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

Abstract

We propose a general quantum circuit based on the swap test for measuring the quantity $⟨{\psi }_1|A|{\psi }_2⟩$ of an arbitrary operator $A$ with respect to two quantum states $|{\psi }_{1,2}⟩$. 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 $(H,-H^{†})$ 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.

arXiv comment6+3 pages, 4+1 figures, published version

References in corpus (49)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities
• Edge states and topological invariants of non-Hermitian systems
• Exceptional Topology of Non-Hermitian Systems
• Non-Hermitian Physics
• Topological phases of non-Hermitian systems
• The physics of exceptional points
• Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances
• Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems
• Quantum fingerprinting
• Topological Band Theory for Non-Hermitian Hamiltonians
• Suppressing quantum errors by scaling a surface code logical qubit
• Anomalous edge state in a non-Hermitian lattice
• Non-Bloch Band Theory of Non-Hermitian Systems
• Correspondence between winding numbers and skin modes in non-hermitian systems
• Non-Hermitian Chern bands
• Non-Hermitian Boundary Modes
• Efficient Light Funneling based on the non-Hermitian Skin Effect
• Anatomy of skin modes and topology in non-Hermitian systems
• Non-Hermitian robust edge states in one-dimension: Anomalous localization and eigenspace condensation at exceptional points
• Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems
• Colloquium: Understanding Quantum Weak Values: Basics and Applications
• Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms
• Non-Hermitian bulk-boundary correspondence and auxiliary generalized Brillouin zone theory
• Observation of parity-time symmetry breaking in a single spin system
• Bulk-Boundary Correspondence in a Non-Hermitian System in One Dimension with Chiral-Inversion Symmetry
• Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems
• Quantum Algorithms for Fermionic Simulations
• Observation of arbitrary topological windings of a non-Hermitian band
• Topological quantum matter with cold atoms
• PT-Symmetry in Non-Hermitian Su-Schrieffer-Heeger model with complex boundary potentials
• Tunable non-reciprocal quantum transport through a dissipative Aharonov-Bohm ring in ultracold atoms
• Bloch state tomography using Wilson lines
• Dynamically encircling an exceptional point in a real quantum system
• Non-Hermitian Topological Anderson Insulators
• Localization and topological transitions in non-Hermitian quasiperiodic lattices
• Observation of non-Hermitian topology with non-unitary dynamics of solid-state spins
• Topological Anderson insulators in two-dimensional non-Hermitian disordered systems
• Efficient Quantum Algorithm for Computing n-time Correlation Functions
• Quantum computation of magnon spectra
• Quantum simulation of parity-time symmetry breaking with a superconducting quantum processor
• Dynamic winding number for exploring band topology
• Time-dependent $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics in generic non-Hermitian systems
• Non-Bloch band theory and bulk-edge correspondence in non-Hermitian systems
• Direct Probe of Topological Order for Cold Atoms
• Topological Anderson insulators with different bulk states in quasiperiodic chains
• Machine learning topological invariants of non-Hermitian systems
• Non-Hermitian dynamics of slowly-varying Hamiltonians
• Observation of exceptional point in a PT broken non-Hermitian system simulated using a quantum circuit

Cited by in corpus (2)

• Dimension-independent weak value estimation via controlled SWAP operations
• Direct measurement of the quantum geometric tensor in pseudo-Hermitian systems