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arXiv:1703.02542 · Journal of Physics A Mathematical and Theoretical

Topological classification of time-asymmetry in unitary quantum processes

Jacob Turner, Jacob Biamonte

Published Apr 21, 2021• 7 citations
arXivHTMLPDF

Abstract

Abstract Understanding which physical processes are symmetric with respect to time inversion is a ubiquitous problem in physics. In quantum physics, effective gauge fields allow emulation of matter under strong magnetic fields, realizing the Harper–Hofstadter, the Haldane models, demonstrating one-way waveguides and topologically protected edge states. Central to these discoveries is the chirality induced by time-symmetry breaking. In quantum walk algorithms, recent work has discovered implications time-reversal symmetry breaking has on the transport of quantum states which has enabled a host of new experimental implementations. We provide a full topological classification of Hamiltonian operators that do not exhibit symmetry under time-reversal with respect to the induced transition probabilities between elements in a preferred site-basis, i.e. the nodes of the graph on which the walk takes place. We prove that a quantum process is necessarily time-symmetric for any choice of time-independent Hamiltonian precisely when the underlying support graph is bipartite or no Aharonov–Bohm phases are present in the gauge field. We further prove that certain bipartite graphs exhibit transition probability suppression, but not broken time-reversal symmetry. Furthermore, our development of a general framework characterizes gauge potentials on combinatorial graphs. These results and techniques fill an important missing gap in understanding the role this omnipresent effect has in quantum information and computation.

DOI
10.1088/1751-8121/abf9d0

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Abstract Understanding which physical processes are symmetric with respect to time inversion is a ubiquitous problem in physics. In quantum physics, effective gauge fields allow emulation of matter under strong magnetic fields, realizing the Harper–Hofstadter, the Haldane models, demonstrating one-way waveguides and topologically protected edge states. Central to these discoveries is the chirality induced by time-symmetry breaking. In quantum walk algorithms, recent work has discovered implications time-reversal symmetry breaking has on the transport of quantum states which has enabled a host of new experimental implementations. We provide a full topological classification of Hamiltonian operators that do not exhibit symmetry under time-reversal with respect to the induced transition probabilities between elements in a preferred site-basis, i.e. the nodes of the graph on which the walk takes place. We prove that a quantum process is necessarily time-symmetric for any choice of time-independent Hamiltonian precisely when the underlying support graph is bipartite or no Aharonov–Bohm phases are present in the gauge field. We further prove that certain bipartite graphs exhibit transition probability suppression, but not broken time-reversal symmetry. Furthermore, our development of a general framework characterizes gauge potentials on combinatorial graphs. These results and techniques fill an important missing gap in understanding the role this omnipresent effect has in quantum information and computation.
Given its importance to many other areas of physics, from condensed-matter physics to thermodynamics, time-reversal symmetry has had relatively little influence on quantum information science. Here we develop a network-based picture of time-reversal theory, classifying Hamiltonians and quantum circuits as time symmetric or not in terms of the elements and geometries of their underlying networks. Many of the typical circuits of quantum information science are found to exhibit time asymmetry. Moreover, we show that time asymmetry in circuits can be controlled using local gates only and can simulate time asymmetry in Hamiltonian evolution. We experimentally implement a fundamental example in which controlled time-reversal asymmetry in a palindromic quantum circuit leads to near-perfect transport. Our results pave the way for using time-symmetry breaking to control coherent transport and imply that time asymmetry represents an omnipresent yet poorly understood effect in quantum information science.
Particle transport in quantum systems, which can be modeled by quantum walks on graphs, demonstrates a faster propagation advantage over the corresponding transport in classical systems. As known from several graph examples, achieving further advantages is possible by adding directional control in quantum walks. One way to introduce directional bias is via time-reversal symmetry breaking that can be achieved with chiral quantum walks, where complex phases are added to the edge weights. However, it is not known for which complex phases values and on which graphs quantum transport can be enhanced. Therefore, the classification of graph properties remains an open problem. Here we tackle this graph classification problem with a graph convolutional neural network trained on a set of simulated examples. We find that chiral quantum-walk dynamics leads to almost always faster transport on hypercubes compared to nonchiral dynamics. We connect our paper to physical implementations of quantum walks in superconducting qubits and optical waveguides. Our results open the possibility and flexibility of experimental implementations in demonstrating quantum-walk advantage.
Discrete-time quantum walks (DTQW) have topological phases that are richer than those of time-independent lattice Hamiltonians. Even the basic symmetries, on which the standard classification of topological insulators hinges, have not yet been properly defined for quantum walks. We introduce the key tool of time frames, i.e., we describe a DTQW by the ensemble of time-shifted unitary time-step operators belonging to the walk. This gives us a way to consistently define chiral symmetry (CS) for DTQW's. We show that CS can be ensured by using an ``inversion symmetric'' pulse sequence. For one-dimensional DTQW's with CS, we identify the bulk $\mathbb{Z}\ifmmode\times\else\texttimes\fi{}\mathbb{Z}$ topological invariant that controls the number of topologically protected 0 and $\ensuremath{\pi}$ energy edge states at the interfaces between different domains, and give simple formulas for these invariants. We illustrate this bulk-boundary correspondence for DTQW's on the example of the ``4-step quantum walk,'' where tuning CS and particle-hole symmetry realizes edge states in various symmetry classes.
Symmetry under time-reversal appears in the microscopic description of many physical systems. In a quantum mechanical setting it acts as an anti-unitary operator, so does not fall under general analyses based on unitary symmetries. In classifying zero temperature phases of matter in (1+1)d lattice models, the role of anti-unitary symmetries is, however, well-understood. In recent years, the Symmetry Topological Field Theory (SymTFT) approach to this classification has given a general framework to understand symmetries as topological defects, but does not naturally include anti-unitary symmetries. Following recent proposals in the literature, we adopt a symmetry-enriched SymTFT for a theory with both internal and time-reversal symmetry. In particular, we take a standard SymTFT associated with an internal unitary symmetry that is then enriched by a background time-reversal symmetry. A detailed analysis of the topological boundary conditions of this enriched SymTFT allows us to characterize the corresponding (1+1)d gapped phases that preserve the enriching symmetry (i.e. those that do not spontaneously break this symmetry in the ground state). Line operators in the SymTFT approach are related to non-local string-order parameters (with charged end-point operators) for SPT phases. These are subtle in the anti-unitary case and we explore them both on the lattice and in the continuum. We include an analysis of unitary string order parameters that reveal the Klein bottle SPT invariant. On the lattice, we show that the correct end-point charge coincides with the time-reversal-charge only when the end-point operator is hermitian.
Symmetry plays an important role in the topological band theory to remedy the eigenstates' gauge obstruction at the cost of a symmetry anomaly and zero-energy boundary modes. One can also make use of the symmetry to enumerate the topological invariants---giving a symmetry classification table. Here we consider various topological phases protected by different symmetries and examine how the corresponding topological invariants evolve once the protecting symmetry is spontaneously lost. To our surprise, we find that the topological invariants and edge states can sometimes be robust to symmetry-breaking quantum orders. This topological robustness persists as long as the mean-field Hamiltonian in a symmetry-breaking ordered phase maintains its adiabatic continuity to the noninteracting Hamiltonian. For example, for a time-reversal symmetric topological phase in 2+12+12+1 dimensions, we show that the Z2{\mathbb{Z}}_{2}Z2​ time-reversal polarization continues to be a good topological invariant even after including distinct time-reversal breaking order parameters. Similar conclusions are drawn for various other symmetry-breaking cases. Finally, we discuss that the change in the internal symmetry associated with the spontaneous symmetry breaking has to be accounted for to reinstate the topological invariants into the expected classification table.
Topological phases of matter are understood to be characterized by particular configurations of entanglement encoded within the ground states of many-body quantum systems. In this thesis, we discuss how novel phenomena associated with topological phases can arise in systems that are driven out of equilibrium or coupled to their surroundings, which are not described by ground state physics. Firstly, we consider systems undergoing unitary time evolution due to some external driving. We show how the entanglement features of the time-dependent wavefunction can be topologically characterized, and demonstrate that these topological properties are reflected in the dynamics of various quantities usually associated with topological phases in equilibrium. We introduce a new non-equilibrium topological classification scheme which can be used to predict whether or not the topological features of a given system will be preserved under time evolution. In brief, this classification captures the fact that certain symmetries (namely antiunitary symmetries) are inevitably broken once the system is driven out of equilibrium; thus any topological phenomena protected by such symmetries cannot be expected to persist in non-equilibrium scenarios. Secondly, we investigate how topologically non-trivial systems are affected by dissipative effects, i.e.~coupling to an external environment. We demonstrate that system-environment interactions facilitate processes in the system that effectively break any antiunitary symmetries, regardless of the symmetries of the microscopic Hamiltonian. Accordingly, those phases that were shown to be unstable against time-dependent driving are also fragile against coupling to an environment. To illustrate the consequences of this fragility, we consider the effects of dissipation on the coherence properties of topological bound states, as well as the conductance properties of chiral and helical topological edge modes. We find that the decoherence rate of the former and the deviation from quantized conductivity of the latter are (not) exponentially suppressed in the inverse temperature when the phase in question is protected by unitary (antiunitary) symmetries. These results regarding open systems can be connected to the same non-equilibrium classification developed in the context of isolated systems undergoing unitary dynamics. Our findings highlight a distinction between topological phases that are either robust or fragile against non-equilibrium effects and/or system-environment coupling. The ramifications for the use of such systems in quantum technologies are discussed.