we identify locations of topological phase transitions in the system, and reveal the breaking of pseudo-unitary near topological phase boundaries. As the winding numbers are associated with non-unitary time evolution, our investigation enriches understanding of topological phenomena in non-unitary settings.

Detecting Topological Invariants in Nonunitary Discrete-Time Quantum Walks.

We report the experimental detection of bulk topological invariants in nonunitary discrete-time quantum walks with single photons. The nonunitarity of the quantum dynamics is enforced by periodically performing partial measurements on the polarization of the walker photon, which effectively introduces loss to the dynamics. The topological invariant of the nonunitary quantum walk is manifested in the quantized average displacement of the walker, which is probed by monitoring the photon loss. We confirm the topological properties of the system by observing localized edge states at the boundary of regions with different topological invariants. We further demonstrate the robustness of both the topological properties and the measurement scheme of the topological invariants against disorder.

Effects of Disorder on Non-unitary <i>PT</i> Symmetric Quantum Walks

$\mathcal{PT}$ symmetry, namely, a combined parity and time-reversal symmetry can make non-unitary quantum walks exhibit entirely real eigenenergy. However, it is known that the concept of $\mathcal{PT}$ symmetry can be generalized and an arbitrary anti-unitary symmetry has a possibility to substitute $\mathcal{PT}$ symmetry. The aim of the present work is to seek such non-unitary quantum walks with generalized $\mathcal{PT}$ symmetry by focusing on effects of spatially random disorder which breaks $\mathcal{PT}$ symmetry. We numerically find non-unitary quantum walks whose quasi-energy is entirely real despite $\mathcal{PT}$ symmetry is broken.

One-dimensional quantum walks subject to next-nearest-neighbour hopping decoherence

It is believed that the decoherence will lead to a crossover from quantum to classical for the diffusive behaviour of discrete quantum walk in the long time limit. However, a few systems with sub-ballistic diffusive behaviours have been found in some non-unitary quantum walks. In this paper, we study the one-dimensional discrete quantum walks subject to a short-range hopping decoherence, i.e. with a probability the walker could hop to next-nearest-neighbour lattices unilaterally and/or bilaterally in one time step. We find that, when the decoherence effects only come from the bilateral hopping operation, the diffusive behaviours of quantum walks are sub-ballistic and the distributions of position exhibit three peaks. These results are quite different from those of the previous non-unitary quantum walks. Our results could be used to improve the algorithmic properties of quantum walk due to its faster diffusive speed and more uniform spreading.

Bounds for mixing time of quantum walks on finite graphs

Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov et al (2002 Proc. 33rd STOC (2001) (New York: ACM) pp 50–9 arXiv:quant-ph/0012090v2) and it is found that for particular examples of walks on a cycle, a hypercube and a complete graph, quantum walks provide no speedup in mixing over the classical counterparts. In addition, non-unitary quantum walks (i.e. walks with decoherence) are considered and a criterion for their convergence to the unique stationary distribution is derived.

Optimal computation with non-unitary quantum walks

Quantum versions of random walks on the line and the cycle show a quadratic improvement over classical random walks in their spreading rates and mixing times, respectively. Non-unitary quantum walks can provide a useful optimisation of these properties, producing a more uniform distribution on the line, and faster mixing times on the cycle. We investigate the interplay between quantum and random dynamics by comparing the resources required, and examining numerically how the level of quantum correlations varies during the walk. We show numerically that the optimal non-unitary quantum walk proceeds such that the quantum correlations are nearly all removed at the point of the final measurement. This requires only O ( log T ) random bits for a quantum walk of T steps.

Nonunitary quantum walks on hypercycles

We present analytical treatment of quantum walks on multidimensional hypercycle graphs. We derive the analytical expression of the probability distribution for strong and weak decoherence regimes. Upper bound to mixing time is obtained.