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arXiv:2210.12732 · Physical Review A

Quantum circuit for measuring an operator's generalized expectation values and its applications to non-Hermitian winding numbers

Ze-Hao Huang, Peng He, Li-Jun Lang, Shi-Liang Zhu

Published May 8, 2023• 7 citations
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Abstract

We propose a general quantum circuit based on the swap test for measuring the quantity $\ensuremath{\langle}{\ensuremath{\psi}}_{1}|A|{\ensuremath{\psi}}_{2}\ensuremath{\rangle}$ of an arbitrary operator $A$ with respect to two quantum states $|{\ensuremath{\psi}}_{1,2}\ensuremath{\rangle}$. This quantity is frequently encountered in many fields of physics, and we dub it the generalized expectation as a two-state generalization of the conventional expectation. We apply the circuit, in the field of non-Hermitian physics, to the measurement of generalized expectations with respect to left and right eigenstates of a given non-Hermitian Hamiltonian. To efficiently prepare the left and right eigenstates as the input to the general circuit, we also develop a quantum circuit via effectively rotating the Hamiltonian pair $(H,\ensuremath{-}{H}^{\ifmmode\dagger\else\textdagger\fi{}})$ in the complex plane. As applications, we demonstrate the validity of these circuits in the prototypical Su-Schrieffer-Heeger model with nonreciprocal hopping by measuring the Bloch and non-Bloch spin textures and the corresponding winding numbers under periodic and open boundary conditions (PBCs and OBCs), respectively. The numerical simulation shows that non-Hermitian spin textures building up these winding numbers can be well captured with high fidelity, and the distinct topological phase transitions between PBCs and OBCs are clearly characterized. We may expect that other non-Hermitian topological invariants composed of non-Hermitian spin textures, such as non-Hermitian Chern numbers, and even significant generalized expectations in other branches of physics would also be measured by our general circuit, providing a different perspective to study novel properties in non-Hermitian as well as other physics realized in qubit systems.

DOI
10.1103/physreva.107.052205

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The quantum geometric tensor (QGT) fundamentally encodes the geometry and topology of quantum states in both Hermitian and non-Hermitian regimes. While adiabatic perturbation theory links its real part (quantum metric) and imaginary part (Berry curvature) to energy fluctuations and generalized forces, respectively, in Hermitian systems, direct measurement of the QGT, which is defined using both left and right eigenstates of a non-Hermitian Hamiltonian, remains challenging. Here we develop two quantum simulation schemes to directly extract all components of the QGT in pseudo-Hermitian systems with real spectra. Each scheme independently determines the complete QGT using generalized expectation values of either the energy fluctuation operator or the generalized force operator with respect to two time-evolved states prepared through distinct nonadiabatic evolutions, thereby establishing two self-contained measurement protocols. We illustrate the validity of these schemes on two qqq-deformed two-band models: one with nontrivial topology and the other with a nonvanishing off-diagonal quantum metric. Numerical simulations demonstrate that, for suitably chosen nonadiabatic ramp velocities, both schemes achieve high-fidelity agreement with theoretical predictions for measuring the QGT in both models and successfully capture the topological phase transition of the first model using Chern numbers calculated from Berry curvatures. For larger velocities, the generalized force scheme yields greater accuracy for the real part of the QGT, while the energy fluctuation scheme better captures its imaginary part. This work establishes a framework for extending dynamical measurement schemes from Hermitian to pseudo-Hermitian systems with real spectra.
Quantum simulation at fixed particle number admits two equivalent descriptions, a first-quantized (particle) representation and a second-quantized (occupation-number) representation. Their quantum resource costs differ sharply across computational tasks, so the ability to convert coherently between them is valuable. We construct an explicit unitary QQQ, with inverse Q†Q^\daggerQ†, that maps a first-quantized state to its fixed-NNN occupation-number form while diagnosing the input's particle-exchange symmetry. The conversion is therefore symmetry-agnostic at the input yet fully resolved at the output, and it applies uniformly to bosonic, fermionic, and parastatistical sectors. At its foundation lies a structural identification that we place at the center of this work: the quantum Schur transform supplied by Schur-Weyl duality is the non-abelian Fourier transform of the commuting pair (SN,U(d))(S_N,U(d))(SN​,U(d)), and the occupation-number representation is its weight basis, retaining only the labels shared by both factors, the irrep λλλ and the u(d)\mathfrak{u}(d)u(d) weight. This reduction is lossless for bosons and fermions, while a canonical Gelfand-Tsetlin promise renders it one-to-one for the remaining sectors. Algorithmically, QQQ composes the strong Schur transform with reversible arithmetic that computes occupations as successive row-sum differences of the Gelfand-Tsetlin pattern, yielding gate complexity poly(N,d,log⁡(1/ε))\mathrm{poly}(N,d,\log(1/ε))poly(N,d,log(1/ε)). The converted state is prepared efficiently in quantum memory. Any classical algorithm that outputs it explicitly, however, pays a cost set by the sector dimension, which is polynomial of degree NNN in ddd at fixed NNN and exponential in NNN when d=Θ(N)d=Θ(N)d=Θ(N). Finally, an efficient classical sampler for the induced occupation-number distribution would yield one for arbitrary quantum circuits, contrary to standard complexity assumptions.
Topological transitions in non-Hermitian systems are generally boundary sensitive: a point-gap winding transition under periodic boundary condition (PBC) and a non-Bloch bulk real-line-gap transition under open boundary condition (OBC) at Re(E)=0\mathrm{Re}(E)=0Re(E)=0 are governed by different spectra and therefore need not coincide. Here we show, for a class of chiral non-Hermitian Su--Schrieffer--Heeger (SSH)-type lattices, that these two criticalities can be locked by an exceptional-point-constrained (EP-constrained) parameter evolution. The key requirement is not the occurrence of isolated exceptional points, but the persistence of a zero-energy Bloch degeneracy along the entire sweep, which is generically exceptional in the non-Hermitian regime. In an analytically tractable limit of an extended non-Hermitian SSH chain, the EP-constrained manifolds and both transition boundaries are obtained in closed form, making the locking explicit. Away from this limit, numerical generalized-Brillouin-zone (GBZ) calculations confirm the correspondence for representative constrained sweeps, whereas unconstrained paths show that isolated exceptional points or Hermitian degeneracies do not enforce locking. We further verify the mechanism in a spinful four-band extension with branch-resolved GBZs, including strongly branch-imbalanced regimes. These results establish a path-dependent diagnostic principle: along EP-constrained sweeps in this SSH-type class, changes in PBC point-gap winding can indicate OBC non-Bloch bulk real-line-gap transitions and the corresponding changes in zero-energy boundary modes.
We consider a Su-Schrieffer-Heeger(SSH)-type trimer model with next-nearest-neighbor(NNN) interaction and balanced loss-gain(BLG) to study the combined effect of lattice symmetries, topology, non-hermiticity and general boundary conditions(GBC)on the existence of flat band and the nature of Bulk-Boundary correspondence(BBC). We derive the necessary and sufficient conditions for the existence of an entirely real spectrum under the periodic boundary condition(PBC). The exact expressions for the compact localized states(CLS) and energy eigenvalues corresponding to flat bands are derived analytically under the PBC. We establish topological phase transitions(TPT) for PT-symmetry and pseudo-chiral symmetry through the computation of the Zak phase and sub-lattice Zak phase, respectively. The Hamiltonian under the open boundary condition(OBC) is studied numerically, and edge states are observed in the topologically non-trivial phase, thereby establishing the non-hermitian BBC. The CLS exists in both bulk and the boundary for systems having only pseudo-chiral symmetry, and an additional PT-symmetry destroys the CLS at the boundary. We generalize a known formalism to study the same Hamiltonian under GBC, and derive analytic expressions for the energy and eigenstates for a class of boundary conditions in parametric ranges which admit flat band under the PBC. The edge states for these boundary conditions, including the OBC, are obtained analytically in the topologically non-trivial phase, thereby establishing BBC. The non-hermitian skin effect(NHSE) is seen in the model with reciprocal bulk interaction and strongly non-reciprocal boundary terms. The winding number based on spectral topology is computed analytically.
Topological states of quantum matter have inspired both fascinating physics findings and exciting opportunities for applications. Due to the over-complicated structure of, as well as interactions between, real materials, a faithful quantum simulation of topological matter is very important in deepening our understanding of these states. This requirement puts the quantum superconducting circuits system as a good option for mimicking topological materials, owing to their flexible tunability and fine controllability. As a typical example herein, we realize a Z2\mathbb{Z}_2Z2​-type topological insulator featuring the quantum spin Hall effect state, using a coupled system of transmission-line resonators and transmons. The single-excitation eigenstates of each unit cell are used as a pseudo-spin 1/2 system. The boundary of the topological phase transition is fixed in the phase diagram. Topological edge states are shown, which can be experimentally verified by detecting the population at the boundary of the plane. Compared to the previous simulations, this compositional system is fairly controllable, stable and less limited. Therefore, our scheme provides a reliable platform for faithful quantum simulations of topological matter.